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The Official Journal of the BC Association of Mathematics Teachers

FALL 2016 Volume 57 Issue 2

Vec tor

MATHEMATICAL EXPLORATION

The Power of By Jordan Forseth

SHORT STORY

The Pigeon Hole Caper By Isaac Torres

PROFILES

Emily By Leo Neufeld

34

44

46

CONTENTS

IN EVERY ISSUE

President’s Message

Book Review

Problem Sets

Math Links

BCAMT

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57

54

60

Fall 2016 | Volume 57 | Issue 2

22

RESEARCH REPORT

The Silent Partner: Reading to Learn Mathematics By Sandra Hughes

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4851

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49

SOCIAL JUSTICE IN MATHEMATICS

A Lesson Plan Journey By Ariel U’Chong Muirhead

26

29 Math for All! Needs, Prospects and Possibilities By Latika Raisinghani

Fraction Activity By Hwie Lie Johns

Assessing and Teaching Through the Competencies By Deanna Brajcich

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19

IMPLEMENTATION

To Blog or Not to Blog? An exploration of the effectiveness of using a blog By Alex Sabell

09

GRANTS

Improving Our Practice: A Journey of Pedagogical Change By Cynthia Lai and Sara Lai

58

NOTICE TO CONTRIBUTORS We invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable materials written by BC authors on BC curriculum items. In some instances, we may publish articles written by persons outside the province if the materials are of particular interest in BC.

Submit articles by email to the editors. Authors must also include a short biographical statement of 40 words or less.

Articles must be in Microsoft Word (Mac or Windows). All diagrams must be in TIFF, GIF, JPEG, BMP, or PICT formats. Photographs must be high print quality (min. 300 dpi).

The editors reserve the right to edit for clarity, brevity and grammar.

MEMBERSHIP ENQUIRIES If you have questions regarding membership status or have a change of address, please contact Brad Epp, Membership Chair: [email protected]

2015/16 MEMBERSHIP RATES $40 + GST (BCTF Member) $20 + GST Student (full time university only) $65.52 + GST Subscription (non-BCTF)

NOTICE TO ADVERTISERS Vector is published two times a year: spring and fall. Circulation is approximately 1400 members in BC, across Canada and in other countries around the world.

Advertising printed in Vector may be of various sizes, and all materials must be camera ready.

Usable page size is 6.75 x 10 inches.

ADVERTISING RATES PER ISSUE $300 Full Page $160 Half Page $90 Quarter Page

The views expressed in each Vector article are those of its author(s), and not necessarily those of the editors or of the British Columbia Association of Mathematics Teachers.

Articles appearing in Vector may not be reprinted without the explicit written permission of the editors. Once written permission is obtained, credit must be given to the author(s) and to Vector, citing the year, volume number, issue number and page numbers.

BCAMT EXECUTIVEMichael Pruner, President Windsor Secondary School [email protected]

Deanna Brajcich, Vice President Sooke School District [email protected]

Ron Coleborn, Past President Math & Science Helping Teacher K-12 Burnaby School District [email protected]

Colin McLellan, Secretary McNair Secondary School [email protected]

Debbie Loo, Treasurer Burnaby South Secondary School [email protected]

Brad Epp, Membership Chair South Kamloops Secondary [email protected]

Colin McLellan, Listserv Manager McNair Secondary School

VECTOR EDITORSSean Chorney Simon Fraser University [email protected]

Peter Liljedahl Simon Fraser University [email protected]

NCTM REPRESENTATIVE Marc Garneau Numeracy Helping Teacher K-12 Surrey School District [email protected]

ELEMENTARY REPRESENTATIVESDeanna Lightbody Instructional Services, District Teacher K-8 Langley School District [email protected]

Jennifer Barker Surrey School District [email protected]

Jennifer Carter Vernon School District [email protected]

Sandra Ball Numeracy Helping Teacher K-12 Surrey School District [email protected]

SECONDARY REPRESENTATIVES Chris Becker Princess Margaret Secondary School [email protected]

Chris Hunter Surrey School District [email protected]

Robert Sidley Burnaby Mountain Secondary School [email protected]

Christine Younghusband Sunshine Coast School District [email protected]

INDEPENDENT SCHOOL REPRESENTATIVES Richard DeMerchant St. Michaels University School [email protected]

Darien Allan Collingwood School [email protected]

POST-SECONDARY REPRESENTATIVE Peter Liljedahl Simon Fraser University [email protected] Cover art was created by Kaliq Pillai-Monsanto. Kaliq is a grade 4 student

at KB Woodward. What follows is what Kaliq says about this piece: First we had to get a blank piece of paper and paint it different colours. Then we got another blank piece of paper and we had to cut out the triangle shape. A triangle has three corners and three sides. Then we fit it all on a piece of paper. I wanted to make it so detailed and it took me a lot of hard work and a long time. I felt like at the end it was going to look very good and it did. I want to dedicate this to my teacher, Mrs Nelson.

Vector • Fall 2016 5

CONTRIBUTORS

HWIE LIE JOHNS

Hwie Lie teaches mathematics at Sutherland Secondary School in North Vancouver. Her interests include incorporating problems, puzzles, and games into the classroom. She completed a Master of Education at Simon Fraser University in 2008 and is currently an executive member of the BCAMT.

JORDAN FORSETH

Jordan Forseth has been teaching high school mathematics for eight years and is a graduate of the University of British Columbia. He currently teaches in Vancouver, BC, and is an avid supporter of the ‘flipped’ classroom.

SANDRA HUGHES

Sandra currently teaches secondary mathematics in Burnaby. Her previous teaching experiences range from the intermediate grades to secondary mathematics and science in both the public and private sectors. Sandra is completing her Masters in Mathematics Education at Simon Fraser University where she is exploring the impact of reading on the learning of mathematics.

SANDRA BALL

Sandra works as the “Inner-City Early Learning Helping Teacher” in the Surrey School District. She has taught for over 30 years, with most of her experience focused on working with early learners.

DEANNA BRAICICH

Deanna Brajcich has been teaching for the Sooke School District for over twenty years educating students in grades K-10 and currently teaching at the elementary level. For the last ten years she has been providing mentorship and professional learning opportunities in Mathematics. Deanna is currently the Vice-President of the BC Association of Mathematics Teachers and a representative on the Island Numeracy Network.

DAVID WEES

David started his teaching career in a Brooklyn inner city school. He has taught in IB schools in London, England and Bangkok, Thailand. In recent years, he began working for New Visions for Public Schools as a Formative Assessment Specialist.

JANICE NOVAKOWSKI

Janice works for the Richmond School District as a teacher consultant supporting teachers and students in the areas of mathematics, science and technology. She has taught elementary grades, been a resource teacher and a teacher-librarian, and worked with University of British Columbia (UBC) pre-service teachers.

BETH BALDWIN

Beth has been teaching mathematics at West Vancouver Secondary School since 2010. She is the acting department chair and teaches grades 9 through 12. Beth is currently working on the completion of her Master’s Degree in Mathematics Education at Simon Fraser University with a research focus on mathematical self-efficacy.

KANWAL SINGH NEEL

Dr. Kanwal Singh Neel is a career educator who has served as a teacher in Richmond and as an associate Director of Professional Programs at SFU. He is currently a Project Coordinator with SFU’s Friends of Simon Tutoring Program which serves as an outreach to immigrant and refugee children and providing university students as tutors and mentors. Kanwal is an internationally acclaimed mathematics educator, host of the award winning television series Mathematics Shop and one of the authors of Mathematics Makes Sense textbook series published by Pearson Canada.

LATIKA RAISINGHANI

Latika Raisinghani is a PhD Candidate at the Department of Curriculum and Pedagogy, Faculty of Education, University of British Columbia, Vancouver, British Columbia. Latika’s research focuses on cultural diversity and culturally responsive teaching in mathematics and science classrooms. Her aspirations to bring “education for life” into today’s diverse classrooms are informed through living, learning and teaching as a teacher and teacher educator in multiple cultural contexts for more than twelve years.

ALEX SABELL

Alex Sabell recently completely her Masters of Education in Curriculum and Instruction, with a focus on Numeracy, at Simon Fraser University (SFU). She works for the Surrey School District as a grades 4/5 teacher. In her classroom, she strives to provide opportunities for her students to learn mathematical concepts and skills through problem solving, discovery, and exploration. She loves sharing her excitement and enthusiasm for mathematics and hopes to continue her professional development in this area.

ARIEL U’CHONG MUIRHEAD

Ariel U’Chong Muirhead is a pre-service Biology and English teacher in Simon Fraser University’s Professional Development Program. Ariel is currently doing her long practicum in the Burnaby school district and is excited to share her passion for learning with her students.

CYNTHIA CLARKE

Cynthia Clarke has been teaching primary students in the Richmond School District (#38) for 37 years. She has a strong interest in teaching Science and Mathematics and has presented workshops throughout B.C. and Alberta.

SARA LAI

Sara Lai has been working in the Richmond School District (#38) for 17 years. She has taught both primary and intermediate grades. Presently, she has a keen interest in digital learning and curriculum and enjoys integrating technology in all subject areas.

ISAAC TORRES

Isaac Torres is a programmer, writer, translator and designer. Their work ranges from writing to coding and translating. Together with their partner and a fluffy cat, they make art to make the world a better place

LEO NEUFELD

Leo Neufeld taught mathematics and chaired the Mathematics Department of Camosun College in Victoria for over 26 years. He loved especially to introduce students to the structure and applications of matrix and linear algebra. In retirement, he continues to be involved in mathematics-related activities (he recently received a lifetime achievement award for these) and, of course, in the Mathematics Challengers competition.

I am excited to begin another year of teaching mathematics,

especially in a time when so many changes are taking place in our province. The BCAMT is busy working to support all mathematics teachers through conferences, district pro-d

events, committee work, and as an advocate with respect to the

new curriculum and assessment. I am one year in on a two-year term in my role as president, and I have to say that it has been a very eye opening experience. Seeing all the hard work and dedication that all of the BCAMT executive put in regularly in order to make conferences run well, bring fresh ideas to the field through Vector articles, and visiting far off places in BC to lead pro-d workshops and regional meetings has been truly inspiring. All of these activities and more are implemented through our goal of promoting excellence in mathematics education.

I am writing this message in the Fall, just after our Mathematics Conference, knowing that by the time you read this, we will all be a few months into implementation with the new curriculum for K – 9. For myself, I am excited to begin trying a couple of new things in my Math 8 and 9 classes. Full implementation of this curriculum is not expected by the ministry, and it should not be expected by school districts or administration. I have been assured by all parties, that the implementation is expected to be slow and careful so that teachers are able to bring in new practices, assessments, and content with care, collaboration, and reflection. For myself, I am interested in highlighting the curricular competencies, communicating and reflecting, in my classroom. I am going to make another attempt at daily math journals using FreshGrade as the means for collecting data on student reflecting and communicating of their learning. I am interested to hear about what you are doing differently this year under the new curriculum framework. The BCAMT listserv is a great place for a public discussion on this, but you may choose to use twitter as well; @BCAMT is our twitter handle.

The BCAMT is continuing with its goal of providing professional development workshops and regional meetings to smaller BC communities. Last year, we visited the Gulf Islands, North Island, Nelson, Smithers, Prince George and Prince Rupert. Our executive members and others not only provided teachers in these communities with up-to-date information on the Math Curriculum and BCAMT initiatives, but they also organized workshop sessions on assessment, problem solving, thinking classrooms, and more. We plan to continue with these activities in the 2016-2017 school year, so if you would like to organize a BCAMT workshop in your community, please contact me at [email protected].

I like to start all of my classes with non-curricular problem solving tasks (like the president’s problem on the next page). I find these


PRESIDENT’S MESSAGE

tasks useful for improving student confidence in their problem solving, building a thinking culture in my classroom, and for starting classes with a lighter and more enjoyable tone. I have spoken with many teachers who also like to use these tasks in their classrooms, and the common concern is with the difficulty of finding appropriate tasks to use. The BCAMT will be releasing two non-curricular tasks every Sunday throughout the school year via the listserv, Twitter and Facebook (#weeklymathtask). I would love to hear from you regarding your experience in using these tasks, so please respond back with photos of student’s work or personal anecdotes. The tasks will also be collected and posted on our website. If you have any suggestions of tasks, please send them to me through the email given above.

The Fall Conference this year was held at Gladstone Secondary in Vancouver, and it was a huge success. I would like to thank the conference chairs, Debbie Loo and Colin McLellan, for organizing such a great conference. The theme was ‘Math is Social’ and we had plenty of outstanding speakers including our keynote speaker, Fawn Nguyen. Fawn came all the way from Southern California to open the conference, and she shared about the importance of taking time to learn, teaching our students like they were our own children, and not being afraid to challenge all of our students. We were also able to recognize two award winners. Katie McCormack, from Brock Middle School in Kamloops, was the recipient of the BCAMT Outstanding Elementary Teacher award. Maria Nicolidakis is from Burnaby Central Secondary School in Burnaby, and she was the winner of the Ivan L. Johnson Memorial Award. Congratulations to both of these outstanding teachers.

That’s about all for now. I wish you all the best as you continue through your school year, and hopefully you will find some good ideas and inspiration in this edition of Vector.

Micheal Pruner BCAMT president


Solve the President’s ProblemPICTURE STORY

Adapted from http://nrich.maths.org/325

How does this picture illustrate the following:

13+ 23+33+ … + 63 =(1+2+3+ … +6)2

Could you draw a similar picture to represent the sum of the first seven cube numbers?

What about other sums of cubes?

Suggest an expression for the sum of the first n cube numbers. Can you prove that your expression works, using diagrams and explanations? Send us your thoughts!

Vector • Fall 2016

IMPLEMENTATION

9

BY ALEX SABELL

To Blog or Not to Blog? An exploration of the effectiveness of using a blog

Teachers often try to find new ways to engage students and create learning environments with the potential to capture their interest and make a lasting impression. In just such an attempt to create a learning opportunity in my grade 4 class of 27 students, I decided to experiment with the use of a blog as a medium for building mathematical knowledge and improving my students’ ability to communicate their thinking. It seemed a natural choice to harness the power of social media to help students relate to mathematics in a more personal way. I also wanted to take advantage of a technology already ever-present in their daily lives—one that has the ability to give even the most reticent student a confident voice.

To engage my students I considered the implementation of mathematics journaling. I saw this as an opportunity to create a learning environment where my students felt supported and better connected to both the subject and to me within the larger “community of learning” that I believe is so vital to student success. I have always found it challenging to divide my time among 27 individuals, each requiring allotments of my attention, but I thought mathematics journaling offered a potential solution to this challenge that I was eager to explore. As well, I hoped that through regular teacher-student correspondence my students would form new understandings as they received/applied feedback, and acquire the skills to improve their communication.

While considering how I might implement mathematics journaling, I surmised that a mathematics blog could do much more than a journal thanks to its “forum” format. A blog would generate a shared experience that encouraged learning from each other’s successes and mistakes. It also had the potential to facilitate a multi-layered structure due to a quicker turnaround time for peer-to-peer and teacher-to-student feedback. Through this experience I learned a great deal about managing a blog, the benefits and challenges of incorporating this form of technology, and ultimately about myself as a teacher.

LEARNING TO MANAGE A BLOGAs someone new to the blogging world, the idea of expanding the mathematical conversations from the classroom and mathematics journal to the online domain was not a decision that came without reservations. Taking on such a project was daunting; however,

I could not ignore the possible benefits that could come from exploring this opportunity.

Vision and Set Up

I decided to try Kidblog because of its organization and security features. Kidblog allowed me to make a post whereby all students respond in one place and I could avoid going into each student’s private page to read their ideas. I also appreciated the ability to approve all posts before they went onto the page where they could be seen by other students. The blog itself was password protected and could only be accessed by the students and myself.

First Post

I attempted my first blog post in December 2015. My first question was titled “Divergent Thinking” and students were asked how many ways they could count the dots in the picture.

I chose this question because it connected to the curriculum content and competencies. This problem required students to communicate their thinking in many ways, use multiple strategies to engage in problem solving and develop mental mathematics strategies to make sense of the quantities. Because blogging was brand new to many of my students, I chose to have them start this entry in class. They were all given a copy of the dot orientation so that they could do some of their thinking on paper. This post ended up with 119 entries made by the students and me. In all, 19 of my 27 students made at least one post. All students were able to participate in the problem; some solving it by making addition sentences while others used


related to our lesson on area and multiplication.

Amanat

Area!!!!!!!!

Area is the amount of small squares inside a rectangle, say I have a rectangle (6 by 8) you would count the amount of small squares inside the rectangle (that’s a square I did on my rectangle work making thingamajig) .

Connection

The connection between area & multiplication is that when you are finding the area of a rectangle (6 by 8) you could count all the small squares in the rectangle or you could count the the first row of squares from the top and the first row from the side & multiply the 2 together & get 48 & if you were going to use multiplication you would use the strategy of counting the first and the first row from the side & get 48.

Mrs. Sabell

No problem Amanat! Something for you to think about...do you think that that definition of area would apply to other shapes? For example a triangle?

This student demonstrated a clear understanding of area as it applied to rectangles. My response was meant to extend her thinking and have her consider whether or not this definition fit with other shapes.

multiplication equations to describe what they saw. This prompted discussions about the connection between multiplication and division, which then led to an introduction to factors.

Josie (see above) seemed to have no problem making a variety of number combinations during class and had moved to focusing on multiplication-type equations, I felt the blog reinforced language that we had discussed in class. During the lesson, I had inserted the term “factor,” so I was hoping to see her incorporate this in her post. When she did, I used the blog to determine if she truly understood what a factor was and supported her in refining her definition.

Initial Excitement

This first experience yielded a number of issues I had not anticipated. Firstly, there was my initial excitement about the discussions that I was able to respond to each student thoughtfully without the numerous distractions that occur in a regular class that make it impossible to equally divide time among all students. I was also able to catch minor errors in thinking and it offered a nice opportunity for formative assessment. These points emphasized the value of continuing my exploration with the blog.

A side benefit of this was that I became increasingly attuned to any learning gaps or minor misconceptions among the participants. Often I would sit down to respond to such posts and think about how my student had walked out of my classroom lesson with an incorrect understanding and how the blog was giving me a second chance to correct it. I also liked that it was not immediately after the lesson so I could see what they retained after some time had passed. For those students who did grasp the concepts or learn the skills, I felt the blog cemented their learning and I was able to extend the question to challenge them further. The following blog exchange

Example of post from this lesson: (Note: I have kept grammar, punctuation and spelling as close as possible to how they appear on the blog)

Josie I found 19 different ways to get to 32, here are some of them: 2 times 6 plus 9 times 2 plus 2 times 1; 2 times 14 plus 1 times 2 plus 2 times 1; 16 times 2; 128 divided by 4; 224 divided by 7; 320 divided by 10. I also learnt about factors, factors are numbers that make up other numbers, for example 32’s factors are 1, 2, 4, 8, 12, 16, and 32 or 45’s factors are 1, 3, 5, 9, 15, and 45.

Mrs. Sabell Thanks Josie. In your definition of factors, you say that they are “numbers that make up other numbers.” Could we revise that definition a bit? If I were to say that the numbers 17 and 15 make up 32 (17 +15=32), does that mean 17 and 15 are factors? Why or why not?

Josie What I mean by numbers that make up other numbers is a number divided by another number evenly with no remainder. For example, 20 can be divided by 1, 2, 4, 5, 10, 20 so that means these numbers are factors of 20.

To answer your question - no, 15 and 17 are not factors of 32 because they both have remainders (32 divided by 17 = 1r15 and 32 divided by 15 = 2r2).

Mrs. Sabell Well said. Thanks for clarifying your definition :)


Shift In Conversational StructureThe conversational structure of the blog changed from being a back and forth exchange between teacher and student to one that involved multiple participants. As much as I enjoyed the social nature of the dialogue, I realized that my time was not being managed as well as it could be. My responses to students were often repetitious and, intermittently, I had to ask for more detail or a review of the original questions. Fortunately, this struggle happened alongside my students’ emerging interest to interact with each other on the blog.

To encourage constructive feedback and comments on the blog, I created a number of lessons that established rules for posting, discussed reasons to give a response and allowed students time to practice using sentence prompts appropriately. Initially, students were just posting “good job” everywhere which was certainly polite, but not beneficial to anyone’s learning. The rules were meant to give them some structure when posting. It actually worked extremely well as comments became far more thoughtful and it gave them some basic vocabulary to use when trying to respond respectfully pointing out errors (see appendix 2 for rules and sentence prompts).

As soon as I encouraged students to start responding to each other I was impressed by the care and thought that went into their responses. To my delight, they often responded in the same way I might have. This shift resulted in the students commenting a number of times before I inserted my own questions and comments, many of which were mainly to clarify concepts and extend the learning.

Examples of student responses:

Gurisha Sami, I think I noticed a mistake in your explaining, when you said, “Long division has some steps like multiply, divide, bring the number down, and take away.”

I noticed the steps are out of order. I think the steps are supposed to be divide, multiply, take away, bring the number down.

AmanatI really liked how you explained how that in the last question the answer NEED’S to be a two digit number and I really liked how you explained that in each question you need to count how many pieces of fruit there is actually in each group!

Josie I was wondering if you could give me any of your own examples of fractions that would look the same when represented in a picture.

EXPERIENCING THE BLOG

Affordances of the Blog In my exploration of mathematics blogging, I identified many advantages to incorporating this form of technology into my mathematics program. However, there were three main ways in

which the blog significantly impacted my classroom: it provided time for uninterrupted, reflective responses from the teacher; it served as a venue for formative assessment and to differentiate instruction; and it provided a forum for discussion of mathematical ideas outside of classroom time that would have been non-existent otherwise.

Time for Uninterrupted, Reflective Responses

A great frustration for many teachers, including myself, is the challenge around equitable division of quality teacher time in a busy classroom. The rarity of opportunities to check in and hear the ideas of every student without interruption is an unfortunate reality of our job. What I found useful about the blog was that it allowed me to step outside of time and provide unique, single attention to each student. For this reason, I was able to catch minor mistakes and misunderstandings, push students to elaborate on their ideas, make connections between old and new ideas and extend their thinking. The blog differed from classroom conversations—where I attempted to achieve the same goals—in that I had time to process and thoughtfully consider what they were saying and then respond in the most meaningful way possibly. Moreover, unlike interactions that occurred in person, feedback and ideas were not lost in the moment and could be revisited at a later time.

Formative Assessment and Differentiating Instruction

Another affordance of the blog was that it acted as a formative assessment, giving me the information I needed to respond to the various learning levels in my class. In many cases, I indentified learning gaps that I was able to address on the blog or later in the classroom. For the first time in my career, I really felt that I was able to thoroughly attend to the learning needs of my higher achievers by providing them ongoing opportunities to challenge their thinking. Also, the blog provided a great setting for self and peer assessment. Examples were provided and students could compare their work to the work of others. Below is an example of how I used the blog to inform myself about how my students were making sense of an image and question provided that asked them what they noticed about 4 equivalent fractions (see Appendix 1, March 4th, Fractions).

Gurisha

PART 1: How Are The Circles Similar?

What the circles have in common is the second half of every circle is not coloured in. Another thing I noticed about the circles is the coloured parts of the circles ( Or even the white parts) are counting by 1’s. Example: The red circle has 1 part coloured in. The blue has 2 parts coloured in. The yellow circle has 3 parts coloured in and the brown circle has 4 parts coloured in. So the pattern is 1, 2, 3, and 4. I also discovered there is another pattern. The first circle is split into 2 pieces. The second circle is split into 4 pieces. The third circle is split into 6 pieces, and the fourth circle is split into 8 pieces. So the pattern is 2, 4, 6, and 8. You could also look at these patterns in a different way instead of pictures. If you look at the numerator on every fraction, the


numbers will be 1, 2, 3, and 4. If you look at the denominator on every fraction, the numbers will be 2, 4, 6, and 8. The circles are also similar because all of the circles show one fraction. If you add the numerator by itself, the answer is going to be the denominator. Example: The first circle’s fraction is 1 over 2. If I add 1 with 1 the answer is 2 (our denominator). 1+1=2. I also noticed the denominator for each fraction is even. 2, 4, 6, and 8. (I added patterns in my comment because I wanted to share the patterns I discovered.)

Mrs.Sabell

I think it is wonderful that you are incorporating our fraction vocabulary into your post (numerator and denominator)! It looks like you and Sami noticed some of the same things. The fractions above are what we call equivalent fractions. Equivalent fractions are different fractions that name the same number. For example, if we look at the images above, the circles are all broken into a different number of pieces and are described by different fractions but they are all representative of half the circle. Can you think of any other way to break the circle into pieces, colouring part of them, where the image will still show half of the circle coloured?

While quite lengthy, this example demonstrates the time and effort some of the students put in. It also captures a subtle cultural change that occurred in our class, which was an increased excitement around “noticing” things in the math. In Gurisha’s case, she noticed the patterns that were occurring but did not extend those patterns in order to predict other fractions that would create half a circle. She also lacked the vocabulary to describe the relationships between circles. My intention was to introduce the idea of equivalent fractions, strengthen her understanding and still encourage all the things she did notice.

Mathematical Discussions Outside of the Classroom

Another undeniable advantage to the blog was that it promoted mathematical thinking and conversations outside of class time. Often I found myself grateful for the extra exposure that my students were being provided due to the blog. The blog was an ongoing conversation, happening at home, where students had to read, think, ask questions and share their ideas. They were able to receive encouragement, help and feedback as they needed, which supported their learning outside of the classroom.

GREATEST CHALLENGESAlthough the benefits overshadowed the less desirable parts of running a blog, there were a number of challenges which I believe are worth discussing: specifically, the issues of sustainability, ensuring and maintaining student participation and the unfortunate risks associated with social media.

Sustainability

The blog was teacher-driven and, by virtue of its dynamic nature, necessitated a fairly substantial allotment of teacher time.

Throughout the course of the week, I would make one class post and spend approximately three hours responding to students. Because the blog provided a wealth of invaluable feedback, this time replaced marking and other forms of formative assessment.

Ensuring and Maintaining Student Participation

Another challenge was that a number of students consistently forgot or did not contribute to the blog. These students were broken up into two groups: students with parents who valued responsibility around homework but whose parents were unaware of what was happening in class and the students who did not have that kind of parental support at home. I attempted to deal with this challenge by eliciting student feedback, involving parents, offering class time, and providing positive encouragement. In the end, I was successful in increasing participation of the first group but unable to get the students in the latter group consistently participating on the blog.

Security and Social Media

I approached the implementation of the blog with some naivety around the possible drawbacks of social media. Although our blog was not accessible to strangers, I had given all my students the same password. Unfortunately we had an incident where an older sibling acquired access to the blog through her sister’s account and made a number of inappropriate entries. Due to the extreme nature of the posts, this issue was dealt with by notifying our administration and the parents, assigning students new passwords that were personalized, and using the incident as a teachable moment by following up with lessons on privacy and internet safety.

Conclusion

This entire experience has left a significant, lasting impression on me. The blog has enhanced my teaching practice by providing me with an alternate method of communicating and sharing mathematics ideas with my students and I have been able to hear their voice in a way that I had not been able to do before. The most enjoyable part of this experience has been the increased connection I have had with my students and the opportunity it provided to share my excitement about math. I have derived immense pleasure from seeing my students cultivate new understanding and develop an eagerness and proficiency in communicating their ideas.

I have included a number of the questions that I used on my blog in Appendix 1. As well, Appendix 2 is the handout I gave students to establish rules for responding and contains the sentence prompts that we used to practice.


APPENDICES

Appendix 1

BLOG QUESTIONS

Dec 20th- Square and triangle numbers

Above are two sets of numbers. On the left are triangle numbers and on the right are square numbers. Investigate the following:

1. Can you find the pattern for each? What do you notice?

2. If the pattern continued, could you predict the 10th number for each? What would it look like?

3. CHALLENGE: What connections are there between the triangle numbers and the square number sets?

Have fun and be sure to tell me about what you find!

Jan 15- Learn and Teach- “Best of times”

Choose a strategy from “The Best of Times” to learn about and teach the rest of us.

Your post should include the following:

1. The strategy you are teaching. Example: I am teaching 3 times tables using the strategy Double and add 3.

2. A step by step example of how to do one of the challenge questions (using the book as an example).

3. Your opinion on the strategy. Do you think that it is a helpful strategy why or why not? When might it make sense to use? Do you already use it?

Jan 30- Multistep Problems

In your feedback forms many of you said that you enjoyed our challenging thinking problems so here are a couple to work on this week! Keep in mind that I am interested in your thinking and the things that you try in order to solve the problem. I can›t wait to hear your ideas about how you attempted to solve them! :)

1. LEVEL 1: Mrs. Sabell and Mrs. Peters love to read. Mrs. Peters reads twice as many books a week as Mrs. Sabell. If Mrs. Peters reads 6 books, how many books does Mrs. Sabell read? What about if Mrs. Peter’s reads 8 books? What other possible solutions could there be ?

2. LEVEL 2: Robbie, Ryan and Joseph love playing Minecraft and have all built a number of castles. Robbie has three times the number of castles as Ryan and Joseph has half the number of castles as Ryan. Find as many possible combinations of castles that they could have in their world!

Feb 12- Sentence starters

This week we are going to practice using our sentence starters in order to provide each other with helpful feedback. As we talked about today, feedback can have different purposes. These include the following: helping someone or correcting misunderstandings, asking a question to make sense of something that will help your learning or theirs, asking someone to provide more detail about their thinking, encouraging the thinking of others (must be specific), offering a new idea that builds on the idea of someone else.

Provide feedback to the following:

1. I am going to use the doubling strategy to multiply 3 × 8. I know that 3 × 4 is 12 so if I double the 12 (12 × 2), I will get the answer 24. I know that this is correct because 3 × 8= 24.

2. I am going to use the doubling strategy to multiply 3 × 8. The answer is 24. I know because I used doubling.

3. I know that 64 divided by 8= 8 because 8 × 8 = 64.

4. 32 divided by 4 = 8 because I can take take 32 smarties and put them into 8 piles and each pile will have 4.

Jan 22- Connecting Area and Multiplication

Today we explored the connection between area and multiplication! In your own words, answer the following questions:

1. What is area? If you can’t remember exactly what it is tell me something that you do remember about it.

2. What is the connection between area and multiplication? Use an example to support your explanation.

3. Share anything else you found interesting about the lesson today.

Feb 18- Exploring divisibility rules

Choose one of the following to try:

LEVEL 1: Give an example of one 5 digit number that is divisible by ALL of the following numbers: 1, 2, 3, 5, 10. Tell us how you can prove that your number is divisible by 1, 2, 3, 5 and 10. (Hint: remember the patterns that helped us come up with divisibility rules yesterday).

LEVEL 2: Give an example of one 10 digit number that is divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Tell us how you can prove it.


Feb 25- Long division- show me what you learnedThis week you have been learning long division with Mrs. Peters. Tell me what you have learned. Include an example to describe (in words) the steps to divide. Let me know if there are any questions that you have or things that didn’t quite make sense.

***Also, feel free to respond to each other using thoughtful sentence prompts. The purpose of your comment should be to encourage them to elaborate on their thinking, clarify misunderstandings, answer questions that help your understanding and/or tell them something that are doing well.

March 4: Fractions

Today we did some exploring and made fractions in a number of ways.

PART 1: Below is an image of 4 circles that look similar but are represented by different fractions. Can you tell me what they have in common? How are the different?

PART 2: Can you find 2 fractions that are different but would look similar when represented by a picture (like in the case below)?

April 15th: Comparing Fractions

LEVEL 1: Choose 2 fractions to compare. Tell me which is greater and how much greater. Tell me how you know.

LEVEL 2: Choose 2 fractions that have different denominators to compare. Tell me which is greater and how much greater. Tell me how you know.

Appendix 2

RULES FOR RESPONSES

I notice that some of you are responding to others post and I think that this is fantastic. However, in order to make sure that our posts are valuable to each other I am only going to be “publishing” or approving comments to others that follow our rules for responses.

1. No “Good job” – as nice as this is, it does not inform the personof what was good.

2. Uses a sentence start or equally thoughtful comment that either helps the person correct a misunderstanding or encouragesthem by pointing out what they did that was interesting oradmirable to you.

BLOG SENTENCE STARTERS

I see that you are struggling with __________________. Have you tried _____________________?

I really liked when you said __________________because _________________________.

I can tell that you understand __________________because __________________.

I can see that you put effort into describing __________________.

I am a bit confused about _________________. Could you explain __________________?

I noticed that ____________________. I was wondering/thinking _________________.

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IMPLEMENTATION

16

BY HWIE LIE JOHNS

Fraction Activity

The purpose of this activity is to help students develop a deeper understanding of fractions in a collaborative, hands-on and engaging way. In choosing activities that do not involve computations, these activities not only become more accessible to students of all abilities, but they also focus on the meaning of fractions instead of computation with fractions.

Many of the core competencies in the new curriculum are also being addressed through these activities, as will be elaborated later.

I have done this activity with six different classes of grade 6 and 7 students this year. The activity takes about 60-75 minutes. To begin, I randomly placed students into groups of three. (If you have leftover students, they can be put in a group of two). I had three or four stations set up around the classroom, depending on how many students were in the class. The students then proceeded to work through each station. I gave the students about 15-20 minutes to work on each station before I asked them to move to the next one. At the end, there should be time left for discussion/questions/answers.

STATION #1Materials:

Each student will need a double-sided sheet with six 8x8 grids on each side:

Table Instructions:

Show at least 6 ways to divide a square into quarters

STATION #2

(From Marian Small’s website onetwoinfinity.ca)

Materials:

Pattern blocks and one answer sheet per group (see below)

Table Instructions:

1. Construct a design where:

• the area of the red section is 2.5 times the area of the yellow section AND

• the area of blue is 2/3 the area of green

2. Answer Questions on sheet provided

Station #2 Questions: Names _____________________

• What fraction of the pieces are yellow? _______ Green? _______

• What fraction of the area is yellow? _______ Green? _______

Is this the smallest # of pieces you can use? How can you use more pieces but keep the general rule the same?

STATION #3

(From Jo Boaler’s website youcubed.org)

Materials:

At most 150 pieces of square paper. (I found that even with students working in groups, many students like to fold their own piece of paper. If they reuse their paper for some of the constructions, then you can save paper). You will also need to photocopy instructions for each station.

Instructions:

For each question, start with a square sheet of paper and make folds to construct a new shape. Then, explain to your partner how you know the shape you constructed has the specified area.

1. Construct a square with exactly ¼ the area of the original square. Convince yourself and others that it is a square and has ¼ of the area.

2. Construct a triangle with exactly ¼ the area of the original square. Convince yourself and others that it has ¼ of the area.


multiple solutions and creativity. This hands-on activity provides students a concrete visual for representing fractions and equivalent fractions. It also helps reinforce the difference between the fraction of pieces compared to the fraction of the area.

Comments on Station #3:

By starting off with an easy example and asking students to “convince yourself, ” students get practice communicating and reasoning. The act of folding paper reinforces “how” you can get a fraction from a whole. The last example has been changed from Jo Boaler’s original example. The new example has students connecting square numbers to geometry.

POSSIBLE SOLUTIONS

Station #1

Station #2

Using the least amount of pieces: 1 yellow piece, 5 red, 3 green and 1 blue. 1/10 pieces are yellow and 3/10 pieces are green. The fraction of area that yellow occupies is 3/13 and the fraction of area that green occupies is 3/26.

Station #3

Fold paper on the dotted line.

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3. Construct another triangle, also with ¼ the area, that is not congruent to the first one you constructed. Convince yourself and others that it has ¼ of the area.

4. Construct a square with exactly ½ the area of the original square. Convince yourself and others that it is a square and has ½ of the area.

5. Construct a square with exactly 9/16 the area of the original square. Convince yourself and others that it is a square and has 9/16 of the area.


By asking students to draw many different solutions, we are able to tackle common misconceptions. For instance, I found that most students thought that each quarter had to be congruent:

As well, many students thought that the following were different solutions:

At this point, some students may get stuck. If they require guidance, try to help them as little as possible without giving them the punch-line. For example, show the group a non-example and engage them in a discussion about whether or not it is a solution and why.

As soon as one student catches on, the rest will follow quickly.

With this task, students are problem solving, representing fractions, and communicating with their peers. Mathematics language, involving area, congruence, rotations, and symmetry were just a few of the words that students were using.


By asking students to construct “a” design, this allows room for

Lottery tickets that would never sell.


Lottery tickets that would never sell.

The organization of BC’s curriculum endeavors to change more than the placement of concepts; it includes a pedagogical shift for which many teachers may be unprepared. Education systems around the world are challenged to change designs and approaches to teaching and learning in order to meet the needs of students. Flexible learning approaches, embedded inquiry experiences and a new focus on competency-based instruction.

The Island Numeracy Network (INN) aspires to address these transformations by creating useful and straightforward resources accessible to all educational stakeholders. The INN offers support and mentorship to teachers spanning the island and has created the Diagnostic Mathematics Assessment used all over the province. Comprised of representatives from most island districts, the INN has identified the key areas of teacher concerns are all attempts to make evaluation more personal and relevant.

Exploring Principles of Learning

The journey began with our goal of identifying important learning principles which reinforce particular competency areas. We started with the Principles to Action resource published by the National Council of Teachers of Mathematics (NCTM). Over the period of five meetings, the INN explored the beliefs behind productive mathematics practices. We looked at the important principles identified by the NCTM as imperative to effective mathematics pedagogy. Not surprisingly, the assertions behind the research supported the changes in our BC curriculum and aligned with the curricular competencies.

Exploring the Principles to Action resource demanded most of our meetings in the first year. In year two we began to look globally. We directed our attentions to the OECD research document: The Nature of Learning: Using Research to Inspire Practice. The document includes the Fundamentals of Learning, the seven Principles of Learning, key shifts and an emphasis on innovative learning environments, which reinforced particular curricular and core competencies. We then matched these to curricular competency areas.

At this point, we contemplated the principles closer to home: such as the First Peoples Principles of Learning (FPPL). We endeavoured to link these principles to certain competencies from the NCTM and

OECD principles. It became quite apparent that the FPPL cannot be fit into one particular competency area but should be woven throughout all the competencies.

By linking all these principles to the curricular competencies, and by highlighting the connections to the FPPL, our hope is teachers will recognize and understand the purpose and advantage of teaching and learning through competency-based experience.

Activities and Thinking Prompts

Many teachers have asked “What do these competencies look like in my mathematics class?” as well as “How will I know if my students are learning the content through the competency-based experiences?” Our intention was to create a resource that answered these two important questions.

Creation of the thinking prompts began with our members brainstorming and researching which questions would reveal the expectations of the competencies. Secondly, we also contacted Sandra Ball in Surrey School District; she had just created a list of mathematics question prompts linked to the competencies! These prompts grew, shrunk and changed numerous times each time we met in the fall.

Activities were our next challenge: which types of activities would best highlight each competency area? Much in the same fashion as the thinking prompts, we used resources and our own experiences to create a long list of tried and true teaching and learning activities.

Creating the Resource Tool

BCAMT’s grant provided the INN with the most important thing we needed to see our resource to completion: time. Many of our INN members were able to meet over a period of two days without interruptions of work and home.

The Final Product

With the help of Teri Ingram at the print shop in school district 71, our members experimented with different layouts and decided on the one we have today: a rising sun representation in the style of of Roy Henry Vickers. This style allowed for the connections between the competency areas and the supporting activities, thinking prompts and principles of learning. At the centre, the First Peoples Principles reach out to all the components of our resource.

We look back fondly at the numerous hours of professional development behind its creation and hope it will contribute to the teaching and learning opportunities of which we know educators are capable. Most of all, we hope it will provide a place to turn while educators grapple with the much needed changes in how and what we teach and assess.

Please find the resource poster available for grades K-5 and grades 6-9 on the next two pages. Please contact the Island NumeracyNetwork if you have any questions.

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BY DEANNA BRAJCICH

Assessing and Teaching Through the CompetenciesIsland Numeracy Network’s New Resource


First Peoples Principles

OECD & NCTM Principles

ThinkingProm

ptsLearningActivities

Mathematics

BC Ministry of Education M

athematics Curriculum

2016 • Grades K-5

Island Num

eracy Netw

ork • [email protected]

BCAM

T

Connecting & Re�ecting

Communicating & Representing

Understanding & SolvingReasoning & Analyzing

COMPETENCIESCURRICULAR

Learning ultimately supports the w

ell‐being of the self, the family and the community.

Learning involves recognizing the consequences of one’s actions.

Learning involves recog nizing that some knowledge is sacred.

Learning is holistic, re�ective, experiential and relational.

Learning involves generational roles and responsibilities.

Learning recognizes the role of indigenous knowledge.

Learning is embedded in m

emory, history, and story.

Learning requires exploration of one’s identity.

Learning involves patience and time.

Assessm

ent through curricular competencies poster k-5


First Peoples Principles

OECD & NCTM Principles

ThinkingProm

ptsLearningActivities

Mathematics

BC Ministry of Education M

athematics Curriculum

2016 • Grades 6-9

Island Num

eracy Netw

ork • [email protected]

BCAM

T

Connecting & Re�ecting

Communicating & Representing

Understanding & SolvingReasoning & Analyzing

COMPETENCIESCURRICULAR

Learning ultimately supports the w

ell‐being of the self, the family and the community.

Learning involves recognizing the consequences of one’s actions.

Learning involves recog nizing that some knowledge is sacred.

Learning is holistic, re�ective, experiential and relational.

Learning involves generational roles and responsibilities.

Learning recognizes the role of indigenous knowledge.

Learning is embedded in m

emory, history, and story.

Learning requires exploration of one’s identity.

Learning involves patience and time.

Assessm

ent through curricular competencies poster 6-9


RESEARCH REPORT

22

The Silent Partner: Reading to Learn Mathematics

Recently, I had the pleasure of teaching mathematics to a Grade 9 Preview class in summer school. In the context of our British Columbia school system, that is a whirlwind seventeen-day experience at the beginning of July with classes being nearly four hours in length. There are benefits in working for a concentrated period of time with the same group of students, but also difficulties in ‘doing mathematics’ for that length of time with only a fifteen-minute Nutrition break. On the last day of class, one of my students gave me a note that said, “When your bum is numb, your brain is too.” However correct this statement, the fact remained that this is the nature of summer school and the time given to investigate the several topics that comprise the Grade 9 mathematics curriculum. My pedagogic goal was to expose these students both to the content and to the strategies that would make their upcoming credit Grade 9 mathematics experience a little less daunting and frightening for some, and hopefully more successful for all.

My personal goal was to explore the impact of reading on the learning of mathematics. I have been interested in the connection between reading and mathematics for quite some time. Particularly when working one-on-one with my mathematics students, I have watched them stare and tilt their heads as if puzzling to try to make sense of the mathematics that is in front of them. I have been curious as to what and how they are ‘reading.’ I am not referring solely to their comprehension of words or word problems, but about reading mathematics. In reviewing literature relating reading and mathematics, there exists a wide range of connections between the two disciplines. Along this spectrum there are topics from how to teach decoding strategies or how to teach reading in the mathematics classroom to reading to learn mathematics. In the subtleties lie vast differences. I wanted to explore them. How, then, to proceed?

The most common text in mathematics classrooms is the mathematics textbook. Students that I have worked with and watched struggle to read mathematics are most often working from a mathematics textbook. For this reason, I decided to explore reading through the Mathematics Makes Sense 9 textbook that my summer school students would be using. I planned to use the textbook in the sense of a flipped classroom. Rather than give traditional lecture notes every class, I would have my students read and ‘translate’ a section from their text the night before we engaged in tasks based on this reading. One difficulty with summer school, however, was that

more content was investigated every day than in a regular-length mathematics class. Therefore, my goal was to try to do this twice a week and on alternate days do a textbook reading activity in class. I knew I would have to monitor and adjust often. In what follows, I describe a few of the activities we undertook, how I adapted and what I noticed.

Setting the Scene: Day 1

It was a gut feeling, but I wanted to make reading an accepted part of my mathematics classroom activities, not a special case or focus. I did not want to recreate Teach Reading in Mathematics (Barton & Heidema, 2002), a book that fosters reading strategies in the mathematics classroom, as this could plausibly segregate the two disciplines by accentuating one in the other. Nor did I want to emphasize teaching decoding skills through word problems or in any way make mathematics subservient to reading.

High school teacher Monte Else also wanted his students to read to learn mathematics because he was concerned about students missing class and being able to remain current. However, he prescribed a set guide that his students had to complete as nightly textbook reading homework (2008, p. 23). Each one of his homework assignments was out of ten marks and, for example, his students would receive one mark for describing what they would learn and two marks for defining key terms. My belief is that reading is a personal, meaning-making, interpretive and generative activity and thus the sense-making had to be autonomous and not as a directed worksheet (Weaver, 2009).

When I distributed the highly anticipated textbooks to the Grade nines – and the students did look at these texts as their window of advantage to next year – the first task was a textbook scavenger hunt. On Day 1 we were reading to find information, but also to become comfortable and familiar with reading and looking through our textbook and talking to other classmates about what we were reading. The task was a printed page of directed questions such as “What is the title of Section 5.1?” I did write a few more challenging and less directed questions on the board such as “Find a word or mathematical symbol that is new to you,” but overall it was a directed activity. Only one student found a famous mathematician or attempted the extra questions.

For homework that night I gave them a journal assignment. This assignment asked them to look at their textbook independently and answer prompts such as “What do you look at most/least?” and “When you read through a section of your textbook, what confuses you most/least?” This student’s response summed up what I read from the majority of written submissions:

When I look at the textbook, I look mostly at the questions as they are what is needed to be completed. I look least at the featured mathematicians as they teach about history, not mathematics. New mathematical language sometimes confuses me when I read the book. I enjoy the pictures and the graphs because they clarify my confusion on a subject or a problem.

Overall what these students were most comfortable with were the illustrations and visuals, but the explanations, technical words or

BY SANDRA HUGHES


equations were more difficult and they wrote that they avoided these. Homework questions were also good, because that is what was normally required, or in the words of the following student:

The thing I look at the least is the Example because before the teacher gives us any homework she gives us a lesson by telling us how to do the chapter and solve problems so when it is time to do the homework I know so much that I go straight to do the homework.

While I had anticipated that most students would use their textbooks for homework questions, they had confirmed that they did not read the textbook for information or, for what I felt, was to learn. Was it possible to get them to read to learn mathematics?

Rising Action: Flipped Textbooks

On two occasions during the first full week of summer session, I asked my students to read pages from their textbook for homework and then create notes for a friend that had missed class. I asked them to write these notes in their own words and with attention to explaining the mathematics. I chose the ‘Connect’ sections of the textbook which explain mathematical ideas or processes.

The first reading assignment from page 74 concerned multiplying and dividing powers. When I read their notes the next day, one phrase in particular caught my attention: ‘Patterns emerge.’ There are certain phrases I felt the typical Grade 9 student would not normally use and this was one. Because it caught my eye, I checked the textbook and counted the papers. There were variations such as ‘Patterns will appear’ but of the nineteen homework papers submitted to me, ten used this textbook phrase. Many students also copied the same numerical examples as the textbook.

For the second reading, I assigned two separate pages: page 107 on adding rational numbers and page 115 on subtracting them. I was hoping my students would connect the two separate sections from the textbook. Overall, they did not. This assignment was eerily similar to the first in that there was one sentence that jumped out at me when I read their work and it was common to every paper except two. From the writing of one student, ‘To subtract rational numbers do the same as subtracting integers.’ What surprised me this time was that most students noted that adding rational numbers was like adding fractions, while subtracting rational numbers was like subtracting integers. These comparisons were taken directly from the textbook. There was no connection between adding and subtracting rational numbers and most students kept the two sections separate just as the textbook had. Since almost all students incorporated the sentence comparing rational subtraction to integer subtraction, I had to wonder why.

An Unwitting Protagonist: Pressurized Reading

I wanted to draw attention to the amount of information on a textbook page of mathematics as compared to a novel. I borrowed a simple strategy from Diane Kahle’s article “Getting to Know Your Middle Grades Mathematics Textbook” (2006). I asked my students to read a page from their textbook for approximately a minute and

then to close it and write down what they remembered. Next I asked them to draw a line under this writing and reread the page to record what they felt was important or what they had missed. What I noticed about this work was the wide variety and diversity in how they represented what they read. The range extended from one student only listing key words such as ‘rational numbers, opposites, ...’ to another who simply drew a number line with several different rational numbers upon it. In other words, during the minute of reading everyone saw or read something different and these differences were reflected in the written record. This variety in interpretation indicated to me that the previous two journal entries were not completely personal meaning-making.

Support Characters: Simple Strategies Used in Class

When I read mathematics, I am aware that it is more complicated than reading a novel or literature. It involves more than the comprehension and interpretation of words. There are words, and there is the mathematical register; the implied understanding that comes with the use of certain terms and their organization. There are also diagrams, charts, equations and number lines, and the reading does not solely occur in a left-to-right, top-to-bottom format. My students work and questions suggested that they were reading only one format at a time and not using one to support the other. This appeared dependent on the focus of the task. For example, students would predominately use written words for journal entries because these they perceive to be writing tasks. In class however, they appeared to focus on computations and numbers because the context was a mathematics classroom. One of my students could not follow the numeric explanation for a surface area problem in class solely because she did not read the written word ‘front’ nor locate its placement on the diagram. She was not connecting the illustration, diagram and numbers.

Between reading assignments, I engaged my class in a few simple activities designed to draw attention to the complexity of reading mathematics and to the idea that meaning-making in mathematics, as in reading, is personal and unique to the individual. I wish I had had the time to repeat some of these activities, but the limited duration of summer school was unfortunately prohibitive. One very simple activity I used was based on the idea of peer editing from my elementary teaching experiences in Readers’ Workshop. I had my mathematics students read each others’ summaries and compare ideas. I asked them to record what they noticed on their work. I was impressed with how serious they took this activity and how much they read each others mathematicsematics. One boy wrote, “He thinks the original shape is going to be minimized or enlarged. For this to be possible, he believes that the shape must be of right corresponding lengths and right proportion. Mine (organization of information) was fairly straight down but my partner has multiple connections.” Another student wrote, “I tryed [sic] to make my own examples that kind of failed and my partner just went with the textbook.”

I also incorporated a strategy called ‘Cloning an Author’ (Borasi & Siegel, 2000, p. 64). Borasi and Siegel have researched extensively on reading to learn mathematics. While I was focusing on the mathematics textbook, their work focused predominantly on the use


of non-traditional text such as historical essays. However, cloning an author is an activity I felt would work with the mathematics textbook as students record important points or terms and definitions on separate cue cards. Then they organize these cards into a structure or organization that makes sense to them. We used rulers to tear 8 ½ x 11” sheets of blank paper into small cue-card-sized rectangles.

For this activity I assigned two different sections of the textbook on scale factor, from the enlargements section on page 319 and the reductions section on page 326. After they recorded one idea per card, I asked them to organize their facts in a way that made sense to them. I was very clear that they did not have to use the format of the textbook nor use every card they wrote if it was redundant. I was curious to see how they would link their points, particularly as they had not connected the similarities between adding and subtracting rational numbers previously.

While many students still kept the sections separate, they were starting to compare them (example–see Fig 1). Some were linear in presentation while others were tabular. One student placed related ideas from the two sections on cloud-like pieces and used colour to indicate which section the idea came from. As they were finishing and comparing their work with their neighbour, I walked around and commented on the diversity of presentation and the different ways of making sense. I held up examples as I talked. One boy thought the colour accents were a great way of connecting facts, and added colour to his work. After, I posted a variety on the bulletin board as examples of the different ways we think.

Conflict: Reading a Mathematics Story in Mathematics Class

The Cloning-an-Author activity had focused attention on personal meaning-making and had generated dialogue and investigations into scale factor, but many students were still struggling with scale factor. Serendipitously, I had been flipping through mathematics resources at the summer school and located a book I was familiar with titled MATHEMATICIANS ARE PEOPLE TOO (Reimer, 1990). The first chapter is on Thales and the first few pages describes how he calculated the height of the Cheops pyramid. Proportions! I photocopied those few pages. To start, I asked the class to read the story with their neighbour. Perhaps it was the age group, but the class happily read aloud to one another. The class was full of story. This time for their journal homework, I loosely used a different strategy from Borasi and Siegel titled ‘Sketch to Stretch’ (p. 64). I asked them to draw a picture of the mathematics in the story and explain how this was similar to what we had been doing in class.

I was amazed at the images and explanations this class produced, and this for a non-credit course! Interestingly, every student submitted their homework to me. Interestingly, everyone used their own words. The few examples attached here represent the diversity of presentations, yet all students connected proportions, scale factor and similarity as well as history to what we were learning (example –see Fig 2).

Climax: How Can I Explain My Thinking?

We were in last few days of class. I wanted to try another flipped textbook activity to see if there would be any difference in the written reading responses of my students. Spontaneously I decided to have them read their textbook in class. I had noticed two pages in the textbook titled “How Can I Explain My Thinking?” (p. 152–153) After a brief discussion, they set out to independently read and explain how to write equations to describe patterns.

I was impressed by students’ personal meaning-making and voice. Their work showed creativity, personal voice, personal preference, and thought. One student used metaphors to remember the ‘three ways’ of explaining your thinking. Another student generated their own numeric examples, complete with a miscalculation. Still another student, who always wrote very little in class, described, “I think diagrams are my favourite thing to show my thinking” and wrote nearly a page of text.

Most astonishing for me from all of this was the question one boy asked while I was circulating the classroom. While he was writing his journal entry, he called me over to ask, “Why did they put the 2’s here?” He was looking at the numeric representation next to the diagram. He was reading the mathematics and trying to make sense of where the numbers came from.

Resolution

As with any good story, there is something that leaves you wanting more. In three weeks of reading the mathematics textbook for explanations, students were beginning to notice and question. The student above was noticing the mathematics that was embedded in the illustrations and was trying to connect the visual and the numeric representations.

Reflecting on this experience, I know that I cannot continue to be the only individual in my classroom interpreting and translating the textbook. What became enjoyable and important was that I was not the only one in the classroom asking questions: my students were beginning to ask me the ‘why’ rather than only the ‘how’ questions. I do not expect that my students will read the textbook and miraculously understand mathematics, but I know that I will give them the opportunity to consider and interpret it rather than continually tell them the meaning they should know. In addition, from my professional readings and from conversations with both high school teachers and university professors, students entering university struggle to read and understand their textbooks. While not every high school student is university bound, every high school student should have the opportunity to become a mathematically literate citizen.


Fig 1

Fig 2

ReferencesBaron, L. et al. (2009). Math Makes Sense 9. Pearson Education Canada.

Barton, M.L. & Heidema, C. (2002). Teaching Reading in Mathematics (2nd ed.). Aurora, Colorado: Mid-continent Research for Education and Learning.

Borasi, Raffaella, and Marjorie Siegel.(2000). Reading Counts: Expanding the Role of Reading in Mathematics Classrooms. New York: Teachers College Press.

Else, Monte. (2008). Reading as a Learning Strategy for Mathematics. Retrieved from http://scimath.unl.edu/MIM/files/research/ElseM.pdf

Kahle, Diane. (2006). Getting to Know Your Middle Grades Mathematics Textbook. Retrieved from www.ohiorc.org/adlit/inperspective/.../vignette3.aspx

Reimer, Luetta & Wilbert Reimer. (1990). Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians. Indianna: Dale Seymour Publishers.

Weaver, Constance. (2002). Reading Process and Practice. Portsmouth, NH: Heinemann.

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26

ARIEL U’CHONG MUIRHEAD

A Lesson Plan Journey

This reflective journal-like document outlines a pro-cess of creating a mathematics project that brings social justice into the classroom.

July 2, 2016

While working on an assignment in my mathematics methods course in the Professional Development Program in the Faculty of Education at SFU, I came up with an idea for a project that combines student-based inquiry and the history of Aboriginal land in Canada. My idea was that groups of students could look at different Aboriginal nations’ land claims and investigate how the area of the land changes over time. From a mathematics perspective, students would be able to investigate how to calculate areas using maps and legends, research methods of measuring land on ground level (surveying), and become familiar with different units of measurement for land area and how they are related to one another (square kilometres, acres, hectares). From the social justice side, students would be able to learn what reservations are and their significance to Canadian history, to compare the area of the land over time (calculate percentage of land lost), to evaluate the impact of treaties, and to learn about the Aboriginal nations in their region. I think that using mathematics to investigate the changes in land area is a good way to critically examine the effects of government treaties on Aboriginal nations. Students will work in groups of 3-4 and present their work as a poster. Ideally, each group would be able to research the land of a different nation in South Western British Columbia, and the project could culminate with a “fair” type class, where the posters are displayed and students are able to walk around and look at each other’s research projects.

I see this project fitting in nicely with a unit on area. The draft of the grade 10 mathematics curriculum cites “area and volume” under content, and I think grade 10 would be a good year to do this project, because the students will be old enough to gain some appreciation for what their research shows and means.

I have two immediate problems. First of all, I’ve never taught mathematics or designed a mathematics unit plan before. I think for this assignment, I would like to focus on this inquiry-based project and develop it well. The final product of my assignment will be a

unit plan, but I plan on keeping the lessons leading up to and after the project fairly simple so I can focus my main efforts on creating a good project. My second problem is that I have no knowledge on Aboriginal land or treaties. My next step is to do some research on my own and see what kind of resources are out there that would work well with this project.

July 7, 2016

I have done some googling and it’s difficult, but not impossible, to find resources like the ones I’m looking for. I think that the difficulty in finding these resources, such as explicit maps of Aboriginal land claims, even current ones is very telling about the general public’s interest in these matters. Let me discuss some of my findings.

I found it very difficult to research some of the Aboriginal nations on the mainland for a few reasons. First of all, I’m not from the mainland, so when landmarks are described, they generally mean nothing to me. Secondly, many of the nations have large pieces of land and much of the land is scattered into parcels. I decided to look into a nation I know more about, so I chose the Snuneymuxw First Nation which is located in Nanaimo, my hometown. I have learned that Snuneymuxw First Nation has land claims over 6 small reserves (only one of which is actually habitable, by the way), but I’ve been unable to find a map (http://pse5-esd5.ainc-inac.gc.ca/fnp/Main/Search/FNReserves.aspx?BAND_NUMBER=648&lang=eng). I was hoping that Google maps may have a feature that outlined First Nations reserves, but when I type in “Snuneymuxw reserve” in Google maps, it gives me the location of the Snuneymuxw First Nation office. I have found some poorly drawn maps, or some maps that have nothing to do with First Nations but have the one or two of the reserves included on them, so I think it would be possible to find these maps with enough diligence. One would think that the Aboriginal Affairs and Northern Development Canada would be an excellent resource for this kind of project, but it’s a difficult site to navigate (http://www.aadnc-aandc.gc.ca/eng). Their “interactive map” of First Nations of Canada and their “geography” links are startlingly unhelpful. Through this website, though, I was able to find the names of the reserves and the area of land they occupy, which is valuable information. I’m starting to believe that finding physical copies of maps at public libraries or city historical centers may be easier than finding them online.

Seeking out physical maps is something that I’m not going to do at this point in time. I will design my lessons so that if students are not able to find the resources they need, I will provide alternative options.

One very good resource I was able to find was for Stó:lō First Nation. They published a massive atlas called the A Stó:lō-Coast Salish Historical Atlas. I’ve been unable to access the book itself, but this pdf provided me a glimpse of the table of contents and I believe it will contain everything needed for the project (http://artsandscience.usask.ca/keithcarlson/Publications/1-Books/A%20Stolo%20Coast%20Salish%20Historical%20Atlas.pdf). There is also a copy of the book at Vancouver Public Library (Call #:970.3 C65S875c).


Over the years, as more and more students complete this project, my resources can accumulate. For the first year, I can provide my student with more computer lab research time and provide alternative options for the project so that the project still fulfils the mathematical requirement for the project. I am going to start designing the unit plan, and I will develop the project based on information I can find about Snuneymuxw land claims.

July 21, 2016

I want to start drafting out my unit plan now, so my first step is going to be figuring out what area concepts are considered most important to cover in grade 10, since the newly revised curriculum simply states “area and volume” under content. I am going to look up some grade 10 mathematics text books and teaching resources to see what kind of problems grade 10 students are expected to be working with.

I just saw that the revised mathematics 10 curriculum also lists “metric and imperial measurement conversions” as content that students are meant to know. This fits in very well with my conversion between square meters, acres, and hectares. I now see this project fitting into a Measurement unit. This is my idea of the possible outline for the Measurement Unit:

• Estimating measurements using unconventional units (i.e., how many pencils high is the desk?)

• Measuring in metric units• Metric unit conversion• Area of rectangles, circles, and irregular shapes• How to calculate large areas• Large area unit conversion• Project introduction with computer lab time• Calculating volumes of right cylinders, right cones, right

pyramids, right prisms, and spheres (taken from the old BC Mathematics 10 curriculum)

• Project conclusion and presentation

Also in the grade 10 revised curriculum are the following big ideas: We can describe, measure, and compare spatial relationships and Analyzing data and chance helps us to compare and interpret. These can be incorporated into my unit plan.

July 22, 2016

Worked on my unit plan a bit. Started trying to connect the unit and the project to curricular content, competencies, and big ideas.

July 25 2016

I’m working on my unit plan and realizing that I’m planning out lessons that don’t really incorporate anything to do with social justice. At first I thought that I was doing a whole bunch of work for nothing, but I’m enjoying creating a mathematics unit plan and I think it’s a good experience. The social justice aspect of this assignment will be present in the final unit project.

Here’s a first draft of the assignment criteria:

Aboriginal Land Inquiry Project

Purpose: to mathematically examine how the area of land belonging to an Aboriginal nation has changed over time.

In a small group (3-4 people), you will research how the area of land belonging to a First Nation of British Columbia has changed over time. The findings from your research will be presented in poster form at a poster fair on [due date]. You will have two full classes in the computer lab and one full class to assemble your poster. Use the following questions to guide your research:

• What is the history of the First Nation you are researching in association with the land? What is the importance of land in their culture? How is land perceived?

• What is an Aboriginal land reserve and an Aboriginal land claim? What are the differences between the two?

• Discuss different ways that large areas of land are measured.

• How much total land is claimed by all First Nations groups in Canada? What would this total area look like grouped together on a map of Canada? Report this area is square meters, square kilometers, acres, and hectares. What is the population density compared to the population density of all of Canada? What percentage of the Canadian population is Aboriginal? What percentage of Canada’s total area belongs to First Nations groups?

• What is the current area of land claimed by the First Nation group you are researching? Try to find a map of this area with the border of the Aboriginal land. If after extensive research you are unable to find a map, draw what this amount of land would look like on a map of the community. Report this area is square meters, square kilometers, acres, and hectares. What is the population density of this area compared to the population density of the municipality?

• How has this area changed over time? Research any important treaties pertaining to the land belonging to the First Nation and report whether the treaty increased or decreased the amount of land. Try to find maps, but if you’re unable to, represent the changes in land area on a map of the community.

• Are there any current land disputes? If yes, how much land is being disputed and where is the land located?

• What challenges did you face researching this project?

Your poster should include a paragraph as an introduction (describing the purpose of the inquiry and what you set out to learn about) and a paragraph for a conclusion (discussing your major findings and their implications). Also include a bibliography listing the books and websites you used in your research. Ensure that all the maps on your poster have a title, a legend, and a scale.


Your group will informally present your poster at the poster fair on [due date]. The short presentation should highlight the important findings and their implications as well as any challenges you faced while completing the project. All members are expected to contribute to the presentation.

Comments: I got all the lessons planned out. Now it’s a matter of thinking critically about the unit plan and filling in the remaining boxes. I think I should provide a more in depth rationale for the purpose of my assignment and how it ties into the curriculum. Also, I need to develop a self and peer assessment rubric for the students to fill out. (See above)

July 27, 2016

All right, just starting to fill in the boxes of my unit plan now.

Now that I have the boxes filled out, I think I have a better understanding of the rationale of including my inquiry project in this unit. I also think that my unit plan does a good job of providing this rationale so I don’t need to write a separate one. For a while I was worried that this project is not very “mathematics heavy”, meaning that it doesn’t require a ton of computation, but I have now realized that the project requires a strong understanding of area and it also makes area less of a formulaic concept (i.e., length x width) and makes it concrete and culturally important.

All I need to do now is develop the self and peer assessment rubric and decide how much weight they will have on the final mark.

Final reflection: Overall, I’m really happy with the unit plan I produced. Even though the social justice aspect of my unit plan really only comes into play for the inquiry project, I think that designing a mathematics unit plan was a good exercise for me. I was really worried at first considering that I don’t know a lot about teaching mathematics for social justice, but I am happy with the final outcome. I think that the inquiry project I developed introduces a valuable conversation to the classroom. It is a way to look at the loss of land that Aboriginal communities have faced over the years and the project will highlight the unfairness of Canadian history by focusing on the Aboriginal culture’s views of land and how much land was lost through treaties and negotiations. I believe the inquiry project also creates a connection between a mathematical concept and a concrete applications in the real world.

V

4 3 2 1

Introduction and conclusion (4) Introduction and conclusion are complete paragraphs. Introduction clearly and insightfully discusses the purpose of the inquiry and what you set out to learn about. Conclusion clearly and insightfully discusses the major findings and their implications.

Introduction and conclusion are missing or show minimal understanding of the project and the research done.

Content: (8)

• History of First Nation with land• Description of reserve and land claim• Map of Canada with Aboriginal land and

associated calculations• Map of research First Nation land and

associated calculations• Evidence of research into history of land

claims and land disputes• Challenges with the project

All content is clearly and insightfully described. Content is supported by maps or diagrams if necessary.

Information is incorrect/missing. No evidence of research is provided when information is absent.

Maps/images (4) All maps are clearly labeled and include a title, legend, and scale. The reason behind including the map or image on the poster is clear.

Maps do not have a title, legend, and scale. The reason for a map or image being included is unclear.

Poster (4) Poster is polished and professional. Poster layout is easy to read and appealing.

Poster is incomplete/done with obvious haste.

Presentation (4) All members of the group contribute to the presentation and demonstrate an understanding of the research done and the importance and implications of their findings.

Group members are unclear of the purpose or findings of the project and/or unprepared for the presentation. Not all group members contribute to the presentation.

Bibliography (1) Present/Absent

Total mark /25

Aboriginal Land Inquiry Project Marking Rubric



29

Mathematics for All!Needs, Prospects and Possibilities

LATIKA RAISINGHANI

IntroductionWhy do we teach mathematics to all? Is the sole purpose of teaching mathematics to prepare students to perform at a certain achievement level so that they can get entry into higher education institutes, and thereby, secure higher paying jobs and higher social status? Or, is there a greater cause for the teaching and learning of mathematics? How and what should we teach as mathematics to accomplish this greater purpose? I have always been drawn to these types of questions because as a student, teacher and teacher educator in multiple cultural contexts: I have witnessed several students feeling devastated, and succumbing to standardized, Westernized modes of teaching, learning and thinking.

I believe that the teaching and learning of mathematics should create opportunities that could empower diverse students in identifying, embracing, and contributing to a mathematics that is part of their lived experiences. It has been a continued quest for me to (re)search for ways that could help in making mathematics approachable, relevant, meaningful and rewarding learning experience for diverse students. In this paper, I continue this pursuit by first navigating and questioning the present realities of mathematics teaching. Then, I attempt to offer some pedagogical insights that might guide us, as teachers and learners of mathematics, to engage in mathematics in ways that can help us invite mathematics for all students.

Mathematics Teaching: Present Realities

In today’s schooling, mathematics is often taught as a neutral, value-free and acultural body of knowledge, which is disconnected from real-life contexts and lived experiences of students. Rendering curriculum as compartmentalized and fragmented, such teaching practices typically divide mathematical activity into pure and applied mathematics. When students are given a problem, they are often expected to classify it by drawing boundaries and distinguishing between “useful” and “extraneous” information. Unfortunately, in this process, mathematics becomes decontextualized as we enculture students to ignore the interconnectedness of mathematics by disjointing the connections between problems and real-life contexts.

The situation becomes worse when educators are forced to teach mathematics as a “ritual of alignment” wherein instructions are designed to align with standardized assessments with a mere focus of preparing all students to perform at a certain achievement level. Measuring students’ learning solely in terms of mastery and competence in formal, “classical knowledge,” such standardized, test-driven, text-based teaching of mathematics demands students to be “encultured” into a “culture-free” or “Eurocentric” mathematics which is essentially divorced from everyday realities that diverse students face (Gutstein, 2007/2012, p. 302). How genuine are such efforts of teaching mathematics, which disconnect mathematics from the daily socio-cultural political realities of life and emphasize the teaching of neutral, abstract, content-focused mathematics in hopes that it would provide full opportunities for life, education, and career choices to all students? How could we transcend the present modes of teaching and bring inherent interconnectedness of life and mathematics into today’s classrooms?

I suggest that we (re)think and decide critically—should we merely continue enculturating all students into the theoretical world of mathematics to comply with how mathematics is perceived in its ideological sense, or should we care about how mathematics could serve students and society? How can we create a “mathematics” that is really for all, and how should we teach it so that it empowers all students, and does not create status hierarchies? Are there any educational alternatives? The following section describes some approaches that, I suggest, will guide us towards what and how we should teach as mathematics so that it becomes “Mathematics for All!”

Promote Contextualized and Open-ended Inquiry Learning

Noddings (1994) has argued for the need to “differentiate learning of mathematics by interests and try to reduce differentiation by traditional status hierarchies” (p. 95). She has asked us to recognize the need to respect all students’ interests and provide choices—“at least by high school, we should provide different mathematics courses for students with different interests” (p. 97). She has recommended including various aspects of mathematics such as sociology, psychology, politics, history into the teaching and learning of mathematics to ensure “excellent programs for all our children” (p. 95). By suggesting the integrated instruction of mathematics, Noddings has echoed Boaler (1993), who hoped that by teaching mathematics “in context”, students would be able to perceive the links between problems encountered in school and the “real world” (p. 13). The use of contexts which are more subjective and personal would allow students to become involved in mathematics and recognize it as a means to understand reality and break their perceived image of mathematics as a remote body of knowledge.

However, this connection-making within mathematics would require us to move away from “traditional text-book based teaching” that usually presents mathematics as a procedural knowledge required for solving theoretical problems with no explicit connections to real-life contexts. To enable students to understand mathematics and utilize it in unfamiliar situations, we need to create opportunities for students to engage in processes of mathematics so that they can understand how mathematical knowledge can be


applied to solve particular problems, know if they are applying the knowledge correctly, and perceive the role mathematics plays in different daily-life practices (Skovsmose, 2007). One good resource to begin such open-ended mathematical inquiry in our own classrooms is John Mason, Leone Burton and Kaye Stacey’s (1986) book, Thinking Mathematically, which generates opportunities for students to understand the processes of mathematical discovery and think about mathematics differently. Integrating open-ended questions or problems which often have more than one “correct” answer allow students to utilize different strategies to approach the same problem in more than one way, and learn through the diversity of their shared mathematical ideas. An online available resource designed by Ron Pelfrey could also be helpful for introducing open-ended mathematics problems (please see Pelfrey, 2000).

This connection-making within mathematics can further be strengthened through problem-based and project-based inquiry learning. Rather than simply providing the prescribed procedures, problem-based learning begins with presenting problems that require students to unfold and understand mathematical procedures through the processes of solving problems. Holton and Lovitt’s (2015) Creative Activities in Mathematics, which is designed for problem-based mathematics investigations at the lower and middle secondary levels as per the Australian curriculum, could serve as an exemplary resource for incorporating problem-based learning in Canadian classrooms. One of the activities suggested in this book is the Nim like games, which ask students to alternatively pick one or two coins/tokens stacked as one or more piles (as per the difficulty level of the game) by following specific rules. These games create hands-on opportunities for students to continue and make sequences involving whole numbers, and understand concepts of divisibility and variables. Project-based inquiry learning encourages students to apply their mathematical knowledge in solving real-world problems. To allow students to see how their perceived roles of mathematics can be made real, such projects can be developed around students’ school settings and/or extended to wider community contexts by connecting mathematics to the community, culture, and place.

Embrace Culturally Relevant Pedagogies and Acknowledge Ethnomathematics

One way to connect mathematics with the community, place, and culture, is to acknowledge the relationship between mathematical aspects situated in a culture through ethnomathematics, which draws attention to the cultural nature of mathematics, and opens up the possibility for classroom conversations about how people in a particular culture use(d) mathematics for understanding and explaining their natural, social and political environments. Looking at mathematics through an ethnomathematical perspective recognizes the connections of mathematical ideas and particular cultural contexts, which can prompt active and flexible culture-based understandings among students and help them perceive mathematics as a human activity rather than a rule-bound, memory-based subject that often results in inflexible, classroom learning of limited use.

The University of Hawaii’s Project Mathematics and Culture in Micronesia: Integrating Societal Experiences (Macimise) is one such attempt to investigate ethnomathematical perspectives embedded

in the social practices of Pacific Island cultures and invite these into formal mathematics taught in local schools. As a participant in this Macimise project, I looked at how mathematics is embedded in the weaving of various Kosraean cultural artifacts such as baskets and food plates which are woven with coconut fronds, and suggested how the weaving of these could be utilized to introduce and strengthen the understandings of various mathematical concepts such as odd and even numbers, lines, intersections, angles, shapes, patterns, translations, and many other geometrical concepts in elementary classrooms (please see Mathematics in Kosraean Weaving in 2D designs under Resources at macimise.prel.org). Informed through my these experiences of investigating mathematics in Kosraean weaving practices, I suggest that the weaving practices of first nations peoples could be a starting point for elementary students in Canada: A teacher might invite elders to demonstrate and/or teach weaving to students and thereby, invite students to explore geometrical concepts such as patterns in real-life contexts before they are introduced to more complex and abstract concepts in higher grades.

However, such integration of cultural practices into the learning of mathematics would require us to respect and acknowledge multiple ways of knowing and not to see ethnomathematics narrowly— only as mathematics of “others, exotic, non-Western, nonliterate, underdeveloped people,” (Mukhopadhyay, Powell, and Frankenstein, 2009, p. 69) but as an inclusive mathematics that embraces a wealth of human activity beyond academic mathematics—the “historical and contemporary mathematical knowledge and practices of all peoples” (p.68). Mathematics in a Cultural Context (MCC) is another example of a successful ethnomathematical project that was initiated in response to the persistent exclusion of native language, culture, and pedagogy from the practices and norms of schooling in Alaska (Lipka et al., 2005). In one of the MCC projects, elementary students built fish racks with the guidance of Yup’ik elders. Asking students to optimize the space for drying fish in their fish rack, this project prompted deeper explorations of mathematical concepts related with shapes, measurements, area, perimeter, and volume among students by stimulating them to connect “school-based knowledge with everyday [community] knowledge” (p. 370).

Donald (2011) takes this cultural integration through ethnomathematics further by emphasizing the value of historical relationships and seeing culture (and culture practices) not as an artifact, but as a way of connecting the past to the future through the present by walking through the place, listening to the stories reflected in the memories of elders while developing and implementing culturally responsive curriculum. One good example of such an approach is Jungic and Maclean’s (n.d.) Math Catcher at Simon Fraser University, which utilizes multiple resources such as First Nations imagery and storytelling, models, pictures and hands-on activities to enhance elementary and secondary students’ motivation and interest to learn mathematics by creating opportunities to explore how mathematics is integrated in the Aboriginal cultural contexts. Indeed, storytelling provides an alternative way of learning, which helps in presenting mathematics in an interesting, relational manner, and in contexts that students could imagine, visualize, and relate to. The context and places of stories can also offer lessons to reconnect mathematics with the history, culture, and nature, and thus, motivate students to explore and extend their mathematics’


learning by critically analyzing the role mathematics might play in various societal issues.

Indeed, appreciation of the value of ethnomathematics as a people’s mathematics, and understanding it as a basic to all school mathematics, can lead students further to discuss about their “intentions [of using mathematics] and how they relate to the mathematics they may choose to do”. Investigating and (re)learning ethnomathematics can actually create a “place” that opens up another possibility of learning mathematics through place-conscious mathematics.

Create a Place for Place-conscious Mathematics

A place-based or place-conscious education is an emergent movement that allows educators to recognize the power of place and utilize it to contextualize the lived experiences of students (Green, 2005). Adopting from Cajete’s (1994) ecology of the indigenous education where the main purpose of education is educating “for life’s sake,” Grunewald has stated that the goal of place-conscious education is, how to become native to a place. Indeed, place-conscious mathematics allows us to see “place” as a concretization of abstract notion of culture and immediate contexts of our lives. By embracing it as a possible alternative to the classical mathematics dominant in today’s schooling, we could move beyond the present supremacy of standardized, test-focused, textbook-based teaching of Eurocentric mathematics.

Then what is stopping us from breaking this monopoly of Eurocentric knowledge in our mathematics classrooms? Gruenewald has shown us the path, but it is ironic to see that he himself is not very optimistic about place-conscious education becoming an ultimate reality in today’s schooling—“with its focus on “accountability,” the discourse, the language, or the grammar of school reform lacks a vocabulary for place” (Gruenewald, 2003, p. 642). So should we lose hope, or dare to create a dynamic place for the learning of mathematics through creating place-conscious mathematics? The challenge for us as mathematics educators lies in how do we create a vibrant and viable place that is informed and shaped by the critical, community knowledge of students within the stationary, undifferentiated space occupied by the “classical, dominant, Eurocentric knowledge” of school mathematics (Gutstein, 2007/2012, p. 302)? How do we transform this seemingly neutral, abstract space of acultural mathematics into a place that is familiar to students’ cultural ways of knowing and lived experiences?

Certainly, by acknowledging our relationship to “place,” we can “strengthen children’s connections to others and to the regions in which they live” (Gruenewald, 2003, p. 645). One attempt to establish such place-based cultural connections is reflected in the culturally responsive curricular investigation and model developed by Nicol, Archibald and Baker (2010, 2013). An example of a suggested activity in this model is: Taking students to a beach and inviting them to explore natural patterns and build their own patterns by collecting and organizing objects found on the beach, and then taking a “mathematicswalk” to see, identify, and extend patterns made by their peers (Nicol et al., 2010, p. 47). The same beach exploration could be adapted for higher grade students by showing specific objects (a razor clam shell, in this case) and asking, “What

is the ratio of the length to the width of a razor clam shell? Does this ratio hold for different size shells? Where else do you see this ratio?” (p. 45). Indeed, the integration of cultural, place-based contexts in our mathematics teaching would allow us to move away from the “mechanistic, compartmentalised view of teaching,” and embrace an “interconnected, wholistic, and synergistic” mathematics in our classroom contexts (p. 53). Acknowledging and embedding such place-based understandings in our teaching and learning of mathematics is crucial because the way we describe the place of teaching and learning of mathematics, will change the very meaning of mathematics education.

Teach mathematics to bring social justice and peace

Thinking about the meaning and purpose of education, one aim of teaching mathematics should be to develop “citizens who respect, and value contributions from all, regardless of race, class, gender, and act with a sense of social justice” (Boaler, 2006, p. 74). But how many of us are fully aware of the role, mathematics has, or could play, in bringing social justice (and thus peace)?

Aikenhead (2002) is concerned about cognitive imperialism pervading today’s school science, where Aboriginal children and also children coming from other cultural backgrounds (non-Euro-Westernized cultures) are expected to be “assimilated” or “colonized” into thinking like a “Western scientist” in their science classes” (p. 288). Unfortunately, similar is the case in many mathematics classrooms, where we often expect students to develop a “mathematician’s world view” (Noddings, 1994, p. 96) without worrying about whether or not it connects with their culture-based world views. Such controlled and closed mathematics which does not respect diverse views and favors only one way of “right” answers is dangerous because it marginalizes and fails people who do not “get it correct”.

Should we expect all students to be proficient in classical mathematics, and force them to possess “uniform skills” as demanded by today’s “gap-gazing”, “standardized, test-based schooling” ? Will we only then be considered accountable? No, in the name of accountability and standardized uniformity, we should not confuse equality with equity because by teaching every student equally, in the same manner, we compromise the equity which demands understanding individual students’ educational needs, and teaching responsively to meet the specific needs of the particular student. Understanding equity requires us to recognize that rather than pushing all students to think like a “mathematician”, we should create opportunities for them to learn and use mathematics for their own purposes.

The challenge for us, as educators, is to create an inclusive, equitable, safe and peaceful learning environment where diverse students are allowed to (re)discover and (re)learn mathematics of their choice. How do we encourage the unlearning of individual competency and mastery of content focused, text-based mathematics of dominant classical curriculum, emphasized so far in our schooling, and make the learning of mathematics a collective deed to invite and empower all students as learners of mathematics? How do we encourage “relational equity” (Boaler, 2006, p. 76), — respectful intellectual


relationships where each person is committed to the learning of others, respects the ideas of others, and is learning to communicate mathematics that relates with socio-cultural contexts and creates awareness about social justice? One resource that can be helpful in beginning such mathematical discourse is Stocker’s (2006) book Maththatmatters which includes mathematics lessons designed to empower students in civic and mathematical literacy. Definitely, by cultivating critical, culturally responsive, communal understandings of mathematics in our teaching and learning moments, we can make mathematics meaningful for diverse students and bring equity, social justice and peace in our mathematics classrooms.

Conclusion and future directions

Let us admit what is “our truth” and how we are using mathematics. Is it not true that in today’s economy-driven society, mathematics is perceived and taught as a “gatekeeper” for higher education (Noddings, 1994, p. 90), a ticket for entry into high-status jobs, and a winning token for higher social status? Are not we (mis)using the “power of mathematics”?

Acknowledging the power of mathematics, D’Ambrosio, the Father of Ethnomathematics, admitted that “human use of mathematics is capable of both horrors and wonders. Although mathematics may seem innocent and neutral, it is a powerful tool that enables technologies which could be used to bring peace, or breed violence. Indeed, the controlled and predicable procedural way of presenting standardized mathematics in today’s schooling leads to hidden socio-cultural and political prejudices. In our standardized efforts to ensure we achieve education for all, have we not forgotten the quality of education which is essential for sustaining the dignity of all students? Have we not forgotten to answer the key curriculum question: “What knowledge is most worthy?”

Ensuring “Mathematics for All”— the task is quite challenging, but not impossible! To begin, let us take this first step and think critically how, while engaging ourselves in teaching and learning of mathematics, we can challenge oppression and transform material and historical reality of mathematics teaching that is currently being practiced: Definitely, by identifying, accepting and utilizing the role that mathematics could play in bringing social justice, and by accepting that the teaching of mathematics is not a politically neutral activity. Indeed, by developing socio-political consciousness, a sense of agency, and positive socio-cultural identities in our mathematics classrooms, we can make our students capable of not only “reading the world” but also “writing the world”— capable of not only seeing what is happening in the world but also empowering them to transform their world to bring equity, justice and democracy.

Moving further, we can engage children in place-conscious mathematics that utilizes the power of place to contextualize the learning of mathematics and creates a dynamic space that helps in reinserting the land into children’s lived experiences while simultaneously empowering them to employ dominant classical knowledge to answer questions that are critical for the community not only from anthropocentric or biocentric perspectives, but also from an ecocentric perspective.

Thus, by promoting understandings of what could/should be done with the power of mathematics, we could become agents for peace and teach mathematics for social justice to bring universal dignity and world peace. By learning to embrace the tensions that are a natural part of equity practice, we can begin the process of privileging equity over equality, education over schooling, and power/identity over mere access and achievement, and engage ourselves in constant reframing of mathematics education that occurs in everyday practices and classrooms.

The Crucial Decision

What does the future hold? Time is calling us to stop painting a gloomy and pessimistic picture of the future (of mathematics and mathematics education) by blaming past circumstances and present realities. It is urging us to take thoughtful action in the present and (re)imagine the future through (re)visiting the past. If we really want “Mathematics for All,” we must (re)think and move beyond merely finding and justifying the answers to the teacherly, rhetorical, and hermeneutic questions: Why (do/would/should) we teach mathematics to all students? As mathematics educators, we must become and remain cognizant of, what are we teaching when we claim to be teaching mathematics?

If we want to “live math,” we must begin constructing a counter/parallel narrative of “Mathematics for All” against the dominant narrative of today’s standardized test-based schooling. This narrative is demanding from us as teachers, parents and students to choose: What stories will we tell and what stories will we live during our mathematical discourses of living, learning and teaching? The choice is ours!

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Nicol, C., Archibald, J., & Baker, J. (2013). Designing a model of

culturally responsive mathematics education: Place, relationships and storywork. Mathematics Education Research Journal, 25(1), 73-89. doi:10.1007/s13394-012-0062-3

Noddings, N. (1994). Does everybody count? Reflections on reforms in school mathematics. Journal of Mathematical Behavior, 13(1), 89-104. doi: 10.1016/0732-3123(94)90040-X

Pelfrey, R. (2000). Open-ended questions for mathematics. ARSI Resource Collaborative, University of Kentucky. Retrieved from http://www.uky.edu/OtherOrgs/ARSI/www.uky.edu/pub/arsi/openresponsequestions/mathorq.pdf

Skovsmose, O. (2007). Doubtful rationality. ZDM Mathematics Education, 39, 215–224. doi 10.1007/s11858-007-0024-5

Stocker, D. (2006). Maththatmatters: A teacher resource linking math and social justice. Toranto, ON, Canada. Canadian Centre for Policy Alternatives.

V


MATHEMATICAL EXPLORATION

Think back to your mathematical days of high school. I bet you never looked at a triangle and said, “Oh, I should use complex numbers

to solve for that missing angle.” Really–who would? I also highly doubt you ever thought that taking the logarithm of a negative number

would actually help you solve that problem. Little did I know that when my good friend and colleague, Emek Benny, drew a seemingly

innocent diagram on my whiteboard, this opportunity would present itself.

Consider the following triangle.

Given: AB= AC, ∠ABD = ∠CBD , AD+BD= BC

Question: What is the measure of ΔBAC?

At first glance I thought, “I’m sure he already has a great solution, and I bet it involves a circle. Can this be done without one?” I re-expressed

the problem numerically and symbolically, such that AD =1, BD= x, BC= x+1, ∠ABD = ∠CBD=θ , and ∠BAC =φ . Since ΔABC is isosceles,

∠ACB = 2θ

BY JORDAN FORSETH

The Power of PART I: A TRIANGLE BEYOND THE “REAL”


After deciding to find the value of θ first, and then solve for angle φ , I applied the Sine Law to generate the following two proportions

(just to see what happens).

Given that φ =π − 4θ, and using the identity sin(A−B) = sinAcosB−cosAsinB, I quickly simplified each proportion.

Isolating x in equation (II) and then substituting into equation (I) gave the following result.

Though pleased there was only one equation remaining, I was a little disappointed. I wanted an analytic solution to the problem, yet the

thought of applying more trigonometric identities to expand and simplify did not seem appealing. Nevertheless, I meticulously worked

it out and simplified to produce the quartic polynomial equation (2y −1)(8y3 −6y −1) = 0, where y = cosθ.

The first observable solution is not valid, (and any coterminal angle) is outside the domain . Hence

our desired value of θ must be a solution to the depressed cubic equation 8y3 − 6y −1= 0. Given a cubic equation of the form

y3 + ay +b = 0, we can apply Cardano’s method1 (Gerolamo Cardano, Ars Magna, 1545)

(2)

1 Although Cardano did not discover this formula himself, he was the first to publish.

2 If the discriminant , the equation has three distinct real roots. Either way, the above formula will only produce one root, the principal root (within the context of complex numbers, which Cardano himself did not fully understand).

(2)


With a little complex analysis (not shown here), we get (the principal root), as well as

and , where k∈ Z. The only value of θ within the domain is .Thus the solution to our triangle

problem is

What fascinated me was not the answer itself, nor the use of Cardano’s method. Despite having solved the problem (in a rather obscure

way), I was quite taken by how complex numbers played a role in obtaining the answer–a “real” answer; it was like they appeared for the

sole purpose of disappearing again. Of course I knew there was a purely geometric solution to our triangle problem; elegant and more

straight forward than this approach. However, I am often keen to venture outside the natural realm of a particular problem so as to bring

seemingly unrelated theorems and ideas into the mix. Consider Fermat’s Last Theorem:

If n is a positive integer greater than 2, and x, y, z are integers, then xn + yn ≠ zn.

Pierre de Fermat, a 17th century French mathematician, said he could prove it–but never gave an explanation3. What you might not

know is that this simple problem has yet to be proven directly (that we know of). It was proven by Andrew Wiles in 1995, though it was

his unification of extremely complicated and abstract ideas, specific to modern mathematics, which finally put Fermat’s problem to rest.

Specifically, he showed the Taniyama-Shimura Conjecture to be true for all semistable elliptic curves; hardly an obvious relationship

between that and xn + yn ≠ zn.

By proving this conjecture, Wiles not only solved Fermat’s Last Theorem by default (after more than 350 years), he also paved the way to new

discoveries and ideas relevant to the mathematics of today. Had Fermat’s Last Theorem been solved during Fermat’s time, it likely wouldn’t

have spawned all of the new mathematics that arose simply because it hadn’t yet been resolved. After all, it wasn’t until mathematician

Gerhard Frey considered what would happen if Fermat was wrong that anyone even suspected a connection between Fermat’s Last Theorem

and elliptic curves (Fermat’s Last Theorem, Simon Singh, 1997).

.

3 It is widely believed that Fermat did not in fact have a valid proof. Fermat made only one known reference to such a proof as a mere scribble within the margin of his personal copy of Bachet de Méziriac’s translation of Diophantus’ Arithemtica (1621), an ancient Greek text. Fermat made mention of the specific cases n=3 and n=4 in his correspondences with other mathematicians, but never openly discussed his general solution or expanded upon it. It is possible that he came to realize his proof was false, and never felt the need to remove it from his notes. However, we will never truly know for certain.


I then asked myself, “Since complex numbers appeared unexpectedly in my first solution, what would happen if I intentionally incorporated

them earlier?” By means of a special technique, which Emek and I had both used in the past4, I realized that this triangle could not only

utilize complex numbers in a very powerful way, but could also make use of logarithms of negative numbers. Yes–logarithms of negative

numbers! But hang on, how can this be done? This triangle problem is a “real” problem with a “real” solution. Are we allowed to bring

in complex numbers, just because? In actuality, every real number can be thought of as complex–we simply don’t bother to mention it,

and for good reason (imagine young children having to learn 2+3=5 within the context 2 + 0i + 3+ 0i = 5 + 0i ). Nevertheless, they’re

there. Why not use them, just as other mathematicians have done in the past?

One might say, “If there’s a simpler and more efficient way to solve a problem, it must always be the best way!” I disagree. Sometimes the

simpler method merely solves the problem faster–but then it’s over. If you wish to dig deeper, often a more drawn-out approach can create

opportunities to explore other areas of mathematics allowing you to make connections you didn’t think were possible.

To set the stage, we will make use of the following two complex identities.

4 Emek used the identity as an alternative method for solving the equation cos(mx) = cos(nx). I had previously used the same identity as an alternative method for finding the integral ∫cos4 x.dx


Proof of identity (1), using the following Maclaurin series:

Substitute iθ in place of x in series (i).

Proof of identity (2):

5 Andrew Jones (Department Head) explored the analogous hyperbolic identities and , as well as the identity for further enrichment in his Calculus BC class


OR

It is here we will use identity (2) to transform equation (III) – allowing us to break free of the real number system!

Simplify, multiply by e7iθ (eliminating all negative exponents), and then factor. Also, since cosθ and sinθ both have a period of 2π,

eiθ = ei(θ+2k j π ) for all k j ∈ Z.

Two of the three factors generate obvious solutions, but none are relevant to the problem.

None of these solutions satisfy

Hence we must consider the third case, when ei(9θ +2k3π ) +1=0 (i.e. when ei(9θ +2k3π ) =-1).

To solve for , we can express the above equation in logarithmic form (just as x3= 9 can be written as x=log39).

for any k ʹj∈ Z.

6 In complex analysis (and higher mathematics in general), “log z” is understood as logez, not log10z, and is multi-valued (we’ll see this later). We will reserve the familiar “in x” notation for positive real values of x only.

(6)


7 We refer to θ 0 as the principal argument, Arg z,, if and only if -π <θ0 ≤ π. The general argument, arg z, consisting of θ 0 and all coterminal angles θ0 ± 2π, θ0 ± 4π, etc., can be represented by Arg z + 2kπ, where k ∈ Ζ.

Wait a minute... how can we take the logarithm of a negative number? Believe it or not, logarithms are not only applicable to positive

real numbers (like we saw in high-school), but they exist for complex numbers as well as negative real numbers. In fact the only value for

which log z has no meaning is when z=0. But in order to make sense of them we first need to understand how complex numbers behave.

Any complex number of the form z=x+iy is said to be in rectangular form,

where x and iy represent the real and imaginary parts of z, respectively, and

. Any real number x can also be expressed as the complex number z = x+ 0i

(as mentioned before). Within a two-dimensional space, one can visualize a

complex number much like they would an ordered pair (x,y).

Alternatively, the location of a complex number can be described in terms of

i) its distance from the origin: the modulus (denoted by |z|, a real number

defined by ), and ii) the angle between the terminal arm and the

positive real number line: the principal argument, Argz (denoted by θ0)7. This

is known as polar form, where z=|z|eiθ0 (recall that

E.g. Consider the complex number z=3+4i.

The modulus is 5, and . Hence the polar form (approx.) of 3 + 4i is 5ei(0.2952π ) , where 0.2952π is the

principal argument.

).


Now consider the real number –1 (also understood as the complex number z = −1+ 0i).

The modulus is 1, and θ 0 =π. Hence the polar form of –1 is e iπ, where π is the principal argument. This marks the appearance of Euler’s

Identity: eiπ + 1 = 0. Wow! I never would have thought a question about a triangle would lead to this amazing result.

Given −π <θ0 ≤π , the complex logarithm of z = |z| eiθ can be understood in two different ways. [Note the capitalized “Log”.]

Principal logarithm: Multi-valued logarithm (k ∈ Ζ):

Again, since cosθ and sinθ both have a period of 2π, eiπ = ei(π +2kπ ) = for all k ∈ Ζ.


Now we can solve for θ explicitly in equation (IV). [It is here that all references to complex numbers “disappear”, bringing us back full

circle to the real number system.]

The only value of θ within the domain , where kʹ = 0 .

Hence our final solution is as it should be.

____________________________________________________________________________________________________________

Who knew such a simple triangle could be so… complex! Yes, it’s a lot of work for a problem that can be quickly solved using circle

geometry. But consider what we’ve accomplished:

i) An unanticipated journey outside the real number system and back, ii) the algebraic identity

iii) a real-world occurrence of log(-1), and iv) a genuine instance of Euler’s Identity: (often

regarded as purely theoretical).

[Acknowledgments: Emek Benny, for your inspiration; Andrew Jones, for your support.]

“The definition of a good mathematical problem is the mathematics it generates, rather than

the problem itself.” – Sir Andrew Wiles

Secondary Mathematics Education M.Sc., M.Ed.Develop insights into the nature of mathematics and its place in the school curriculum.

Apply Now - start as early as Fall 2017

Classes scheduled for working teachers

Admission Information: [email protected]

YOUR CAREERA D V A N C E

Qualify forTQS upgrade

www.sfu.ca/education/sec-math2017


Secondary Mathematics Education M.Sc., M.Ed.Develop insights into the nature of mathematics and its place in the school curriculum.

Apply Now - start as early as Fall 2017

Classes scheduled for working teachers

Admission Information: [email protected]

YOUR CAREERA D V A N C E

Qualify forTQS upgrade

www.sfu.ca/education/sec-math2017


SHORT STORY

BY ISAAC TORRES

Three criminals sat around a circular wooden table. The cheap, wood chip kind that stains never really wash out of. One of them pulled three stacks of paper and spread them on the table.

The boss, a woman of charm and brains, waved a hand in front of her, presenting the papers “Alright, friends. I’ve got here some good intel. There are three banks with storage vaults in town, and each of these three banks has fifty VIP-level clients, each giving the bank one high value item. Here I managed to collect the policies of each bank regarding how they store the shinnies; and have come up with a plan to hit one of them. But I don’t want your heads to be perpetually stuffed with wool, so I want you to read up, and figure out which of the banks we should hit. Let’s make this an exercise”

The muscle, a big intimidating lug with a golden tooth, leaned over to examine one of the bundles. “Boss, ya know I don’t read.” he complained, thumbing through a bunch of company policies that flew over his head. “Why don’t ya give me the cliff notes, and we can get onto the breakin’? I ain’t no thinker, never will be.”

The boss glared at him for a second, more than enough for the muscle to put his head down and start reading.

The climber, a lanky person of indeterminate gender and a devious grin, reached for a leaflet on top of another stack and started scanning it. “The cliff notes are on the leaflet, big guy. Boss didn’t actually expect us to read the entire employee manual in a single night.”

“Oh. Sorry, boss.”

The Crimson Box made it a policy to put all their most valuable goods into one of their 30 vaults; keeping all other vaults still highly guarded, yet mostly empty. Every week, the vaults were cycled randomly.

The Viridian Lock had instituted a rule that each of their 30 vaults had to have something of high value stored within it; for

the discerning elite that wanted to make sure their goods had their privacy.

The Lavender Silo had little in the way of policy for how to store the goods they had been entrusted with, keeping most of their vaults in use, but with a wide variation of value within each.

“It wasn’t easy getting this info,” the boss started once the muscle and the climber had read the leaflets. “But I know you can probably figure out which should be our target.”

The muscle sat deep in thought for a few seconds. “The Lavender seems choice, boss. We could get sum good stuff if we luck out.”

The boss nodded, and looked at the climber. “And what do you think? Do you agree with the big boy?”

The climber shook their head, shifting from their slouch to leaning toward their two accomplices. “No. If we want to get it good, we should hit the Crimson. With some more research we can hit the good vault, haul everything, and retire.”

The boss nodded again. “That’s good. However, not the answer I was looking for.” The boss pointed at the Viridian stack, and stood up.

“Viridian?” the muscle asked. “But Lavender can have more stuff on the good vaults! We just need ta be lucky!”

“And that’s the key word, big guy: Lucky. I’d rather not leave my future riches up to chance.”

“Well, in that case…” the climber interjected “We take out luck from that part and go hit Crimson. We do our research, find out the good vault of the week and hit before they do the switch. We get a ton of goods and never have to risk our necks again.”

The boss turned her attention to the climber, sighed and started pacing the room.

The Pigeon Hole Caper


“You see, the problem with that,” she started, waving her finger in the air with dramatic purpose as she walked back and forth, “Is that it’s only three of us. We can’t just bring a crane in to drag everything out. Ideally, with a larger crew, it would be feasible to empty it in a timely fashion. However, I’d rather not risk it as we are. To top it off, if our intel is wrong, we’ll hit an empty box and we’ll have wasted a perfectly good heist.”

She turned to her two sidekicks and put her hands on the table.

“Think about it. 50 clients, each with a valuable item they trust the bank with, a bank with 30 vaults...?” she looked from the muscle to the climber and back, waiting for the dots to connect in their heads.

“The bank’s policy IS to ensure each vault has some minimum value…” the climber started, “probably why only crackpots store their stuff in there.”

“And… if each box has one crackpot’s loot…” the muscle continued, “that leaves out… 20 more that would have to share vault with others.”

“Good thinking! I knew you had it in you!” the boss exclaimed, patting the muscle in the back. “So, we have at least one good thing in whatever vault we hit, probably some loose change, and if we luck out, it’s a shared vault with another goodie for us to grab. No need to overwork ourselves physically, and we still get a good haul.”

The peculiar grin on the climber spread wider. “Yeah, I like the sound of that.”

“Good. Now, here’s what we are going to do…”

_____________________________________________________

The headline in the next week’s local newspaper read:“Trio of Crooks severely underestimate security staff in Viridian Lock; barely make it past entrance before being apprehended.”

V


PROFILES

EmilyBY LEO NEUFELD

The 2016 roll of graduates at Mount Douglas Secondary School in Victoria includes the name Emily Tsao.

In 2010 a very young, bright-eyed Grade 7 student at Arbutus Middle School had heard of Mathematics Challengers (MC) and was urging teachers to form a Grade 8 team for the MC competition the following February. Her persistence went unheeded. So, in desperation, Emily convinced her mother to enter her as an individual in the upcoming event. Mom would have to be registered as her Coach. The last-minute appeal to have Emily enrolled succeeded and there she was at Camosun College on February 11, 2011, confident in her mathematics skills and eager to test them against a group consisting entirely of Grade 8 students.

Well, she not only held her own, Emily took home the Third Place Individual trophy that day. Her teachers at Arbutus were flabbergasted, but mom and the rest of the family, while proud, were not surprised.

The next year, Emily worked hard to convince four of her fellow Grade 8s to form an MC team and Arbutus wisely found a teacher to sponsor them. At the Regional that year, Emily led the team by winning top place in both Individual standings and in the Face-off Stage. Her Team took first place over all in the Grade 8 category, advancing them to the Provincial competition held at UBC in Vancouver.

Mathematics Challengers provides Grade 8 and Grade 9 students with the wonderful opportunity of show-casing their mathematics abilities for an afternoon while enjoying a college campus with

school-mates and having fun winning medals and trophies.

To take Grade 9, Emily needed to enrol at Mount Douglas Secondary School. There, to her delight, she discovered that a teacher was already taking Grade 9 teams to the MC competition despite the fact that he did not actually teach these students until a few days prior to the February date of the Regional. Emily bravely undertook to organize prep sessions for a group of other mathematics lovers and joined one of three teams entered from Mount Doug that February. The results? Mount Douglas took first place for Grade 9 and Emily was awarded the First Place Individual trophy. Apparently, she had prepared her fellow, Mount Doug competitor too well—Lily Yang, after an exciting battle, narrowly bested her in the final round of the Face-off Stage.

There is no Mathematics Challengers competition for students after Grade 9. So, remaining undaunted, Emily simply changed her focus. With the help of the Mount Douglas MC Coach, she began giving MC prep sessions to Grade 9 students as they entered the school each fall. It was exactly what she was continuing to do for kids at Arbutus. In addition to her studies and musical interests, she found satisfaction in sharpening the competitive edge of willing students in two schools year after year. Emily would anxiously await news of how her charges had fared after each event. This, her final year at Mount Douglas, she was asked to be the Coach of their MC contingent, but had to forego this challenge because of a scheduling conflict. However, she was ecstatic when she heard that Mount Douglas had placed first overall at the Vancouver Island Regional.

Emily has been a remarkable story and will continue to impress her colleagues and teachers as she moves on full-time to university. We wish her well and express our sincerely deep-felt appreciation for her indomitable spirit, unyielding tenacity and selfless generosity especially as it was directed towards her cherished Mathematics Challengers cause.

Emily, we proudly salute you!

V

Emily Tsao accepting the Individual Third Place Trophy at Camosun College in 2011.

Vector • Spring 2016 47

REPORT


BOOK REVIEWMathematics Education Across Time and Place:

BY KANWAL SINGH NEEL

Over Two Millennia from Athens to Zimbabwe

Mathematics Education Across Time and Place is a window into different places around the world spanning two centuries in how mathematics education has been influenced by history, education, politics, and society. Fictional biographies of mathematics educators are used to illuminate what was happening in history, what were some of the political and societal norms of that day, and about some of the Big Questions such as: “What mathematics should be taught in schools? To whom should mathematics be taught? How should mathematics be taught?

The book is organized in eight chapters starting with the Greeks and Romans followed by a chapter on the Islamic Influence. The next three chapters take us from the Italian Renaissance to the Mathematical Practitioners of England to the French Revolution. The last three chapters move to Early America, Canada and the Twentieth Century. The book has twenty-one essays in the form of stories, biographies, autobiographies, dialogues, speeches and letters, which give the reader the experience and knowledge about mathematics education at different times and places around the world.

Though the essays are fictional, they are scholarly writings based on historical documents and professional sources. O’Shea skillfully prefaces each essay with a brief commentary that provides a deeper context about the influence of historical developments and educational reforms on how and what mathematics was taught. So the narrative provides a thread that extends across 23 centuries and winds through some lived experiences of mathematics educators from Athens to Zimbabwe.

As I read about the lives and the issues the mathematics educators faced, I find that a number of the issues are still relevant in today’s context. Learning mathematics in context with manipulatives is still pedagogically sound as it was a few thousand years ago. “We learnt division by dividing piles of apples between us, and he even seemed to see some mathematical purpose to those wonderful games we

played with dice and knucklebones, not to mention the more complex game of chess” (p. 10). Examples show how mathematics was used to solve problems in daily lives such as the development of maps, the use of arithmetic in commerce, calculation of latitude and longitude with adoption of astronomy.

Using mathematics to solve problems; appreciate and value mathematics; communicate and reason mathematically; and use mathematics to better understand the world has been a common thread over the centuries. A problem from the 14th century is still intriguing today: “There is a man who is seriously ill and finally makes his will. He has a wife who is pregnant and leaves 1000 lire to her in this manner: if the woman bears a daughter, 1/3 will go to the daughter for her dowry, and if she bears a son, ¾ will go to the son and ¼ to the wife. The good man died and his wife delivered twins, a son and a daughter. How much should each one have?”(page 89)

Most of the autobiographies included in this book come from an Euro-centric worldview. The inclusion of autobiographies from the Islamic world provides a unique opportunity to bring together Greek mathematics with its emphasis on geometry and number theory, and Hindu mathematics with its numeration system and focus on algebra and trigonometry (page 38). O’Shea acknowledges, “not included is any history pertaining to China, Japan, or India, as few students chose those areas, and reliable sources were difficult to find” (page xviii). We need to know about different worldviews, as “all

O’Shea, T. (2016). Mathematics Education Across Time and Place: Over Two Millennia from Athens to Zimbabwe.

Victoria, BC: Friesen Press


people do mathematics because all cultures count, locate, measure, design, play, and explain” (p. 309). Worldview is an expression of the creative process that connects all things. Indigenous peoples have historically applied the thought process of mathematics within cultural contexts, which are holistic. Most Indigenous cultures have an orientation to learning that is metaphorically represented in its art forms, its way of community, its language, and its way of understanding itself in relationship to its natural environment.

Thomas O’Shea has been retired for a number of years but his passion for mathematics education continues. This book is his labour of love; he truly is a maestro dei maestri (teacher of teachers). He achieves the purpose of the book to help mathematics teachers, teacher educators, and interested members of the public appreciate the path that we have followed to the present state of mathematics education. His use of imagination and story telling lets one understand the historical and social context for schools, schooling, and the place of mathematics in schools. The book is a pleasure to read; it enables one to reflect on the critical role of mathematical understanding in our personal and professional lives, in our history, and in our culture. I wholeheartedly recommend it to those interested in the teaching and learning of mathematics. The book is bound to expand the horizons of anyone interested in mathematics as a human endeavor.

V

How do we get the most out of daily life? Is happiness simply enough, or are challenges necessary to help find a deeper experience? How can we affect our own experiences to help get more out of each day? These are all questions discussed in Finding Flow by Mihaly Csikszentmihalyi (pronounced “chick-sent-mi-high”). The author presents many compelling arguments of how people can find flow in their every day–that is, finding activities that are a balance of appropriate challenge to skill. According to Csikszentmihalyi, the state of flow is one in which an individual is so focused that he or she will lose sense of time and enter a serene and clear state of ecstasy –a feeling of being outside of one’s self. To instill flow an activity must have clear achievable goals and give the individual continuous feedback motivating the activity further. The concept of flow is not a new one–it was first formally proposed by the author in a 1975 publication. Flow theory describes individuals pursuing activities through intrinsic motivations rather than extrinsic. According to the author, finding this experience of flow will help people find an overall improved quality of life as his research has indicated that people are generally happier when experiencing flow.

Although this book is relevant to the general reader the ideas presented have many useful applications to the teaching of mathematics. As a teacher, the goal of any successful classroom may just be to have our students experience flow. Hopefully, we’ve all experienced those classes that seem to just fly by with a buzz in the room of students eagerly working through tasks or engaging in interesting projects. Although not directed at educators, the viewpoint of the author can be applied to teaching by taking his points and applying them to our classrooms.

One means of specifically addressing this phenomenon in any classroom is to simply give students the physical space and freedom to experience flow. Motion and activity are traditionally not something experience out in a classroom, but Csikszentmihalyi discusses how sports and physical activities have some of the highest frequency of flow experience. Having students moving during class, while engaging in thought-provoking activities, has been shown to increase focus and engagement. If focus is improved then students are more likely to experience flow. However, the activities provided for students must be of suitable challenge, as another important feature of flow is that the individual sees their task as achievable. It is very challenging for most people to stay engaged in any activity while passively sitting for a lengthy period of time. Another means

Finding Flow: The Psychology of Engagement with Everyday LifeBY BETH BALDWIN


My only criticism of the book is that it felt somewhat outdated –it was published in 1997, nearly twenty years ago. With the technological advances of the twenty-first century, many of the activities that the author discusses have either changed or in some cases no longer exist. The main leisure activity the author often criticizes is television, but with the internet age and the vastness of online communities it seems to me that Csikszentmihalyi would have other opinions of leisure as well as work with the rise of computers.

Overall, Finding Flow was an interesting and enjoyable read with applications to both people’s everyday activities as well as education. Encouraging and fostering flow in our classrooms will lead to active learners who will begin to self-regulate through their experience. If we can find ways to introduce flow into those everyday activities that we must do–the mundane, then perhaps they don’t have to feel like a chore, but can be a way of meeting an interesting challenge and increasing our skills simultaneously. I believe Csikszentmihalyi put it best when he said that if “we can focus consciousness on the tasks of everyday life in the knowledge that when we act in the fullness of the flow experience, we are also building a bridge to the future of our universe” (p. 137).

V

of encouraging flow specific to the mathematics classroom, is by presenting interesting and challenging problems. Giving time for students to struggle with these problems, discuss, reassess, and guide them to those moments of discovery will increase flow. Redundant and repetitive problem sets or worksheets probably do not encourage flow and will more likely dissuade students from being a part of a thinking classroom as they do not generally present a challenge, but rather repeat a task again and again. Although the student’s skill is perhaps improving with worksheets, the challenge is not increasing appropriately and they are likely to slip away from the experience of flow and possibly experience boredom.

A student in a state of boredom is a signal to the teacher to increase the challenge level of their activity, or perhaps decrease student resources (for example change the rules, take away a group member, etc.) in order to maintain flow. As teachers, we must monitor our students and become aware of their change in state from engagement to boredom, or perhaps frustration. Finding activities that have a low-entry skill requirement, but that can be incrementally increased will enable all students to constantly be challenged appropriately. On the other hand, we as teachers must also observe when a challenge is too great for a student (a state of frustration) and be able to offer some strategies to help them improve their skill–either by offering assistance ourselves or from other students or decreasing the challenge level. Most importantly we must constantly be surveying our classrooms for students that are out of flow and finding a way to ensure that both challenge and skill are appropriate for all learners.

Csikszentmihalyi states that once flow has been experienced people will try to seek out the same feeling again. This is beneficial in the classroom as we may start to see students make their own changes to engage in whatever they are experiencing over time. This could promote student self-regulation and help students push problems and concepts further through their desire to stay in flow. Classrooms are easier to manage when students are in flow.

The book is organized into three sections (three chapters each) and each looks at a specific aspect of everyday life. The first section looks at how people tend to spend their time as well as how they tend to enjoy it. The next section looks specifically at how we feel about work, leisure and relationships, as well as ways to introduce flow within each of those contexts. The book finishes with a section that discusses how to potentially change one’s patterns to encourage flow and discusses potential reasons why or why not one is able to experience flow frequently.

Throughout the book Csikszentmihalyi presents research indicating how the average person spends their time divided into 3 categories; productive activities (work or school), maintenance activities (e.g. eating, driving, bathing) and leisure activities. His studies of flow indicate that although the average person spends the least time in leisure activities this is most likely where they are to find flow. The author also argues that although people generally state that they dislike working, it is another likely source of flow given that there is an appropriate level of challenge. This “paradox of work” is well explained in the fourth chapter of the book. Many people feel valuable through work and are able to better themselves, thus making it likely to be more engaging despite feeling that it is not something we may choose to do.

ISBN-13: 978-0465024117, published in 1997 by Basic Books in New York, NY


PROBLEM SETS

Financial Literacy is being introduced in Kindergarten with the revised mathematics curriculum. It is far more than naming and recognizing the Canadian coins.

“Financial literacy is the application of number sense to contexts involving earning saving, spending and giving money. Financially literate citizens make thoughtful decisions about what they want and what they need. They set realistic goals, establish priorities and make plans for saving and spending money.”

Carole Fullerton and Sandra Ball 2015

The learning standards which focus on Financial Literacy are intended to look more deeply at what it means to be financially responsible (structured around the notions of earning, saving, spending and giving). Our youngest learners need support and explicit teaching to reach these goals.

The following problems involve the concept of trading. Students need to know that we can trade money for objects of work and that some things have more value than others.

GRADE 1- 2 PROBLEM – TRADE YA!

1. Use pattern blocks for this task.

2. Explain that each of the pattern blocks has a different value. (Note: the prices, like the areas of the shapes are proportional)

K - GRADE 1 PROBLEM – TRADE YA!

Read the book “Balancing Act” by Ellen Stoll Walsh. Explore the concept of equality and inequality with a balance scale. Discuss the importance of the scale being balanced when both sides are equal.

Show the students a collection of natural materials (shell, pinecones, rocks, sticks…whatever is available).

1. Explain that the value of each of the items is different. Tell the students that you like the shells the most and they would be the same as the value of five rocks or ten sticks or two pinecones.

2. Draw a balance scale and ask: If I put two shells on one side, what would make it balance on the other side? What could you trade two shells for?

3. Ask the students to suggest ideas that could balance the scale with the two shells (i.e. ten rocks or five pinecones). It is not about the weight of the objects, but it is about the value. NOTE: You may need to lower the value of objects for some students (start with five).

4. Have the students explore other ways to trade objects for other values. (i.e. five rocks can be traded for two pinecones)

PRIMARY PROBLEMS BY SANDRA BALL GRADE 4+ PROBLEMS BY KEVIN WELLS


GRADE 2- 3 PROBLEM – TRADE YA!

1. Repeat the above problem using the following values:

a. Yellow hexagons are worth $3. b. Red trapezoids are worth $1.50.c. Blue rhombi are worth $1d. Green triangles are worth $0.50

2. Challenge the students to create as many designs as possible with

3. the value of $10.

4. Encourage them to find a design that uses the fewest or most amount of blocks.

5. Ask: What did you notice about trading the blocks for other blocks?

GRADE 4-7 PROBLEM

Explain what is wrong with the pictures above. How should the displays read?

(adapted from http://www.huffingtonpost.com/2013/06/18/work-math-fails_n_3442834.html)

a. Yellow hexagons are worth $6.b. Red trapezoids are worth $3c. Blue rhombi are worth $2d. Green triangles are worth $1

3. Challenge the students to find as many designs as possible with the value of $20.

4. Ask: What relationships did you notice? How did you trade pattern blocks to create other possibilities.

5. Present other values for the students to explore.


PROBLEM SETSGRADE 6-7 PROBLEM

The perimeter of the rectangular park shown is 42 km.

1. A ranger estimates that there are 9 deer in each square km of the park. If this estimate is correct, how many total deer are in the park? Explain your answer using numbers, symbols and words.

2. What is there in the park that might make this calculation inaccurate? Can you determine what might be a better estimate? Make it clear how you go about this.

(adapted from http://blogs.edweek.org/edweek/curriculum/2014/08/a_closer_look_at_a_math_perfor.html )

GRADE 7-8 PROBLEM

1. The area of the square is 100 square centimeters. What is the area of the circle?

2. The diameter of all four circles is 3 cm. What is the shaded area in square cm?

3. If the shaded ball I now removed and the remaining three balls formed into a triangular shape, what would be the area enclosed in the centre space by these three balls?

(adapted from http://mathtop10.com/7th_grade_math_challenge_free/7th_grade_math_contest%20P1.htm )


GRADE 8-9 PROBLEM

Sharing Chocolate

Charlie and Liz have each started nibbling at their chocolate bars. They bump into each other in the park and this is what they have:

Charlie has 16 squares. Liz has 10 squares. Would they be able to share them equally?

The following week, Charlie and Liz meet again. Here is the chocolate they plan to share this time:

Would they be able to share their chocolate equally this time?

In the weeks that follow, Charlie and Liz meet up regularly at the park. Here are the numbers of squares of chocolate they have each time they meet:

9 and 5

4 and 7

12 and 10

5 and 6 When will they be able to share their chocolate equally? Can you create an image (diagram) that helps you decide? If Liz and Charlie are sharing a large number of Lego bricks, creating an image to represent all the bricks will take too long...

Can you create simple diagrams to help you decide if they will be able to share their bricks equally, if

• Liz had 41 and Charlie had 72 bricks?• Charlie had 4 621 457 and Liz had 659 201 bricks?!

Can you explain how you decide if any two large numbers of bricks can be shared equally?

(from http://wild.maths.org/sharing-chocolate )


PROBLEM SETSGRADE 9-10 PROBLEM

Got It Can you be the first to get to 23?

Got It is a game for two players. The first player chooses a whole number from 1 to 4. After that players take turns to add a whole number from 1 to 4 to the running total. The player who hits the target of 23 wins the game.

Have a go at playing the game as many times as you like. You could use the interactive at http://wild.maths.org/got-it.

GRADE 10-11 PROBLEM

Two ladders in an alley Two ladders are placed cross-wise in an alley to form a lopsided X-shape. The walls of the alley are not quite vertical, but are parallel to each other. The ground is flat and horizontal. The bottom of each ladder is placed against the opposite wall. The top of the longer ladder touches the alley wall 5 feet vertically higher than the top of the shorter ladder touches the opposite wall, which in turn is 4 feet vertically higher than the intersection of the two ladders. How high vertically above the ground is that intersection?

(from http://www.qbyte.org/puzzles/puzzle01.html )

GRADE 11-12 PROBLEM

Coin triplets Two players play the following game with a fair coin. Player 1 chooses (and announces) a triplet (HHH, HHT, HTH, HTT, THH, THT, TTH, or TTT) that might result from three successive tosses of the coin. Player 2 then chooses a different triplet. The players toss the coin until one of the two named triplets appears. The triplets may appear in any three consecutive tosses: (1st, 2nd, 3rd), (2nd, 3rd, 4th), and so on. The winner is the player whose triplet appears first.

a. What is the optimal strategy for each player? With best play, who is most likely to win?b. Suppose the triplets were chosen in secret? What then would be the optimal strategy?c. What would be the optimal strategy against a randomly selected triplet?


GRADE 12+ PROBLEM

The absentminded professor

An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, where 0 ≤ k ≤ n, what is the probability that the other box currently contains k matches?



MATH LINKS

http://wodb.ca/

WHICH ONE DOESN’T BELONG? is a website dedicated to providing thought-provoking puzzles for mathematics teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn’t belong. Enjoy!”

http://bedtimemath.org/

Quoted from their website: “We make mathematics part of the family routine. Every day, we serve up a quick bite of wacky mathematics just for fun. It’s nothing like school. Parents can sign up by email, on our website, and on our free app. Whether it’s flamingos, ninjas or pillow forts, kids can see the mathematics in their favorite topics. No logins. No drilling. No scores. It takes only 5 minutes a day, and kids clamor for it.”

http://bigmarker.com/communities/GlobalMathDept

This site hosts an online community of mathematics educators that regularly sponsor webinars on topics in mathematics education, drawing from a wide variety of speakers. All webinars are recorded so you can explore topics from previous sessions. You can also use the discussion forums to engage in ideas with other mathematics educators.

http://mathed.podomatic.com/

This site hosts Podcasts that engage in conversations with mathematics education researchers. There are currently 1616 interviews posted.

https://talkingmathwithkids.com/

Quoted from their website: “This website is dedicated to helping parents support their children’s mathematical development. We know we need to read with our children every day, but what should we do for math? Answer: Talk about mathematics with them as we and they encounter numbers and shapes in our everyday lives.”

Spring 2016 Mathematics Websites

Links selected and described by David Wees (http://davidwees.com)


BCAMT

For the past two years, we (authors) have been on a mathematics journey together to study the revised curriculum and to explore ways to make our students (and ourselves) more passionate about mathematics. Last year, we experimented with numerous ways for our children to record their thinking such as drawing and labeling diagrams, taking photographs, and using different iPad applications. We felt that we were on the right track, but we also felt that we had so much more to learn!

So this year, we worked on a number of initiatives. We applied for a week long sabbatical from our district to analyzed the revised curriculum, line by line, until we felt more confident about what we needed to teach. We attended as many workshops as we could by our “mathematics heroes” that included: Janice Novakowski, Kim Sutton, Carole Fullerton, and Trevor Calkins. We went to the NorthWest Mathematics Conference in Whistler and spent our evenings trying different dice and card games while reviewing what we had learned each day. We immersed ourselves in learning the new “lingo” and we are proud to say that we now understand what it means to “subitize” and “decompose!” We talked to colleagues, administrators, students, and parents about how we use numbers in our daily lives. For example: “How many times does Division 8 flush their toilets in one weekend and how much water was used?” The more we learned, the more excited we became about teaching mathematics.

In our pursuit of understanding communication, one of the steps we took was to try to get our students and their parents to think beyond the basic number facts and to be more aware of how important mathematics is in our daily lives. One of the ways we achieved that was to hold a bottle drive after the Winter Break. We encouraged our students to bring in empty water bottles, pop cans, and juice containers for us to recycle. After two weeks of collecting, we took all the students along with the donations outside to our basketball courts where we used hoola hoops to sort, classify and count our “collection”. We recorded the data in a variety of different ways and talked about what we had learned. Then, we calculated how much money our collection was worth and how many ways we could represent that information.

The final part of our process was to make decisions about how we would spend the $44.70 that we had collected from our bottle drive. We used Scholastic Book orders to have the students make “wish lists” of the books they would like to purchase using the money. Then we asked the students to identify their top three choices and tallied the results which helped us to place an order of books for our classrooms. Not only were the students excited to read some new material, but also they learned a lot about financial literacy!

Another one of our goals this year was to bring more mathematics into the homes of our students. First, we produced two videos (with three more along the way) called “Mathematics Moments,” featuring step-by-step instructions for playing mathematics games that parents and students can view at home (we applied for a grant from the BCAMT to fund this project!) This turned out to be a big hit as our students were excited to see their teachers as “YouTube stars”!

QR Code for Mathematics Moment #1.

(from https://www.youtube.com/watch?v=LcRh66a1qWg&feature=youtu.be )

BY CYNTHIA CLARKE AND SARA LAI

Improving Our Practice: A Journey of Pedagogical Change

QR Code for Mathematics Moment #2

(from https://www.youtube.com/watch?v=LVSfaq0iOnM)

We also organized a Mathematics Expo night for students and parents. We invited other staff members to join us and had representation from every grade level K-7. Each teacher set up 3 or 4 tables and showcased different games that families could play at home. The event was attended by more than 50 families and was a great success. Feedback from parents, students, and staff was very positive and everyone learned a lot about mathematics!

This year has been one of tremendous professional growth and personal satisfaction for us. We have learned so much but we are not done yet! The learning never ends. So where do we go from here? We want to continue challenging ourselves and our students next year by creating more inquiry-based mathematics projects. We have already developed (and can’t wait to implement) a series of “Home Mathematics Challenges” for our families to explore together in the coming year. Also, we will be producing and posting more “Mathematics Moments” on the Internet for our families and colleagues to access.

V


BCAMTBritish Columbia Association of Mathematics TeachersGrant Application FormThe BCAMT will be funding several mathematical initiatives throughout BC. These initiatives must meet the goals and objectives of the BCAMT. Please note: this grant is not meant for individual professional development.

BCAMT Goals for 2016-2017:

1. Professional Development: to promote excellence in Mathematics education throughout the province by promoting professional development in all aspects of Mathematics education.

2. Curriculum: to promote the development and implementation of sound curriculum and the selection of appropriate resources.

3. Communication / Public Relations: Promote excellence in Mathematics education throughout the province by promoting good communication with members, other educators, Ministry of Education, parents, students and the community.

4. Membership: Promote excellence in Mathematics education throughout the province by maintaining current BCAMT membership and recruiting new members from all levels of mathematics education.

The BCAMT values the sharing of ideas and requests that successful applicants submit a summary of their initiatives with its highlights for publication in a future Vector or newsletter.

All applications must be postmarked no later than May 5th, 2017. Applicants will be informed about funding after approval by the BCAMT Executive. Successful applicants may wish to re-apply for funding each year but are not guaranteed continued support.

Complete this form and be sure to include:

• a rationale for funding request (maximum of 1 page)• details of your initiative (maximum of 1 page)• a detailed budget. (List expenses, other funding, etc.)


BCAMT MEMBER: [ ] Yes [ ] No

Applicant’s Name

School/District

Address

City/Town Postal Code

Work Phone Home Phone

Email Address

Initiative

Dates(s) Location(s)

Please check the target audience:[ ] BCAMT Members[ ] Teachers[ ] Students[ ] Community[ ] Other ___________________________

Please check the type of initiative:[ ] Workshop[ ] Research[ ] Contest[ ] New LSA[ ] Exhibition[ ] Other ___________________________

Applicant Signature

Send the completed application to: Brad Epp Chair, Funding Application Committee BC Association of Mathematics Teachers #51 – 383 Columbia Street West, Kamloops BC V2C 1K5 or email: [email protected]


BCAMT

AWARDS & CRITERIA

Outstanding Teacher Awards (Elementary; Secondary; New Teacher with less than five years teaching experience)

• shows evidence of significant positive impacts on students, staff and parents

• has initiated innovative and effective programs in their classroom, school, district, or province (teacher research, technology, activelearning, assessment, etc.)

• has and continues to demonstrate excellence in teaching mathematics regularly in British Columbia (teaching style, knowledge ofthe curriculum, current curriculum trends, etc.)

• has made contributions to mathematics education at the school, district or provincial levels (eg. workshops, seminars, conferences,community projects, curriculum development, publishing, etc.)

• is not a current member of the BCAMT Executive

Ivan L. Johnson Memorial Award

The Ivan L. Johnson Memorial Award is awarded in honour of long-time BCAMT executive member Ivan Johnson. Ivan donated money to the BCAMT for an award in which the recipient will receive significant funding to cover costs of attending the NCTM Annual Conference.

• inspires teachers to try new ideas that improve the quality of mathematics education

• consistently seeks ways to innovate practices in the mathematics classroom

• actively engages in professional dialogue involving mathematics pedagogy

• is not a current member of the BCAMT Executive, but is a member of the BCTF

British Columbia Association of Mathematics Teachers Awards InformationThe BCAMT sponsors awards in three categories (Outstanding Teacher, Ivan L. Johnson Memorial, and Service) to celebrate outstanding achievements of its members. Winners are honoured at a BCAMT conference and receive a commemorative plaque.


Note: Nominees for the BCAMT Outstanding Teacher Awards will automatically be considered for this award. Previous winners of BCAMT Outstanding Teacher Awards may also be nominated. Recipients of this award are expected to contribute an article to Vector.

Service Award

• has provided extraordinary service to mathematics education as an active member of the BCAMT for a significant period of time

SELECTION PROCESS

• all nominations are reviewed by the BCAMT Awards committee (consisting of a minimum of five previous award recipients) whorecommend the recipients to the BCAMT Executive for ratification

• each nomination is considered for two years, after which time the application can be re-submitted with updated information

HOW TO NOMINATE

Required documentation:

• a completed nomination form (one person per form);

• nominee’s curriculum vitae which demonstrates evidence of teaching, contribution, innovation, professional involvement and impact;

• nominator’s summary (one page only) explaining concisely the reasons for the nomination;

• two letters of support (one page each) with concise information about how the nominee fulfills the criteria.

Send all required documents listed below in an envelope to:

BCAMT Awards c/o Dave Ellis 2086 Newport Avenue Vancouver, BC V5P 2H8

Deadline: May 5, 2017

vector - bc association of math teachers review problem sets math links ... please contact brad epp,...

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