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Ve c t o rThe Official Journal of the BC Association of Mathematics Teachers

Spring 2012 • Volume 53 • Issue 1

Vector is published by the BC Association of Mathematics Teachers

Articles and Letters to the Editors should be sent to:

Peter Liljedahl, Vector [email protected]

Sean Chorney, Vector [email protected]

Membership Rates for 2010 - 2011$40 + GST BCTF Member$20 + GST Student (full time university only)$58.50 + GST Subscription fee (non-BCTF )

Notice to ContributorsWe invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable mate¬rials 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.

Articles can be submitted by email to the editors listed above. Authors should also include a short biographical statement of 40 words or less.

Articles should be in a common word processing format such as Apple Works, Microsoft Works, Microsoft Word (Mac or Windows), etc.All diagrams should be in TIFF, GIF, JPEG, BMP, or PICT formats. Photographs should be of high quality to facilitate scanning.

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


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 should not be reprinted without the permission of the editor(s). Once written permission is obtained, credit should be given to the author(s) and to Vector, citing the year, volume number, issue number, and page numbers.

Membership EnquiriesIf you have any questions about your membership status or have a change of address, please contact the BCAMT Membership Chair:Dave Ellis ([email protected])

Notice to AdvertisersVector is published three times a year: spring, summer, 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 IssueFull Page $ 300Half Page $ 160Quarter Page $ 90

Technical InformationThe layouts and editing of this issue of Vector were done on a Dell using the following software packages: Adobe Acrobat Professional, Adobe InDesign, and Microsoft Word.

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3 2011-2012 BCAMT Executive

5 From the Editors

C. Norris, S. Chorney, D. Wright, and T. Thielmann 7 Communication on

Communication

Ian deGroot 18 Damage on a Seismic Scale!

Marg McDonough 23 Evaluation FOR Learning - Levels Tests in Math

Duncan McDougall 27 Graphing Polar Coordinates

Andrew Adler, Joshua Keshet, and Dave Ellis 36

Challenging Algebra: Problems from the 2010 Math Challengers Competition

Erica Hopkins and Katja Schuurman 41

A Visual Learner’s Dream: Introducing Document Cameras into the Mathematics Classroom

Werner Liedtke 47 Why Do so Many Canadians Lack Numeracy Skills?

Egan Chernoff 58 Where Have all the Submissions Gone?

Tom O’Shea 63 Ivan Johnson…A Personal Tribute

Simin Chavoshi 65 Book Review: Exploring Probability in School

69 Spring 2012 • Problem Set

72 Spring 2012 • Math Websites

ON THE COVER: Cover art was created by Eliot Aharon, a PDP student at SFU. Eliot undertook professional development after a successful career in information technology industry. As a student, Eliot uses doodling as a technique to maintain focus and engagement in class.

Ve c t o rThe Official Journal of the BC Association of Mathematics Teachers


3 Vector 3 Vector

The 2011–2012 BCAMT Executive

PresidentChris BeckerPrincess Margaret Secondary School (Penticton)Work: [email protected]

Past PresidentDave van BergeykSalmon Arm Secondary SchoolWork: [email protected]

Vice PresidentBrad EppSouth Kamloops SecondaryWork: [email protected]

SecretaryColin McLellanMcNair Secondary School (Richmond)Work: [email protected]

TreasurerKatie Wagner Robert A. McMath Secondary School (Richmond)[email protected]

Membership ChairDave Ellis Home: [email protected]

Elementary Representatives Sandra BallNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Selina MillarNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Carollee NorrisNumeracy Support Teacher School District 60 (Peace River North)Work: 250-262-6028 [email protected]

Donna WrightEcole Sandy Hill Elementary (Abbotsford)Work: 604-850-7131 [email protected]

Middle School RepresentativeDawn DriverH.D. Stafford Middle School (Langley)Work: [email protected]

Spring 2012

The 2011-2012 BCAMT Executive

Secondary RepresentativesMichael FinniganYale Secondary School (Abbotsford)Work: [email protected]

Michèle RoblinHowe Sound Secondary (Squamish)Work: [email protected]

Danny YoungMoscrop Secondary SchoolWork: [email protected]

Christine YounghusbandHome: [email protected]

Bryn WilliamsProgram Consultant: Mathematics and Science (Burnaby)Work: [email protected]

Independent School RepresentativeRichard DeMerchantSt. Michaels University School (Victoria)Work: [email protected]

Listserv ManagerColin McLellanMcNair Secondary School (Richmond)

NCTM RepresentativeMarc GarneauNumeracy Helping Teacher K-12School District 36 (Surrey)Work: [email protected]

Vector EditorsPeter LiljedahlAssociate Professor, Faculty of EducationSimon Fraser University (Burnaby)

Sean ChorneyMagee Secondary School (Vancouver)Work: [email protected]

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Happy New Year. As you may have noticed, our last publication of Vector was dedicated solely to elementary teachers. The BCAMT executive made a decision to focus specifically on elementary teaching and learning for the inclusion and support of our elementary teachers. We hope you enjoyed what the issue had to offer. For future issues, we continue to request articles from an elementary perspective. If you are an elementary teacher or you have any connection to elementary teaching or learning, we invite you to contribute to Vector. Vector supports math teaching from K-12, and beyond.

For this issue, we also announce our new Vector webpage. On the BCAMT website (www.bcamt.ca) under the communication tab, you will find the Vector page. On this page you will gain access to all the old Vector issues, in pdf format, as far back as the late Sixties! Scan some of the old 70’s or 80’s articles and enjoy the changes mathematics education has gone through in the past few decades. There is also a full page of the types of articles we, at Vector, are looking for as well as information for contributors to Vector. We also have a new plan for book reviews, check it out. We hope you enjoy the website.

In this issue, as planned, we present a few excerpts on the process, “Communication”. To initiate a discussion in dealing with the seven processes, we invite responses and/or further thoughts on the processes. In our next issue, we will post some thoughts on the process, “Connections”.

In this issue, Ian de Groot starts us off by connecting some recent “real life” Richter scale data with some grade 12 math content. He presents a fictitious lesson of how a conversation about such data might look like. Dr. Marg McDonough, in addressing, the challenges of testing students of different abilities, presents an alternative type of test, what she terms “level testing”. Different levels of questions provide varying challenges to students of different abilities. She provides a good argument as well as some examples.

Duncan McDougall shares one of his many mathematical explorations in his article about graphing polar coordinates. He offers a six-stage approach as a guide to help plot polar graphs. We also have an update from 2010 MathChallengers competition presented by Andrew Adler, Joshua Keshet, and Dave Ellis. They provide four challenging questions used in the competition, but thankfully they also provide solutions.

Erica Hopkins and Katja Schuurman present us with an experience in using a document camera with their math class. With the support of a BCAMT grant, they were able to purchase and implement the document camera. Their article, in this issue, reports their success. Werner Liedtke challenges us, yet again, to consider the use of language in education. He poses many questions surrounding the use of terminology and catch phrases used in education circles. And Egan Chernoff shares some of his experiences as editor of Viniculum, Saskatchewan’s mathematics teacher’s journal, particularly addressing the challenge of gathering submissions.

Tom O’Shea writes a thoughtful eulogy for the late Ivan Johnson, a past president of the BCAMT.As well, we have a book review from Simin Chavoshi, and some challenging problems for you to look through.

We hope you enjoy the issue.

FROM THE EDITORS

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Some mathematicians find it hard to keep New Year’s resolutions.....

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Communication is one of the processes in the BC IRP. The following excerpts present “communication” as it is seen and implemented from four different teachers’ perspectives. The editors are interested in responses to these pieces as well as excerpts about the next process, connections, of which we will be discussing in the next issue.

How does communication support the intended curriculum? Carollee offers some critical insight into

communicating in the classroom as well as some practical ways

of getting students to communicate.

By Carollee Norris

I was excited to read that the editors of Vector are beginning of series about the mathematical processes that are imbedded the BC curricular

documents (also known as IRP’s or Integrated Resource Packages). My personal experience with the mathematical processes is that, together, they are the means through which students build an understanding of mathematical concepts. When students are doing a set of algorithmic problems, they only need to be concerned about following the required rule(s) correctly and remembering the basic facts needed to execute those particular rules. How very different it is when students are given a word-based problem to solve and then asked to represent their thinking with manipulatives, tools, drawings, and words, and also to verbally explain their thinking to others.

Communication, as a math process, implies the same meaning as the word does for ordinary life. Wikipedia gives this definition: “Communication is the activity of conveying meaningful information. Communication requires a sender, a message, and an intended recipient… The communication process is complete once the receiver has understood the sender.” Wikipedia goes on to list that in human interaction there can be nonverbal, visual oral and written communication.

There are some interesting points to note here. First, the activity requires “conveying meaningful information.” The italics are my own, but I want to highlight the idea of meaning. Rule-based, algorithmic calculations are usually meaningless for students. The learners often have little or no understanding of why the rule works or of the underlying mathematical concepts. Therefore, they are not conveying meaningful information since it has no personal meaning for them. Secondly, Wikipedia refers to the fact that the process has been completed “once the receiver

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has understood the sender.” Just putting information out there is not communicating. The understanding by the receiver must also take place. It has to be clear enough for another to understand what is being communicated.

There are many times when providing the opportunity for students totalk with partners is effective. This is easy to facilitate if partners are designated ahead of time. It could be as easy as designating seat or table partners because of proximity, or the teacher can choose the partner pairs. I often posted a list of talking partners. At different times in the year I partnered students for different reasons. Attributes such as ability, behavior, and gender sometimes mattered. Seating in the classroom can matter as well too, because often I deliberately partnered students who were are not sitting near each other. Thus partner talk gave gives me an opportunity to have the students get up and move – something recommended by brain researchers to enhance learning. Students might be asked to talk about concepts from the previous day’s lesson, to share what they remember about quadrilaterals, or to estimate the answer to a particular problem. The possibilities are endless!

Problem solving gives a wonderful opportunity for students to talk about mathematics. In the beginning, as students work together in pairs or small groups, they must figure out the what mathematical operation is inherent in the problem, and then they must figure out what strategies and/or tools will best be used to help them solve the problem. Students are encouraged to find multiple strategies and pathways to single solutions and If there is a single correct solution to the problem, then when a solution is found using a particular tool or strategy, the students begin looking for another method. Iif there were are multiple correct solutions to the problem, then students continue looking for more solutions, as long as the working time permits.

Once the working time is up, then communication can take centre stage with the whole class involved. The focus is on sharing solutions, discussing strategies and tools, and wrestling with making sense of all the mathematics involved in the solution(s). As students attempt to articulate their thoughts and ideas with each other, they cannot help but think about the mathematical concepts on a deeper level.

Last year I was facilitating such a discussion in a grade 5 class. The students had worked in pairs for 20 minutes on a particular problem, and now were sharing strategies and solutions. As one girl was attempting to explain the thinking process that she and her partner had gone through, she suddenly dropped her train of thought, put her hand on her hip, and loudly announced, “This is hard!” I knew exactly what she was talking about. She was finding it difficult to make her thinking clear as she spoke, and her frustration was evident. My response was to agree

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with her, that, yes, it indeed was difficult for her to talk about her math thinking, but that the process would only get easier, as in mastering any task, with practice. It is only as students practice talking about, writing about, describing, and explaining their thinking, that they get good at it. If they are not given repeated opportunities to practice, they don’t grow much in that area.

One thing I have done to help students with communication in math,(although it is certainly not a brand new idea) is to use a word chart of mathematical terms. Many primary classes use a word wall to help build general vocabulary with students, and I have seen some such word walls that are very elaborate in their set-up. Personally, that kind of word wall is not manageable for me. So instead, I developed a simple system that worked for me and was effective for the students.

I would tape up on the wall a piece of chart paper, label it “Math Words” and then use different coloured marker for each new word I added. If I wrote a phrase (e.g. “least common multiple” or “ten frames”, the whole phrase was in a single colour to distinguish it as a phrase. I added words rather haphazardly, not trying to group words according to strand or topic, writing as many on a line as will fit. Sometimes I would will add a visual cue (e.g., a plus sign beside the word “sum”, a drawing of a trapezoid beside that term) to help students connect the word to its meaning. Once the page fillsed up, I would will just add another piece of chart paper beside the first one and continue. I was am always delighted with how often I would will see the students using the math word chart. They would be talkingwill talk to a partner or writing write something on their page about how they had solved a problem, and then stop to look over at the chart. I could watch them scan the lines until they came come to the word they needed. It seemsmed to jog there their thinking, and off they would go, back to their talking or writing. The math word chart was as effective in my primary grade 2 class as it was in my intermediate classes.

What it all boils down to is this: if students are to learn to communicate effectively in mathematics, then they absolutely must have many opportunities to practice. There is no short cut, no “two-week unit and they have it”. What we choose for students to do, day after day, in our mathematics classrooms matters! I’ll close with this, one of my favourite quotes: “There is no other decision that teachers make that has a greater impact on students’ opportunity to learn and on their perceptions about what mathematics is than the selection or creation of the tasks with which the teacher engages students in studying mathematics.” (Glenda Lappan & Diane Briars. “How Should Mathematics Be Taught?” Prospects for School Mathematics. NCTM, 1995).

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What is the process when students communicate? Sean offers a perspective that communication is not a process of transferring knowledge but a form of development.

By Sean ChorneyCommunication is one of the seven processes in the British Columbia mathematics integrated resource package (IRP). It, along with the other processes, is described as a necessary and critical component in a mathematics program. Further elaboration states that in order to achieve the goals of mathematics education and encourage life long learning, communication is essential. Clearly the communication act is valued in the curriculum. There are, however, very few guidelines provided in the IRP. Although IRP does state that students should have an opportunity to read about, represent, view, write about, listen to, and discuss in order to express their understanding, the activity of communication is still left open ended for interpretation, implementation and integration. What is communication? How can we implement it into the classroom?

I offer a lesson, I’ve learned, in working through this process of communication. The first thing I notice when I attempt to implement the act of communication in either written or verbal form in the classroom is that it is very messy. The process is challenging both for the teacher and the student. Rarely have I come across an example in my classroom where the communication process went “just right”. When I have attempted to introduce writing or speaking into the classroom it seems to present much more of a challenge than it should. And in my observations, there seems to be more going on than just the organizing of one’s thoughts. There is a sense of a deeper thinking that goes beyond the choosing of the best words or the organizing of one’s presentation. In my observations, there seems to be a development of the content (the message) in the act of communication. That is a very important point for me. This development of content within the process of communication may affect how we, as teachers, implement mathematics into our classrooms. It may be obvious to many of us but I often find myself getting caught up in the model of communication as being an act of representing what exists in the “mind”.

If one were to think of communication as a process of transferring information back and forth, from one to another, the tendency might be to think of communication as a way to determine whether a student has received or understood the message. Once could request that a student write something so as to determine whether they “get it”. But I question this method, for it assumes that mathematical content can exist as a complete coherent “whole” in the mind. Learning, in this case, would be based solely on how good a student is at taking in information and

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accommodating that information. But I believe learning is much more than that.

An alternative view is to adopt the idea that communication is a form of development. That is, in the process of writing or speaking, the content, the mathematical content, is being developed along with the word choices and the organizing of thoughts. It’s like a tri-developmental process. Communication as categorized in the IRP is a process and a process as I read it is ever developing. In communication one does not arrive at a final destination. Each moment in the act of communication is a moment of development. Vygotsky put much emphasis on this coordination of language and thinking. He claimed that they develop dialectically, together.

I offer a short excerpt from Tim Rowland’s “The Pragmatics of Mathematics Education: Vagueness in Mathematical Discourse” (2000). A student is explaining how he found a pattern of squares emerging from a sequence of dot images. It’s a piece that has always made me smile as it reminds me how difficult it is to explain something one is not fully sure of.

Allan (student): But as, if you went round all the dots, it would only come to about, if you did it once it would come to one, uh, less than nine, you got, uh, because, because there’s o… there’s only…cause you only have, y…you can miss out a line exactly, cause you, you can miss out a gap, cause you, um, y’d ‘ave to go all the way round the whole dots. Probably if you minus one from the …, if you square the number you’d probably find that if it was actually, if you minus one from that you’d probably find that that would be the answer… (p. 192).

One could read this as a student finding it hard to express what he knows or one could see the development of mathematical understanding. The difference being, that in the first perspective, the student is using words to describe understanding, while the second perspective he is relaying thinking through words. Communication, that involves mathematical development. Development that is on going and possibly may never be fully complete. I still find myself learning new things about “familiar” mathematical formulations.

Giving this student many more opportunities to speak or write what he is thinking will help him develop his ability to represent his ideas and it will also support mathematical development. And so if his ideas are still being formulated then I suggest that his verbalization may never be perfect. Offering students to be engaged in these activities is important. Not only do they get better at organizing their thoughts and their articulation but also the mathematical content will further develop. This tri-development is, I believe, the process of learning. In the act of communication, verbalizing, writing, thinking and understanding are

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interrelated aspects of mathematical engagement. They are all essential components needed for development. Without one of these, there will be challenges with learning. With this in mind I think it is important not to expect clear unambiguous articulation of ideas. I only have to listen to myself define square root in an ad hoc fashion and I know that simplicity and/or clarity is not always possible. The expectation that communication should be clear and straightforward might be an assumption that inhibits students from thinking freely and consequently learning mathematics.

It is not a natural event to have to articulate mathematics. It takes more than just activities that practice writing or speaking. It also takes “good” mathematics teaching. That is, if mathematics is being developed through communication students need to be given tasks that promote expression without concern for initial formulations and they need to be guided through discussion as to what kinds of formulations might be appropriate. And then, and most importantly, the students should be given another opportunity to re-represent their understanding through communication. This is where, in my experience, the most development will occur. Asking students to communicate is one way to strengthen mathematical development. But we also need to remember that communication is not the “answer” to our challenges, there are six other processes to consider!

How can we assess communication? Donna presents us with a communication rubric and how it might be implemented.

By Donna Wright The exciting rubric presented in this paper can be found in Damian Cooper’s book Talk About Assessment, High School Strategies and Tools. The rubric is a great place to start the communication process. A student’s ability to communicate effectively can either help them advance or it can hold them back. Communication allows us a glimpse into a student’s mind, it allows a teacher to see what they are thinking, or how they understand and reason through a problem. And the communication process also allows us to see what we can to do to further help them. But we must tread carefully as we need to be clear on what it is we are after when we ask students to share with us and what it is they are to communicate. It is important to share with the students what it is we are looking for in their “mathematical communications”.

What is communication? Communication as defined by Webster’s dictionary is the “sending, giving, or exchanging information and ideas,” expressed both verbally and nonverbally. Under the heading Communication [C] in the BC Ministry Integrated Resource Package it states:

Students need opportunities to read about, represent, view, write about, listen to, and discuss mathematical ideas. These opportunities allow students to create links

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between their own language and ideas, and the formal language and symbols of mathematics. Communication is important in clarifying, reinforcing, and modifying ideas, attitudes, and beliefs about mathematics.Students need to be encouraged to use a variety of forms of communication while learning mathematics. Students also need to communicate their learning using mathematical terminology. Communication can play a significant role in helping students make connections among concrete, pictorial, symbolic, verbal, written, and mental representations of mathematical ideas (p. 6)

Cooper’s rubric is presented below:

Generic Rubric for Mathematical CommunicationCriteria Level 0 Level 1 Level 2 Level 3 Level 4Reading and interpreting mathematical language, charts, and graphs

No interpretation present

Misinterprets a major part of the information, but carries on to make some otherwise reasonable statements

Misinterprets part of the information, but carries on to make some otherwise reasonable statements

Correctly interprets the information, and makes reasonable statement

Correctly interpret the information, and makes soluble or insightful statements

Using correct mathematical symbols, labels, units and conventions

No correct use of symbols, labels, units or conventions can be found

Sometimes uses mathematical symbols, labels, and conventions correctly

Usually uses mathematical symbols, labels, and conventions correctly

Consistently uses mathematical symbols, labels and conventions correctly

Consistently and meticulously uses mathematical symbols, labels, and conventions, recognizing novel opportunities for their use

Using appropriate mathematical vocabulary

No use of appropriate mathematical vocabulary

Sometimes use mathematical vocabulary correctly when expected

Usually uses mathematical vocabulary correctly when expected

Consistently uses mathematical vocabulary correctly when expected

Consistently uses mathematical vocabulary correctly, recognizing novel opportunities for its use

Integrating narration and mathematical forms of communication

Neither mathematical or narrative form is present

Provides either mathematical or narrative form, but not both

Provides both mathematical and narrative, but the forms are not integrated

Provides both mathematical and narrative forms and integrates them

Produces a variety of mathematical forms and narrative, integrated and well chosen

Integrating narration and mathematical forms of communication

No explanation of justification is provided

Provides explanation and justification that have limited clarity

Provides explanation and justification that have some clarity

Provides explanations and justifications that are clear for a range or audiences

Provides explanations justifications that are particularly clear and detailed

What is really nice about Cooper’s rubric is that it carefully unpacks the above description of communication into five different criteria areas. This makes it easier for teachers and students to use, score and to have a clear purpose by focusing on one area at a time.

Why do I want to teach/assess communication? After all, I am teaching “mathematics”. Well, first off, it is part of the curriculum. Second, our students will need to be able to problem solve and communicate effectively to experience success in the future, no matter what careers/jobs they choose to do. It is not easy to communicate one’s mathematical thinking and reasoning to others; it needs to be taught. As a teacher of mathematics, I need to be able to teach students how to do this.

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I have found the implementation of the rubric to work best when I share it with students before actually using it. I hand out copies to the class explaining what it is and what it will be used for. The students look at it in small groups and develop questions and comments that can be shared with the class during facilitated discussion. We then look at a problem

that either I, or an anonymous student from another class, has solved. As a group we use the rubric to collaboratively assess the “communication” ability of this student, allowing for discussion and questions. Yes, this takes time, but it well worth it and makes the entire process more successful than without discussion.

After this process, I give the students a problem to solve, usually one that they have already experienced success with. I remind the students that this time they need to explain their thinking, connections and reasoning. Then I have the students use the rubric to self assess their communication skills. This “forces” the students to self-reflect on what the have done, and make improvements that they might notice. It also shows the teacher how the student believes s/he is doing, as opposed to your assessment of their performance. If the two perspectives match, great, if not then this might need to be further investigated.

Finally, with the same rubric, I assess the student’s ability to communicate. Looking at the data for each student, I look at the score for each criteria area, as well as an overall score. After a couple of times doing this, I am able to determine the areas of communication in which the students have strengths and weaknesses. This will allow me to focus on just the specific areas of need for each student. For example, after marking an assignment, I may have noticed that my students are able to use appropriate mathematical vocabulary, yet are having difficulty integrating narrative and mathematical forms of communication. We could then focus on just this one criteria area, providing students with descriptive feedback of what they need to do to improve and sharing samples of what this actually looks or sounds like. Focusing on one criteria area of the rubric allows us to have a clear purpose for what we are doing, as well as makes the job of assessing “communication” less daunting and more efficient.

An example of a problem (Grade 8/9) that works well with this rubric is:

How far will a Matchbox car roll off of a ramp, based on the height the ramp is raised? Students need to enter data in a table and plot it on a graph. They then need to determine an equation that can be used to make predictions. Students will need to explain how they solved this problem, explain the connections they’ve made, identify which variable is the dependent/independent variable etc. It is a fun and engaging activity; the data collection can be done in small groups. This problem and others like it can be found in John Van de Walle’s book Student –Centered Mathematics, Grades 5-8 on page 302.

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This communication rubric could also be easily adapted to use in earlier grades. I invited my colleagues to help me with this, it lead to great discussion about assessing the mathematical processes. In the future I plan on rewriting the rubric with the students.

Research and experience shows that whenever one can involve the students it creates better understanding. This can only lead to greater success for all.

Assessing student communication abilities is a bumpy road, one does get dirty and sometimes falls down. It involves self-reflection, tweaking, frustration and insight. But through it we receive the honour of really getting to know our students, the way in which they learn, think and reason like we never have before. This is very rewarding, as it allows us to target the needs of our students and take them further in their understanding than we might have thought possible. Hopefully you find this rubric helpful and useful in your journey of assessment.

Does communication provide a stronger formation of math concepts? Travis shares his thoughts of communication as a tool in the mathematics classroom.

By Travis ThielmannDuring my studies and readings, a teaching tool has jumped out as having lots of potential to be very effective and helpful as a math teacher. The tool is a relatively simple concept that may have broad consequences in formative and possibly summative assessment. The idea is to have students communicate verbally or in writing what they are doing as they solve a problem. This would include why they make certain steps, why the answer seems correct, or maybe different approaches. It would also include statements about when they are stuck, thoughts on getting out of it, and ideally how they got past that stage. All of this has opened the possibilities of communication, allowing me a deeper window into my own thinking, and helping me process my own problem solving strategies. It also shows the possibilities of such a strategy in the classroom, because now I get a window into what each student is thinking as they are solving problems, even when I am not there to talk with them one-on-one. This may save time for those students who got a question wrong and, as a teacher, I am sifting through their work to try to figure out what they know about the math I’ve taught, and what they do not understand. If they are writing their thoughts, as well as doing the math, then there is opportunity for a stronger sense of what the student is doing.

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I see this as valuable in all marked work as well as non-marked work. In non-marked work, students then have the opportunity to go back and look at that they have done before and why. “Why” is very important to mathematics, and if students have a deeper understanding of why, not only will they have deeper retention of the material, but they will be able to apply it to a deeper and wider set of problems, ones that may not have been directly addressed in class. In fact, I would use this strategy in science classes as well, when feasible. This also allows for marking to occur more on understanding than just achieving right answers.

In marked work, despite the obvious increase in material to read as a teacher, I think it may actually cut down on marking time particularly when trying to mark large problems where a student has given an incorrect answer. It can take a long time to find out where the student went wrong just to give them some marks on their knowledge. Here the knowledge is ideally well laid out with words helping the teacher understand what is happening in the students’ mind as they solve the problems. It seems the benefits have the potential to outweigh the costs once the system is built into the students as normative in the math class.

The freedoms abound, if students make scribal errors, there are explanations with it showing their understanding of correct mathematics despite their mistake. If there is a misunderstanding of the math, then, as a teacher, the misunderstanding is more directly revealed, giving opportunity to address the specific problem rather than hoping to hit a broad section of math to help students along. All of this gives greater communication between teachers and students, allowing for stronger formation of math within the students, because the teacher can more easily address the actual problems. It also relieves some pressure off students in the high stakes game of getting the right answer, while still allowing a check for mathematically understanding in a more nuanced way. This may allow students who were failing due to pressure rather than lack of understanding to succeed, while still giving room for students who were already successful to grow deeper in their own understandings. Overall I am excited about the benefits I personally have experienced using this strategy, as well as the benefits it will give me as a teacher.

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51st NW Mathematics ConferenceVictoria, British ColumbiaOctober 18-20, 2012Victoria Conference Centre

Come to Victoria next fall and… Smell the roses with keynote speakers: Dan Meyer, Catherin Fosnot, and Patrick Vennebush. Restore your enthusiasm and expand your knowledge, and share ideas....Renew acquaintances, build networks, and share ideas….

www.nwmc2012.com

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This story was written as a result of the news reports on recent earthquakes in Haiti, Chile, New Zealand and Japan. When reading several articles comparing strengths of earthquakes in various countries, I used the Richter ratio calculation that is taught in the current Principles Math 12 course to check, but I could never get the same numeric results shown in the articles. As a result, I dreamed up the fictitious lesson that follows.

Peter’s hand shot up from the back of the Math 12 class and I knew right away that I was in for the usual grilling.

“ Yes, Peter, what’s on your mind? ” I asked.

“Do you remember the unit that we recently finished on logarithms?”

“It is not something that I will forget, Peter, especially the class average on that unit. ”

“Well, I was thinking about the magnitude of earthquakes especially since the big ones that recently hit New Zealand and Japan. I remember the part that we covered when you said that the Richter scale compares the intensities of earthquakes, and that the intensity of an earthquake is measured by the amount of ground motion or shaking which is recorded on a seismograph.”

“Super, Peter. You have a good memory, please go on”.

“Well sir I was reading the front page of the Globe the other day and there was a chart comparing the size of recent earthquakes and I tried to use the formula that you gave us during the class, and I guess that the Globe and Mail newspaper is out to lunch!”

“How is that ? ” I asked quizzically.

“Well the chart shows that the 2011 Japan quake with a Richter magnitude of 8.9 is 708 times as strong as the 2010 Haiti quake which had a magnitude of 7.0 ”

Damage on a Seismic Scale!By Ian deGrootIan is a retired math teacher and department head from North Vancouver who still tutors. He was an executive member of the BCAMT for 28 years and president for 3 years and also served on the board of directors of the NCTM in Reston, VA. He was presented with a Prime Ministers Award for excellence in teaching mathematics by Jean Chretien in 1994. “

By this time

most of the

students had

become fully

engaged in

the dialogue

between Peter

and me and

had pulled out

their trusted

calculators and

were busily

crunching the

numbers.

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“So according to our Math 12 calculations how would I calculate the intensity of Japan 2011 compared to Haiti 2010 ? ” I queried.

“ Well I think that we would take 108.9 ÷ 107.0 which gives 101.9 and using my calculator this is about 79 times as intense.

“ Here is the diagram from the newspaper, sir, and it shows that Chile 2010 with a Richter magnitude of 8.8 is 501 times as strong as the Haiti quake, but using the equation from Math 12, 108.8 ÷ 107.0 the intensity is only 63 times as strong, yet the newspaper claims that it is 501 times as strong. I don’t get it”

Peter shared the diagram from the newspaper article with the class.

“Yes I see the problem and I’m puzzled” I responded sheepishly, offering no explanation as I had none to offer.

By this time most of the students had become fully engaged in the dialogue between Peter and me and had pulled out their trusted calculators and were busily crunching the numbers.

The class was unanimous in the conviction that the columnists should all go back to school and redo their Math 12 logs unit!

I naturally became enthusiastic by the level of interest this topic had aroused and decided to pursue it to a conclusion.

The class was

unanimous in

the conviction

that the

columnists

should all go

back to school

and redo their

Math 12 logs

unit!

Globe and Mail, March 14, 2011

Spring 2012 20

I quieted the now very noisy and excited class with the universal timeout gesture.

“I have an idea” I exclaimed.

“As tonight’s homework assignment, I will ask you to do a little research and see what you can come up with that can clarify this question for for tomorrow’s class.”

The following morning’s Math 12 lesson was a teacher’s dream come true. Every student was anxious to describe what he or she had discovered.

I realized that Peter had touched on a topic that was “real” math and close to home at the same time.

Since Peter was the student that had initiated the question in the first place, I let him take over the class and present his findings.

“OK, Pete, you’re on stage” I exclaimed.

“Well, here is what I discovered. The Richter scale is based on the amplitude of seismic waves, the stronger the earthquake the stronger the vibrations it causes. Since Richter magnitude is a logarithmic scale, an increase of one in magnitude corresponds to a factor of ten increase in the amplitude of ground motion. So what we learned in our Math 12 class is how to compare seismic shaking.”

I glanced around the classroom and was pleased with the attention that was being given to Peter’s explanation.

“ Go on Peter ”, I exclaimed. “You’re doing a great job of explaining our Math 12 ratio, but what about the newspaper’s calculation? How did they arrive at their numbers?”

“Well, I googled earthquake energy and sifted through the tons of information available there.

What I discovered is that the ratios given in the newspaper diagram are based on the moment magnitude scale.

This scale is a direct measurement of the amount of energy released by an earthquake. Scientists developed a formula which involves the average displacement of the fault, the rigidity of the crust, and the total area of rupture on the fault ” Peter explained while reading from his scraps of paper notes.

“Terrific work, Peter. You have our attention.

Give us the formula so that we can verify the calculations ourselves.”

“ The formula involving displacement and area and simplifies to 10(1.5(m1-

m2)) where m1 and m2 are the magnitudes of the earthquakes that we are comparing.”

The following

morning’s Math

12 lesson was a

teacher’s dream

come true.

Every student

was anxious to

describe what

he or she had

discovered.

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“So”, I interjected, “ Japan 2011 with a Richter of 8.9 and Haiti 2010 with 7.0, using your formula, would give us?”

Peter confidently continued his explanation.

“It would be 10(1.5(8.9 – 7.0)) which is 10(1.5(1.9)) and voila we have the number that the Globe came up with….. 708, which means that Japan 2011 was 708 times as strong as Haiti 2010 or released 708 times as much energy!”

The class burst into applause and Peter beamed proudly.

“But that surely can’t be the end of the story? ” I ventured.

“I mean where did that formula come from? Did it just drop from the heavens?”

I looked around the room hoping to catch the eager eye of anyone but Peter to respond to my query and, as luck would have it, there was Sara with her hand held high in the air ready to join the fray.

“Yes, Sara, do you have an idea”?

“Yes sir I do. I googled earthquake measurement and many sites came up which explained almost everything clearly”.

“Terrific work”, I replied eagerly “Go on Sara.”

“Can I read from my notes?”

“Certainly, Sara.”

“One site that I liked says that the Richter scale was designed in 1935, by a mathematician named Charles F. Richter who worked for the California Institute of Technology.”

“The Moment Magnitude Scale, which I’ll call MMS, was introduced in 1979 by Tom Hanks who was a scientist for the U.S. Geological Survey and Hiroo Kanamoi, who was an employee of the California Institute of Technology.”

“MMS and Richter both measure the magnitude of an earthquake, but MMS determines the total energy released by an earthquake through a math formula that is based on the area of a fault that is split during an earthquake, while Richter is based on the logarithm of the amplitude of the waves shown on seismographs.”

“The article claimed that the Moment Magnitude Scale is the reason why more geologists and seismologists are using it as a tool to study earthquakes, since it gives researchers a clearer picture of an earthquake’s impact on the earth.”

“Well done, Sara, but where’s the math?”

“Well, the formula that Peter used was obtained from the findings of Tom Hanks and Hiroo Kanamoi who developed the seismic equation:

Spring 2012 22

M = (2 ÷ 3)log(M0) – 10.7

where M is the moment magnitude scale and M0 is the magnitude of the seismic moment.

So solving this equation for log(M0) gives:

(3 ÷ 2) (M +10.7) = log(M0)

then, from our lesson on logs, we know that

M0 = 10(1.5) (M +10.7)

To get the ratio of energy release between two earthquakes we just use:

which gives us 10(1.5(m1- m2)) using what we learned about exponent rules, and this gave us the formula that Peter used in his explanation.”

“Awesome, to use your lingo, if I may. Great work class!

The 2011 earthquake in Christchurch, New Zealand was 6.4 and caused tons of damage. My question is how many times as much energy did Japan 2011 release or how many times as strong was Japan compared to Christchurch?” I continued.

Ravi’s hand was one of the fastest to go up.

“OK Ravi, let’s have your solution.”

“It’s 10(1.5(8.9 – 6.4)) which equals 5623 times as strong” exclaimed Ravi happily.

“My final question would be how would that stack up with what we learned in Math 12 to compare the magnitude of earthquakes?”

“ Our answer would be 108.9 ÷ 106.4 = 316” was the response from a voice at the back of the room.

“Quite a difference. Well we all now know why, I hope.

Perhaps in future when we study earthquake magnitudes in our logs unit, I think that we should use the MMS scale. What do you all think?”

“Yes, then we would be able to understand what the newspapers are writing about.” Bruce loudly exclaimed.

So what we

learned in our

Math 12 class is

how to compare

seismic shaking.

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Evaluation FOR Learning - Levels Tests in MathBy Dr. Marg McDonough Dr. Marg McDonough has been involved in math education at every grade level for over 33 years, as a classroom teacher, mathematics specialist teacher, staff associate at the University of Victoria, sessional instructor in math education at UBC, and as a teaching school principal. In addition, Marg has given many workshops on numerous math topics such as the use of manipulative materials, motivation in the math classroom, problem solving, and starting a math lesson with excitement.

Have you every given a test and found that a third of the students didn’t even attempt most of the questions? At the same time,

another third of the class finished the test in record time and seemed bored or disinterested. For evaluating the progress of these students, this particular test has failed miserably.

The good news is that the test seemed to hit the middle third of the class at about the right level, so you can use this particular measurement instrument to assess the progress of these students. But what about assessing the other two thirds?

Most of us have had this experience, and wondered how we can evaluate the progress of more of the students without sitting with each one individually – an excellent way to gauge how a student is progressing, but prohibitive in terms of time and a typical teacher’s schedule.

The literature on ‘Evaluation for Learning’ gives a lot of ideas and suggestions on how to address this problem. The philosophy of evaluation used as a learning tool is excellent and the resulting progress in students’ learning is amazing. The strategies of discussing the content of the evaluation with the students, allowing them to contribute to the questions, and encouraging them to take ownership of their learning and the evaluation process, leads to greater learning on the part of the students. While including all of these strategies is excellent for student learning, there are times when it is necessary to assign a student a final mark on a topic. This is where a levels test is most valuable.

The whole idea of a levels test is to allow students to choose the questions they wish to answer by providing a range of difficulty levels on the same topic in the same test. Students can focus on what they can do rather than

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what they cannot do, while at the same time the more capable students can choose to challenge themselves. A real celebration of learning.

A levels test is not complicated or difficult to design or administer. Begin by reviewing the intended learning outcomes (ILO), making sure that you know exactly what you expect the students to know at the most basic level. Prepare the students for the evaluation as you would for any other assessment; review the material to be tested, get input from the students as to the content of the test, and so forth. Now it is time to build the test.

A typical levels test may have three or four levels for each topic. An example could be a four levels test containing 200 possible marks. In order to achieve 100% on the test, students must accumulate 120 marks. Students choose the questions they wish to answer at the level they feel most comfortable. All of the questions meet the curriculum requirements and are designed to assess the students’ proficiency with the ILOs. However, by allowing students to decide on the level of difficulty, a levels test gives the less capable students the opportunity to be successful at the most basic level, and the more capable students to demonstrate their abilities at a much higher level.

The LEVEL ONE questions are easier, shorter, faster to complete and are therefore worth fewer points. A student who chooses LEVEL ONE will have to complete many more questions, thus demonstrating his or her proficiency on the topic, or highlighting areas in which he or she is having difficulty.

The higher level questions require more knowledge and understanding, take more time to complete, and are therefore worth more marks; the number of marks increasing as the difficulty level increases.

Here is an example of one topic on a typical four level levels test. The heading and general directions for the entire test are given to show the set up.

Order of Operations – Levels TestThere are four levels in this test and a possible 250 marks. In order to achieve a perfect score (100%) you need to correctly complete questions worth 150 marks.

Evaluate the following expressions:

LEVEL ONE (one point per correct answer)

a) 5 + (6 x 3) b) 14 - 8 ÷ 2 c) 5 x 2 + 3 x 4

Have you every

given a test and

found that a third

of the students

didn’t even

attempt most of

the questions?

25 Vector

LEVEL TWO (one and a half points per correct answer)

a) 6 x 2 + 4(5 – 3) b) (8 – 4) ÷ 2 + 7 x 2 c) 12 x 4 ÷ 4 x 12

LEVEL THREE (two points per correct answer)

a) [3(5 + 2 x 3) – 3] ÷ 10 b) (3 x 2 + 4)(9 – 2 x 2) ÷ 5 + 4 x 2 c) 12 ÷ ([7 + 4 – 2] ÷ 3) x 2

LEVEL FOUR (three points per correct answer)

[9 ÷ (14 – 4 + (5 + 7) ÷ 3 – (8 + 3)] x 2a) -------------------------------------------------- etc. (9 x 4 ÷ 9 x 4 + 5) ÷ (3 + 4)

For the purposes of this example, there are only a few questions at each level – on a levels test, you would include more questions. Also, this example only covers one topic, basic order of operations with whole numbers. The use of exponents, common and decimal fractions, integers, basic algebra and other related topics using order of operations are other topics that might be on this test. In each case, each topic should be broken down into the most basic level (meeting the ILO), and then more difficult levels added for students who exceed expectations.

After using levels tests for many years with my grade seven, eight, nine and ten students I found the following results:

a) students took more responsibility for their performance on the test and thus, their learning,

b) students enjoyed levels tests and were disappointed if I gave them a regular test, They repeatedly told me that they felt in control of their learning, and found the levels test more interesting,

c) students gained confidence in their math skills and their ability to do math,

d) almost the entire class was fully engaged in the test for almost all of the test period. More time spent on working through the questions usually translated to greater understanding and more learning,

e) some students always started at level one and worked through these questions, while other students immediately began with the highest level,

f) I always discussed the test with the students after they had completed it. Students told me that they often challenged themselves to attempt at least a few questions at a level that was outside their comfort zone. Once they had completed enough questions at the

The higher

level questions

require more

knowledge and

understanding,

take more time

to complete, and

are therefore

worth more

marks; the

number of marks

increasing as

the difficulty

level increases.

Spring 2012 26

lower levels to ensure they passed the test, they felt more confident trying some of the more difficult questions,

g) evaluation of the students’ learning and grasp of the concepts became very easy. I could tell at exactly what level each of the students was achieving success, and where they started to have difficulty,

h) since the level one questions were designed to evaluate the ILOs of each topic, I was confident that my evaluation was reliable and valid,

i) even students who were fearful or disinterested in math developed confidence and interest, and their skills improved. As their skills improved, they gained greater confidence and interest, and achieved greater success – an upward spiral in learning for students.

I strongly recommend using levels tests for all strands of the curriculum. I found that the students in my class felt better about doing math, and went on to achieve better results in the later grades and on national and international competitions.

Suggestions for discussion, remembering that we are focusing on using testing to support student learning:

1. Could we allow the student to look at, and discuss, the test with each other for five minutes before they are allowed to begin work independently?

2. Could we give out the test for students to review for 10 minutes the day before the test – no writing or recording of the questions?

3. What about giving the test back and having strong students work with weaker students to review the questions?

4. Would it be worthwhile to mark the test without putting numbers (marks) on the papers, but record the marks, then have students go over the paper and assign the mark?

Almost the entire

class was fully

engaged in the

test for almost

all of the test

period. More

time spent on

working through

the questions

usually

translated

to greater

understanding

and more

learning,

Marg is currently co-creator of Root 7 Educational Resources, a website dedicated to providing high quality math resources for teachers, students and parents, providing workshops and consulting services.

27 Vector

Graphing Polar CoordinatesA six-stage menu for functionscontaining

or

By Duncan McDougall

In his 33rd year as a career teacher, Duncan spent the last fulfilling 19 years as a professional tutor. He became a teacher so as to be in a position to help others. In his current position as a tutor/mentor, coach and counselor, he assists students who have learning difficulties at all levels of mathematics from elementary to 2nd year Calculus. Writing articles for the benefit of students and teachers alike is simply an extension of this work.

“When in Rome, do as the Romans do” aptly applies to sketching curves on the polar plane. As teachers and instructors, we must let the student (and sometimes ourselves) realize that we are no

longer on the Cartesian rectangular plane, but rather on a plane made up of a set of concentric circles and a radius vector. This is the first step towards successfully graphing in polar coordinates. A popular approach to graphing polar coordinates is converting to rectangular coordinates by using cosx r θ= and siny r θ= , and I believe this to be confusing and defeating of the purpose. In this rectangular mode we are then tempted to use rectangular thinking to solve a polar problem. In particular, there is no real need for derivative unless of course we need the angle between the tangent line and the curve. And, blindly making a table of values without plan or forethought is equally disastrous as we totally ignore period, symmetry, and domain of the given function. However, one could attempt to memorize the most common curves then tweak the parameters, but this approach falls apart when given a curve such as / ( sin )r 1 1 2 θ= + , a reciprocal function.

Hence, I propose a more scientific and systematic approach for successful sketching:

1. Identification (ID) of the given function and the quadrants in which the ratio is positive or negative (CAST Rule);

2. Period of the given function especially when we are graphing n-leaved roses;

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3. Symmetry, as all sine and cosine curves are symmetric about the

ray 0θ = or 2θ π= ;4. Domain, as we then know what the playing field is;5. Zeroes for both r and θ ;6. A small table of values (4 to 6 values).

The object of a set of directions or menu is to gather as much data and information before attempting to sketch. Directing our attention to an appropriate domain and consequent table of values is directive, instructive and efficient. The real directing forces behind this domain are the period and symmetry, and therein lies our focus. Once we know the period, we can limit our table of exact values to half a period while we let symmetry take care of points in the other quadrants. A carefully planned set of values gives us the information we need while limiting the amount of number crunching. We’ve all been in the situation where we have no idea what the curve really looks like and so make a table of our favourite values. When it doesn’t turn out the way we planned, we get to re-evaluate and start all over. This becomes frustrating at best. A few select examples of this proposed sequence will justify my premise:

Example 1: cosr 2 θ=

Steps (1) ID A cosine curve which is positive in QI and QIV but negative on QII and QIII

(2) Period

(3) Symmetry For symmetry about the ray θ θ= , we let θ θ= − in our given equation; this gives

cos( ) cosr 2 2 rθ θ= − = = and so there is symmetry about the ray 0θ = .

(4) Domain Since cosr 2 θ= has period 2π and is symmetric about the ray 0θ = , we can limit our table of values to 5 or 6 in QI up to π .

(5) Zeroes

we are no longer

on the Cartesian

rectangular

plane, but rather

on a plane

made up of a

set of concentric

circles and a

radius vector.

2 21π π=

coscos

,

0 20

32 2

θθ

π πθ

===

cos( )

r 2 0r 2 1r 2

===

29 Vector

(6) Table of Values

__ __

Example 2: cosr 3θ=

Steps (1) ID A cosine curve which is positive in QI and QIV

(2) Period

(3) Symmetry For symmetry about the ray 0θ = , we let θ θ= − in our given equation; this gives

( )cos cosr 3 3 rθ θ= − = = and so there is symmetry about the ray 0θ = .

(4) DomainSince the period is 2 3π and there is symmetry about the ray 0θ = , we can limit our table of values to 5 or 6 values up to 3π .

(5) Zeroes

(6) Table of Values

__ __

Notes before graphing cosr 3θ=

We recall that the period is 23

π and so we only need 4 or 5 points

r θ

0

0 23624

132

ππππ

23π

cos

, , , , ,

, , , , ,

0 3

3 5 7 9 113 2 2 2 2 2 2

3 5 7 9 116 6 6 6 6 6

θ

π π π π π πθ

π π π π π πθ

=

=

=

r θ0

181296

ππππ

13

22

21

20

Note: In order to determine values for the table where n, the coefficient of θ is greater than 1, we divide

, and6 4 3π π π

by this

coefficient.

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up to half the period or 3π . Interestingly enough, we expect to have

6 petals as our graph because 2 3 6× = (two solutions per period). However, a more detailed look at an extended table of values reveals that the petals double up on themselves and so 6 petals become 3. This effect will happen for all odd coefficients of θ but does not happen for even coefficients of θ . Let’s examine the following table for

cosr 3θ= : __ __

__ __

What we observe is that the would-be petal in QI falls into QIII, the would-be petal in QIV falls into QII and the would-be petal in θ π=

falls into 0θ = . To help understand this a bit more, 2r 2= − simply

means 22 in the opposite direction; r 1= − is r 1= in the opposite

direction, etc.

As for sinr 3θ= , we have the same effect as above but a rotational

and shift of 6π . In order to calculate the “sine shift”, recall that the

principal difference between a sine and cosine curve is 2π , hence,

r θ0

6432

23

34

56

ππππππππ

10

22

1

0

12

20

1

−−

−

r θ

-

-

7 0625

4 24 133 025 13

274 2

11 06

π

ππππ

ππ

The dotted lines or curves represent the would-be leaves. The solid lines or curves represent the actual curve.

31 Vector

32π ÷

(our coefficient of θ ) is 6

π . The sine shift for ( )sin 2n 1 θ− then is ( )2n 12

π ÷ − .

Example 3: sinr 4θ=

Steps (1) ID A sine curve which is positive in QI and QII but negative in QIII and QIV

(2) Period

(3) SymmetrySince ( )sin sin4 4θ θ− ≠ and

( )sin sin4 4π θ θ− ≠ , there is no

symmetry about the rays 0θ = or 2πθ =

(4) DomainSince the period is 2

π but there is no

symmetry, our table will have to contain enough values in order to see how many leaves appear in QI. We will then use the fact that there are 4 2 8× = leaves to be drawn and that we have 2 leavers per quadrant.

(5) Zeroes sin, , , , , , , ,

, , , , , , , ,

0 44 0 2 3 4 5 6 7 8

3 5 3 70 24 2 4 4 2 4

θθ π π π π π π π π

π π π π π πθ π π

==

=

(6) Table of Values

__ __

24 2π π=

( )sinr 4 0r 0==

r θ0

24161286

316

524

4

ππππππππ

01

22

23

21

32

22

12

0

, , , , , ,, , , , , ,

3 54 6 4 3 2 4 63 5

24 16 12 8 16 24 4

π π π π π πθ ππ π π π π π πθ=

=

Note: we let:

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Example 4: A cardiod form ( )( )

sincos

r 2 1r a 1

θθ

= −= +

Steps (1) ID A sine curve which is positive in QI and QII but negative in QIII and QIV

(2) Period 2 21π π=

(3) Symmetry For symmetry about the ray

2πθ = , we let θ π θ= − in

our given equation; this gives

( )( ) ( )sin sinr 2 1 2 1 rπ θ θ= − − = − = and so there is symmetry about the ray

0 2π=

(4) DomainSince the period is 2

π and there is

symmetry about the ray 2πθ = , we can

limit our Table 1 values from QII and QIII as the zeroes of the function make this extension necessary.

(5) Zeroes

(6) Table of Values

__ __

( )sinsin

sin,

0 2 10 11

32 2

θθ

θπ πθ

= −= −==

( )sin )( )

r 2 1 02 1 0

r 2

= −= −=

r θ 0

2- 2- -

-

22 333 245 2 16

2 07 2 165 2 244 2 333 42

πππππππππ

+

+

+

33 Vector

Example 5: A reciprocal sin2r

1 θ=

−Steps (1) ID A sine curve which is positive in QI and

QII but negative in QIII and QIV.

(2) Period

(3) SymmetryFor symmetry about the ray 2

πθ = , we let θ π θ= − in our given equation; this

gives: sin( ) sin

2 2r r1 1π θ θ

= = =− − −

and so there is symmetry about the ray

2πθ = .

(4) Domain Since the period is 2π and there is

symmetry about the ray 2πθ = , we can

limit our table to values from QII and QIII.

(5) Zeroes

Since 2 0≠ , there is no value for which r 0=

(6) Table of Values

__ __r θ

2 21π π=

sin2

1θ

θ=

− sin2r

1 0

2r1

r 2

=−

=

=

22

33

45

6

76

54

43

32

πππππππππ

( )( )

( )( )

no value/

/

/

/

4 2 32 2 24

124

32 2 24 2 31

−

−

+

+

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Example 6: cos2r 4 2θ=

Steps (1) ID A cosine curve which is positive in QI and QIV but negative in QII and QIII

(2) Period 22π π=

(3) Symmetry For symmetry about the ray 0θ = , we let 0θ = − in our given equation; this gives cos ( ) cos2 2r 2 4 2 rθ θ= − = = . Therefore,

there is symmetry about the ray 0θ = .

For symmetry about the ray 2πθ =

, we let θ π θ= − in our given equation;

this gives

Therefore, there is symmetry about the

ray 2πθ = and for symmetry about

the origin, we let r r= − in our given equation; this gives ( ) cos2 2r r 4 2θ− = =. Therefore there is symmetry about the origin.

(4) Domain Since cos 2θ has period π and cos2r 4 2θ= is symmetric about the rays

and 0 2πθ θ= = and the origin, we can

limit our table of values to 5 or 6 values in

QI up to 4π

.

(5) Zeroes

We’ve all been

in the situation

where we have

no idea what

the curve really

looks like and

so make a table

of our favourite

values

coscos

, , ,, , ,

0 4 20 2

3 5 72 2 2 2 23 5 7

4 4 4 4

θθ

π π π πθπ π π πθ

===

=

( )( )cos2

2r 4 2 0r 4 1r 2

===

35 Vector

(6) Table of Values

__ __

The above examples have all been done with the same menu or sequence and so the beginner now has a method or systematic approach which extracts as much information as possible and directs our focus to an appropriate domain and table of values. It gives the person new to this topic a chance to understand the nature of the topic and to be successful at sketching polar coordinates.

r θ0

2864

ππππ

...

21 861 681 410

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Math Challengers is a mathematics competition for B.C. students in Grades 8 and 9 (or lower) who compete as school teams at

regional (Lower Mainland, Vancouver Island, and Okanagan this year) and provincial tournaments (at SFU this year). In 2010 Math Challengers involved about 500 students. The competition builds skills, promotes strategic problem-solving, and exposes students to some complex problems that require creativity and persistence in order to be solved. Students have opportunities to exchange mathematical ideas through the competition. Teachers, volunteers and former student participants coach competitors beginning each Fall and continuing until the competitions (February and March), either as part of in-class instruction or as an extracurricular activity. The stronger teams from the regional tournaments advance to the provincial competition, where the most successful school teams and the individual champions in each grade are recognized with trophies and medals. At the grade 8 level, an Intramural Competition with Washington and Oregon Math Counts teams occurs with the top 4 or 8 individual competitors.

Since we believe that the study of algebra, number theory and combinatorics is very important, in Math Challengers competitions students are exposed to questions that involve these vital branches of mathematics. We provide here four such problems along with their

Challenging Algebra Problemsfrom the 2010 Math Challengers Competition

By Andrew Adler, Joshua Keshet, and Dave EllisMath Challengers is a school team competition for grades 8 & 9 students. It is open to all schools in the province (both public and independent). There are regional competitions held in Kelowna, Victoria and Greater Vancouver, as well as a provincial competition hosted by either SFU or UBC. MathChallengers has been in existence for about seven years. In 2011 about 550 students from over 50 schools were involved. Math Challengers promotes both greater interest in mathematics and improving achievement in mathematics. For further information visit the website: www.apeg.bc.ca/mathchallengers/ or email [email protected].

The competition

exposes

students to

some complex

problems that

require creativity

and persistence

in order to be

solved.

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solutions. These are among the harder problems that were posed in the Math Challengers 2010 Provincial Competition.

Problem 1:

You play the “Lucky 4” game as follows. You roll a fair standard die and receive in pennies the number you rolled. You keep doing this until either you have accumulated exactly 4 pennies, in which case you win, and the game is over, or your last roll pushes you over 4 pennies, in which case you lose. What is the probability that you win? Express the answer as a common fraction.

Solution.

Maybe you are very lucky, and immediately toss a 4 (probability: 1/6). Game over, you have won.

You also win if you toss a 3, then a 1(probability: 2)6/1(6/16/1 =× ); or

a 1, then a 3 (probability: 2)6/1( ); or a 2, then a 2 (probability: 2)6/1( ).

You can also win with 2, then 1, then 1; or with 1, 2, 1; or with 1, 1, 2

(each has probability 3)6/1( ).

Finally, you win the least likely way if you toss 1, 1, 1, 1 (probability: 4)6/1( ).

Add up, we get 432 )6/1()6/1(3)6/1(36/1 +++ , which “simplifies” to

1296343

.

Another way.

The final answer turns out to be 43 6/7 . Such a nice-looking answer makes one suspect there may be a more structured approach.

What is the probability of winning in exactly k tosses? It is not hard to

see that this is k)6/1( times the number of ways that we can get a sum of 4 in exactly k tosses.

Let ),( rnC be the number of ways of choosing r objects from n distinct objects. Now, imagine putting 4 identical doughnuts in a row. Then there are 3 “inter-doughnut” gaps. The number of ways to get a total of 4 in exactly k tosses is the same as the number of ways of putting 1−k “dividers” in the 3 gaps (each of the 3 gaps is allowed to have at most one divider). The number of doughnuts up to the first divider represents the result of the first toss, the number of doughnuts between the first two dividers represents the result of the second toss,

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and so on. The same idea shows that the number of ways of distributing n identical doughnuts between k people so that everyone gets at least one doughnut is )1,1( −− knC .

Thus the number of ways of winning in one toss is )0,3(C , the number of ways of winning in two tosses is )1,3(C , and so on. Thus the probability of winning in “Lucky 4” is

432 )6/1()3,3()6/1()2,3()6/1()1,3()6/1()0,3( ×+×+×+× CCCC .

Take out the common factor 6/1 . We are left with the binomial

expansion of 361 )1( + ; so, the required probability is .

Essentially the same argument shows that the probability of winning in

the “Lucky 5” game is , and that the probability of winning

“Lucky 6” is . These would be much lengthier to deal with using our first approach. A small modification works for “Lucky 7”. Beyond 7, things get more complicated.

Problem 2:

What is the smallest positive integer n such that the leftmost digit in the decimal representation of n2 is equal to 7? Hint:

1024210 = .

Solution.

The first ten powers of 2, starting with the 0-th power, are 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512.

The next ten powers of 2 are 210, 211, and so on up to 219. These are 10241× , 10242× , 10244× , and so on up to 1024512× .

And the next ten powers of 2 are 220 to 229. These are 2)1024(1× , 2)1024(2× , and so on up to 2)1024(512× .

Continuing, for each successive collection of ten powers of 2, we multiply the numbers 1, 2, 4, 8, , 512 by a suitable power of 1024. These powers of 1024, for a while, have a decimal expansion that has

shape (1...) x (10)j. So, if n is small, multiplying K by n)1024( does nothing dramatic to the leftmost digit of the decimal expansion of K .

We want to “push” one of the numbers 1 to 512 into having a decimal

What is the

smallest positive

integer n

such that the

leftmost digit

in the decimal

representation of n2 is equal to 7?

Hint:

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expansion that begins with 7, by multiplying by a suitable power of 1024. The number 64 requires the smallest push so that the leftmost digit is 7. More precisely, note that 70/64 = 1.09375; so, the needed push should be just under 1.1.

Experiment: 1024 is definitely not enough of a push; neither is 2)1024( .

It is a bit less obvious that 3)1024( is not enough. We need to check that 64 x (1024)3 still starts with the digit 6. This is not too hard. Since no calculators were allowed, the work can be delegated to a team member:

2)024.1( is approximately 1.05, and (1.05)(1.024) is still well under

70/64. But 4)024.1( is about 1.1, and we need to increase 6.4 by only around 9.4% to get to 7. So, the first power of 2 that has the leftmost digit 7 is 64x(1024)4, that is, 26 x 10244 = 246, and n = 46.

(It takes even longer to get a power of 2 that starts with 9. We can in fact find a power of 2 whose decimal expansion starts with any specified list of d digits.)

Problem 3:

There is a group of 7 women and m men arranged around a circular table so that the number of people whose right-hand neighbour is of the same sex is the same as the number of people whose right-hand neighbour is of the opposite sex. What is the largest possible value of m ?

Solution.

It helps to draw a little sketch while following the argument. First of all, imagine that women A and B are next to each other (i.e. no men sit between them), and that the number of people whose right-hand neighbour is of the same sex is the same as the number of people whose right-hand neighbour is of the opposite sex. Now, put four new men between A and B. It is easy to see that the number of people whose right-hand neighbour is of the same sex has increased by 2, and the number of people whose right-hand neighbour is of the opposite sex has also increased by 2. Thus, if we have a seating configuration that satisfies the condition in which there are no men between A and B, then adding 4 more men between A and B is also a valid configuration in which the number of men is increased by 4. Since we want to maximize the number of men, we can assume from now on that no two women are next to each other.

If no two women are next to each other, the number of people whose right-hand neighbour is of the opposite sex is 14 (all the women, and

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all their male left-hand neighbours). Let 721 ,,, mmm be the number of men in the 7 gaps between women. Since no two women are next to each other, the number of people whose right-hand neighbour is of the

same sex is )1()1()1( 721 −++−+− mmm .

Set this equal to 14. We find that m1 + m2 + ... + m7 = 27 ; so, the maximum number of men is 21. (Exactly the same argument shows that if there are w women, the maximum possible number of men is w3 .)

Problem 4:

Let 2)13()( −

=nnnA . What is the smallest integer 1>n such

that )(nA is a perfect square?

Solution.

It turns out that n = 81. So, if a team’s search is suitably organized, there is a fair chance of finding the answer in time. But the search territory can be cut down a lot. Let )(nA be the perfect square 2x . Then

22)13( xnn =− . Since any prime factor of n is also a factor of n3 , then it is clear that n and 13 −n can have no prime factor in common. So, either (i) n is a perfect square and 13 −n is twice a perfect square or (ii) n is twice a perfect square and 13 −n is a perfect square.

In case (i), 2kn = , and we want 13 2 −k to be twice a perfect square. This forces k to be odd. So, try 3=k , 5=k , and so on. Quickly, we

find that 9=k works, since then 22 )11(224213 ==−k .

In the meantime, perhaps, another team member can be dealing with case (ii). There n is twice a perfect square, say 22kn = ; so, we want

1613 2 −=− kn to be a perfect square. One can search; but the search will be futile, for 13 −n can never be a perfect square. The reason is that

13 −n leaves a remainder of 2 on division by 3. But a perfect square always leaves a remainder of 0 or 1 on division by 3. (Any integer a is

of the form t3 , 13 +t , or 23 +t , where t is an integer. Clearly, 2)3( t is divisible by 3. By expanding 2)13( +t and 2)23( +t we can see that each is 1 more than a multiple of 3.)

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A Visual Learners Dream:Introducing Document Cameras into the Mathematics Classroom

By Erica Hopkins and Katja Schuurman

Erica Hopkins is an intermediate teacher and facilitator at Godson Elementary School in Abbotsford. Erica’s educational interests include increasing student achievement based on the research of Dr. Robert Marzano, 21st century learning strategies and educational leadership.Katja Schuurman is a grade 3 teacher and technology leader at Godson Elementary School in Abbotsford. Katja’s educational interests include classroom technology integration, 21st century learning skills and building students’ math fluency.

Godson Elementary is an inner-city school located in Abbotsford, B.C. As with many inner-city schools, the population is a large mix

of students with identified special needs, Aboriginal heritage, E.S.L., low socio-economic status and a high level of transience. Due to this unique school composition, academic achievement in core subjects, including mathematics is of the utmost concern and achievement in these areas is consistently low. Throughout the 2010-2011 school year, two teachers at Godson Elementary have focused their professional development and resources on increasing student achievement within the mathematics curriculum through the increased use of technology during teaching time. Research has shown that when students are involved and actively engaged in the curriculum, an increase in student achievement and deeper conceptual understanding takes place. Research has also shown that the increased use of technology within the classroom is having a large impact on students’ engagement and learning. Students are learning 21st century skills while increasing their understanding of key curricular concepts. This tie between student engagement using 21st century teaching materials and the resulting increase in student achievement is by no means incidental.

The incorporation of technology in mathematics classrooms at Godson

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Elementary was facilitated through the use of the document camera. Document cameras, also known as image presenters, digital overheads or visualizers, are real-time image capturing devices used to display documents or objects to an audience. Essentially, they are high-resolution web cameras, which are able to magnify and project images of actual 3D objects, as well as transparencies. In theory, any object can be displayed under the document camera. The camera takes the picture and produces a live image onto a screen or whiteboard using a projector. There are numerous features of document cameras that are practical in a classroom setting. The particular model of document camera used at Godson, the Epson DC11, has a 5 mega pixel sensor, 10X digital zoom, 30 frames per second video and is capable of creating audio recordings. Additionally, it is able to take photos, freeze frame documents or objects and comes with a microscope attachment. It can be used on any flat surface and does not require the dimming of lights that is needed when using an overhead projector. It is compatible with scanners, laptops, SMARTboards and other interactive whiteboards. While the cost of the document camera, ranging from $500 to over $1000, as well as the requirement of a projector to operate the device, could be seen as a drawback to purchasing the document camera for a classroom, the fact that it does not require much maintenance or upgrading is a strong positive. Additionally, in comparison to other popular technological devices used in the classroom, document cameras require very little training and minimal setup.

The document camera was shared between a primary (Grade 2/3) classroom and an intermediate (Grade 4/5) classroom, and was used daily for each math lesson taught in the respective classrooms. It was used for teacher instruction of concepts, student demonstration and assessment, in the hopes that it would increase student engagement and participation, allowing for a deeper understanding of the mathematical concepts being taught.

During teacher instruction time, the camera was used for demonstrating processes and the use of manipulatives, completing question examples, sharing read alouds picture and text books, and highlighting concepts in the text or on worksheets. Student demonstrations included completing questions, showing work samples, using manipulatives, completing guided demonstrations, and demonstrating math games. Assessment was done through the use of student work samples, observations of students assisting peers, and students explaining the learning strategies they used to complete computations.

In both classrooms, the document camera had a vast impact on student engagement and participation. As soon as the camera was turned on, students were excited to interact and engage with any materials that were presented under the camera lens. Both primary and intermediate

Essentially,

they are high-

resolution web

cameras, which

are able to

magnify and

project images

of actual 3D

objects

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students were fascinated with the life-like colour images, which grabbed the students’ attention immediately and maintained it throughout the lesson.

Manipulatives have always been an effective tool for helping students solidify their understanding of concepts in math. The document camera took this effectiveness to a new level. For example, when introducing the concept of multiplication at a grade three level, the teacher used colour counters. Traditionally, the teacher would have drawn large symbols on the whiteboard (circles etc.) to represent the counters students had in front of them. The students would then be expected to mentally transfer the teacher’s representation to what they saw in front of them. While many students were able to do this, some students were left behind when they could not make the connection. As well, displaying images for the class on the whiteboard could be cumbersome, time consuming and not always easy for students to see. The document camera allowed for the presentation of the very same manipulatives that the students were using at their desks. Students could now more readily connect to what the teacher was demonstrating. The teacher could quickly arrange and rearrange the counters under the document camera to form multiplication arrays and the students were able to easily follow along. Students were also a lot more confident approaching the document camera to demonstrate their use of manipulatives to their peers, than they were when students were expected to draw the arrays on the whiteboard. Using the document camera to compete these examples together resulted in greater student success on independent practice.

As with the miscommunication that sometimes happens between teacher demonstration on the whiteboard and students’ use of manipulatives a similar problem arises when students are expected to transfer between oral explanation of a worksheet and completing the actual assignment. While a solution might be to create a transparency of each worksheet that students were expected to complete, this creates a few problems. The cost involved with creating overheads for each worksheet could deter teachers from doing so. This often results in teachers attempting to hold up and explain an assignment while students are trying, often unsuccessfully, to follow along. Even if each worksheet was indeed made into a transparency, the further away that students are seated from the screen, the more difficult it would be for them to see a small, often blurry, representation of the worksheet. In cases where a classroom was not equipped with an overhead projector, teachers are often made to “create” their own worksheet on the board and students are expected to understand a document that does not necessarily correspond to what they have in front of them.

Through the use of the document camera, teachers can simply

Manipulatives

have always

been an

effective tool for

helping students

solidify their

understanding

of concepts in

math

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slide the worksheet under the lens and the exact replica of the page is projected. For better viewing, the document camera allows the teacher to zoom in on particular questions, which allow for easier viewing, regardless of where a student is seated in the classroom. Additionally, when worksheets have been completed and the teacher wants to check for understanding, the students are able to slide their own copy of the worksheet under the document camera, rather than having to recopy their work onto a transparency or whiteboard. This is especially motivating when the question that is being completed is lengthy or difficult. Rather than focusing attention on the logistics of how to complete the worksheets, students are able to demonstrate their understanding of the actual material that is taught in the lesson.

For example, step-by-step graph construction was very easy to explain and students had no trouble following. Using a student’s ruler and graph paper was far superior to creating a graph on the overhead projector. Students were able to see the actual numbers on the ruler and the entire grid on which to draw. Additionally, if a mistake was made on the paper, it could be erased, as a pencil was being used. Once the graph was setup, the students could come with their own felts and colour in the bars. Students clamored for the opportunity to participate and help fill in the graphs with their peers.

Textbook instruction was also revolutionized with the document camera. As previously mentioned, teacher instruction of assignments tends to be lost with a lot of students. The ability to properly pull meaning from textbooks is a reading skill that many students struggle with. Often, even with explicit instruction how to read and understand text features, students still have a difficult time comprehending their math texts. Textbooks can be very overwhelming to students, especially those weak in reading, because they are very busy. There are so many images, headings, sections, examples and words to process, that students are not sure where to look. Similar to the problems surrounding worksheets, teachers can have a hard time trying to get across what they want their students to be focusing on. If fact, textbooks tend to be even more complicated, which makes the teacher’s task that much more challenging.

With the document camera textbook content is easier to demonstrate and process. Using this tool the teacher is able to zoom into specific parts of the math textbook to highlight the important information. Whether it is a specific question, a heading, or an answer key at the back, the document camera allows for focused attention and easy viewing. Students are able to clearly see what they are supposed to do. The document camera also provides flexibility in instruction by being able to turn to different pages or sections as needed. If a student requires clarification from a previous lesson, the teacher simply needs

Textbooks

can be very

overwhelming

to students,

especially those

weak in reading,

because they

are very busy.

Students are not

sure where to

look

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to flip back a few pages in order to review with the entire class. The document camera also comes in handy when there is a shortage of textbooks available for student use. Again, the ability to project the textbook in clear, full colour, with the added ability to zoom in and out as needed, solves the problem of textbook sharing and unclear black and white photocopies.

The document camera also makes for quick and immediate assessment. While students are working, the teacher can ask them to show the question they have completed under the document camera for other students to see and to help those students who may still be struggling with the content. Students were very eager to not only complete their work, but also ensure that their work was completed correctly and to the standards expected so that they could show their work under the document camera. Great discussions were had based on looking at the student’s work. Their peers were quickly able to point out any errors, what was correct and even asked each other questions based on the information displayed. Marking assignments is also easy. Once the class is finished their worksheets, the teacher can mark the worksheet with the class under the document camera. Not only does this save time for the teacher, but also the students are able to get immediate feedback on their answers and are able to do the corrections on their work following the document camera.

The desired goal of integrating document camera technology into the mathematics classroom was to increase student engagement and participation, and allow for a deeper understanding of the mathematical concepts. Specifically, we wanted to see 95% of all students actively participating in daily math lessons and 90% of students fully meeting learning expectations in mathematics. Prior to using the document camera, 65% of students were actively participating in daily math lessons according to informal teacher observations and student self-assessments. Approximately 75% of students were fully meeting learning expectations based on daily work samples, quizzes and end of unit tests. After using the document camera for a five month period, 98% of students were actively engaged and participating in daily math lessons according to informal teacher observations and student self assessments. 87% of students were fully meeting learning expectations based on daily work samples, quizzes and end of unit tests. These positive results seem to indicate a strong correlation between integrating technology into the classroom (the document camera) and student engagement and achievement.

Overall, student response to the document camera has been unbelievably positive. Not only are they able to better sustain attention during the lesson, but also their work habits have improved tremendously. As for academic achievement, there is no doubt that teaching with the

Students were

very eager to not

only complete

their work, but

also ensure

that their work

was completed

correctly and to

the standards

expected so that

they could show

their work under

the document

camera

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document camera has lead to great gains in student learning. Students are better able to engage with the content being presented, resulting in a higher level of understanding and increased concept retention.

Using the document camera has increased the quality of instruction in general. Not only was the class as a whole more engaged, the document camera targeted the needs of various learning styles due to its multi-modal delivery method. Visual learners could see the information clearly; auditory learners still heard the teacher’s instructions, while kinesthetic learners had the option to manipulate real life objects under the document camera lens. The relatively simple set up and use makes this a very attractive tool to all teachers, regardless of technological knowledge. After experiencing what the document camera can do, it seems improbable to return to teaching with overhead projectors and other 20th century learning tools.

Not only was the

class as a whole

more engaged,

the document

camera targeted

the needs of

various learning

styles due to

its multi-modal

delivery method

47 Vector

Why Do so Many Canadians Lack Numeracy Skills?Speculations, Opinions and Selective Suggestions

By Werner Liedtke

Werner is Professor Emeritus at the University of Victoria. His current main interests include a search for strategies related to the development/assessment of key aspects of number sense in young children: visualizing, estimating and relating numbers; recognizing numbers without counting; flexible thinking about number; and confidence.

The attempt to map Canada’s math skills by the Canadian Council on Learning (CCL) reported in The Globe and Mail (1) includes

the result that, ’55 per cent of adult Canadians are lacking in the basic numeracy skills they need to navigate their lives.’ Numeracy is defined as, ‘the ability to use and understand numbers in everyday life.’ It is surmised that possible reasons for this lack of numeracy may be attributed to ‘jargon-heavy concepts’ that are not explained and, according to one quote, to a lack of goals.

The data about ‘HOW WE SCORED ACROSS THE COUNTRY’ can be very discouraging, especially for those who have devoted a lot of time and energy, in fact parts of our lives, to the improvement of numeracy. Why have our efforts failed? Why have previous curriculum revisions and the adoptions of new mathematics programs failed our students? My selected reflections about possible answers to these questions will include speculations and opinions about possible reasons for the lack of numeracy by making reference to: ‘jargon-heavy concepts’ and ‘edu-speak’; ‘lack of goals’; number sense; and mathematics programs that have been and are used by students.

About unexplained ‘jargon-heavy concepts’ and ‘edu-speak’:

It is easy to find evidence which shows that it is rather difficult for those who communicate about aspects of education, and I include

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myself in this group, to resist the temptation of using terms, phrases or clichés of a very general nature. Such phrases may have an important ring to them while planning a presentation or authoring a paper. It is quite a challenge to keep every possible listener or reader in mind and then ensure that each one will interpret and visualize the ideas that are presented in the same way. Newspaper reports, articles in professional journals, as well as Ministry documents for teachers and parents that deal with issues related to education and mathematics education provide data that indicate that overcoming this difficulty and meeting this challenge is not an easy task.

Since CCL does not include a definition nor an example for ‘unexplained jargon-heavy concepts’, it can be argued that this phrase could be classified as an ‘unexplained jargon-heavy concept.’ Readers are left to speculate about items that might be included in a list of such concepts. What are some examples that could have a possible negative impact on the development of numeracy skills?

My guesses include a few examples that are related to ideas and topics from the new curriculum. These examples are a little more than mere guesses, because over the years I have collected data that show these terms or expressions can be and are interpreted in different ways. My data come from conversations with members of different committees; members of different audiences; and from responses on many assignments given to my students that involved conducting interviews with parents and teachers. I would like to invite readers to submit their reactions to my examples and at the same time share several of their own.

The three examples may not necessarily be the best that can be chosen, nor may they be in agreement with what CCL has in mind. The intent is to illustrate that the issue raised by CCL is important and its consideration can indeed have a positive effect on fostering the development of numeracy in our students.

Mathematical reasoning: What is meant by mathematical reasoning? It does not take long for those in contact with older students to make reference to inductive and deductive thinking when this question is posed. Many respondents will suggest that fostering mathematical reasoning is for students in the higher grades.

How is mathematical reasoning developed in the lower grades? What are possible criteria of mathematical reasoning? There will be those who make reference to the fact that, ‘the basics have to be taught first.’ This comment could be included in the list since during discussions about ‘basics’, almost everyone will agree that ‘they’ are important, but a request for a definition or for examples will result in responses that vary greatly. Many people will make reference to ‘basic facts’ – and

Why have

previous

curriculum

revisions and

the adoptions

of new

mathematics

programs failed

our students?

49 Vector

the definitions for these will also vary.

Visualization Skills: While serving on a Ministry committee quite a few years ago, the members were informed that the term visualization was not to be used. The reason given made reference to the fact that some people might attribute religious meanings to this term.

The ability to visualize is an important goal of many aspects of mathematics teaching/learning. What are visualization skills and some possible examples of these skills? When a request is made to a group of teachers, teachers-to-be or adults for some examples, the responses can be surprising.

Patterns – an important part of pre-algebraic thinking: I have heard this said and have seen it in print, but have been unable to find the origin of this statement. What exactly did the author or the authors of this statement have in mind?

The statement has had an impact on program development. In some programs students in grade one spend about one tenth of the school year on activities with patterns. My questions about this include: What is the main purpose of many of the activities? How do these activities connect to ongoing and future learning of mathematics? What types of activities would be of greater benefit as far as ongoing and future learning of mathematics is concerned? (Many students in our schools repeat very similar tasks with patterns in Kindergarten and grades one and two. Perhaps someone would consider the suggestion to ask students in grade three what they think a pattern is, and why they think they look at and study patterns. The responses can be informative and interesting and could make for some fascinating reading in a future issue of Vector.)

About ‘Edu-speak’:

Statements related to mathematics teaching, learning or assessment that are difficult or impossible to interpret or may be interpreted differently by different readers or listeners are sometimes referred to as ‘edu-speak.’

In 2003, the then Minister of Education Clark, was quoted as stating, ‘Sugar coated report cards for students in BC schools are on the way out.’(2) The article includes the Minister’s goal that, ‘report cards in all public schools next year will get rid of “edu-speak” in an attempt to give parents the straight goods.’ Reaching such an admirable goal is not easy, but could make a contribution to fostering the development of numeracy in students.

There could be errors in reporting, but I believe the examples attributed to Clark to illustrate her point are inappropriate. The comment about a

It is quite a

challenge to

keep every

possible listener

or reader in

mind and then

ensure that each

one will interpret

and visualize the

ideas that are

presented in the

same way

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student, “… understands addition and subtraction of fractions and is working hard in other areas” is given as an example of ‘edu-speak.’ An example of a statement intended to ‘give parents the straight goods’ reads, “… needs help with fractions.” I think the latter statement is even less specific and more inappropriate than the previous one since it is impossible to interpret. Such a conclusion about a student cannot be translated into any type of effective intervention for this student. For me this observation is another typical example of ‘edu-speak.’

‘Edu-speak’ is prevalent. As examples from my collection illustrate, it can be part of:

- assessment comments: ‘… does not understand place value’;

- requests made of student teachers: ‘teach the concept of division’; ‘teach the value of money’;

- advertisements in newspapers: ‘… rate of progress is determined by the student, not by the teacher’;

- advertisements for so-called educational toys: ‘… will develop math skills.’

Some of the questions that came to mind as I read two recent articles that dealt with some issues related to mathematics education are included in parenthesis (sometimes I was tempted to react by asking, “Really?”) My intent is not to be facetious, but to try to suggest possible questions readers could have as they try to interpret the information that is presented to them. I believe different backgrounds of readers will result in interpretations that may not agree with what the authors had in mind – and this could also be true for readers who specialize in the subject area that is discussed. I must admit that I do not know the answers to some of the questions I have posed. I think some of the comments from the articles can be classified as ‘edu-speak’ or ‘edu-babble.’

(i) An article about confronting ‘edu-babble’ (3) includes the following statements. ‘…learning in the lower grades has more to do with acquiring existing knowledge than constructing completely new knowledge. There is a core base of knowledge and skills that all students need to acquire …’ (How is learning defined? What is an example of existing knowledge? How is it to be acquired? What is an example of a core of - , a base of - , and a core base of knowledge?); ‘teachers should recognize the value of traditional methods’ (What is a traditional method? What are traditional methods? – responses from adults to these questions yield some fascinating and diverse results); ‘…students who memorize their basic math facts are far better positioned to

How is

mathematical

reasoning

developed in the

lower grades?

What are

possible criteria

of mathematical

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51 Vector

master complex mathematical concepts than those who never learn them.’ (What happens should memory fail these students? What are these basic math facts? How is mastery defined? What is an example of a complex mathematical concept and how a basic math fact relates to it? Are there students who, ‘never learn the basic math facts’ – whatever these facts are? Where does this conclusion originate? Did these students learn and then forget?)

(ii) An article about developmental dyscalculia (DD) and the inability to distinguish between groups of objects (4) includes the statement that this disorder ‘makes it difficult to master addition, subtraction, multiplication and division.’ (How is mastery defined? What understandings and skills about the four operations does this statement refer to?).

Readers are told that DD leads to, ‘trouble recognizing whether one image has more dots or pieces of fruit in it than another. There is not a complete breakdown but the system is much weaker. This system, our ability to represent magnitude, is really the building block of everything else.’ (How is the degree of weakness assessed? What is the ‘system’ that is much weaker – much weaker than what? Does this conclusion hold true for continuous quantity? What is a building block? What does ‘everything else’ refer to?)

I believe that indicators of a lack of number sense can be a result of a program students are using. The heavy emphasis for some programs is on the manipulation of symbols (numerals) rather than on the development of the important aspects of number sense, which include: number recognition, visualizing number, flexible thinking about number and estimating number (use of a referent). I think that before a student’s inability is attributed to DD, the program that was used by the student needs to be examined and it may also be necessary to have a conversation with the student’s teacher or previous teacher(s).

Advice given to parents includes: ‘using a number line for counting’ or, ’drawing a horizontal line and writing numbers on it at evenly spaced intervals’; and ‘build more number talk into daily life.’(How does a continuous number line or a ruler assist with the ability distinguish between groups of discrete objects? For students in the early grades it is too abstract, especially for someone with DD. In fact, the number line is detrimental to fostering the ability to recognize and visualize number – sets of discrete objects? How would any parent know what is meant by ‘number talk’ – a list of possible definitions collected from parents could make for some fascinating reading – thinking about my what parents might say makes me smile? What are some examples of this talk? Should parents make a distinction between number and name

Perhaps

someone would

consider the

suggestion to

ask students

in grade three

what they think

a pattern is, and

why they think

they look at and

study patterns

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Spring 2012 52

for a number? Why or why not?)

I agree with the CCL observation that, ‘a better explanation of jargon-heavy concepts’ as well as specificity of language (5;6) – trying to get rid of ‘edu-speak’ – will improve our communication about teaching, learning and assessment and can therefore result in a positive contribution to fostering the development of numeracy in our students.

About ‘Lack of Goals’:

The quote in the CCL report that, “We’re not taking the right steps because we don’t have any goal” is more than a little puzzling. I think that without an explanation or some sort of elaboration the statement can at best be classified as a ‘jargon-heavy concept’ that requires an explanation.

The new mathematics curriculum includes many goals that hold great promise for our students. Critical components that students must encounter in a mathematics program are identified. General goals for students are listed. Key goals that relate to fostering the development of number sense and conceptual understanding are identified. General and specific learning outcomes are identified for each topic and matching suggestions for assessment procedures are made. Students who leave our schools having reached the majority of these goals will not be included in a group lacking numeracy skills in future mappings of Canada’s math skills. It is impossible to speak of a lack of a goal, and the author of the quote must have had something else in mind. What that may be, we do not know.

Something that could be beneficial for having students reach the goals that are identified in the curriculum would be easy to administer survey instruments for the key components, ideas and goals related to the development of numeracy. If administered at the beginning of a school year, the results of such a survey can provide teachers with information about specific strengths and weaknesses for a group of students. Such data would make it possible to make adjustments to instruction and can result in effective intervention before it might be too late. This availability and usage could have a positive impact on the development of numeracy skills and future mappings.

About Number Sense:

Several ministry documents include the statement that, ‘Number sense is the key foundation for numeracy.’ As it stands, this statement could be labelled as an example of a ‘jargon-heavy concept’ that requires explanation.

My interpretation or translation of the reported CCL mapping result that

I believe

different

backgrounds

of readers

will result in

interpretations

that may not

agree with what

the authors had

in mind

53 Vector

’55 per cent of adult Canadians are lacking in basic numeracy skills’ is that the responses the people belonging to this category produced are indicators of lack of number sense (some of the key aspects of this number sense are identified in the previous section). That result should not be too surprising, since it is safe to state that the mathematics many of these people studied is unlikely to have focused on the development of number sense.

Presently, and this was true for the previous curriculum (IRP), the development of number sense is to be part of the program for every grade level. This development requires a special focus when new sets on numbers are introduced to students and it needs to be part of ongoing instruction. Special instructional settings that include discussions, as well as exchanges of ideas and opinions are required. This implies that teachers play a key role in the development of number sense. It is unlikely that number sense will develop in students who do not experience the special instructional settings.

Sir Isaac Newton stated that it is possible to teach anyone to repeat, but to teach for understanding is a challenge. Rules and practice settings can be created for students that can result in making these student s ‘test-wise’ – for certain types of tests, but not numerate (7). What happens when the rules become too numerous to remember for some students, or when rules are forgotten, or when ‘mapping’ problems go beyond what has been memorized?

Without number sense students will not reach the important goals of the new curriculum and future CCL mappings of Canada’s math skills will yield the same results. It is number sense that enables students to reinvent strategies and facts that may be forgotten. Number sense can make it possible to come up with alternate solution strategies. It is number sense that will enable students to reach the goals of the curriculum and become numerate.

About Mathematics Programs:

Over the years I have met and conversed with many people who belong to the 55% who lack numeracy skills. When people find out that I am interested in mathematics education, many will make statements like: “I was never good at math”; “I hated math”; “I couldn’t do math”; “I never did understand math”; “There was too much to memorize”; “It was boring” – and some seem to be proud of the fact that they were not good at math. (I have yet to hear someone say something similar about reading.) Many of the teachers-to-be in my courses have admitted to not understanding mathematics, and their scores on items competency tests related conceptual understanding and number sense supported their claim. The mathematical language they used was frequently inappropriate or incorrect i.e., confusing number and

In fact, the

number line

is detrimental

to fostering

the ability to

recognize and

visualize number

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Spring 2012 54

amount; assuming guessing and estimating are synonymous (many people writing for newspapers make the same errors). These students thanked their teachers of mathematics for coaching them to pass tests. People in sales will tell us that members of the 55% group do not want to look at the figures related to a sale which appear on a page, because these are meaningless to them. All these people care about is the lowest possible payment.

I have also encountered people who have declared that they liked math and that they were good at it. I believe that this type of declaration does not guarantee exclusion from the 55% who lack numeracy skills. Reasons for liking math that I have listened to include: “I knew when I was right”; “It was a matter of either being right or wrong”; “I got high scores on tests”; “I could calculate answers quickly.” I have yet to hear someone say, “I am a good problem solver”; “I am a reflective problem solver” or, “Everything made sense to me.”

The mapping population learned mathematics in our schools and used mathematics programs authorized by the Ministries of Education. What are possible reasons for results of the mapping that might be related to mathematics programs that are used in school?

Over the years I have listened to teachers, to directors of instruction and to people who are in charge of instructional programs for districts and territories. When questions about accommodating the development of aspects of numeracy and number sense are posed, the assumption seems to be made by most of these people that the adoption of a certain mathematics program will help students reach the goals they are aiming for.

Districts or schools that have data which show that intervention is required and that scores on tests of mathematics are too low will look for a mathematics program as a solution. Someone will convince them that there exists a program that meets the need. When attempts are made to narrow the gap between aboriginal and non-aboriginal students, someone will jump in and suggest that there exists a program that will help to solve this problem.

Is it possible for any mathematics to meet these needs and solve these sorts of problems? I have very serious doubts and will try to explain why I think that is the case.

As a graduate student I had the opportunity to visit several grade two classrooms that were using a program of individualized instruction (IPI). Specific objectives were identified for all aspects of the elementary mathematics program. Published pamphlets of tasks for each objective were available for each of these objectives. Students selected a pamphlet, completed the tasks and gave them to a marker.

a better

explanation of

jargon-heavy

concepts will

improve our

communication

about teaching,

learning and

assessment

55 Vector

A score of 80% or higher allowed the students to go on to a higher level for the same objective(s) or to the next objectives and tasks in the hierarchy of objectives. A lower score implied a repetition of similar tasks until at least the 80% level was reached. There were five, six or seven adults in the classrooms that were observed. Each one was marking papers, not one was teaching.

Learning about mathematics is not a solitary activity. I believe that students who attempt to go through a mathematics program on their own will not develop number sense and they will lack the numeracy skills they need.

I recall listening to a presentation where the teachers in attendance were told that the session will tell them, ‘how to teach and how to accommodate all of the psychological principles that are required for effective teaching.’ Lesson planning would be a thing of the past, since all lessons are prepared. Teachers were told what to say during each lesson. One of these programs not only told teachers what to say, but also how to say it, i.e., extra vowels were included in words to indicate special emphasis – “Isn’t that greaaaaaaaaaaat!” The students in these types of programs learn how to repeat and to memorize steps and procedures. They do not develop number sense and the required numeracy skills

I had the opportunity to visit with students in the BC Youth Detention Center. (I was ignorant about this type of setting and it was a very sobering experience to sit beside young people who thought they could do anything they wanted and had done it – it resulted in a feeling very difficult to describe.) I observed two students trying to learn something about mathematics from a program called ‘Success-Maker’. The students told me that to them this was very boring and their non-verbal reactions as they responded to requests supported this claim. My observations led me to conclude that the program has the same limitation that holds for any individualized instruction. These students will not develop conceptual understanding or number sense required for basic numeracy skills.

An article appeared in the Vancouver Sun (8) inform readers that one elementary school had raised money to buy iPads for the students. Readers are told that this technological device can, ‘tackle basic addition and subtraction with colourful applications that make learning feel like fun’ and the use of this device requires a, ‘change in the traditional power structure that gave teachers control over learning. We have to concede the fact that they don’t have all the knowledge.’ Some of my questions about the information in the article (some may be predictable by now) and reactions pertaining to these quotes include:

- What is meant by basic addition and subtraction?

the results of

such a survey

can provide

teachers with

information

about specific

strengths and

weaknesses

for a group of

students.

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Spring 2012 56

- What is an example of a colourful application for this ‘basic’ learning?

- How is ‘fun’ defined and what role does it play in this ‘basic’ learning? Is it suggested that teachers are unable to create such a ‘fun’ environment?

- What is meant by ’teacher control over learning’? What type of learning?

- What knowledge or type of knowledge are teachers lacking?

My conclusion that an iPad, or similar devices, cannot contribute to the essential components of numeracy, the development of conceptual understanding and number sense are based on the following reasons:

• Only teachers have all of the knowledge required to create settings and employ strategies deemed conducive to the development of the required aspects of numeracy i.e., mathematical reasoning, ability to visualize, number sense.

- Only teachers know how to pose the many high-order-thinking questions that are required to develop mathematical reasoning and to accommodate and/or react appropriately to all types of responses.

- Only teachers know how to orchestrate the discussions and oral exchanges that are required for the development of number sense – the key foundation of numeracy.

- Only teachers know how to accommodate all types of responses, correct, incorrect, or partially correct, during discussions. This skill of accommodating all responses can result in having students clarify their thinking; modify their thinking; and/ or become more flexible in their thinking.

- Only teachers know how to use students’ questions and responses to make necessary adjustments in questioning and/or instruction.

- Only teachers know how to motivate students and to employ strategies that foster confidence, curiosity and use of imagination.

There is only one success-maker, a teacher. A teacher as a ‘coach of thinking’, not a mathematics program nor a new device of technology, holds the key to reducing the percentage of Canadians who lack numeracy skills because, ‘there is not now, never has been, and is hoped never will be a genuine substitute for a good teacher who knows how and what children need to learn and when they need to learn it’ (9). However, mathematics programs can make an important contribution if teachers are explicitly shown how their contribution as

Sir Isaac

Newton stated

that it is possible

to teach anyone

to repeat, but

to teach for

understanding is

a challenge

57 Vector

part of the program accommodates the critical components and goals of the curriculum and if strategies are explained and illustrated which are conducive to the development of number sense, mathematical reasoning and visualization (10). Hints for collecting assessment indicators for these important aspects of the curriculum need to be provided.

References:

1. Bradshaw, J. (2011). Folio: Mapping Canada’s math skills. The Globe and Mail. Mar. 5, A10-A11.

2. Steffenhagen, J. (2003). Report cards revamped. Times Colonist. Aug. 27, A1-A2.

3. Zwaagstra, M. (2011). Purdue University study confronts edu-babble. Vancouver Sun. Feb.8, A11.

4. McILROY, A. (2011). Why things just don’t add up for some students. The Globe and Mail. Jan.15, A7.

5. Liedtke, W. (2002). Watch your (our) language. Vector, 43(2), 15-19.

6. Liedtke, W. (2000). How Meaningful are Conversations about Basics? Vector, 41(2), 27-30.

7. Liedtke, W. (2003). Wise/Numerate and/or Test-wise and/or Otherwise. Vector, 44(1), 13-15.

8. Steffenhagen, J. (2011). No more pencils, no more books: this Vancouver school has embraced iPads, iPods and apps. Vancouver Sun, Mar.18, A1.

9. Reys, R. (1971). Consideration for teachers using manipulative materials. The Arithmetic Teacher, 18(8), 551-558.

10. Liedtke, W. (2007) Why so Many Joans and Johnnies Can’t Visualize. Vector, 48(2), 47-55.

Learning about

mathematics is

not a solitary

activity

Spring 2012 58

Recently, I looked back over my first two years as editor of vinculum. I would like to take this opportunity and elaborate on some of the

trials, tribulations, and triumphs I have encountered and, in doing so, hopefully, help to generate some discussion within the mathematics education community.

In October 2008, as the newly appointed liaison between the University of Saskatchewan and the Saskatchewan Mathematics Teachers’ Society (SMTS), I found myself at the society’s annual general meeting. During the meeting it was announced that the SMTS was looking for a new editor for their journal. Once the meeting was adjourned, I mentioned, casually, to a few of the individuals I was sitting with (at the back of the room), that I had, at a point prior, thought about the position of editor. (To be honest, what happened next is still sort of a blur.) Unintentionally, and unexpectedly I had volunteered for the position, had been quickly vetted, and, subsequently, right then and there, was appointed the new editor of the journal! I left the meeting an accidental-editor.

In dealing with the task of producing our first issue, I set three goals. First, I wanted to set up an editorial board. After a few emails to particular members of the SMTS, I quickly had an editorial board filled with local individuals heavily invested in the teaching and learning of mathematics. When given the reins of the journal, I was also given the opportunity (if I wanted) to start from scratch. After deliberating for quite some time, I did decide to start over. Starting with (if you will) a blank slate, I spent a great deal of time reading other journals, however, in a much different fashion than I had before. Instead of “reading the articles,” my focus turned to which types of sections to include and, for the majority of time, layouts. After finally putting together a structure and layout that I considered aesthetically pleasing and adaptable for subsequent issues, I had but one last task left: getting people to write

Where Have all the Submissions Gone?By Egan ChernoffEgan Chernoff is a former high school mathematics teacher with the Vancouver School Board. Currently, he is an Assistant Professor at the University of Saskatchewan working with prospective elementary, middle, and secondary mathematics teachers.This article originally appeared in the October 2010 issue of vinculum: Journal of the Saskatchewan Mathematics Teachers’ Society and is reprinted with the permission of the Saskatchewan Teachers’ Federation.

I left the meeting

an accidental-

editor.

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59 Vector

for our journal.

Although I had been told that, in the past, previous editors had difficulties procuring submissions from members of the SMTS, I was not concerned. After all, I had a back-up plan. Over the past five years, while attending a number of local, provincial, national and international conferences, I consistently received the following message: The mathematics education community (i.e., mathematics educators, mathematicians, mathematics teachers, and other individuals with a vested interest in mathematics education) was a unique collaboration of individuals interested in the teaching and learning of mathematics and, further, mathematics teachers. Having received the message loud and clear, I thought, if I had any difficulties in getting submissions from local members, I would simply rely on the mathematics education community (and their interest in the teaching and learning of mathematics and mathematics teachers) to contribute articles to our journal; if the local mathematics teachers were not going to submit, I would still be able to get submissions from mathematicians, mathematics educators, graduate students, and others.

After putting together the fist call for papers, I excitedly faxed and emailed the call for papers everywhere. I contacted school districts, provincial associations of mathematics teachers, list-serves, mathematical organizations, and mathematics and education departments, colleges, and faculties in a variety of universities and colleges. In short, if you were, at that time, a member of the mathematics education community, I tried to get our call for papers to you. All I had to do now was sit back and wait for the submissions to start pouring in.

I waited and waited, to no avail. With the original deadline a thing of the past and only a few submissions from members of the SMTS, I extended the deadline. Further, I sent out special invitations to certain individuals of the mathematics education community who I knew, for sure, were interested in the teaching and learning of mathematics and mathematics teachers. Once again, I sat back and waited for the submissions to start pouring in. Once again, I waited and waited, to no avail. With the extended deadline a thing of the past, we started to scramble. Down, but definitely not out, we rallied. As seen in the author list of our first issue, members of our editorial board stepped up to the plate and wrote some interesting pieces for our first issue. Anecdotally judging from emails we received, our fist issue was a success and definitely of interest to the members of the SMTS. However, and even with the SMTS satisfied, I began feeling a sense of remorse.

Even long after our first issue was published, I was unable to shake the fact that there were no submissions from certain members (e.g., post-secondary) of the mathematics education community. As such, I hedged my bet for the second issue (which I’ll explain shortly) and decided to focus my time and efforts on explaining, at least to myself, the lack of submissions.

I sat back and

waited for the

submissions to

start pouring in

Spring 2012 60

To determine whether or not the low submission rate for our first issue was, perhaps, an isolated incident, I took a similar, yet much more subdued, approach to advertising our second call for papers. For example, placed on the inside of the back cover of our first issue was the call for papers for our second issue, which meant that members of the SMTS had been notified well in advance of the deadline for our second issue. While I did send out our second call for papers through similar channels, I did very little to further promote (e.g., extra faxes, posters) our upcoming issue. This time, while waiting to see if the response from the mathematics education community would be different, all the while expecting the worst (i.e., no submissions), I had a special issue – written by a diverse group of mathematics teachers taking a summer graduate course – already canned, which, if necessary, would become our second issue. Having hedged my bet, I was able to, without worry, sit back and wait for submissions, which also meant I could now investigate a few explanatory hypotheses I had created over the past few months.

Explanatory hypothesis one: The mathematics education community was writing articles for the journals of their respective (i.e., local or regional) mathematics teachers’ association / society. Examining the publication records of mathematicians, mathematics educators, and a variety of other individuals interested in mathematics education debunked my hypothesis. Curriculum vitas, those available, were not peppered and were definitely not littered with professional journal articles, as I had expected. Although distraught, I was able, through my investigation, to find a few individuals who had invested the time and the effort – often as a graduate student or very early on in their academic careers – and had written an article (or two) for a professional journal. Examining these particular publications led me to my second explanatory hypothesis.

Explanatory hypothesis two: The mathematics education community was only writing articles for refereed or peer reviewed professional journals. To investigate my new hypothesis, I first categorized the existing (North American) professional journals into three categories: refereed (i.e., perhaps blinded and reviewed by experts in a particular topic area), non-refereed (i.e., reviewed by an editor or an editorial board before publication), and other (i.e., not belonging to the previous two categories). Having sorted a large number of journals, it became quite clear that the majority of them fell into the non-refereed or other category and only a small number of the journals were refereed. I became convinced that the lack of submissions from the mathematics education community was a direct result of the non-refereed status of our journal. After all, given that the journal was not receiving any submissions and the non-refereed status of our journal, one would expect a lack of submissions from individuals involved in a publish or perish environment (where articles categorized as non-refereed or other

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61 Vector

are not recognized). For me, this meant that the non-refereed status of vinculum was at the root cause of our lack of submissions.

Unfortunately, but now expectedly, we received one submission for the second issue of our journal. Fortunately, through hedging my bets, as detailed above, I was well prepared for the lack of submissions and our special issue went to press as our second issue. Unfortunately, I was not well prepared for the now overwhelming feeling of remorse, which manifested from the following realization: I was, at the very beginning of my academic career, devoting the majority of my time, efforts, and resources to a non-refereed publication, which had value for the members of the SMTS, but held little to no value in the academic setting I was now a part of. Thinking about my path to tenure and promotion, the remorse became debilitating. I was convinced, by becoming an accidental-editor I had made a tactical error in my young academic career.

Once the initial debilitating wave of remorse had subsided, I spent my time attempting to reduce the dissonance (i.e., accidental-editor’s remorse) I was experiencing. But, no matter how hard I tried, I was unable to adjust my beliefs to align with my editorial actions. In a desperate attempt to cope, I thought: why not remove my dissonance entirely and change the status of our journal from non-refereed to refereed.

While revisiting my earlier investigation of professional journals, I noticed a certain vagueness associated with declarations of refereed status. To be clear, there are a few journals, but definitely not the majority, which are traditional refereed journals. However, there are also a large number of journals that simply self-declare refereed status. Attempting to remove my accidental-editor’s remorse would have led to an entirely new sense of remorse derived from anointing refereed status to our non-refereed journal. While vinculum does possess certain elements of a refereed journal (e.g., acceptance rate less than 1, peer reviews of submissions), I was and still am struggling with the notion of refereed status for professional journals. After all, refereed status, as I had determined, was the root cause for our lack of submissions from the mathematics education community.

Attending a conference in May of 2010 provided me with an (ad-hoc) opportunity to sit and discuss issues (e.g., refereed status, submission and acceptance rates) with other journal editors. From this conversation, a number of themes emerged (see Chorney, Chernoff, & Liljedahl, in press, for full details), including: drawing in post-secondary members of the mathematics education community (e.g., mathematics educators and mathematicians) to be further involved with all aspects of professional journals; recognizing the need for continued dialogue between all members of the mathematics education community; and journal structure and organization (e.g., the development of two-tiered journals). The message that emerged from our group’s session was clear:

I was convinced,

by becoming

an accidental-

editor I had

made a tactical

error in my

young academic

career.

Spring 2012 62

professional journals should be a collective effort of the mathematics education community specifically for mathematics teachers. I left the session somewhat more comfortable in my role as accidental-editor of a non-refereed journal.

Revisiting our third issue, which came out just after the May 2010 conference, I noticed that vinculum was beginning to take shape. For example, we had, despite maintaining our non-refereed status, a much stronger post-secondary influence in our submissions. Nevertheless, even with the evolution witnessed in the third issue of our journal and even with the recent community support I had received, I could not entirely remove the remorse I continued to feel. While I had somewhat reduced the dissonance, in order to cope, I was still an individual, early in my career, focusing my time and efforts on a non-refereed publication.

After reading some recent retrospections by prominent members of the mathematics education community, I was able to send, this, the fourth issue of our journal to press without an ounce of accidental-editor’s remorse; it was completely gone. Sure, it still took a tremendous amount of effort the get submissions for our latest issue; sure, the mathematics education community still does not see professional journals as a priority venue for publication; sure, I have, with the latest issue, added more lines to the non-refereed sections of my curriculum vitae; and, sure, I could go on. However, as Mason (2010) asserted (and as I now understand), “it is incumbent upon us to remain steadfast that the purpose of our work is to understand and contribute to student learning of mathematics” (p. 3). After reading this resonant quotation, I now feel quite embarrassed for once having accidental-editor’s remorse. After all, being the editor of vinculum is not about me, it is about the members of the SMTS. Part of being a journal editor is contribute to different areas of the mathematics education community and to contribute to the student learning of mathematics, which can be achieved by contributing to a professional journal, like vinculum. As such, I ask you, as a member of the mathematics education community, to contribute an article; however, do not think of it as a line on your CV (do not make the same mistake I did), but purely for the learning of mathematics.

References

Chorney, S., Chernoff, E. J., & Liljedahl, P. (in press). Editing mathematics teachers’ journals in Canada: Bridging the gap between researchers and teachers, in P. Liljedahl, S. Oesterle, D. Allen (Eds.), Proceedings of the 2010 annual meeting of the Canadian mathematics education study group.

Mason, J. (2010). Mathematics education: Theory, practice & memories over 50 years. For the Learning of Mathematics, 30(3), 3-9.

professional

journals should

be a collective

effort of the

mathematics

education

community

specifically for

mathematics

teachers

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63 Vector

Ivan Johnson ... A Personal Tribute(November 20, 1924 - October 14, 2010)By Tom O’Shea

Tom O’Shea is a retired mathematics education faculty member from SFU’s Faculty of Education.

MEM

OR

IUM Earlier this year, Ivan Johnson passed away at his home in Whistler. For younger math

teachers who didn’t know Ivan I’d like to introduce you to the most wonderful role model you could find in mathematics education. For those who knew him, I’d like to share with you some memories and recount some of his contributions to mathematics education at all levels...local, provincial, and national. Ivan was born in BC. He spent some early years as a labourer and, as a result, became a life-long union supporter. After a first stint teaching in Quesnel he moved to Burnaby where he remained for rest of his teaching career. In Burnaby he served as a classroom math teacher for ten years, a math department head for another ten years, the district math consultant for two years, the head of Burnaby’s Schou Education Centre for three years, and the math department head of the newly established Burnaby South school for the final four years until his retirement in 1996. I first met Ivan when I was serving as Vector co-editor in the 1980s. Ivan had a beef (I can’t remember what it was) with the BCAMT and Ian deGroot had invited him to attend an executive meeting to make his case. Ivan was so persuasive that we said here’s a teacher that we have to get onto the executive. We did, and Ivan subsequently served many years, including as co-editor of Vector and ultimately as President. He maintained his strong professional association with the BCAMT after retirement and served as informal ambassador to the 2009 Northwest Math Conference in Whistler. Throughout his career, Ivan was a strong proponent of cooperative learning, modelling the process in his own classroom and presenting his ideas at professional con-ferences. In 1990, he organized BC’s first Math Camp on behalf of the BCAMT to show students how people in business and industry use mathematics to solve problems. He was also an early proponent of using technology to teach mathematics and was a strong influence in this regard in Burnaby South as Department Head and in Burnaby School District as the district math consultant. In addition, he served as a mathematics textbook consultant, a member of the NCTM’s Regional Services Committee, and as one of the key writers for the Western Canada Consortium that developed the WNCP K-12 math-ematics curriculum. Ivan was recognized at the national level in 1995 for his skills as a mathematics teacher and for his contributions to mathematics education by winning the Prime Minister’s Teaching Award for British Columbia. I was delighted to write a strong letter of support for his nomination and that he was recognized in this way. In 1996, Ivan was the winner of the BCAMT’s Secondary Teacher of the Year award.

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In the early 1990s, as I got to know Ivan better through our professional connections, I encouraged him to apply for the position of Faculty Associate for SFU’s PDP program. He did, and was the unanimous choice for the job. We worked together as a team for a year, along with Jane Turner, and it was a delight to see how he interacted with potential teachers and shared his knowledge of mathematics and the practicalities of working with school learners. His first love, however, was his own classroom and when Burnaby offered him the position of Department Head at the new Burnaby South secondary school, Ivan returned to the district after only one year at SFU. In 1994-95, I had a sabbatical year at SFU and, goaded by late night talks at various con-ferences with Ivan and Ian deGroot that my vision for mathematics teaching was unrealistic giv-en that I hadn’t taught in a high school classroom for twenty years, I applied to teach at Burnaby South for the year. Ivan, as department head, was strongly supportive and, overcoming a number of hurdles at the district level and with the College of Teachers, we constructed an assignment in which I taught four classes, every second day for the year. In this way I came to know and appreciate Ivan at a very personal level. Students worked in pairs in his classroom. He relied on the innovative use of technology. He provided enrichment opportunities. His students clustered around him at end of each class; their affection and respect for him was tangible. In many ways, Ivan was an ideal department head, taking advantage of the strengths of individuals in constructing a effective teaching group: one teacher took the lead in computer-based learning, another as contest coordinator, another as leader for district math professional development workshops. He was wonderful with new teachers, recognizing talent and help-ing others facing difficulties with teaching or classroom management. Ron Colburn, a future BCAMT president, began his career under Ivan’s guidance that year. I appreciated his fairness in administrative matters. At the meeting to decide teaching assignments the following year, each teacher in turn started by choosing the one class they would most like to teach. This was fol-lowed by a list of least desirable classes, from which each teacher had to make a selection. The result did not, for example, favour senior teachers and ensured that all had a balanced load of new preps, repeated classes, desirable and difficult classes. Ivan even allowed me to serve as acting department head for four days while he was away on a professional release. He circulated an email to teachers in the department to this ef-fect saying that this would probably be the only opportunity I would ever have to serve as head, but they should not worry because I was there only every second day so the damage I could do would be minimal. Ivan…you were wonderful. After his retirement, Ivan continued to contribute to math education (for example, by teaching methods courses at UBC) and to the community. Ivan moved to Whistler where he volunteered as a hiking and skiing guide and was instrumental in forming a Seniors Skiing group recognized by Intrawest. He was an active member of the Mature Action Committee which ad-vocated for seniors’ housing in Whistler. This eventually was constructed and Ivan moved into it for the last few months of his life. In summary, Ivan exemplified everything that was admirable in a mathematics educator. How does that come about? Certainly through the force of personality. Early experiences and a family history of social commitment. A broad educational base (Ivan’s master’s degree was in Religious Studies). A solid base of mathematical understanding and a continuing exploration of new topics. Professional dedication. Respect for students and colleagues. Who knows? Wheth-er a great teacher is born or made, Ivan you were one, and we are all the better for it.

65 Vector

Exploring Probability in SchoolReviewed By Simin ChavoshiAuthor: Graham A. JonesISBN: 0-38724-529-4© 2005 SpringerPaperback, 390 pages

Exploring Probability in School is an edited book written by a group of 20 authors with considerable

background in mathematics and mathematics education. Graham Jones the editor of the book has organized the book into five sections and fourteen chapters. In the first three chapters, the historical roots and different interpretations of chance and probability are introduced and the notion of probability literacy is discussed. Characteristics of probabilistic reasoning among elementary, middle school and high school students are presented separately through chapters 4-10. The research that examines children’s conceptions of core probability notions such as combinatorial reasoning, randomness, sample space, relative frequency, compound events, independence and so on is presented throughout these chapters. The three final chapters of the book proceed to issues that relate to teachers in the sense of the political, social and pedagogical challenges they face both in teaching and assessing students probability knowledge.

Mathematical and philosophical journey of Probability:

Chapter 1 gives an account of how the ideas of randomness, determinism and chance underwent an evolution from the ancient Greek to the time that the concepts were modeled into probability. One of the main ideas in the chapter is given by the authors’ statement: “Whatever our philosophical conceptions of chance and necessity and our epistemological conceptions of probabilities are, they are compatible with the contemporary mathematical theory of probability. In developing an axiomatic theory that was adequate to support these different interpretations, mathematics does not enter these philosophical or epistemological debates”(p. 20). A contradictory remark as it is, one may guess what it is supposed to mean is that mathematical development of probability takes place first and then it is followed by a cluster of epistemological and philosophical

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controversies.

The historical chronicle of randomness as perceived, encountered and defined up to the present time adds an invaluable part to this chapter followed by an adequately emphasized role of mathematical formalism in axiomatization of probability. Since the authors bring up the relation between probability theory and measure theory in this chapter, it might be useful to add that today it is believed that measure theory is an indispensable tool to deal with problems in probability theory.

Why teach probability?

This important question is being addressed several times throughout the book and appears as a main discussion topic for chapter two. The whole discussion around this question could be divided into two categories. First: probability as a part of mathematics should be learnt by its own right or in other words for the same reason that other parts of mathematics appear in curriculum. This category also includes probability education aimed at facilitated understanding of more advanced probabilistic concepts later at college level (e.g. p 80)

The Second group of rationales for probability education is geared toward literacy: probability is an essential empowerment tool for real life since random events and chance are parts of life. Probability education aimed at literacy is discussed in chapters two and twelve, great examples of impacts of probabilistic innumeracy (e.g. p. 313) are provided and clear suggestions for what could be done are made. Emphasizing the modeling aspect of probability to the K-12 students and making sure they grasp the idea as to what extent a particular modeling is relevant to real life is the heart and soul of this set of suggestions. That being said, one can see that the importance of probability in modeling more advanced aspects of life is not looked into, the way it is convoluted into financial theories and stock markets modeling, and the role it plays in trades and decision-making theories are not exemplified adequately in the book.

Where are the devil’s advocates?

The purpose of the book according to its authors is to meet the “need for a book that presents a coherent body of research-based knowledge on probability teaching and learning” (p. 2). The book takes the responsibility of delving into “the extensive and diverse research literature” (p. 3) on probability education and come up with fresh insights into probabilistic teaching and learning. In so doing a vast number of researches is being addressed all through the chapters and a substantial body of literature is being reviewed. However the lack of opposing points of views and the rare evidence of controversy and critical approach toward the literature makes the reader skeptical of the fairness of judgment or the scope of

mathematical

development

of probability

takes place

first and then

it is followed

by a cluster of

epistemological

and

philosophical

controversies

67 Vector

the literature under review. For example there are certain aspects of probability education towards literacy that could be examined more carefully if one takes into account some opposing arguments. Here I provide few examples:

Some believe that probability is not understood well enough even by experts to be applied to common life problems. Taleb in his 2007 book brings up details of such lack of understandings and gives examples of huge implications that make any educator think again before discussing probability with students. Devlin provides another brilliant example of the so-called independent events (a concept which is massively investigated in the book) that shows how the common probability theory can lead to ridiculously wrong results. Another important aspect of putting probability into curriculum, that the book doesn’t look into, is the behavioral impacts of being introduced to concepts that have direct links with gambling. The question of whether there is a connection between the students’ probability education (considering the age, the approach and the content of probability education they receive) and their later degrees of risk awareness and the extent of making wise decisions in non-deterministic situations is not addressed in the book.

Probabilistic thinking, what is special about it?

In chapter 10 in an attempt to explain the existence of misconceptions and learning difficulties that still remain at high school level, it is said that in other branches of mathematics counterintuitive results are encountered only when working at a high degree of abstraction. Probability is distinguished from other parts of mathematics by virtue of “probabilistic reasoning is different from logical or causal reasoning and that counterintuitive results are found in probability even at very elementary levels”. I personally agree that the probabilistic reasoning and the way it interacts with logic, individual experience of randomness and chance is different with the way algebra does, for example. But it doesn’t explain why the probability misconceptions last longer than others.

What was going on in the math education world?

Jones suggests that the book will appeal to curriculum developers, teacher educators and teachers will find more to capture their interest in the learning activities and teaching implications presented at different levels of schooling. As well, researchers and graduate students in fields like mathematics education, mathematics, and psychology will also find lots of interesting information. A typical graduate student in mathematics education might find some of their expectations unmet due to lack of ties that connects the research in probability theory with the bigger body of research in mathematics education. For example recent theories of learning could be brought up when looking into students’

probability is

an essential

empowerment

tool for real life

since random

events and

chance are parts

of life

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conception of probability. As well, the language related issues in written and spoken probability communication from the point of view of theories of discourse would be an excellent choice to be included in the book. In addition, addressing the research in use of tools in mathematics education would make a remarkable addition to chapter four.

Concluding remarks:

The book is generally well written and easy to read. The assumption about readers’ knowledge is overall consistent all along the book with few exceptions. Definitions are adequately provided and the metadiscourse structure of the book makes each chapter more readable than a research paper (not as easy as a book though). There is a noticeable amount of overlap between the materials discussed among different chapters; the same theoretical frameworks are being explained more than once, the same research pieces are being reported and discussed in different chapters, and some definitions and descriptions are visited several times. However I should admit that for those who choose to read only the article that suits their particular interests or needs, this overlap will serve as a means to provide more information and comprehensibility.

Taleb gives

examples

of huge

implications

that make any

educator think

again before

discussing

probability with

students

69 Vector

GRADESK-1

How many triangles are in the following shape?

GRADES2-3

What is the number in the box so that the equation is correct?

? 3 3 2

1 0 5+

? 5 1 7

6 8−

One person can eat two hamburgers if they eat alone. Whereas, if two people eat together, their appetites increase and they can eat five hamburgers together. If there are five people in a car, what is the maximum number of hamburgers they can eat? (Note: you must finish the entire burger to be given credit for it)

GRADES4-6

If “A” and “B” are single digit numbers and 11A B+ = , how many different values for “A” are

there?

Spring 2012 • Problem SetPR

OBL

EM SET

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GRADES7-9

Given that 2AB BC CD= = , and the area of is 256cm , what is the area of ?

GRADES10-12

You are one of 100 prisoners. Tomorrow, the guard will place, at random, a black or a white hat on each prisoner’s head (including yours of course). Prisoners will be unable to see their own hats. All the prisoners will then be led to the courtyard where they cannot communicate with each other in any way. At random each prisoner will then be asked to come up to the guard at the gate and declare the colour of the hat on his/her own head. If the answer given is correct that prisoner is free to go, if the answer is wrong the prisoner is killed. This is done in plain sight of the other remaining prisoners.

Your task is to devise a strategy with the other prisoners tonight, to save the most lives tomorrow, bearing in mind that tomorrow you will not be able to communicate with each other.

71 Vector

CALCULUS

Suppose the addition within the fraction never stops, following equation continues on towards infinity

33 33 33 333 .....

k+ =+

++

+

and beyond

There are five cunning pirates that are very good at math trying to split 100 gold coins amongst them. Each pirate wants to maximize their share and get as much gold as possible. (hint: gold will not be divided evenly). Each pirate will get a number from 1 to 5 and they take turns to make a proposal on how to split the gold. For instance, pirate number 1 will make his proposal and the group will vote “for” or “against” it. If they vote “for” it then the gold will be split accordingly. However, if they vote “against” it then pirate 1 will be kicked off the boat and gets nothing, and then pirate 2 goes next...and so on. In order for a proposal to be accepted half or more of the people alive need to vote for it. Q: what proposal should pirate 1 make to maximize his share?

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Pi is (still) WrongA look into an alternative measure of the unit circles circumference.http://www.youtube.com/watch?v=jG7vhMMXagQ

Infinity Elephants

A mixture of art and geometric sequences

http://www.youtube.com/watch?v=jG7vhMMXagQ

Origami Proof of the Pythagorean Theorem

Get some paper and try this out

http://www.youtube.com/watch?v=z6lL83wl31E&feature=related

The Science and Mathematics of Sound, Frequency, and Pitch

Pythagorus is not just about triangles

http://www.youtube.com/watch?v=i_0DXxNeaQ0&feature=youtu.be

Spring 2012 • Math Web SitesM

AT

H

WEBSIT

ESThese websites are brought to you via two extremely powerful resources: one of them is youtube, the other, videos created by Vi Hart. These are outstanding videos. We recommend you take a look at one of them because once you’ve watched one, you won’t be able to ignore the others.

Peter Liljedahl Faculty of Education Simon Fraser University 8888 University Drive Burnaby, BC V5A 1S6

vector - bc association of math teachers · 2015. 11. 18. · vector is published by the bc...

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