Sciematics: The Changing Face of Education. Saskatoon, May 9-11, 2012, College of Agriculture and Biosciences, U of S. http://www.sciematics.com/

SUM conference: May 3-4, Saskatoon. Featuring Dan Meyer and Marian Small. http://www.smts.ca/sum-conference/

“Posing conjectures and trying to justify them is an expected part of students’ mathematical activity. “(NCTM, 2000, p. 191)

“Teaching mathematics as an exercise in reasoning should also be commonplace in the classroom. Students should have frequent opportunities to engage in mathematical discussions in which reasoning is valued. Students should be encouraged to explain their reasoning process for reaching a given conclusion or to justify why their particular approach to a problem is appropriate. The goal of emphasizing reasoning in the teaching of mathematics is to empower students to reach conclusions and justify statements on their own rather than to rely solely on the authority of a teacher or textbook.” http://www.nctm.org/standards/content.aspx?id=26596

Mathematical Process of the Month: Reasoning R

Reasoning describes our ability to make sense of things. Our philosophy of teaching mathematics for deep understanding is founded on the premise that students must make sense of the mathematics, not just memorize procedures. Providing students with opportunities to explore, manipulate, demonstrate, make conjectures and explain approaches and understandings are ways that we help students to reason for themselves.

Mathematical reasoning underlies logical thinking and sense making and is the mental process of making connections and drawing conclusions based on what is already known. Mathematical reasoning involves knowing which procedures apply to which purpose, and being able to identify useful strategies. Reasoning is a natural component of mathematical proof, as an explanation of one’s reasoning and understanding is the premise of mathematical proof. Logic, reasoning and proof are of pivotal importance in developing mathematical literacy. Students who learn math through understanding will be more confident in applied mathematics in further education and in life.

The rigorous demand for proof is the essence of mathematics and the cornerstone of humanity’s advances in accumulating knowledge and understanding of our world. If we are helping students gain an understanding of Math as a Human Endeavour (one of the goals of math education), then we must also reveal mathematics as understanding relationships, not just procedural knowledge. Procedural fluency is important but cannot be sought at the expense of conceptual understanding. Because students today need to be able to think critically, analyze and interpret information, math instruction needs to encourage prediction, reflection and justification of results. Students that can communicate their reasoning will develop confidence in their thinking.

Math instruction must involve tasks that engage all students in thinking about, discussing, and making sense of mathematics, not just practicing procedures. By organizing the curriculum around central ideas, students see a coherence that contributes to their reasoning and helps them make sense of the mathematics.

Focus in High School Mathematics, Reasoning and Sense Making, National Council of Teachers of Mathematics.

Saskatchewan Renewed Mathematics Curriculum

“Classroom instruction in mathematics should always foster critical thinking – that is, an organized, analytical, well -reasoned approach to learning mathematical concepts and processes and to solving problems.” Ontario Grade 9 and 10 Curriculum http://www.edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf

Possible Curriculum connections that support some discussion of Pi Math 7 SS 7.1 Circles, radii, diameter Math 8 SS 8.2 Surface area of shapes including cylinders Math 9 SS 9.2 Surface areas of shapes including cylinders

Great Pi Day Ideas

I teach 7th grade Honors Pre-Algebra. I had each student measure the circumference and diameter of a circular object to the nearest tenth of a cm the day before pi day. They then recorded the class data on a chart. I had them calculate C/d. They also found the class average which they discovered was very close to pi. Then they created a scatter plot to see the correlation between diameter and circumference.On pi day we ate pie and discussed our findings. I also read them the book “Sir Cumference and the Dragon of Pi”. However the big hit was the photo story I made for them with picture of our class set to the music “Lose Yourself (In The Digits)” This song is great (clean) and a free download. By S. Browning http://www.piday.org/2008/2008-pi-day-activities-for-teachers/

www.exploratorium.edu/learning_studio/pi/

Send a pi day greeting card http://www.123greetings.com/events/pi_day/

Some examples of Pi day activities: Younger grades: Create a Pi caterpillar, with 3 on the head, each paper circle segment created by students has a separate digit of pi. This caterpillar can be added to year after year. Having students write their name on each digit provides a legacy of past students. For older students, try graphing the circumference of circles vs diameter. Guess what the slope will be!

Sir Cumference, his wife, Lady Di of Ameter, and their son Radius use geometry and problem-solving techniques to help King Arthur. A math adventure by Cindy Neuschwander

You tube http://www.youtube.com/watch?v=OU_O8PdDJpI

Formative Assessment Feature Always, Sometimes, or Never True: This formative assessment strategy involves having the students examine a set of statements, and decide if they are always, sometimes or never true This formative assessment task promotes reasoning because students must justify their answer in writing. Allow students to respond to the statements individually, then have them discuss and compare in small group. There will be debating, and this is to be encouraged! Example of an Always, Sometimes, Never activity: A right triangle is isosceles

□Always □Sometimes□Never

Justify your answer:

An isosceles triangle is an equilateral triangle

□Always □Sometimes□Never

Justify your answer:

A triangle can have only one obtuse angle

□Always □Sometimes□Never

Justify your answer:

A triangle can have more than one right angle

□Always □Sometimes□Never

Justify your answer:

The angles in a triangle sum to 180 degrees

□Always □Sometimes□Never

Justify your answer:

A triangle can have three acute angles

□Always □Sometimes□Never

Justify your answer:

A right triangle is an isosceles triangle

□Always □Sometimes□Never

Justify your answer:

A right triangle is an acute triangle

□Always □Sometimes□Never

Justify your answer:

In this activity, students must understand that conjectures need to be justified. It encourages mathematical thinking because students imagine examples and non-examples to support their answer. Dialogue and mathematical argument are encouraged. Always, Sometimes, Never can be used as a pre-assessment, to gain understanding of students’ prior knowledge, or to check for understanding after instruction. It may uncover misconceptions and reveal the extent to which students understand the concepts.

Create the Problem This is the reverse of problem solving. The teacher provides students with a simple equation or math fact, such as or ½ of 10 is 5, and students create a problem around the mathematics. For example: I have a three sided rectangular pen with a perimeter of 5. If the front edge of the pen is three meters, how long are the two congruent sides of the pen? Or: If I doubled the number of loonies in my pocket then added the three from my wallet, I would have $15. Students share their stories and discuss how well the story matches the equation or expression. Having students write the wording to a problem to match presented mathematics is a powerful way to help them become better problem solvers because they must reason through the wording translation themselves. This is an example of a problem where students can really focus on the process, because the mathematical answer is already provided. It encourages mathematical communication because students must write logical sentences that require mathematical thinking, and then discuss their ideas with each other. This assessment activity allows insight into student understanding because it reveals if they know why a certain procedure may be required, rather than just the process used to perform the computation.

“Students recognize that reasoning is a fundamental aspect of mathematics. Reasoning can be nurtured at a very early age by asking students to explain and justify their observations with questions such as ‘Why do you think that’s true?’ and helping students distinguish between real evidence and non-evidence.”

“Students should develop the habit of providing an argument, reason, rationale, or justification for every answer they provide.”

Florence Glanfield, (2007). Building Capacity in Teaching and Learning. Reflections on Research in Mathematics. Pearson Education Canada, p. 24,25

Web Resources:

https://www.khanacademy.org/exercisedashboard If you haven’t checked this out yet, take a look. There are thousands of video lessons on this site. http://www.livebinders.com/play/play?id=598492 This is a livebinder with resources on how to teach with manipulatives, virtual manipulatives, and SMARTboard lessons.

I just downloaded this webmix of high school math sites. By the way, Symbaloo is an awesome way to keep your bookmarks and links organized. National Library of Virtual Manipulatives: http://nlvm.usu.edu/en/nav/grade_g_2.html

http://www.symbaloo.com/mix/highschoolmath

Fostering Reasoning through Open Ended Questions: Open ended questions have multiple correct solutions, such as when students are asked to find different methods or discuss approaches, but openness is lost if the teacher proceeds through only one method. Categorizing items is an open ended task when the categories are developed by the students. Asking open ended questions promotes higher level thinking in mathematics, promotes collaboration and communication, and requires students to justify their reasoning. Open ended tasks allow all students to participate by providing multiple entry points, and in this way they are naturally differentiated. Students can choose their own approach, employ their own learning style, and make personal choices. They provide opportunity for success and challenge for all students. Because open ended problems require students to explain their reasoning, the teacher gains insight into student learning styles, their understanding, and their use of mathematical language. Allowing students to discuss open ended tasks can help students develop confidence in their mathematical ability. http://books.heinemann.com/math/reasons.cf http://www.hightechhigh.org/unboxed/issue2/opening_up_to_math Becker & Shimada, (1997). The Open Ended Approach. NCTM, Reston, VA. Arlene Prestie, Preeceville School

“Reasoning

mathematically is a

habit of mind, and

like all habits, it

must be developed

through consistent

use in many

contexts.”

-NCTM