Math Congress: A Strategy to Promote Math Talk

This week we had the opportunity to participate in a math congress. This experience reminded me of the importance of encouraging student discourse in math class. I want to create a learning environment that encourages rich dialogue, sharing, and thinking so that my students can develop a deeper, more meaningful understanding of the concepts they are exploring.

Math Congress: What is it?

In Chapter 4 of Making Math Meaningful, Marian Small discusses the lesson strategy of a math congress. The goal of a math congress is to debrief the various strategies or big concepts that students use to solve a math problem. Before the actual congress takes place, students work in pairs or small groups to solve a math problem and write their solution on a chart-sized poster. Each group then holds a mini-congress with another group in order to share their work, check their answers, explain their thinking, and ask questions and give feedback about their peers’ strategies. For each group, one student is chosen as the leader. The leader’s responsibility is to facilitate the discussion and to ensure that every member of their group has a clear understanding of the strategies used. Once the mini-congresses have taken place, the whole class gathers for a large congress. In this session, the teacher strategically selects two or three groups to present their solutions and explain their thinking. The teacher chooses a variety of strategies that encourage students to see the key concepts, make mathematical generalizations, and understand the reasoning behind various mathematical strategies. Thus, the final congress session helps students to consolidate the information that has been presented throughout the various steps of the math congress.

Seeing Different Strategies of Problem-Solving

As we participated in the math congress, I was truly amazed to see all the different strategies or approaches to solving the math problem we were given about determining which store had the better deal on cat food. When we started the problem, I honestly thought there would only be two or three different ways to solve the problem. To my amazement, we saw at least six different ways of doing the math problem! One method involved creating fractions and finding a common denominator in order to compare the two fractions. Some groups did this by finding equivalent fractions that had a larger common denominator while other groups did this by reducing the fractions so that they had a smaller common denominator. Some students chose to found a common number of cans while other students chose to find a common dollar amount. Another method involved physically drawing the money out to determine how much money each can cost. Another method involved creating a ratio chart. It was an empowering and eye-opening experience that reminded me that I need to reinforce in my students the idea that there is never only one right method of solving a math problem.

Olij, B. © 2016

Olij, B. © 2016

Olij, B. © 2016

Olij, B. © 2016

Promoting Discourse and Deeper Understanding

I love the idea of incorporating a math congress into my math lessons! As a student in elementary school and high school, I had the tendency to stick to the strategy that worked for me without making a conscious effort to consider alternative methods. I also struggled with explaining my math reasoning; while I knew what I did, I did not always know why I did it. Students who participate in a math congress are not only presented with various strategies, they also have the opportunity to ask questions, seek clarifications, and provide feedback about these strategies. The sharing, reasoning, questioning, and explaining involved in a math congress promote an incredible amount of math talk! I love that the group leader has the responsibility to ensure that every member of their group has a clear understanding of the strategies used. This means that every student leaves the math congress having learned at least one other method of approaching a math problem. Every student leaves the congress feeling empowered as they understand not only every way a solution was found but also the reasoning and mathematical concepts behind these solutions. 

Keeping Fractions and Ratios Fun and Feasible

When you were a student, did you dread math problems that involved fractions? Did you find math operations with fractions frustrating or intimidating? If you did, you certainly were not alone! This week’s math class demonstrated how students should not need to fear fractions and how we as teachers can helps our students to see fractions and ratios as fun and doable!

Incorporating Stories and Manipulatives

In class, we heard a story about a man named Mr. Tan whose most treasured tile shattered into seven pieces. Mr. Tan was able to piece the tile back together, and in the process discovered many different interesting shapes. At the end of the story, each of us was given seven tangram pieces which represented Mr. Tan’s tile. We were given the opportunity to explore different shapes using the tiles and to see whether we could piece the square tile back together like Mr. Tan did. We were able to combine different tangram pieces and determine their relation to the whole tangram square.

Exploring different shapes.Olij, B. © 2016

Recreating Mr. Tan's tile.
Olij, B. ©2016

I love the idea of teaching math lessons through the use of stories and corresponding manipulatives! Often times students have the misconception that math is boring or distant from their own daily lives. Stories encourage students to use their imaginations and explore the world of math. As a student, my highlight of the day was when the teacher read me a story. In my placement, I see that my students have the same joy and excitement when listening to stories. Why not tap into this love of literature and integrate stories and math to help bring math to life for our students?

Introducing a “New” Way to Divide Fractions

The moment that stands out to me most from this week’s class was when we looked at an alternative way to divide fractions. The traditional method of dividing fractions, the method that I learned as a student in elementary school, is to multiply the first fraction by the reciprocal of the second fraction. While I was able to use this method as a student, I distinctly remember that I never understood why we had to invert the second fraction and change the division sign to a multiplication sign. I simply did what I was taught without questioning the logic behind it.

This week, I was truly amazed to learn that students do not have to use the algorithm of inverting and multiplying in order to divide fractions. A simpler, more logical method is to simply divide the first nominator by the second nominator and to divide the first denominator by the second denominator:

This method follows the typical rules for division and is therefore more natural and far less intimidating for students. I cannot wait to show this alternative method to my students! 

Using Relatable Examples

This week I was also reminded of the importance of using relevant examples that students will find engaging. For example, we started off the class working through a math problem that incorporated the game “Red Light Green Light.” The resource Paying Attention to Proportional Reasoning provides examples of various math problems that involve dogs, bicycles, and baseball cards. These examples demonstrate that, as a teacher, it is important that I tune into the interests and experiences of my students and incorporate them into the explanations and word problems that I develop. This will help my students to see math as fun, engaging, and applicable to their daily lives.  

The Value of Manipulatives

The week’s math class, which focused primarily on fractions, demonstrated the value of manipulatives in learning. Manipulatives allow students to think and reason in more meaningful ways as students view and physically interact with physical objects that represent abstract mathematical ideas. Chapter 1 of Making Math Meaningful to Canadian Students, K-8 (Third Edition) discusses how research since the mid-1960s has shown that “the use of manipulative materials – concrete representations of mathematical ideas – is powerful in developing mathematical understanding” (pg. 4). In this blog post, I would like to focus on three reasons why manipulatives are powerful learning tools.

Manipulatives Make Abstract Concepts Concrete

Manipulatives allow students to interact with physical objects that illustrate abstract mathematical concepts. By viewing and manipulating physical objects, students are able to connect abstract mathematical ideas to the real world. Manipulatives help students to understand how math relates to their personal lives. For example, in class we read The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Polatta which helps students to discover and understand equivalent fractions.  As the teacher reads the book, students use their own chocolate bars  as a manipulative by breaking  the chocolate bar into pieces and organizing those pieces into various groups to create different fractions. Thus, students are able to visually see and physically create a fraction such as one-quarter or ten-twelfths in order to understand what these fractions really mean. Chocolate bars are also a fun (and delicious!) manipulative as they help students to see how fractions are found in their daily life when they divide and share food.

Image retrieved from www.secondgrademathematics.weebly.com/books3.html.

Olij, B. ©2016

Manipulatives Allow Students to Actively Explore

It is important to recognize that children are active learners. Manipulatives allow students to explore math concepts by actively engaging with concrete objects. Manipulatives encourage students to take risks, ask questions, test their ideas, and discover new knowledge about math processes and procedures as they experiment with physical objects. From my own personal experience, I have noticed that I am much more willing to take risks and experiment with different ideas when I am working with manipulatives. In elementary school, I loved to work with manipulatives such as Cuisenaire rods, tangrams, and fraction circles as it was easy to move them around and try different things. I found this easier and far less intimidating than trying to test something on paper.

Manipulatives Encourage Math Talk

Manipulatives are also valuable as they encourage collaboration and discussion. During my experiences as a volunteer in a Grade 3 classroom, I noticed that many student struggled  to discuss the more abstract ideas of math as they felt pressured to use the precise terminology. They were, however, able to verbalize their thinking and discuss mathematical procedures and processes when they described what they did with their manipulatives. Similarly, the teacher was able to promote math talk when she asked students to prove a math idea by using their manipulatives. Given the fact that each student might experiment with different manipulatives, there can be valuable discussions as students share their ideas and findings. For example, one student might use the manipulative of paper clocks to add fractions while another student might use fraction strips. These students could compare and discuss their strategies and results.

This week, therefore, I was reminded of the importance of manipulatives. Before this course, I had the misconception that manipulatives were only helpful for students who struggled with math. I now see that manipulatives are a fantastic resource for all students in the classroom as they encourage students to connect math to their daily life, explore new ideas, and verbalize their thinking. 

Math: Sensible, Useful, and Doable!

“When am I ever going to use this in real life?” “This is too hard!” “This doesn’t make any sense… it’s just a jumble of numbers!” “I don’t know how to do this the right way.” These are typical phrases that math teachers might hear from frustrated and disengaged students. This week in class, we briefly discussed the notion that students need to see math as sensible, useful, and doable. In this blog post, I would like to further reflect on the meaning of these three adjectives.

Sensible

Simply put, math needs to make sense to students. In her video entitled “Brain Crossing,” Jo Boaler discusses how students learn math most powerfully when they both think about numbers as symbols and visualize and draw those numbers as this causes different pathways in the brain to cross and connect. As a teacher, it is important for me to understand how students learn. If I know that students find math to be more sensible when they can visualize the numbers, I need to ensure I am incorporating resources that help students to visualize math (e.g. diagrams, manipulatives, and pictures).

Boaler, Jo. [S Lamb] 2015, August 25. "Day 2 - Brain Crossing" 

Retrieved from https://youtu.be/qZBjub36Bvs

Useful

In order to be engaged and to understand why math matters, students need to see how math connects to their own lives in the real world. As a teacher, I need to ensure I provide students with the opportunity to work with relevant math problems that pertain to the real world. Rather than always give the students questions involving strings of numbers (e.g. “16 x 3 = ?”), I can develop a word problem that is relevant to my students’ lives. For example, I could give the following problem: “Thomas is at the toy store and he wants to buy 16 Pokémon card packs. Each card pack costs $3. How many dollars would Thomas need to have in order to buy 16 card packs?” Thus, as a teacher, it is important for me to create problems and lesson plans that show students how math is useful in their lives.

Doable

It is also critical that students feel that they are capable of doing math. As a teacher, I cannot teach my students one way of doing math and expect them to only ever follow that one method. That is not fair to my students. Rather, I need to respect the fact that there are several different ways of carrying out a math problem and that these alternative algorithms are just as acceptable as the algorithm I personally might choose. This is not something that I personally experienced as a student in elementary school or high school. If I did not know how to do a math problem the way it was taught in class, I felt helpless because I had the mindset that there was only one correct method. Therefore, I understand the need to instill in my students the confidence that there are many different ways of making math doable and that they are not limited to one particular method. One algorithm might make more sense to a particular student or one algorithm might be more helpful for a certain set of numbers. Thus, it is helpful for students to see a variety of methods so they can find the methods that work for them. For example, here are just three examples of different methods we could use to solve an addition equation:

Olij, B. ©2016

Thus, this week’s class has inspired me to facilitate a learning environment where students see math as sensible, useful, and doable!