Transitions in the Conception of Number: From Whole Numbers to Rational Numbers to Algebra

Teaching Strategies

Big Picture

In my classroom I use the question “Why?” and challenge students to “defend your position” throughout each class period. Reasoning is “the process of thinking about something in a logical way in order to form a conclusion or judgment.”¹⁴ Proportional reasoning entails analyzing problems which include proportions in a logical manner and forming a conclusion. Students are required to explain their thinking and demonstrate to their classmates and to me that their problem solving is logical.

If my students have learned but not mastered content from previous grades, then in theory I should be able to scaffold with a review. This should “jog” student memories. The reality is that this approach at scaffolding is not effective because multi-year conceptual understanding has not been developed in earlier grades.¹⁵ There is no memory to "jog”. Without conceptual understanding students know a random assortment of rules and do not know when or why to apply them. For this reason, my unit has a primary goal of developing conceptual understanding. Procedural fluency will be a secondary concern and will come with time as we revisit these concepts throughout the year.

Use of “Thinking Tasks”

“If we want our students to think, we need to give them something to think about—something that will not only require thinking but will also encourage thinking.”¹⁶ My goal when introducing any new concept is to get students to engage in problem solving. In my classroom, problem solving is not applying an algorithm. It is not following a rule. It is not looking at an example and doing your best to apply that example to a new question. Problem solving is what we do when we do not know what to do. It is messy. It usually does not follow a linear or efficient path. Students can get stuck. And they can apply their existing knowledge in creative ways to get unstuck.

If a math task is carefully curated and you have a little bit of luck, students will engage and think. When I curate a thinking task for my classroom, I consider what I want students to discover, what misconceptions I expect them to bring to the task, what prompting questions I can ask if students get “stuck”, and how to release the task to students. My goal is a task which allows every student to participate, a task which challenges the most advanced student, and a task with more than one way to reach a solution. I want students to trudge through the messiness. I want them to have ah-ha moments. I want them to have moments when the ah-ha light bulb flickers off as they realize they are off track. I want them to try and fail. Sometime students will give up, so I want a task that is intriguing enough that students will return. And during the most effective and joyful class periods, I want students to argue about and learn all the math!

I first considered the difference between a general problem or math task and what Peter Liljedahl calls “thinking tasks” when I read Building Thinking Classrooms for Mathematics (BTC) a couple of years ago. BTC helped me realize that when I had a red-letter teaching day, it was the result of thinking tasks. Sometimes tasks that I thought would be great, were ho-hum. Other days the task I had chosen or developed was a huge success. Those successes were always the result of what Liljedahl would identify as a thinking task.

To decide whether a task qualifies as a thinking task, I look to what the problem does. I want it to make students think. The task should require students to draw upon a wide range of mathematical knowledge and to put this knowledge together in different ways. If students can discover a new algorithm or mathematical truth that is added value.

In this unit I will plan to at least two thinking tasks. I discuss the implementation of a thinking task in more detail in the Activities section. Each time the task will give students an opportunity to dive deep into a proportion problem. The goal with this unit is for these tasks to stretch student knowledge and to bring together a group of concepts that students want to “silo”. I want to force students to dig into their mathematical toolbox, go outside their comfort zone, and use some of the tools they avoid when the problems are simpler.

Collaborative Groups

Most of my instruction is through activities in which students can work together to solve problems and discover patterns and rules. Students sit at desks arranged in groups of 3 or 4. Almost all work done in my class is collaborative in nature. Students are encouraged to seek help from peers.

Problem analysis is well suited to group discussion. I regularly assign what I call collaborative group work. These are problem sets or thinking tasks with an expectation that students will work with their assigned group during the allotted class time. Collaborative group work is done at the white boards around the room. Standing up increases student engagement and somehow it minimizes cell phone issues. It also seems to make students feel more visible, which motivates them to engage more actively than when sitting at desks. Class norms include one marker per group. If you are talking, you are not writing. The marker is rotated on a regular basis, so every voice is heard and everyone has the responsibility of careful listening and recording their group’s thoughts.

The benefit of collaborative group work is clear and the research supports what I see in my classroom.¹⁷ My favorite part of collaborative group work is when I notice students stepping back from their section of whiteboard to look around the room. Sometimes this is an indication that the group is “stuck”. Students have learned to look for inspiration by looking at their classmate’s work. To find that inspiration the students must be thinking and analyzing work by other students. I love seeing the engagement and desire to understand. Of course, sometimes this move means the group is feeling confident in their solution and want to check out the “competition.”

Notice / Wonder / Discover

Many teachers at all grade levels use the Notice and Wonder protocol to encourage student thinking and noticing. (If you are not familiar with Notice and Wonder in the math classroom, I encourage you to check out the resources and research on the NCTM website related to Notice and Wonder.) I have built upon the Notice and Wonder model and added “Discover”. This has become one of my favorite teaching strategies. The goal of this unit is to further student proportional reasoning. This unit is remedial in nature. Students will enter my classroom with varying degrees of number sense and different levels of proportional reasoning skill. I expect that some students who bring more prior knowledge with them will draw conclusions and make connections as we work through the unit. And I expect that other students may engage in the activities and not see any connections on their own, but will learn from their peers, or from the whole class discussion.

During the school year, at key points during each unit, I allot a significant portion of class time for Notice / Wonder / Discover. Students learn the routine. Notes are out. Students usually choose to have an individual whiteboard to work on as needed. I have collected student work that demonstrates common misconceptions. I also have a collection of student work solving the same problem in more efficient and less efficient ways. Sometimes I use work I create if I want to make a specific point that is not visible in student work. Day by day we have shared our thinking and work with each other, so the point of Notice / Wonder / Discover is not so much new discoveries. The point is to bring together related ideas and concepts so we can see the big picture. The goal is for students to make observations about very carefully chosen work samples; to ask questions about those work samples; and then to discuss together with my guidance what we have learned.

When I make the time to carefully choose sample work and I have a clear goal of what I want students to discover, this is always a successful teaching strategy. Most of my students will make the connections, clear up misconceptions, and walk away with a sense of understanding.