- Anticipating student responses prior to the lesson
- Monitoring students' work on and engagement with the tasks
- Selecting particular students to present their mathematical work
- Sequencing students' responses in a specific order for discussion
- Connecting different students' responses and connecting the responses to key mathematical ideas
In this post I want to take these five practices and apply them to a problem introducing the concept of the variable that my Math 8 students will be working on next week. The problem comes from Ana Stephens's article called "Developing Students' Understanding of Variable" The rest of this post is a reflection on how I would apply the five practices to Stephen's problem.
Ricardo has 8 pet mice. He keeps them in two cages that are connected so that the mice can go back and forth between the cages. One of the cages is blue and the other is green. Show all the ways that 8 mice can be in two cages.
In my opinion the point of anticipating is so that you are prepared for what your students are about to do. This can range from anticipating student mistakes or misconceptions, to anticipating different strategies that students will use to solve the problem. For this activity I had two groups of anticipations, one was related directly to the problem the other were anticipations about my students' understanding of variable in general.
Anticipations about Variables:
- I anticipate that some students will view variables as labels rather than a symbol that represents a quantity. So in this problem some students might use b and g for variables just because they are both the first letters of the words blue and green.
- I anticipate that some students will not understand that variables are changing quantities and they do not have only one specific value. In the context of this problem students might not realize that b (the number of mice in the blue cage) can range from 0-8 at any given time.
- I anticipate that some students will believe that two different variables must have two different values. So in this problem students might think that b (the number of mice in the blue cage) and g (the number of mice in the green cage) can never have the same value.
Anticipations about Students' Strategies:
- I anticipate that students likely begin this problem by drawing a picture or creating a table. Some students may also use a graph or an equation but I don't expect either of this to be their first step.
- I anticipate that some students might not consider having zero in the blue or green cage.
During this monitoring time it is my job to be walking around and trying to understand my students' thinking, I will most likely do this in the form of questioning. In the book Principles to Actions there are four major questions types. I will list those types and they provide a couple examples of questions I might ask students during this activity.
Gathering Information: students recall facts, definitions, or procedures.
Examples: "So if there are ___ mice in the blue cage, how many are in the green cage?", "How many mice will there always be?", "Is it possible to have zero mice in a cage?"
Probing Thinking: students explain, elaborate, or clarify their thinking, including articulating the steps in solution methods or the completion of a task.
Examples: "When you drew this first picture, what does this situation represent?", "What made you decide to create a table for this problem?", "I noticed you wrote down b=3, what do you mean by that?"
Making the Mathematics Visible: students discuss mathematical structures and make connections among mathematical ideas and relationships.
Examples: "How does the table you made relate to how many mice are in each cage?", "Can you use this information you generated in this picture to create a graph or equation?"
Encouraging Reflection and Justification: students reveal deeper understaning of their reasoning and actions, including making an argument for the validy of their work.
Examples: "So I see you wrote an equation b+g=8, can b ever be equal to g. Why/why not?", "How can you check your equation to make sure it is true?"
What students' work you select to show all depends on what your mathematical goals, or learning targets are for the activity. Your goal is what is going to drive your selection process and decide whether you will select a misconception to show, or maybe a unique way of solving the problem. For this activity my mathematical goal is to show how variables are changing quantities, so therefore I would select students' work that represents this idea. I would want to show a student who represented this problem with a graph, a student who did it using a picture, a student who used a table, and finally a student who wrote an equation or expression.
Now that I have decided what student work I want to show, I have to decide what order I want to present it in. Again, the order that I choose should reflect my mathematical goal(s) of the lesson. Sometimes you might want to start with a common misconception and explain why it does not work that way, other times you might want to start with the strategy that the majority of your students used and then go on to the more unique strategies. For this activity I want to sequence the student responses like building blocks, so I would start with the picture, then show the table, then the graph, and finally the equation. This way I am working from more concrete thinking to abstract thinking. In the article "Orchestrating Discussion" they give a great suggestion of creating a table that can organize your sequencing. I have taken that idea and created a table that I would use while I am monitoring this activity.
This is arguably the most important practice of the five practices. This is where the students consolidate what they just did and they can see the "big picture" of what it is important. This normally will take the form of a whole class discussion after presenting the sequenced answers. As I was thinking about this activity these are some of the discussion points I would want to present to the class:
"So what I am noticing from a lot of your work is that the number of mice in both cages isn't always the same. How can we represent the number of mice in each cage in a general way?"
" So if we let b equal the number of mice in the blue cage and g equal the number of mice in the green cage, how can we write relationship between the two quantities? Is there more than one way?"
"Are b and g always, sometimes, or never equal? What makes you say that?"
"So in math we call b and g variables. Based on what we have discussed what is a definition we can come up with for variable?"
At the end of this activity I would hope that my students would be able to understand that variables are changing quantities, that we can represent with letters or symbols. Each of the five practices above reflects my goal, and every move I would make in this activity would be intentional. I think that the five practices are a great way to structure mathematical discussion in a classroom. They do require some planning in advance but it will help to direct the discussion towards the mathematical goal of the lesson. I believe that by incorporating these practices into my teacher assisting placement I will begin to see better whole-class discussions in my classes.