Concrete Representations that Give Students a Way to Get Started

This blog post is the third in a series of four blog posts exploring the student experience of problem-based learning. The first two posts are available here: “How Do Students Perceive Problem-Based Learning?” and “Inviting Students to the Mathematics.”

By Max Ray-Riek

Once students have an invitation to the mathematics and understand the situation, how do they get started answering questions?

In this series of blog posts, we’re exploring what it takes for students to be successful in a problem-based classroom, specifically the shifts in what being a good math student looks and sounds like. You were invited to think about how students are invited to the mathematics, and how they learn that they have ideas for getting started. With these ideas in mind, what do you notice about how units, lessons, and activities are launched in IM’s curriculum? What are you curious about now, as you think about what it takes for students to be successful in a problem-based classroom?

Here is a reminder of one model for problem-based instruction that we use at IM:

In this post, I’m going to focus on transitioning from the launch of the activity and listening to students’ initial ideas about the situation, to students actually being ready to engage in the problem at hand.

First of all, I think it’s important to think about what the goal of the “problem” part of problem-based learning is. What do your students believe to be true about working on problems in math class? Here are a few questions to consider. How would students answer them? 

  • Should I know how to do the problem by the time the launch is over?
  • If other people are finished and I’m still working, am I slowing everyone down? Should I just stay quiet when we go over the problem, and ask for help later?
  • If I get the right answer, does that mean I can relax until everyone catches up with me? 
  • Can I expect that the teacher will come over and help me individually if I don’t know what to do or I’m behind? Is it okay to do other stuff (like talk or read a book) while I wait for help?

Student beliefs are informed by previous experiences. If students’ previous experiences are from a math class where the teacher only released students to work after it was clear that everyone knew how to get to the answer—or, if they didn’t, the teacher planned to come over and work with them in a small group—then students might struggle in a problem-based setting, because our norms are the opposite of those. In a problem-based classroom, you’ll often see that:

  • Not everyone is quite sure how they’ll do the problem when the launch is over but everyone understands the context and each student has some different ideas to try.
  • We’ll pause and talk about the problem as a whole group even though one group has a wrong answer, another group is only part-way through thinking about the problem, and another group has solved the problem one way and is working on a second strategy for the same problem, but we are all at a point where we can understand and learn from each other’s ideas.
  • When we get an answer, the burden is on us to check our work, and to be ready to compare our thinking to others’ ways of thinking. The learning has just begun!
  • The teacher rarely says more than a sentence to any one group, because the students have the job of doing as much sense-making and showing their thinking as they possibly can (even if that thinking is just a list of everything that makes this problem hard) and the teacher is jotting notes on their clipboard about what each group is doing and what aspects of their work they are going to share, compare, or correct during the discussion.

So how does the magic happen, where the teacher releases students to work on their own and students have ideas to try? Some of it comes from the invitation to the mathematics and the launch, in which teachers and students have activated their prior knowledge and shared what they noticed, so that the teacher can make sure that everyone understands the story and what the question is asking. This is when the teacher can make sure that students know that a problem about the size of “pig pens” is about fenced-off areas for animals to play in, and not ink pens with pictures of pigs on them (true story).

The rest of it comes from students having reliable methods for reasoning about math, that they are confident trying on these problems. I think one of the hardest shifts for teachers to help students make is a shift away from expecting an algorithm or a method to solve problems, and towards believing that problems are solved by representing, reasoning, re-representing, and reasoning some more.

Consider Unit 2 of Grade 6, which is all about ratios. Very early in the unit, there’s a lesson in which we ask students to reason about mixtures that would look or taste the same as an original mixture. Early in the lesson we ask students about making a bigger batch of juice mix that would taste the same, and then about whether two mixtures of paint would be the same color. At the end of the lesson, we ask students for a batch of birdseed that tastes the same but uses less of each ingredient. 

If students are thinking algorithmically, and looking for steps that always work to convert a recipe to another batch size, they might feel like we’ve asked them three really different problems. The first problem requires us to multiply each ingredient by the same number to make more batches of juice that taste the same. The second problem requires us to compare two batches, so we might have to guess what number to multiply by to get one batch to match the other, or we might even do some dividing. The third problem requires us to divide the ingredients by the same number to make a smaller batch, which is totally different from multiplying. 

If students don’t see these problems as related because you do different steps to get to the answer, they’re going to have a hard time learning about the underlying mathematical structure. So how does IM help students see the structure and reason, and actually maybe find the problems to be easier, or at least easier to think about all at once? They can all be represented by the exact same kind of very concrete diagram:

Sometimes students who are used to a more Gradual Release of Responsibility model are also used to a curriculum that focuses a lot on steps or algorithms. Those students learn to ignore diagrams, expecting that they will fade over time as they are expected to move quickly to calculating with numbers. They sometimes decide that diagrams are “baby math,” or that representations are confusing because they’ve never tried to make their own or actively make sense of someone else’s. They’ve learned diagrams at best as another set of steps or rules.

In the IM curriculum, we were very choosy about which representations we introduced when, how we expected students to use them and learn about them, and which ones we never introduced at all.

The representations we introduce are the tools students can almost always fall back on to help them make sense of more complicated situations. We always start with the diagrams that are most closely aligned to how students naturally reason about and represent the stories to themselves, and scaffold students up to more and more abstract and powerful representations. For example, in this same ratio unit (Grade 6, Unit 2), students start with ratio diagrams like the one above, then move to double number lines, and then to tables. Each is more abstract and powerful than the last, and each is made sense of by comparing it to the previous.

But if students aren’t used to starting with their own drawings and ideas, and they aren’t used to thinking about representations and structures, and instead want calculations and steps, it can be hard to make use of diagrams.

Diagrams work best when students’ first task is “How would you draw a diagram to help you think about this?”

And then they are asked, “Here is Jada’s diagram. What was she seeing and thinking?”

And then eventually, “Can you show me where the 2:6 is in your diagram? Can you show me where the 4:12 is? How does your diagram help me see that those ratios are equivalent?”

If a student has connected a representation to their own ideas, and connected a representation to other representations, then that representation becomes something they can use to get started on other, even harder problems in the future. If I see students stuck on a fraction problem, I like to see them draw a tape diagram. If they’re having trouble with percents, I hope they draw a double number line. If I see students stuck on a problem involving integers, I want to see them sketch a number line. When students aren’t sure if the steps they are taking to write an equivalent equation make sense, I hope they are visualizing a hanger diagram. Concrete or visual representations help students see structure, make sense, and have something besides a list of steps to fall back on when they’re stuck, or to get started with when they first see a tough problem.

Next Steps

I invite you to look at the unit you’re currently on. What visual representations or concrete tools are students given to help them see structure, make sense, and get started when they are stuck? Then notice (or ask) if your students see those representations or tools as useful. When students start a problem, do they draw a picture or go for their concrete tools (like patty paper or a foldable number line)? When something doesn’t make sense, or students disagree on an answer, can they use another representation to check their thinking?

If you students aren’t used to diagrams being helpful thinking tools, you might:

  • Share that mathematicians almost always start with a doodle or diagram (true).
  • Take away the problem from a story and give students the task of just representing what they hear, and then have students compare each other’s representations.
  • Present a diagram and support students to notice and wonder about it, and then connect it to the steps or procedure they are more familiar with.
  • Whenever students are stuck, lead with, “Could you draw a diagram to help you understand the situation?”
  • Make a big deal of giving partial credit or not accepting work without a representation, if students are stuck.