Over the years I’ve had the chance to work with all sorts of clever mathematicians. Most of them were under age 11. They had great perspective, unique insight, and novel strategies. I only wish that I could always say the same about myself.
There was Robert. He was a second grader who looked at the world in such cool ways. After many errors comparing two-digit numbers, I asked him to tell me what he was thinking. It turned out that he understood comparison just fine. He just saw the symbol differently. He taught me that the greater than sign (>) wasn’t an alligator at all. Instead, it was an arrow that “pointed” at the greater number. It was a relatively easy fix.
There was Brooke. She showed remarkable skill with number lines. She could place any fraction on a number line. She could, that is, until I accidentally introduced a number line with endpoints of 0 and 2. My number line whiz continued to place fractions (less than one) in the same locations on the number line with these new endpoints. In fact, she put a fraction in the same location regardless of any endpoint I would use.
There was Brian. He was a fourth grader who compared fractions in all sorts of ways. He had exceptional reasoning. He relied on it to confirm his procedures and occasionally to avoid doing them altogether. He reasoned about the numbers in the fraction and the relationship between those numbers. For a time, he believed that fractions could easily be compared. To him, 3/4 was greater than 1/3 because it was only missing 1 piece. 7/9 was greater than 1/4 because it was missing fewer pieces. As we know, these comparisons are correct. But did his “reasoning” always work?
These are just a few of their stories. I share them so that I can highlight the things they taught me about checking for understanding.
#1 Know the Math
This move sounds straightforward. It may even sound insulting. But there’s much more to know about mathematics than what the answer is. This is especially true when we are trying to determine our students’ understanding. Take Brian for example. He compares fractions by considering the number of parts. To him, 7/9 is greater than 1/4 because 7/9 is missing 2 pieces whereas 1/4 is missing 3. Considering the number of pieces is a strategy for comparing fractions. However, it doesn’t always work. We must also consider the size of the pieces. Brian’s thinking would lead him to believe that 8/10 is equivalent to 18/20 because both are missing 2 pieces. Simply, it is critical that we understand the mathematics and strategies as well as the limitations of strategies when examining student understanding. Knowing the math, including the progression of concepts, also enables us to make decisions about what to do next.
#2 Select Good Tasks for Good Information
This idea addresses the old adage “quality over quantity.” There is a notion that many, many tasks or prompts are needed to confirm student understanding. I would prefer 1 or 2 meaty prompts over a worksheet of basic recall prompts for most any topic. Think about Robert. Twenty-five basic comparison problems didn’t tell me much. Instead, it could have reinforced his misunderstanding so much so that I could have had considerable challenges undoing his “understanding.” Instead, what if we asked him to compare 2 sets of numbers and then explain how he compared? What if we asked him what are some numbers greater than 34 and asked him to justify his solutions? What if we asked him to describe how he compares numbers and how the symbols help him compare the amounts quickly? What if Brooke worked with tasks like the one below rather than simply identifying a point on a number line?
#3 Avoid bias
We all have strategies and representations that we prefer. We like the look of the representation. Or, they are easier for us to make sense of. Strategy bias and representational bias can be really hard to avoid. But, bias is really important to keep in mind. Expecting my students to use my strategies may create misunderstanding. Bias can also apply to representations. Relying on the same representation of a concept or strategy may develop understanding of a tool or representation more so than the mathematics. Brooke’s misunderstanding about placement on the number line was rooted in the fact that she always worked with fractions on number lines that featured endpoints of 0 and 1. She developed a misconception about fractions and number lines though her work was usually accurate. Consider this task featured in my new book Mine the Gap for Mathematical Understanding. How do you think Brooke would have done when she had only experienced number lines with endpoints of 0 and 1?
#4 Stay A Step Ahead – Anticipate
We all have been trained to write lesson plans. For many of us, our planning structure didn’t ask us to anticipate what might happen. But anticipation is critical for diagnosing and reacting to our students’ reasoning. When we anticipate what might happen and why it might happen, we are better positioned to react thoughtfully and intentionally. This can help us address the misunderstanding rather than have our instruction shift to direct, procedural instruction.
#5 Expand Our Horizons – Talk with Others
Anticipating strategies and misconceptions, even limiting our bias, can be improved by talking with our colleagues. Collaborative planning can be much more than sharing activities and identifying what needs copied. Together, we can talk about
- What do we think will happen with this task?
- Why might it happen?
- What ways can we represent the concept?
- What tools or models might we use or might our students use?
- What strategies might our students use?
- What experiences have we had with other students learning this concept?
- How does this skill/concept connect with previous or upcoming concepts?
#6 Deter the Infer
I just like the way it sounds. Actually, there is much more to it. This move reminds us to avoid reading too much into what is or isn’t on the paper. We can be suckers for diagrams and drawings. But we have to ask, “Do they connect with the problem?” We have to consider if the drawings are even relevant. The same holds true for written responses. Is more writing better? Does the written response convey how or why a solution is correct? Or, do we have to connect some of the dots for the student? And when we do, are we connecting too many dots that really aren’t there?
#7 Keep the Pencil in Their Hand
This step connects with inferring. Students learn how to “play school.” Some are masters at finding ways to have us do or carryout the work for them. After all is said and done, we can be left thinking that they understand the mathematics. We might think that they just needed some help getting unstuck. But there are also times when we unwittingly do the math and believe that they had the same understanding that WE demonstrated.
#8 Don’t Settle for Right Answers
This move might seem counterintuitive. Isn’t the point that our students find correct answers? But what about the student who gets a correct answer for the wrong reason. Brian is a great example. In some ways, so is Brooke. This move is closely connected to some of the others. We can’t be on the lookout for flawed reasoning behind correct answers if we don’t know the math (#1). Simple tasks may not give us an opportunity to expose the misunderstanding (#2). We might overlook correct answers if we don’t anticipate faulty strategies or misconceptions that create them (#3). We might be unaware of faulty strategies or misconceptions if we don’t talk with our colleagues (#5).
My “understanding moves” above come from all sorts of experiences. There are experiences with students who understand but don’t have the correct answer, like Robert. There are students who have flawed or incomplete understanding like Brian. There are students with misconceptions like Brooke. There are cases of miscalculations when problems were understood; conceptual understanding when problems weren’t understood; representations that didn’t match context; and muddled procedures.
These moves have helped me understand what they understand. I’m sure there are more moves for me to discover. I’m sure I have more to learn about understanding, misunderstanding, and all of the gray in between. I’m sure they have more to teach me.