At Math ANEX, we love to deeply analyze every bit of student thinking that we can. I just wanted to give a little sneak peak at some of the things we look for when uncovering how students think.

Let’s take a look at an example problem.

Plenty of students solve this question by saying that you need at least 10 spaceships for all the students to ride, but does that mean they fully understand the problem? Or that there isn’t more to learn? Let’s dive a little deeper.

One thing we can look at is the precision in their numbers. Some students will write that 65 / 7 = 9.29, others 9.28 or 9.285714 or 9 2/7 or 9 R2. These all provide a little information about how students interact with numbers. For example, 9.28 takes the first couple digits after the decimal, but then they stop, they don’t round up to 9.29. And 9.285714 is probably as far as their calculator tells them. 9 and 2/7 is technically the most precise of the answers and shows that students are nervous about fractions.

We looked at the data to see if there was any difference in general student performance across the entire assessment.

It’s very interesting to see that students who are technically being the most precise (using remainders or fractions) scored better overall than the other students, on average getting almost 1 more answer correct than their peers.

Even more interesting is if we look at how students performed on this question. Given they had the number in their explanation, how likely was it for them to get the answer 10?

It seems almost as if the less students focus on the numerical value itself, the more likely they were to reason that 9.29 spaceships for example doesn’t make much sense and that you would need to round up.

But this is just one instance of what you can find out from a student explanation. Let’s look at a couple more.

## Units

In the problem above, you might divide 65 students by 7 students per ride to get the number of rides you need. (9.29 rides perhaps). But not all students get that unit.

“As I tried to solve 65/7, I realised that the number (9.285…) was going into fractions. Because you can’t split a person into tenths, they needed an extra spaceship to hold the extra people, even if it wouldn’t completley fill.”

Student Response

This student, and many others, talk about that unit resulting in splitting people apart. But that number, 9.285, doesn’t represent people, it represents rides. So, even though this student does quite well in solving the problem, their understanding isn’t complete.

## Tools

What computational tools students use makes a big difference. Counting, Repeat Addition, Repeat Subtraction, Multiplication, Division, Equations, Exponentials, Logs, Trig, etc. really makes a difference in students ability to easily solve problems. Counting can be a valid strategy, but an outdated tool. And students are usually happy to explain what tool they used to solve their problem.

And outdated tools often have small mistakes as you’ll see below.

“counting by 7: 7,14,21,28,35,42,49,56,63,70”

Student Response

“I did 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 which equals 66 and then i counted how many 7’s there were which was 8 sevens. there’s also 1 space left over.”

Student Response

“so l had 65 so l did -7-7-7-7-7-7-7-7-7 and that how l got 9 spaceship”

Student Response

“I stated of multiply 7×6 to low 7×7 to low 7×8 to low 7×9 was 63 and its not 7×10 to high so im basting my answer around nine.”

Student Response

“Because you have to do 65 divided by 7 and then it was 9.2857145 and 9.2857145 people cant fit so i rounded up to ten. Therefore we need 10 spaceships”

Student Response

All of these tools make a difference in a student’s ability to more easily solve problems and take on more complex mathematical situations. For example, we found that students who still relied on counting strategies in 8th grade were far more likely to struggle or fail in Algebra in 9th grade, even though they were able to solve 8th grade math problems just fine.

Below is a graph showing overall performance of students with the use of different tools. This is filtered to just students who got the answer 10 for the science museum problem shown earlier.

As you can see, using the right tool for the job can make a significant difference. And knowing which of your students are using outdated tools can make a difference in your instruction.

## Computational Errors

“Because 65 divided by 7 is 15”

Student Response

Here we have an example of a student who knew that they were supposed to divide, great! However, they ended up with a very different answer than we’d expect. This student could be asked a few questions about how they divide to uncover what is going on so we could address a computational error. In addition, we can start by applauding them on knowing to use division.

## Conclusion

All in all, there is a mountain of information that you can uncover about your students by having them explain their answers. At Math ANEX, we dive deeply into every facet of how students think, but even just taking a quick read can help you gain a lot of insight.

If you need any help in figuring out how your school can start to look into understanding student thinking through their mathematical explanations, let us know!