With the traditional method of math instruction -- the way many of us were taught -- the goal was to get the right answer. Our instructional model, however, requires students to explain and show their thinking. Today we'd like to share some of the cognitive and learning science research behind this approach, and why it's so important that students explain their reasoning.
1. Teachers need to know how students arrive at their answers.
Right and wrong answers don't reveal much about student thinking. We also need to be able to connect what students do with what they know. For example, let's look at a real piece of student work:
If the only information the teacher had was this answer, s/he might think the student doesn't know anything about fractions. Click here to hear the student explain his thinking.
He said, "I have to find a multiple of 10, so half would go to 5/10 and 1/5 would go to 2/10, and multiply that to make one whole."
Now we know that he worked to find the common denominator and knew right away that it was 10. He also quickly found that 1/2 = 5/10 and 1/5 = 2/10.
So we know that this student knows how to find a common denominator (but not when to find a common denominator). We also know that the student understands equivalent fractions. His mistake was in applying the process for adding fractions to multiplication.
Why would a student do this? The answer is the next advantage of having students explain their reasoning:
2. Explanations encourage students to explain the why and when, not just the how.
This student likely learned to add fractions by following a rote procedure: find common denominators, add (or subtract) the numerators, and simplify. If you learn that procedure without knowing why to find common denominators in addition or subtraction, you might, like this student, assume that all operations with fractions require common denominators.
Asking students to explain their reasoning can make a connection between the procedure and the underlying conceptual knowledge, and that connection helps students know when to apply procedures like common denominators.
In addition, students solve problems in different ways, and if you don’t know the way they’re solving it, you don’t know what the student is capable of doing. For example, one student might use a formula to solve a problem while another uses context clues from the word problem. If a test question only asks for the answer, all you know is that they got it right, and you might assume that they all have the same knowledge when they don't.
3. Students learn better when they self-explain.
Several research studies have shown that self-explaining can have a positive impact on student learning. In one study, students who were prompted to self-explain demonstrated a more robust mental model of what they learned than students who were not.*
Self-explaining establishes connections between conceptual and procedural knowledge. These connections both contextualize the knowledge (providing the why and when) and make it easier to remember. Self-explaining can be a powerful tool for students both when they learn new concepts and when they access that knowledge while solving a problem, so giving them opportunities to do so can have an impact on their success.
The simplest way to start having your students explain their thinking is to ask them questions like:
You can also build a student’s ability to explain their thinking by providing opportunities for students to analyze the work of their peers and explain what's correct or incorrect about their thinking. You can, for example, provide several samples of student work and ask students to figure out which is correct and explain why.
We want to improve not just test scores, but real understanding of mathematics, which is why our textbook provides countless opportunities for students to demonstrate their thinking and MATHia analyzes and adapts to how students solve problems, not just the answer they give.
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