Teaching for Conceptual Understanding: Math
In traditional math classrooms that breed math anxiety and hate, students often view math as a performative subject. Get the answer fast and right! We often want to understand why so many students and people come to view math in such a negative light. Why do people think they are not “math people”? Why do young kids LOVE to count and work on problems but older kids dread word problem time? What are we doing to our math students?
Let’s look at the logical progression of a traditional performative math class:
If we want to avoid this, we need to start helping students love math again. Turns out, math is actually quite natural. Our brains seek out patterns and ways to solve problems. Our very number system was created by human brains - so why couldn’t all of our brains understand it? They can! We need to get out of our students way and let their brains do what they can do - SOLVE.
In order to do this we need to focus on conceptual understanding - deep understanding of concepts, not steps. To build conceptual understanding you need:
The open-ended problems don’t even necessarily need to be “word problems”, although they can be.
Take this equal grouping math activity for example:
Students need to create equal groups of leaves. This is an open-ended problem because there are multiple ways to make equal groups. All students need to know is the term “equal” but that can come from a good discussion OR you can start with a kid friendly term - FAIR.
With this open-ended problem, students are not copying any thinking or steps I gave them. They are thinking and showing their thinking. There is also great visual representation. We can look at everyone’s groups and discuss how we made them differently but they are all equal. Even if a student doesn’t make theirs equal, it leads to a great discussion about how to make it equal.
The beauty of conceptual understanding is in this next step. Now that students have DONE the concept of multiplication (equal groups) they can EXPLAIN and you can show them how their thinking IS math.
So, if a student made equal groups of 4 and there were 6 groups they will probably say something like “I made 6 groups and the groups have 4” and as they talk you can write 6 x 4 - this is exactly what they are saying but in math terms. NOW students SEE their thinking as math! THIS is the ticket.
Traditionally, this lesson may have happened in the opposite way. The teacher would show the student the problem, show and make them draw it out (good visual, though) together, and then tell them that’s how the concept works. This completely skips any student thinking and deep connection to the concept.
This is a simple example, but math concept can be understood by students with a quality open-ended problem. Always start with the students’ language and translate it into math words and symbols. Students need to see how THEIR thinking IS the math. This will help students see that answers aren’t the end all-be all in math. Their thoughts and problem-solving matter and their explanations are important.
Good discussion is fueled by great questioning. Make it a point to ask more than you tell when you are teaching a math lesson. For everything a kid tells you, ask a question about it - whether it is correct or wrong.
Let’s look at the progression in a conceptual understanding lesson:
So, I hope you can begin to transition your math block out of “watch me” math and into “try it!'“ math and help your students love problem-solving again!
If you are interested in the equal group problems, this one came out of my Fall Themed Equal Group slides pack (printable or digital form) with 18 fall themed grouping slides. Click below to buy! :)
