Helping students develop a conceptual understanding of math completely changed my math mindset. I now have the answer to the dreaded question, “Why don’t you just teach math the old way? That’s the way we were taught, and we all turned out fine.” True. We turned out FINE. Our math skills are FINE. But is FINE our goal? Wouldn’t we rather have excellent math skills? Math skills that are transferrable and applicable. FINE is not good enough for my students.
It’s time to move past fine. It’s time to move past the statement of “I was never good at math, so my child probably won’t be either.” Every child has the ability to be a mathematician. Let’s get them there! We can give them a conceptual understanding that they can then apply and use as a mathematician.
What Does Conceptual Understanding Mean?
Conceptual understanding of math means that students have a deeper understanding of math concepts, beyond just having memorized facts. While facts are important, in a conceptual approach these facts and algorithms are integrated to help deepen the knowledge of mathematical concepts.
Why is a Conceptual Understanding of Math Important?
Think back to your math experience in school. Were you ever taught why something works? Even how something works? Or was it more of here is the algorithm, now practice it over and over again?
I’m going to be honest here. I hated when my teachers would spend extra time explaining something. I was the student that wanted to be taught the algorithm and move on. That’s all I needed to be able to solve the problems successfully. I didn’t want or care to know the why behind the algorithm. That felt like a waste of time to me. I was REALLY good at math. Numbers were, and still are, my jam!
After I graduated from high school, younger students would ask me for help with their math homework. I knew that I could help them because of my success in math class, but as time went on, I forgot more and more of the algorithms. Let’s be honest, if someone asked me for help with high school or college-level math now, I would have to spend some time on YouTube reteaching myself the algorithms.
This is why developing a conceptual understanding of math is important. The connections made between the why and the how help with the retention and application of the content. Math conceptual understanding allows for ideas and knowledge to be organized in a way that our brains can find them when needed.
Avoid Teaching Students Simple Math Tricks
Have you ever used the butterfly method when working with fractions? How about the Lucky Seven model when dividing large numbers? Do they work? You bet they do! But, what happens when you forget a small part of the trick? More often than not, you get stuck. Without the trick, you don’t have any idea of how to solve the problem.
I’ve taught the tricks. I’ve seen them help students be successful. Here is the problem I always seemed to run into, though. Students were efficient and could use the trick pretty seamlessly while we were learning that specific content. But, as time went on and other content was taught, the trick was not in the front of their minds anymore. They knew there was a trick, but they couldn’t remember how to do it successfully anymore. It was too hard to retain without knowing why the trick worked.
Tricks are okay to teach, but they need to come after students have developed a deep understanding of the content and why things work. By giving students this knowledge before teaching these simple tricks, they have something to fall back on. They do not have to rely on memorization of some type of procedure that they learned six months ago.
Math Conceptual Understanding Helps Students Retain Information
Retention is a battle that every math teacher fights day in and day out. My 4th-grade team had this conversation of retention way too often. It went something like this, “We spent 4 weeks teaching our students how to multiply multi-digit numbers. Now, when I give them a large multiplication problem, they look at me with that deer in the headlights look.” Conversations like this always made me want to bang my head against a wall. Why did we spend so much time on a concept or skill, just for the kids to forget everything we had taught them?
This is where conceptual understanding of math comes in. When students understand the why, they have a way to get the answer, even if they forget the process. Students retain the information and can apply it because they truly understand it.
Let’s Compare Conceptual Understanding vs Procedural Understanding
My guess is that you learned math through a procedural understanding approach. You were most likely taught some type of algorithm (or trick), practiced it over and over, and then took a test. You very likely even did well on that test. But, could you explain what was happening? Could you have taught someone else? That is the difference between conceptual understanding vs procedural understanding.
Conceptual Understanding
I feel like I am repeating myself here, but I cannot emphasize enough the importance of math conceptual understanding. Students need to understand the why through investigation and discussion.
How about an example of a conceptual math lesson? Picture this: A teacher poses a question that requires students to solve a real-world problem involving the math concept that the teacher is teaching. Students solve the posed problem in any way that makes sense to them. This may include using manipulatives, drawing a picture, creating a model, using a number sentence, or even writing down the process of using mental math. Then, select students will share their strategy and the answer they got. A mathematical discussion then occurs among all of the students and the teacher.
Procedural Understanding
Now, let’s contrast a conceptual approach with a procedural approach. With this approach, a teacher shows students a problem. They then show students the step-by-step way to solve the problem. They might practice a few problems in front of the class. Then, the class will solve several problems with the teacher. And last, students will solve problems independently.
With this approach, there is very little investigation or choice of strategy. There is also minimal discussion. Students usually come out of this approach with knowledge of the process, but not why it works.
Developing a Conceptual Understanding of Math Through Rigorous Math Tasks
Math tasks help build math conceptual understanding because of their rigor. Using math tasks allows all students to start where they are and still reach the goal. We like to call this “low entry, high ceiling”. Real-world math tasks give each student a starting point. Some students may draw a picture to solve a math task, while other students may draw a model, use a number sentence, and come up with a rule (or algorithm). The beauty is, all students have the chance to think deeply, investigate the problem, and come up with a solution in their way.
Math tasks are not all created equal. There are essential steps that need to be taken to have effective, rigorous math tasks that meet the math standards and develop conceptual knowledge of a concept.
Dig Deep into Those Core Math Standards
Math standards can feel so broad! I remember being in college and reading through standards to write a lesson plan. There were times when I had no idea what the standard meant. Let’s be real, sometimes I still feel that way! There is so much to be interpreted in a standard. This is why it is so important to unpack and standard and dig deep into what is being asked of students. When unpacking a standard think: “What are students expected to know?” “What are they expected to do?” “How are students expected to represent their understanding?”
Once you have dug into the standard and understand what students are expected to know, do, and represent, then you are ready to start with a math task. Math tasks should be aligned to the standard. Often, it takes several different tasks to hit every piece of the standard.
For example, when I wrote my Rounding Olympics math task unit, it took 4 different math tasks to hit the entire standard. Each of these tasks had multiple problems for the students to work through and discuss. The tasks progressed and developed upon one another to incorporate all pieces of the standard.
Allow For Productive Struggle
Productive Struggle: One of the hardest things for a teacher to let students do. In my first year teaching, my coach would come into my classroom to teach lessons every so often. Every time she taught a math lesson, she did it with math tasks. And, every time she let my students struggle to find a way to solve the problem. I wanted so badly to just jump in and help them. I now see the power of productive struggle. It takes perseverance to develop a conceptual understanding of math.
Let your students struggle. Not to the point of frustration, necessarily, but the challenge is good for the brain. When students face a challenge and persevere through it, it stretches their brains. It helps create a deeper understanding of a concept. You don’t always need to save your students and give them the answer. Struggle is good.
Mathematical Discourse is Critical
I would dare say the most crucial piece of teaching math conceptually is using mathematical discourse. There is so much power in conversation.
Our students are amazing. The ideas and strategies that they come up with are remarkable. Sharing these is powerful. Let’s be clear, though. Mathematical discourse is more than just having a casual conversation about math. It uses a set of practices and tools.
Asking the right questions have a huge impact on the success of mathematical discourse. When conducting a mathematical discussion, it’s important that teachers know what questions they are going to ask. Scripting questions beforehand can increase the effectiveness of mathematical discussions.
Here are some generic questions that work for almost every lesson for you to try:
- How did you solve this problem?
- What similarities and differences do you notice?
- How do you know?
- Can someone restate what he/she said?
- Is the answer reasonable?
- Do you notice a pattern?
TIP: If (more like when) you get stuck and aren’t sure what to ask during a discussion, use my go-to: WHY? This question works in just about any context and forces students to think deeply and explain their thinking.
More Ideas for Increasing Conceptual Understanding of Math
Complete Units Using Math Tasks
Building Conceptual Understanding Podcast
5 Tips to Develop Conceptual Understanding in Math
Teaching students in a way that develops conceptual understanding is thrilling. You will see students who previously had little success in math and often were unengaged change. These students soon become teachers. They get to share their strategies and have discussions with their classmates about their work.
I’m pretty sure that increasing student engagement is a goal of just about every teacher. Math tasks not only increase student engagement but also increase the conceptual understanding of math.
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