Skills vs. Concepts In Mathematics
How skills and concepts support student academic outcomes
Learning math is a balance between two essential ideas: skills and concepts. They are both vitally important to learning mathematics and compliment each other. However, many educators and curriculums do not have a quality mix of both. Some just do not know how to compliment skills and concepts with each other, leading students towards having gaps in their learning. Today, we will explore these two ideas and how they can be trained and intertwined in our math curriculum.
Research on Skills and Concepts in Mathematics
There is a whole breadth of literature on skills and concepts contributions towards learning mathematics, with much of it spear headed by Psychologist Bethany Rittle-Johnson. I highly recommend checking out the lab website that has a ton of resources for student learning ran by her.
What are Skills and Concepts in Mathematics?
Let us start by defining exactly what skills and concepts are in mathematics, and their research appropriate names.
Procedural Knowledge: The Doing of Math
Skills in mathematics are known as procedural knowledge, which are the action steps you need to know to complete problems. For instance, if I want to solve an expression like this:
I would need to know the steps of PEMDAS and how to perform them. Furthermore, I would need knowledge of how to perform the operations of arithmetic and doing arithmetic with negatives as well. Thus, procedural knowledge is two fold: it is needing to know the steps to solve a problem and performing the action of solving the problem.
Conceptual Knowledge: The Why of Math
Concepts in math are known as conceptual knowledge, which is the understanding of how math works and why procedural knowledge works. This is the understanding of the underpinnings of mathematics and having the knowledge to be able to find what procedural knowledge applies to each problem. Let us look at another example of solving an expression like this:
When we see a problem like this, we have a conceptual understanding that this is an expression and we need to use order of operations to solve. We recall a concept that we learned to solve order of operations called PEMDAS, and we use conceptual knowledge that it stands for parenthesis, exponents, multiplication, division, addition, and subtraction. We also have to recall more conceptual knowledge that we solve these operations from left to right and need to group together multiplication and division, and addition and subtraction. Then we begin solving the problem, where we switch over to procedural knowledge to complete the arithmetic.
Intertwinement of Conceptual and Procedural Knowledge
Conceptual and Procedural Knowledge are not mutually exclusive concepts and can be hard to separate in some instances. For example, when we talked about order of operations and PEMDAS, we needed to recall what PEMDAS stood for before performing the arithmetic. Therefore, it is important that, as educators, we are providing students with both forms of knowledge.
It is easy to focus on only providing students with the steps of solving problems in mathematics. In-fact, I would argue that most curriculum and K-12 math education is designed around “solve this equation for x” or “solve this expression”. However, students need to understand the “why” of how mathematics works in order to be able to apply their knowledge to a broader set of scenarios.
Using Conceptual and Procedural Knowledge in your Classroom
Now that we know the types of knowledge that students are learning in a math classroom, how can we apply it to our own classroom?
I will be honest, it would be a waste of time to explain to our students the differences in conceptual and procedural knowledge. These ideas are abstract and not useful for students to understand how it fits into their learning. However, we can explain in more simple terms how skills (procedural knowledge) and concepts (conceptual knowledge) impact their learning. Let’s look at some implementations that could be used in your classroom.
1. Make Sure your Models Provide Conceptual and Procedural Knowledge
When you teach a new concept in mathematics, it is important to provide students with how and why the math works. For instance, in your model you can structure it such that you present conceptual understanding first and then go into how to solve the problem. You can also mix and match the order if you believe it is easier to grasp. Try:
Answering the questions as you create your models for your lessons:
Why does this work?
How do we solve it?
How does this fit into our previous learning?
2. Create a knowledge and skill T chart for each unit you teach
I find this tip highly effective when teaching a new unit. When you begin a new unit of content, create a T chart and label one side knowledge and the other skills. Then begin listing all of the knowledge and skills students need for the entire unit and post them somewhere in the classroom. First, this will help you know all of the content you need to cover over the unit and keep you on track. Secondly, students will be able to see all the information they need to retain and the skills they need to know. Here is an example from a 6th grade Unit on Rates that I have used:
They do not need to be in as much detail as outlined above, and I highly recommend that you keep it in student facing language as much as possible. Make it work for you.
3. Use Math Vocabulary Correctly with Students
When we teach math, it is extremely easy to use incorrect vocabulary. For example, whenever I have talked about variables in the past I would just call them “letters.” Another instance is when talking about functions I would call them “lines.” The downfall of this is that is now how students remember them going forward. Every time you reference a function or variable, their minds will go blank until you utter the words “letter” or “line”. This can be detrimental when students engage in problems that actually use similar vocabulary in the problems.
When you first introduce a concept, it is completely okay to use simple terminology to help students understand what a new concept is. However, right after they learn the definition begin using the new term and never go back. Try:
Always speaking to students with mathematical terminology
Correct students when they use a term incorrectly
Reward students for using excellent mathematical vocabulary
4. At the beginning of the year, test students on some basic procedural knowledge
It can be extremely difficult teaching a 7th grader who does not know their multiplication tables and factoring. Therefore, at the beginning of the year, make sure you create a simple assessment to test students on basic procedural knowledge of arithmetic. Once you identify the group of students who are lacking on their fundamental procedural knowledge, you can support them to learn it.
This is an important idea because as I discussed in a previous article, students can only retain a specific amount of information before they are overwhelmed. Therefore, if students are stuck on basic arithmetic, it can be difficult to teach them more complex concepts in mathematics. Try:
Testing students on multiplication, division, fractions, and solving basic equations at the beginning of the year. A short quiz of 10 to 15 questions is sufficient
Once you identify the group, immediately begin pulling them for tutoring or small groups to implement learning these ideas
Give flash cards and homework assignments to support students
Get families involved in helping students cover information that is essential to their learning
Utilize tech programs that help teach basic arithmetic skills for students to practice with
I hope that some of the tips above help elevate the conceptual and procedural knowledge of your students. As always, if you have any ideas to add or that you use in your own classroom let me know.