- Episode Highlights
- Transcript
You’ve probably heard the phrase “reason abstractly and quantitatively” more times than you can count, but what does it really look like with young learners?
In this episode, we dig into Math Practice 2 and explore what it means to help students make sense of quantities, symbols, and problems in a deep and meaningful way. If you’ve ever felt torn between focusing on conceptual understanding versus getting through all the content, you’re not alone and this conversation will help you find a better balance.
We’ll talk about the power of hands-on experiences, the role of drawings and visuals, and why students need to interact with math in concrete, representational, and abstract ways at the same time. Ready to move beyond memorization and give your students the tools to truly understand math? You won’t want to miss this one.
Links mentioned in this episode:
Math Practice 2 – Reason Abstractly and Quantitatively
Hello Meaning-Makers! Welcome to the Meaningful Math Podcast. In the previous episode, we introduced the Math Practices – the eight long-term goals for students from kindergarten through twelfth grade that are the “thinking and doing” we want our students to experience as they develop their understanding of math concepts.
We dove deep into the first math practice as we discussed the importance of providing space for our students to experience productive struggle as they make sense of problems and persevere in solving them.
Today I’m excited to continue this discussion about the Math Practices as we examine Math Practice 2.
What Does It Mean to Reason in Math?
Let’s start by thinking about what it means to reason. It’s such a simple term, but there is so much depth to it.
To reason means “to make sense of”. When we reason, we gather information to assess the situation, we ask questions and go back and forth between ideas, and we apply prior knowledge.
Eventually, we reach a conclusion or maybe not – maybe we’re still grappling with an idea or maybe new information shared by someone else causes us to change our minds and go back into the learning process.
Reasoning is very cyclical, as it has no finite endpoint. Think for a second about how different this is in comparison to memorizing or being shown a step-by-step process and repeating it yourself. Reasoning involves deep thinking that leads to sense-making and complex understanding.
There’s a reason (no pun intended) why this Math Practice doesn’t say something like “memorize the meaning of quantities “or “follow the computational steps your teacher shows you”. We want our STUDENTS to do the work of making sense of math.
In our last episode about Math Practice 1 we discussed how students won’t know how to persevere if they are never given the opportunity to HAVE to persevere.
Similarly, students won’t have a deep understanding of math if we as teachers do the reasoning and sense-making for them by showing them a step-by-step process or overriding opportunities for deep thinking by just asking them to memorize facts and procedures.
Understanding Quantities, Symbols, and Mathematical Meaning
Now that we have established an understanding of what it means to reason, let’s examine the rest of the practice.
When students are reasoning abstractly and quantitatively, they are making sense of the written and spoken numerals and symbols we use in mathematics, and they are deciding what to do with those quantities.
When I say “symbols” I’m referring to operational symbols like the signs for addition, subtraction, multiplication, and division, the equal sign, a fraction bar, parentheses, etc. These numerals and symbols are assigned meaning in our language.
But in order to have a true understanding of what they mean, you must experience them concretely and work through scenarios where you have to make sense of them for yourself.
Traditional math teaching has focused primarily on the abstract… pages and pages of rote computational practice problems and memorized math facts where to be successful meant you got the answers correct and nothing more or deeper than that.
What Is CRA in Math? A Guide to Concrete, Representational, Abstract Learning
You might have heard of a concept called “CRA”, which stands for “Concrete, Representational, Abstract”. This concept is embedded within Math Practice 2, and as I explain it more I think you will see why.
CRA is a cyclical process where students experience math using all three of those components simultaneously.
“Concrete” can be thought of as hands-on experiences with math ideas – think manipulatives, objects, and real-life situations. “Representational” is the drawings that represent a mathematical idea. Finally, the “abstract” refers to the written and spoken numerals and symbols.
When students interact with a math concept using all three of these components at the same time they are engaging in deep thinking and sense-making.
For example, in thinking about the problem 3 + 2 = 5, the different stages of the CRA model might look like students act out the problem using concrete manipulatives, they draw a picture to represent what they did with the manipulatives, and the written problem and answer is the abstract. There is so much more depth of understanding being developed compared to just simply asking our students to memorize 3 + 2 = 5.
We often think of concrete experiences as being for primary students. Kindergarten and first grade classrooms often have shelves filled with bins of math manipulatives. But it shouldn’t stop there. Manipulatives and hands-on experiences belong in ALL classrooms from kindergarten through twelfth grade and beyond.
It is important to note that when students engage in these three stages, it is not a gradual process. It’s not breaking up the process and doing a few days of concrete, a few days of representational, and a few days of abstract. It’s all three components interacting together at the same time.
Achieving Depth Over Breadth in Math by Focusing on Fewer, Richer Problems
It’s important to know that when focusing on the stages in the CRA model, you will cover fewer problems, but you will go deeper with them. There just isn’t enough time to prioritize the stages with the amount of problems we have historically asked our students to solve, but that’s okay! There is so much value to be had in adopting this “depth over breadth” mentality.
This aligns with the idea that good mathematicians don’t have to be fast, and they don’t have to be the first ones done with a problem. Good mathematicians are deep thinkers who take their time.
Requiring students to solve an excessive amount of problems takes away from this mentality. More problems does not equal more learning.
Reflection Prompts for Implementing Math Practice 2
Here are some prompts that might help you think about how to best utilize Math Practice 2 and the CRA model in your classroom:
- In looking at the sets of problems I typically use, can I select some high-quality problems for my students to engage with in the CRA process, keeping in mind that my students don’t need to do a vast amount of problems in order to develop their understanding?
- What manipulatives can my students engage with that will help them understand this concept? If I don’t have access to manipulatives, how can I get creative with what I do have access to? For example, if you don’t have base ten blocks or pattern blocks, can you download, print, and cut out templates of these blocks?
- What hands-on experiences can I offer my students? In what real-world scenarios or contexts might they encounter this problem, and how can I connect them with these scenarios or contexts?
- Am I providing time and space for my students to draw so that they can represent the concrete? This act of having to translate their concrete actions into drawings deepens their understanding.
- Am I modeling what I want students’ drawings to look like and expecting them to replicate my representations? If so, how can I turn this over to my students and let THEM make the decisions about what these drawings should look like?
- Am I giving students opportunities to work with others during the CRA process so that they have opportunities to share their thinking and learn from others?
And there’s so much more we will discuss with this when we get to Math Practice 3!
Reflecting on Math Practice 2
Meaning-makers, I hope you’ve enjoyed this first part of our journey to understanding the Math Practices. I hope much of it has validated what you’re already doing in the classroom, as well as given you ideas for ways you can prioritize the practices in your teaching. Until next time, have a great one, Meaning-Makers, and thanks for listening!
