The following article is a part of our Signature Pedagogies in the K-12 Classroom series. “Signature pedagogies,” as defined by Schulman in 2005, are “the types of teaching that organize the fundamental ways in which future practitioners are educated for their new professions” and they include three critical aspects: how to think, perform, and act with integrity in the profession. Click here for a primer on the importance of signature pedagogies.
NOTE: Get comfy. This is a longer article. Math is a frequently misunderstood discipline, which often results in it being taught and learned ineffectively—often with long-term consequences for our young people and our society. I kept this as short as I could while still trying to hit (and explain) the major points.
What is Math?
Too often, educators (and therefore students) dilute the purpose of math down to using computational methods to solve posed problems with a single correct answer. This, however, is but a tiny sliver of the actual purpose of the discipline of mathematics, and we do students a disservice when we teach this very limited version of mathematics.
At its heart, math is a logic-based language that we use to describe and/or impact our world. Many of our abstract mathematical concepts come from real-world observations; for instance, there are 360 degrees in a circle because ancient astronomers noted that it took about 360 days for a full calendar year to take place (Ronan, 2020). Consider, for instance, the word “geometry.” The first part, geo, literally means “the earth.” Metry is the “art or process of measuring.” Geometry, in other words, has its roots in practical problems dating all the way back to the 5th century BCE. Ancient Egyptians invented ways to survey land, measure grain, devise crop calendars, etc. (Heilbron, J.L., 2024).
A large part of understanding of being able to use math in everyday life is understanding the language of mathematics itself. Ask any young person who learned that “x” meant times in elementary school, only to have it represent “the unknown” once in pre-algebra—a big portion of math is learning the symbols, what they represent, and how to build number sentences with them. There is even a syntax to math; for instance, whether you believe that 2 + 4 * 3 = 14 or 18 depends on your acceptance of the PEMDAS protocol for reading number sentences.
Once we begin to understand that math is, foremost, a language, we begin to understand the importance of accurately communicating mathematical reasonings to describe our world (the second part of the definition). Indeed, throughout history, a main purpose of math was to ensure effective communication. For instance, we must agree to use the same method of measurement to know where my property line stops and yours begins. There is even an International Committee for Weights and Measures that ensures coordination among nations so that we all define a “kilogram” in the same way.
Consider now the last part of the definition: we use math to impact our world. After reading many, many philosophical tracts on mathematics, I went round and round on whether to use the phrasing “impact our world” or “solve problems.” I finally decided to go with “impact our world” for two reasons: 1) sometimes our “solutions” are not actually effective, even though they have an impact; and more importantly, 2) “solve problems” can connote the idea that there is a single solution that exists, rather than an infinite possibility of “solutions” based on how a person decides to define the problem and appropriate measures of success.
As an example, imagine a wide chasm separating two cliffs that it will take hours to walk around. On one side of the cliff is a person we will call “Jordan” with their fields of wheat, and on the other side is “Taylor” with their apple orchard. Jordan is tired of plain bread and desperately wants to make apple pie; Taylor has no interest in wheat. Jordan sees an obvious problem; Taylor does not. Jordan decides the solution is to make a bridge, and therefore needs to collect wood to do so. But how much wood? Therein enters mathematics. Jordan needs to decide if they want to cross the bridge one time (in which case, a single, very long log could possibly work) or if they aim to travel back and forth often and will need something studier. Does Jordan plan to just grab a couple of Taylor’s apples and put them in a pocket, or does Jordan need enough space for a wheelbarrow of apples? Does the bridge need to be able to bear the weight of an ox carrying a cartload of apples? There are so many ways for Jordan to impact the world according to the problem and how Jordan chooses to define it. Regardless, whatever type of bridge Jordan chooses to build, Taylor’s world will be impacted by Jordan’s sojourns for apples, even though Taylor never saw a problem to begin with.
Two more caveats to the definition of mathematics: The “and/or” part of the “describe and/or impact our world” is important, given that as the discipline of mathematics has progressed, there is a large portion dedicated to theoretical mathematics. Sometimes, we are just looking to describe phenomena so that we can understand them better—we may or may not even be able to impact what we are attempting to describe. Lastly, when I say “our world,” I don’t necessarily mean just the Earth. Rather “our world” is meant to capture the idea of cognitive perception: “our world” encompasses the physical and the ideas we can comprehend. I used the phrase “our world” because “the universe and everything that may or may not be in it, depending on what you believe the limits of human comprehension to be” was clunky.
The Danger in Confusing “Mathematics” with “Computation”
When people say, “I am not a math person,” what they frequently mean is, “I am not quick with computation.” Adding numbers like 4,618 + 47, 342 in their head might give them a headache, or they might struggle to remember whether 3% is written as 0.3 or 0.03. In fact, however, the skill of computation is but one of multiple skills that make up mathematics according to the National Council of Teachers and Mathematics (NCTM; n.d.), and it is categorized under their Number & Operations Content Standard. Even then, it is only one-third of what NCTM says makes up that standard, including that people should be able to:
- Understand numbers, ways of representing numbers, relationships among numbers, and number systems
- Understand meanings of operations and how they relate to one another
- Compute fluently and make reasonable estimates
In other words, if someone isn’t very good at computation, according to the NCTM, they really only struggle with ⅓ of ⅕ of the overall mathematics discipline. If we do the math, that’s only 1/15 of the overall curriculum.
See how much nicer that feels?
The other content standards listed by NCTM include: algebra; geometry; measurement and data analysis, and probability. However, these are just the things that students should know. There are also five Process Standards, which explain what students must be able to do with what they know. The Process Standards include:
- Problem Solving;
- Reasoning and Proof;
- Communication;
- Connections; and
- Representations.
For many of us, the mathematics classes that we took in grade school probably focused heavily on what NCTM now calls the Content Standards. When learning long division, the teacher “taught” the content by writing problems on the board as we watched. If you were lucky, the teacher taught you a helpful mnemonic to say to yourself as you plodded through the steps. Remember “Does McDonalds Serve Cheese Burgers Raw?” for “Divide, Multiply, Subtract, Check, Bring Down, Repeat/Remainder?” A few students intuitively understood what we were doing and the point of each step; the majority memorized it but had no true understanding of why we used those particular steps. As math courses became more and more difficult, laying new understandings over our unsteady previous understandings, many of us became more and more lost, declared in frustration, “I’m just not a math person,” and went off to find subject areas that intuitively made sense to us.
This is one of the reasons for the new Process Standards from NCTM (n.d.). It is often important for students to occasionally do what we like to call “contextless computation,” which is memorizing key facts so that students learn fluency for solving more difficult problems down the road. Math facts and contextless computation should be thought of as basketball players practicing free throws; there’s a reason that coaches both have players practice skills individually and then also put them to use in a real game. A more academic metaphor would be teaching students how to sound out words using phonics, but never actually having them read a story or book.
What Type of Curriculum is Mathematics?
The content portion of mathematics is generally considered a “building block” curriculum. If a student is absent on the day that we teach addition with two-digit numbers, they will undoubtedly struggle during a lesson in addition with regrouping. This makes it crucial that we teach math using appropriate pedagogies; students who either miss or do not truly understand a single concept may find themselves seriously hindered down the line.
That being said, the NCTM process standards are a spiraling curriculum; students continually use the same skills (communications, connections, justifications, etc.) with increasingly more difficult content. This means that students need opportunities to engage in the process standards with each new content topic.
What are Essential Math Signature Pedagogies?
As with all articles in this series, it’s important to remember that these pedagogies, while based on extant research, are not an exhaustive list of everything that can and should happen in a classroom. We use the term “essential” because these pedagogical methods of instruction are foundational to the definition and purpose of mathematics as a discipline.
As a reminder, there are three critical elements of any signature pedagogy, how to think, how to perform, and how to act with integrity in the discipline/profession. Each of the following Essential Signature Pedagogies helps students develop one or more of these critical elements.
Math Essential Signature Pedagogy #1: The Concrete-Representational-Abstract Method
Because math has its foundations in describing and impacting our world, it makes sense that the majority of mathematical concepts should start with concrete materials that students can manipulate to build and show understanding. This also happens to align well with what we know about learning mathematical concepts, especially building number sense. The numeral “2” has no meaning unless students can visualize two objects in their head, which they do by first touching and seeing two objects in real life. Think about how babies and toddlers instinctively want to touch everything: we often learn basic and new concepts through the sense of touch. Therefore, one of the essential signature pedagogies in math is starting students with concrete experiences with topics, then having them represent these experiences, and only then moving to more abstract things like algorithms. (You can read more about this method and the research behind it here).
To the greatest extent possible, teachers should always introduce new concepts with manipulatives, such as unifix cubes, base-10 blocks, fraction circles, algeblocks, etc.. Students should be allowed to play with and experience these manipulatives in order to build better conceptual understanding.
Once students have sufficient concrete experiences, they’ll be ready to represent those understandings, typically with drawings. For instance, first a teacher might have students count these base-10 blocks:
Once the student discovers that there are 15, the teacher might then have the students draw what they see using paper and pencil, like so:
Lastly, a teacher could then have students write the number sentence expressed by these Base-10 blocks:
While using the CRA method is crucial in the younger grade levels to build number sense, this method absolutely can and should be done, to the greatest extent possible, in upper levels of K-12 math as well. In fact, Harvard University believes you can even use manipulatives at the collegiate level to teach calculus. One mathematician has even used knitting patterns to help explain knot theory, which seeks to understand the very nature of three and four dimensional space. So yes, your algebra teachers can and should frequently use the CRA method.
Math Essential Signature Pedagogy #2: Identifying and Solving Real-World Problems with Multiple Correct Answers (AKA, “Big, Messy Math Situations”)
Too often, we see completely context-less math problems like: ⅓ ÷ ¼ = ? Despite the fact that I know the answer is 1 ⅓ (by flipping the divisor), I have to admit that I could not think of a reason I would ever have to divide by a fraction in real life, which means that if I ever did need to divide by a fraction, chances are that I would not even realize it. For this reason, it is important that we not only have students solve problems in context, but we give them practice identifying when a problem needs to be solved. This step is almost always skipped in any math lesson or assessment.
Consider the following scenario: You have started a business selling coffee mugs. It takes ½ cup of paint to paint each coffee mug, which is 1/32 of a gallon. You have ⅝ of a gallon of paint left. How many mugs can you effectively paint, and therefore advertise on your website to sell?
Now, we have a real-life context. Not only that, we can use actual real-life materials (see signature pedagogy #1) to help us determine our answer, as well as to understanding the math behind it (and why we flip the divisor when multiplying by a fraction).
The problem becomes an even richer mathematical task when we allow multiple correct answers. As it stands, this problem has one correct answer. (It’s 20, because ⅝ ÷ 1/32 = ⅝ (32/1) = 160/8 = 20).
Instead, we can add the fact that each gallon of paint costs $15.00. Then we ask: How much should you charge per mug to turn a profit, knowing that the more expensive your mug, the fewer you might sell?
Now this problem has real meat to it. There’s no “correct” answer, and each answer will require justification. These are actual skills people use in real life, and that students could even use to start their own business. We need to normalize having more big, messy math “situations” where students identify the problem, establish their own parameters for success, then propose and justify their solution.
And if you worry that this is going to take “too much time:” if you’re using the CRA method, students are likely going to understand the concepts at a much higher level, meaning that it simply won’t take as much repetition to drill the abstract concepts and algorithms into them. Moreover, teachers don’t need to be using these big, messy math situations every day; even once a week or every other week can be enough. The time evens out in the end.
Math Essential Signature Pedagogy #3: Students Talk (and Write) About Math
If math is a language, then students must not only be reading it, but speaking and writing it–just like with any other language. Too often we see lessons where the teacher stands at the front and lectures students with sample problems. This is akin to a Spanish class where the teacher speaks Spanish at you while students furiously copy whatever is written on the board. The best way to learn a language is to read it, speak it, and write it—in new and varied contexts.
One of my favorite things to do as a math teacher was to ask students if they think their answer to a problem is correct. In the beginning, students would try to study my face to see if I thought their answer was correct. Then, they had to turn and talk to each other, explain their reasoning, and even write it down and not just in numerals and symbols—often in actual words. “If you can’t explain it, you don’t really understand it” became our refrain. If you don’t see students speaking, writing, conjecturing, and justifying in the math class, there’s no way for us to understand if they actually understand or have just been trained to perform some neat math tricks.
Math Essential Signature Pedagogy #4: Differentiation, Differentiation, Differentiation!
Yes, we should see differentiation in every classroom. But because math concepts use the building blocks curriculum approach, it is crucial to ensure that students have access to remediation or challenge activities. The differentiation could look like small groups or it could be the open-ended tasks that allow students to work at a variety of levels.
For instance, I usually taught in math groups for anything computation related, because some groups needed to work with manipulatives longer than others. That being said, I loved to teach geometry using art projects. I would set basic minimum requirements for students, such as include X number of right angles but also challenge goals. Students almost invariably worked to their level of understanding.
Math Not-Essential-But-So-Good Signature Pedagogy #5: Using Art to Teach Math
Linking arts, such as visual arts, music, theatre, movement, etc. and mathematics instruction simply has so many wonderful benefits for students. Firstly, it incorporates multiple other signature pedagogies. It uses the CRA method as students manipulate objects, such as when sculpting, or visually represent objects, such as in a painting. It is automatically open-ended and a bit messy; for instance, there was no “right” way for M.C. Escher to create so many of his wonderful pieces of art. Having students discuss art they’ve created or art they view and the mathematical concepts used to create that art is a fabulous way to get them communicating. Lastly, because of the open-ended nature of art, it allows for so much differentiation in how students complete the task.
Not only all that, however, but using visual arts, music, theatre, dance, poetry, etc. in the mathematics classroom will help engage more areas of the brain as students learn. This helps them learn at a deeper level—and be better able to retain the information.
In Summary
As a math educator, or observer of one, do we expect to see these signature pedagogies in every math class every day? Maybe not, but we should see them more often than we don’t. Mathematics is such an integral part of everyday life, and students deserve to have a strong mathematics literacy that can help them be successful, not just in K-12, but beyond. We ensure this understanding by remembering that mathematics is a logic-based language used to describe and impact the world, extending far beyond mere computation. Effective math education should focus on teaching it as a language, solving real-world problems, and using essential pedagogies like concrete-representational-abstract methods and differentiation. Integrating arts into math can enhance understanding and engagement by making abstract concepts more tangible.
About the Author
Kate Wolfe Maxlow is the Chief Creative Officer at eObservations and DCD Consulting. She has worked as: an elementary school teacher; an instructional coach; a Director of Innovation and Professional Learning; and a Director of Curriculum, Instruction, and Assessment. She can be reached at kate@eobservations.com, kate.maxlow@gmail.com, or at https://bit.ly/kmaxlow.