Mathematical Rigor and High-Quality Instructional Materials

When educators talk about rigor in math, they don’t always mean the same thing. Sometimes they’re referring to the difficulty of the problems, the speed of the pacing, or the depth of understanding. They might even be talking about mathematical justifications being precise. This uncertainty makes sense! Rigor is not easily defined, and when it comes to math, making sense of “rigor” presents a special challenge. In this article, we take a look at what rigor means, both in math education and within the field of mathematics.

What is mathematical rigor in education?

In general, rigor is a term meaning the quality is thorough and accurate. Practically anything can be rigorous—a drawing, a training regimen, a medical diagnosis, or learning standards, to name a few examples—so long as it is exhaustive and precise. In education, mathematical rigor refers to mathematical instruction being thorough and accurate. This applies to both the curricular materials themselves and also how they’re used.

In 2010, the Common Core State Standards Initiative aimed to develop a consistent, nationwide set of standards for math and English language arts. Although not all states use these standards today, their development, whose authors and reviewers included an extensive body of mathematicians, education researchers, teachers, administrators, and other stakeholders, marked a landmark achievement in standards writing.

One of the Initiative’s goals was to clarify the definition of rigor in mathematics instruction. When educators talk about mathematical rigor today, often they are referring to a key shift in math education that pertains to a student’s “deep, authentic command of mathematical concepts.”

Aspects of rigor in math teaching

NWEA’s Mary Resanovich summarizes the Initiative’s findings in an article on prioritizing rigor in education. Rigor was broken down into three interwoven aspects: conceptual understanding, procedural fluency, and application:

• Conceptual understanding refers to a deep and integrated grasp of mathematical ideas.
• Procedural fluency refers to the ability to apply mathematical procedures efficiently, as well as to recognize when to use one procedure over another.
• Application refers to using math concepts and procedures to solve real-world or novel problems.

A rigorous approach to K–12 math teaching considers all three aspects. In Visible Learning for Mathematics, Grades K–12, education researchers John Hattie et al. study what aspects of math education visibly work best. The three aspects of rigor should be given “equal intensity,” they note, and together create “a balance of methods that makes for high-impact instruction” (p. 3). Here is what it might look, for example, to teach adding fractions rigorously:

• Students manipulate physical models, such as plastic fraction bars, alongside visual models, such as drawn area models, to understand adding two fractions conceptually.
• Students practice completing procedures for adding fractions, such as converting denominators to the least common denominator, and can explain why they work.
• Students solve real-world problems that require applying the math skill of fraction addition in order to find a solution.

The meaning of rigor within mathematics

Within the field of mathematics, “rigor” also means something special. Rigor in math (not math education) refers to ensuring that every step of an argument is justified and logical with no details overlooked. In this way, mathematical rigor in education isn’t necessarily the same thing as mathematical rigor in general, although the two ideas are related.

This meaning of rigor is stronger than in other sciences, which will always have built-in experimental error. Math is perhaps the most demanding of rigor of all, as experiments turn to pure thought experiments, and theorems become incontestable truths. The famed German mathematician Gauss is said to have called math the “queen of the sciences.” Here, for example, is a rigorous proof that the sum of any two even numbers is even:

Statement: The sum of any two even numbers is even.

Proof:

1. By definition, an even number is 2 times an integer.
2. Let two even numbers be 2a and 2b, where a and b are integers.
3. Consider the sum 2a + 2b. This can be factored into 2(a + b).
4. Because a and b are both integers, a + b is also an integer. That means the number 2(a + b) is 2 times an integer and therefore even.

A proof like the one above is roughly what rigor looks like to a mathematician. Every step is logical and defensible. There is no metaphor, no hyperbole, just economic and precise language, including numbers and symbols as needed.

Students won’t need to worry about this kind of mathematical rigor until at least classes like algebra or geometry, where problems get increasingly complex. (At times, literally.) But students of all ages can learn with rigor, like how a mathematician would think through assembling a proof in the first place. This sort of rigor can get downright messy as ideas are explored and the math problem solver calls on different mathematical models and procedures.

Common misconceptions about mathematical rigor

In a 1902 article in Bulletin of the American Mathematical Society, German mathematician David Hilbert writes, “It is an error to believe that rigor . . . is the enemy of simplicity.” Hilbert was discussing solving math problems fully and correctly, saying that it’s easy to confuse rigor with difficulty. Seemingly simple mathematical problems can be hard to solve rigorously, and crowding a solution with extra steps and extraneous details is not the same as adding rigor.

“On the contrary,” writes Hilbert, “the very effort for rigor forces us to find out simpler methods.” As math students work through problems, they learn to tame formidable setups. One well-crafted problem can lead to an in-depth practice of skills, concepts, and applications. Below are a few misconceptions that educators may have regarding what it means to teach math rigorously.

Misconception 1: Rigor is about doing more problems.

In a 2024 ASCD article, Nancy Frey and Douglas Fisher write that “assigning 10 more math problems does not automatically increase rigor”; that is, a task does not simply increase in rigor with each problem. However, a task can become more rigorous if the new problems use different models, employ different strategies, or present novel contexts.

Misconception 2: Rigor is about making the problems harder.

Frey and Fisher additionally assert that “rigor does not mean making things hard.” Students spending more effort and time on problems isn’t enough. They should also be learning background knowledge and thinking through different strategies, with difficulty being balanced with complexity.

Misconception 3: Rigor is about introducing topics earlier.

Rigor is not about “introducing concepts and skills at an earlier age,” says Mary Resanovich. Rather, it is about introducing concepts and skills more deeply once the student is ready. Math knowledge is built across grades and concepts, with students learning ideas rigorously along the way.

Math rigor vs. productive struggle

The National Council of Teacher of Mathematics, or NCTM, published the Second Handbook of Research on Mathematics Teaching and Learning in 2007 and commented on how to optimize math learning outcomes: “Research findings suggest that for students to obtain desirable learning outcomes it is important that they are engaged in activities where they have to ‘struggle’ (in a productive sense of that word) with important mathematics. . .” (p. 1304). This is widely considered the first time that “productive struggle” had been named as such, referring to the educational value that comes from feeling stuck. Struggling to figure out a strategy or solution is a necessary condition for successful math learning.

Since the 2007 handbook, productive struggle has emerged as an influential term for thinking about math learning and continues to be studied. In 2024, Jamaal Rashad Young et al. performed a scoping review of the topic. Productive struggle is “essential for deepening understanding, fostering problem-solving skills, and promoting long-term learning,” they write, explaining how when students struggle productively, they grapple with the nuances that come with different math problems and develop the perseverance to master individual concepts.

It can be hard to distinguish between students struggling productively and unproductively. Teachers can look for clues in the way that students confront challenge to help spot productive struggle:

• Students attribute their struggle to outside factors (“this problem is complex”) and not internal ones (“I’m not good at math”).
• Students are able to reflect on their thought processes and evaluate their problem-solving approaches.
• Students persist through challenges and try new strategies instead of giving up.

Rigor in mathematics instruction

The principles of rigorous math instruction apply to all grades, but teaching a 5-year-old is of course very different from teaching teenagers or adults, both in terms of what to focus on and how to teach it. The tables below provide some examples of what rigorous teaching can look like as students advance in their math instruction.

Rigor in elementary school

When students are beginning to learn math in kindergarten and throughout their first years of schooling, rigorous teaching generally avoids timed activities and instead helps students understand math facts conceptually. Automaticity and accuracy are developed by practicing different strategies and applying them to different problems.

Rigor in middle school

At the middle school level, proportional reasoning is key. It’s an important bridge between the concepts of numbers and functions that students generally start to learn deeply around Grade 6. Rigorous instruction has students understanding the “why,” mastering the “how,” and knowing then “when/where” of rates, ratios, and proportions.

Rigor in high school and beyond

By the time students get to high school, mathematical justifications should start to convey expertise as captured in the Standards for Mathematical Practice within the Common Core State Standards Initiative. Models should be chosen strategically and constructed carefully. Complex problems should be broken down, and solutions should have sound reasoning.

Evaluating high-quality instructional materials for mathematical rigor

One way to evaluate mathematical curricular materials is to examine the extent to which they address all three dimensions of rigor: conceptual understanding, procedural fluency, and application. These three aspects sometimes look like integrated, multifaceted problems that apply to real-world situations, and they can also be separated to some extent, such as having students play games that focus on building procedural fluency or having students spend time with a digital simulation to see examples of application.

Reviewing instruction for rigor also involves looking beyond the student-facing materials and paying attention to how those materials are used. It is the teacher’s application of them that ultimately drives rigorous thinking. Look for resources that equip teachers to facilitate math talk and have students compare strategies and explain their thinking.

Mathematical rigor examples and strategies

All said, there are many ways that math instruction can promote mathematical rigor. Some high-level examples and strategies are given below, all of which have the potential to build conceptual understanding, strengthen procedural fluency, and show students where math can be applied.

• Use multiple representations: Use a wide range of representations when explaining a math concept so that students can compare and contrast representations and justify one over another. This helps connect ideas across lessons, deepening understanding.
• Compare student strategies: Students will invariably see and solve problems differently. Call attention to different strategies that students use, encouraging them to try multiple strategies on their own.
• Facilitate student discourse: The math discourse that students use to express mathematical thinking, also known as math talk, helps to not only elucidate the math but also improve social skills and deepen classroom engagement, ultimately leading to better student sense-making.
• Seek open-ended problems: Word problems show how math can apply to the real world, but many students find them difficult and get caught up in language that can be ambiguous or hard to parse. Look for application problems that are open-ended and minimize language barriers so that students don’t just see word problems as convoluted setups to specific procedures, such as Open Middle math problems.
• Teach math vocabulary: Focusing on vocabulary is “bumping up the rigor for our students,” according to Lisa Nelson, the learning director for Liberty Elementary School District in Tulare, California. Using the right vocabulary terms helps students describe concepts with precision and connect ideas in math class to words and ideas they know outside of it.
• Have a low floor and high ceiling: In other words, teach concepts accessibly to all students (“low floor”) while still having space for exploration and no single right answers (“high ceiling”). The high ceiling is especially important towards making a task rigorous, as it enables a task to tap into a wide range of skills and applications. Our article on low floor, high ceiling math tasks describes this idea in detail and offers ways to transform math tasks to be more rigorous in this way.

In the end, there is of course no one right way to teach math. It will depend on the teacher, the students, and the topics being taught. No matter the approach, however, it should be done with rigor if the learning is to truly stick. Students need to learn procedures for doing math alongside the concepts that explain and use them. They also need to see how math applies to the world around them and how the strategies used to solve one problem can be applied to another.

For instructional materials to be high-quality, they must be rigorous, both in how the students do math and in how the teachers teach it. The result? Students not only solve math problems, they become creative problem solvers and lifelong mathematical thinkers.

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