- Several popular ideas in math education are unsupported by high-quality evidence, used at the wrong stage of learning, or harmful to student learning.
- Among these are five common instructional practices: teaching multiple strategies for the same operation before students have mastered one reliable method, allowing technology to replace foundational knowledge and procedural skills, delaying the teaching of procedures until conceptual understanding is secure, teaching primarily through student-led problem-solving, and avoiding repetitive practice.
- Effective math teaching supports new learners through clear, explicit instruction.
Introduction
Math education is plagued with superficially appealing phrases and ideas that aren’t aligned with how students learn. Many students find math difficult or think they are incapable of doing it. Ideas marketed as quick fixes or ways to avoid the hard work necessary for learning math are therefore often attractive to educators and students.
Despite their appeal, many of the most popular ideas in math education are unsupported by high-quality evidence, are applied at the wrong stage of learning, or work against learning.
Here are five popular ideas in math education that should be reconsidered and abandoned.
Misconception 1: New Learners Should Learn Multiple Strategies
When learning basic number operations, children are often taught multiple strategies to solve the same problem or are encouraged to invent their own strategies. For example, they may be shown how to compute 26 × 18 using an area model, concrete materials, the distributive law written horizontally, and the standard vertical algorithm. Often, little emphasis is placed on the most efficient, generalizable, or useful method for future mathematics. All strategies are treated as equal.
This approach is commonly embedded in math curricula, standards, and approved instructional resources. Proponents argue that teaching multiple strategies promotes flexible thinking and conceptual understanding. While comparing different methods and determining which is most appropriate in each context can promote flexibility, this should come only after students have mastered a reliable method. There are several problems with teaching multiple strategies to novice learners.
First, new learners can quickly become overwhelmed and confused when presented with multiple methods at once. Second, students learning foundational skills do not have the expertise to recognize the merits of one method over another or determine which methods generalize best to more complex cases. Exposure to so many approaches also reduces the amount of targeted practice students receive, so they fail to develop fluency with any single method. They become masters of none.
The emphasis on multiple strategies also leads to curriculum bloat. Teachers do not have enough time to teach several approaches for every operation while ensuring sufficient practice. This may be one reason many students are still struggling with basic arithmetic at the end of primary school, when they should be moving on to more advanced topics like fractions or algebra.
Some strategies are also not well suited to computation. Base 10 blocks—physical blocks used to represent ones, tens, hundreds, and thousands—and diagrams can help illustrate place value and explain how the standard algorithms work, but they are time-consuming and impractical for computation. Students need to move beyond these concrete representations and become fluent in symbolic mathematics, which algebra and later math depend on.
Likewise, some strategies that work reasonably well for small numbers become cumbersome when calculations are more complex. For example, computing 26 × 18 as 200 + 160 + 60 + 48 = 468 is less efficient than the standard vertical algorithm and does not generalize well to larger numbers.
Exploring different methods and mental math strategies can be valuable once students have developed accuracy and fluency with a reliable method. For novice learners acquiring foundational skills, however, instruction should focus primarily on efficient, generalizable procedures, such as the standard algorithms, and include sufficient practice for students to master them.
As education researcher Marcy Stein and coauthors write in their book Direct Instruction Mathematics, “Some commercially developed mathematics programs suggest that students generate a number of alternative strategies for the same problem. Rather than developing a conceptual foundation that highlights mathematical relationships [this] confuses instructionally naïve students. Teachers should select the most generalizable, useful and explicit strategies to teach their students.” Here, “instructionally naïve students” refers to novice learners who are still learning the material.
Misconception 2: Students Don’t Need Facts and Procedures Anymore Because Technology Can Do the Work
In math education, the thinking commonly goes like this: Since technology can perform routine calculations, school time should focus less on memorizing facts and procedures than on developing problem-solving and critical-thinking skills. Proponents argue that if basic facts or knowledge are needed to solve a problem, students can lean on technology such as Google Search and AI to fill in the gaps.
To see the problem with this, pick up a textbook on an academic subject that you know nothing about (preferably in a cumulative subject like mathematics or chemistry). For example, if you aren’t familiar with calculus, engage with a calculus book. Turn to a page in the middle of the book and try to understand it. Look up every mathematical symbol you don’t understand, google terms you’ve never heard before, and attempt the problems at the end of the unit. You will likely become overwhelmed and lost, because learning new math concepts depends on layers of prior knowledge that we have practiced and internalized.
The reason the strategy of using technology does not work is that working memory is limited in duration and capacity. When students must constantly look up facts, definitions, or procedures, they are expending their limited mental resources on searching rather than concentrating on absorbing new concepts. Learning mathematics often requires coordinating several pieces of knowledge simultaneously. We need to be able to connect facts, knowledge, and procedures that are readily available in our long-term memory to solve new problems. Without that foundation, learning new math concepts is likely to be frustrating and ineffective.
The other issue is that students who do not know how to carry out a procedure or calculate something themselves are often unaware when technology has provided an incorrect answer, whether due to faulty AI reasoning, data-entry error, or anything else. An expert, on the other hand, has the intuition and experience to tell when something is off with an implausible answer produced by technology.
For a more straightforward example, consider the quadratic equation x2 − 2x – 48 = 0. To factor the quadratic, a student normally searches for two integers whose product is −48 and whose sum is −2. A student who can automatically retrieve the factors of 48 can easily see that 6 and −8 are good candidates. But if basic multiplication facts are not automatic, this becomes a much more difficult problem, and turning to technology or guessing can quickly overwhelm working memory and slow the student down. Students encounter similar issues when searching for common denominators to add fractions. In other words, children are placed at a substantial disadvantage when elementary schools do not ensure that basic math facts are automatic.
Misconception 3: Conceptual Understanding Must Precede Procedural Skill
Nothing in math education is more sought-after or more often used to justify preferred teaching methods than conceptual understanding. Teachers are often given the impression that conceptual understanding must be secure before procedures are introduced. They may even be told that students who achieve procedural fluency without understanding have not truly learned and that procedural fluency is superficial and short-lived. Procedures are portrayed as shallow, while understanding is deep—the holy grail of mathematics education.
One major problem is that the phrase “conceptual understanding” lacks a clear definition. Different people hold different notions of what it is. It is frequently conflated with inquiry-based learning, multiple strategies, use of manipulatives, and “real-world problems.”
Conceptual understanding is also difficult to measure. How can we reliably measure something that lacks a clear definition? As Greg Ashman has noted, this lack of clarity can make it difficult to determine exactly what is being measured in studies that claim to have assessed conceptual understanding. In many cases, the measure used may have actually captured procedural fluency.
This is not intended to be an argument against understanding. Mathematics is not a collection of unconnected arbitrary rules. Every procedure and algorithm has meaning behind it, and every mathematical fact rests on definitions or theorems that can be understood. The standard algorithms, for example, rely on properties of our place-value system, which should be explained when teaching the algorithms to students. The issue is not whether understanding matters. The problem is with the term “conceptual understanding” as it is used in math education: It lacks a clear definition, is difficult to measure, and has been used to de-prioritize procedural skill in ways that evidence does not support.
Definition and measurement problems aside, psychologist Bethany Rittle-Johnson and coauthors found in a 2001 study that conceptual and procedural knowledge influence one another and that gains in one type of knowledge can lead to gains in the other. The authors defined conceptual knowledge as “implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain.” They defined procedural knowledge as “the ability to execute action sequences to solve problems.”
This bidirectional finding, however, doesn’t settle the question of where to start. Since conceptual understanding lacks a clear, measurable definition while procedural fluency is straightforward to measure, procedural skill is often the more practical starting point, particularly when the concept is difficult to understand. Procedural fluency often leads to deeper understanding, because once students become fluent with a procedure, it becomes easier for them to recognize patterns and relationships.
This may partly explain why some adult educators perceive conceptual understanding as more accessible to children than it actually is. Adults possess fluent procedural knowledge built up over years. When someone explains why one-third divided by two equals one-sixth, the explanation feels clear and accessible because they already know the answer. They suffer from the “curse of knowledge”: They have forgotten, or are unaware of, what it feels like to encounter a new conceptual explanation without existing procedural knowledge. But children do not have that procedural foundation yet, so what feels like an aha moment to an experienced adult is often incomprehensible to a child seeing the concept for the first time.
Currently, we seem to have the worst of both worlds: Many students neither understand math procedures nor possess procedural fluency. The available evidence suggests that educators should prioritize developing procedural skills while highlighting key properties of operations, encouraging students to recognize and explore patterns and meaning along the way.
Misconception 4: Problem-Solving Is the Best Way to Teach Mathematics
One of the most influential ideas in math education is that mathematics is best learned through problem-solving. In practice, this often means that students are presented with complex tasks they may not yet have the skills to solve. The assumption is that students will engage with novel problems, often in groups, and come up with their own ideas to solve them. Any foundational skills they were missing will develop in the process, and if not, students can fill in the gaps using technology.
Expert problem solvers are not experts simply because they can reason their way through any complex task. Experts possess mountains of domain-specific knowledge, such as facts, definitions, procedures, and learned problem-solving techniques in long-term memory. When they see a new problem, they call on that knowledge to help them recognize patterns, decide what problem-solving technique to apply, and efficiently solve the problem. Novice learners, who tend to rely on inefficient trial-and-error approaches, are unable to think the same way.
A useful framework is the instructional hierarchy, which describes the stages learners move through when acquiring new knowledge.7 New learners need explicit instruction, modeling, scaffolding, worked examples, and guidance to build accuracy. This is called the acquisition stage. Then they need substantial practice and feedback so the skill becomes accurate and effortless. This is the fluency stage. Complex problem-solving becomes productive only after these earlier stages are solidly in place. When students are asked to engage with problems they don’t have the skills to solve, they become frustrated and confused.
Being able to solve complex, nonroutine math problems is important, but the sequencing matters. Problem-solving does create better problem solvers, but only once students have the foundational skills and fluency to productively engage with the problems in the first place.
Misconception 5: Practice Turns Kids off Math
It’s well accepted that the way you get good at sports or music is through practice. Practice and drills are encouraged, nurtured, and celebrated in those disciplines. Yet in math, practice has increasingly been de-emphasized. There are few more counterproductive messages to send to students and teachers. Students will be more engaged and motivated to do more math when they are good at it, and this comes through well-designed instruction and substantial practice.
Math is extremely hierarchical, meaning that skills build on one another. Solving algebraic equations requires fluency with fractions, which relies on fluency with basic number facts. When children are not given sufficient practice to master concepts, gaps persist and compound, and students can quickly lose confidence in their ability to do math. When teachers incorporate activities that appear engaging but do not build competence or nurture the hard work required to master math concepts, they deprive children of the satisfaction gained by genuine mathematical accomplishment. As with sports or music, repeated exposure and practice in math are imperative to mastering topics and progressing to higher levels.
Teaching Math in Ways That Work
We need to move beyond ideas that simply sound appealing and toward instructional approaches that are supported by evidence. Effective math teaching supports new learners through clear, explicit instruction. In a 2017 article, education researcher Charles A. Hughes and coauthors identified five essential components of explicit instruction based on the research literature: segmenting complex skills, drawing students’ attention to important features of the content through teacher modeling and “think-alouds,” using systematically faded prompts and supports, providing opportunities for students to respond and receive feedback, and creating purposeful practice opportunities for students to build mastery. Barak Rosenshine’s 10 principles of instruction provide an accessible summary of many of these evidence-based principles of explicit instruction and are an ideal starting point for educators seeking to improve their teaching.
Math teachers should also prioritize efficient and generalizable strategies rather than overwhelming new learners with multiple approaches; recognize that mathematical understanding often develops alongside, or after, procedural fluency; and ensure students develop a strong foundation with basic mathematical procedures and math facts to set them up for later success.
Providing children with a strong math education opens doors for future opportunities and allows students to see mathematics as a worthwhile, achievable subject. We owe it to children and their families to get math instruction right. That means moving away from harmful misconceptions and letting the best available evidence guide instructional decisions.
About the Author
Anna Stokke is professor of mathematics at the University of Winnipeg and adjunct professor in the School of Education at La Trobe University. She hosts the Chalk & Talk podcast.
Notes
- Marcy Stein et al., Direct Instruction Mathematics, 5th ed. (Pearson, 2018), 3.
- For additional reading on multiple strategies in mathematics, see Amanda VanDerHeyden, “Why Multiple Strategies Is Not a Fluency Building Technique,” SpringMath Accelerate, https://springmath.org/202602-multiple-strategies.
- For additional reading on techniques for building fluency with math facts, see Anna Stokke, Chalk & Talk, podcast, “How to Build Automaticity with Math Facts: A Practical Guide,” PodBean, November 8, 2024, https://chalkandtalkpodcast.podbean.com/e/how-to-build-automaticity-with-math-facts-a-practical-guide/.
- Greg Ashman, “Conceptual Understanding Is a Myth,” Filling the Pail, May 20, 2026, https://fillingthepail.substack.com/p/conceptual-understanding-is-a-myth.
- Bethany Rittle-Johnson et al., “Developing Conceptual Understanding and Procedural Skill in Mathematics: An Iterative Process,” Journal of Educational Psychology 93, no. 2 (2001): 346–47, https://psycnet.apa.org/doi/10.1037/0022-0663.93.2.346.
- For additional reading on the importance of memorizing math facts, see National Mathematics Advisory Panel, Foundations for Success: The Final Report of the National Mathematics Advisory Panel, US Department of Education, March 13, 2008, https://eric.ed.gov/?id=ED500486.
- Norris G. Haring and Marie D. Eaton, “Systematic Instructional Procedures: An Instructional Hierarchy,” in The Fourth R: Research in the Classroom, ed. Norris G. Harding et al. (C. E. Merrill, 1978).
- For additional reading on the role of working memory in problem-solving, see John Sweller, “Cognitive Load During Problem Solving: Effects on Learning,” Cognitive Science 12, no. 2 (1988): 257–85, https://doi.org/10.1207/s15516709cog1202_4.
- For additional reading on fully guided instruction and problem-solving, see Richard E. Clark et al., “Putting Students on the Path to Learning: The Case for Fully Guided Instruction,” American Educator 36, no. 1 (2012): 6–11, https://www.aft.org/ae/spring2012/clark_kirschner_sweller.
- Charles A. Hughes et al., “Explicit Instruction: Historical and Contemporary Contexts,” Learning Disabilities Research & Practice 32, no. 3 (2017): 140–48, https://doi.org/10.1111/ldrp.12142.
- Barak Rosenshine, “Principles of Instruction: Research-Based Strategies That All Teachers Should Know,” American Educator 36, no. 1 (2012): 12–19, 39, https://www.aft.org/ae/spring2012/rosenshine.