Ep 77. Five math myths that are holding students back
This transcript was created with speech-to-text software. It was reviewed before posting but may contain errors. Credit to Canadian Podcasting Productions.
In this solo episode, Anna Stokke explores five common misconceptions in math education that may sound convincing but aren't supported by high-quality evidence. Based on a chapter she wrote for an essay collection by the American Enterprise Institute and researchED, Anna explains how these ideas can hinder student learning and create lasting gaps in mathematical understanding.
She examines misconceptions such as teaching multiple strategies to new learners, relying on technology instead of facts and procedures, and the belief that conceptual understanding must always come before procedural skill. Throughout the episode, Anna shares what the research actually says and highlights evidence-based approaches that better support student success in mathematics.
This episode is available in video at www.youtube.com/@chalktalk-stokke
The article version of this episode, written by Anna Stokke, ….
TIMESTAMPS
[00:00:00] Introduction
[00:01:21] Preview of what to expect in episode
[00:02:16] Myth #1: New learners should learn multiple math strategies
[00:05:41] What new learners should do instead
[00:06:04] Myth #2: Technology makes facts and procedures unnecessary
[00:09:31] Myth #3: Conceptual understanding must come before procedural skill
[00:13:50] Myth #4: Problem solving is the best way to teach mathematics
[00:15:09] Why Rosenshine’s instructional hierarchy is always relevant
[00:16:29] Myth #5: Practice turns kids off math
[00:18:41] What the evidence says about effective math instruction
[00:20:06] Final thoughts: takeaways for teachers and parents
[00:00:00] Anna Stokke: Welcome to Chalk & Talk, a podcast about education and math. I'm Anna Stocki, a math professor and your host. Welcome back to another episode of Chalk & Talk.
No guest today, just me, and I'm going to talk about math misconceptions, a version of what I'm about to talk about appears as a chapter that I wrote for an essay collection from the American Education Institute and ResearchEd aimed at bringing research-backed teaching practices to educators. I wrote a chapter on math. If you'd rather read the written version, it's linked to in the show notes.
There are some widespread ideas in math education that maybe sound good when you hear them but can seriously hinder student learning. The reason this matters is that time spent on ineffective teaching strategies is lost learning time for many students, and this creates gaps in student math knowledge and holds them back later. I'm going to talk about five popular ideas in math education that are not supported by high-quality evidence.
[00:01:21] Here's a preview of what I'm going to talk about.
Misconception one, new learners should learn multiple math strategies. Misconception two, students don't need facts and procedures anymore because technology can do the work.
Misconception three, conceptual understanding must precede procedural skill. Misconception four, problem solving is the best way to teach mathematics. And misconception five, practice turns kids off math.
I'm going to go through each one of them and explain what's wrong with it, so stick around to hear more. Let's get started. Misconception number one, new learners should learn multiple math strategies.
[00:02:16] So, say a kid is learning to multiply two-digit numbers, something like 26 times 18. Instead of being taught one reliable method and given time to practice it, they're shown four or five, an area model, concrete materials like base 10 blocks, the distributive law written out horizontally, and the standard algorithm. All of these are presented as equally valid methods for calculation, and it's sometimes the case that little emphasis is placed on the most efficient method that would generalize to larger numbers.
The reasoning that's often given is that exposing kids to multiple strategies is supposed to build flexible thinking and conceptual understanding. So, here's the problem with that. Comparing different methods and determining which is best in a certain context can certainly promote flexible thinking, but that's only likely to be effective once students have mastered a reliable method.
You cannot productively compare four strategies if you have not mastered any of them.
[00:03:24] So, there are a few problems with teaching multiple strategies to novice learners. First, new learners get cognitively overloaded when they're given multiple methods all at once.
It's simply confusing. Second, novice learners don't have the background knowledge to evaluate which method is more efficient or which one's going to easily generalize to larger numbers. That kind of judgment requires expertise that they do not have.
Third, spreading instructional time across several methods translates to less time practicing any single one. They can end up not being fluent in any of them. They become masters of none, as I put it in the piece.
Fourth, this approach causes curriculum bloat. There simply isn't enough classroom time to teach several approaches for every operation and still leave room for the practice that kids need to become fluent. This may be part of the reason so many students are struggling with basic arithmetic by the time they should be moving on to fractions or algebra.
And lastly, some of these strategies just are not built for computation in the first place. Concrete materials like base 10 blocks or diagrams can be useful in beginning instruction for illustrating, say, place value, what it means, and why the standard algorithms work in the way they do. But they're slow and inefficient, and they don't scale well.
It is extremely important that kids leave the concrete representations behind and become fluent with symbols because that's what algebra, the gateway to higher level math, and everything after it depends on. Same story with expanded strategies for multiplication or addition. Say, writing 256 plus 752 is 200 plus 50 plus 6 plus 700 plus 50 plus 2 to calculate it.
This becomes extremely cumbersome as numbers get bigger. They do not generalize well, but the standard vertical algorithm does. It's important to get the sequencing right.
[00:05:41] Once students are accurate and fluent with a reliable method, exploring alternate methods and mental math strategies can be genuinely useful. But for novice learners still building foundational skills, instruction should center on one efficient, generalizable procedure with enough practice that it sticks.
[00:06:04] Misconception number two. Students don't need facts and procedures anymore because technology can do the work. This one's been around for a long time, but it's become increasingly relevant with AI in the picture. The logic goes like this.
Computers can calculate, so why should kids memorize facts or learn procedures? Shouldn't school time go toward problem solving and critical thinking instead, with technology filling in the gaps for facts and procedures? To see the problem with this, try this exercise.
Pick up a textbook in a subject you know nothing about, ideally in a cumulative subject like math or chemistry. If you've never learned calculus, pick up a calculus book. Now flip to a page in the middle of the book and try to work through it.
Look up every symbol, google every term that you don't recognize, then try to do the practice problems at the end of the unit. I think you will get overwhelmed quickly. That's because learning new material in math depends on layers of prior knowledge that we've practiced and internalized.
[00:07:21] Remember, our working memory, the mental workspace where we actually think, is limited in both duration and capacity. When a student has to constantly stop and search for facts, definitions, and procedures, they're using up that limited capacity on searching instead of focusing on understanding the new concept. Learning math often requires connecting several pieces of information in the mind at once and connecting them.
And that only works if most of those pieces are already sitting in long-term memory and easily retrievable. And there's a second problem with technology. Students who can't do a calculation or procedure themselves often can't tell when technology gets it wrong.
Whether it's a flawed AI response or a data entry error, if you don't have the underlying skill, you have no way to tell if the output is sensible. Experts have the intuition and experience to tell when something is off with an implausible answer produced by technology, but novices have no choice but to accept it. Now imagine a student who doesn't have their times tables memorized trying to factor a quadratic polynomial like x squared minus 2x minus 48.
To factor it, you're looking for two integers that multiply to negative 48 and add to negative 2. If the student knows the factors of 48 automatically, they can spot the candidates 6 and minus 8 almost instantly. But if basic multiplication facts aren't automatic, this becomes a slow, effortful search and working memory gets overwhelmed fast.
Same story for students learning to find common denominators to add fractions. Kids without automatic recall of basic facts are working at a serious disadvantage for learning later math.
[00:09:31] Misconception number 3. Conceptual understanding must precede procedural skill. If there's one phrase that gets used to justify almost any instructional choice in math education, it's conceptual understanding. Teachers are often told conceptual understanding must be secure before procedures can be introduced.
That fluency without understanding is somehow superficial or short-lived. Procedures are portrayed as shallow and conceptual understanding as deep, the holy grail of math education. So, here's a big problem.
Conceptual understanding does not have a clear definition. In practice, it gets used interchangeably with inquiry-based learning, using manipulatives, with real-world problems, multiple strategies, basically with anything that isn't direct procedural instruction. And if you can't define a thing clearly, then you can't measure it reliably either.
Greg Ashman has made this point well. That vagueness makes it genuinely hard to know what research studies claiming to measure conceptual understanding are actually measuring. In fact, what looks like a measure of conceptual understanding might often be capturing procedural fluency instead.
To be clear, I'm not trying to argue against teaching kids to understand things in math. Math is not just a bunch of arbitrary, disconnected rules. Every procedure has a reason behind it.
[00:11:10] The standard algorithms rest on properties of our place value system, and that should absolutely be explained to students when we teach them. The issue isn't whether understanding matters. It's that the specific term conceptual understanding as it's used in math education is poorly defined, hard to measure, and it has been used to deprioritize procedural skill in ways that are not supported by evidence.
Putting aside the issues with the definition for a moment, there's a well-known study by Rittle-Johnson and colleagues that found conceptual and procedural knowledge, according to their definitions, actually influence each other. Gains in one tend to produce gains in the other. So, it's just not supported by evidence that conceptual understanding must come before procedure.
Further to that, since conceptual understanding is hard to define and hard to measure, while procedural fluency is straightforward to assess, procedural skill may often be the more practical place to start, especially when a concept is genuinely difficult to understand. Keep in mind that fluency with a procedure often makes patterns and relationships easier to see. In other words, understanding frequently follows fluency, and there is nothing wrong with that.
And I think some educators get this backwards because of the curse of knowledge. The curse of knowledge means that it can be difficult to remember how hard it was to learn something when you were a novice learner. Adults have fluent procedural knowledge built up over years.
[00:12:56] So, 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've forgotten what it's like to encounter that explanation without the procedural foundation. But kids don't have that foundation yet.
What feels like an aha moment to an experienced adult is often incomprehensible to a child who's seeing it for the first time. And right now, we seem to have the worst of both worlds, students who neither understand the procedures nor have the procedural fluency. The evidence points toward prioritizing procedural skill development while also highlighting the key properties of operations and emphasizing meaning.
[00:13:50] Misconception number four. Problem solving is the best way to teach math. This is maybe one of the most influential ideas in modern math education.
The notion that math is best learned through what's referred to as problem solving. In practice, this often means students engaging with math problems they don't yet have the skills to solve, often in groups with the expectation that they'll brainstorm solutions. Productive struggle is a term we often hear.
It's sometimes even said that missing foundational skills will develop in the process or that students can fill in the gaps using technology.
[00:14:31] But expert problem solvers aren't experts because they're just better at reasoning. They're experts because they built up enormous amounts of domain-specific knowledge.
Facts, definitions, procedures, problem-solving techniques. These are all in long-term memory. When an expert sees a new problem, they're drawing on that stored knowledge to recognize patterns and select appropriate techniques.
Novices can't do that. They don't have the knowledge base to draw on. So they resort to ineffective techniques or inefficient trial and error.
[00:15:09] I find the instructional hierarchy really useful here as a framework. It describes the stages that learners move through when acquiring new knowledge. First is the acquisition stage, where new learners need explicit instruction, teacher modeling, worked examples, and scaffolding to build accuracy.
[00:15:31] Then comes the fluency stage, where they need substantial practice and feedback so the skill becomes not just accurate but also effortless. Complex problem solving only becomes productive once those earlier stages are solid. When students engage with problems before they have the underlying skills to solve them, they become frustrated and confused.
Problem-solving in math does matter, and it's where we want to get to. Being able to work through non-routine problems and apply our knowledge in unfamiliar contexts is important, but again, the sequencing matters a lot. Engaging with complex problems does build better problem solvers, but only once students have the foundational skills and fluency to actually engage with the problems productively.
[00:16:29] Misconception number five. Practice turns kids off math. This is important.
Nobody questions that you get good at sports or music through practice. In those disciplines, practice is accepted and celebrated. Nobody feels bad for young hockey players doing repetitive drills or music students practicing scales on the piano.
But in math, practice has been de-emphasized and sometimes discouraged. I cannot think of a more counterproductive message to send teachers, and I think that message does serious damage. First of all, the way you get good at math is through a lot of practice.
But also, kids become more engaged and motivated in math when they're actually good at it, and that comes from well-designed instruction paired with substantial practice. Here's why this matters so much in math. Math is really hierarchical.
Each skill builds on skills that came before it. Solving algebraic equations depends on fluency with fractions. Fluency with fractions depends on fluency with basic number facts.
[00:17:51] When kids don't get enough practice to actually master a concept, the gaps persist and compound as the material gets harder. And that's exactly when kids start losing confidence in their ability to do math. Activities that look engaging on the surface but actually don't build competence rob kids of the satisfaction that comes from genuine mathematical accomplishment.
Just like with music or sports, repeated exposure and deliberate practice are what get students to mastery to help them move to the next level with confidence. So let's abandon the message that practice turns kids off math and embrace the idea that practice is the vehicle that gets them to enjoy math.
[00:18:41] Okay, so we've talked about some misconceptions, but we need to talk about what works. First, let's move away from ideas that just sound appealing and toward instruction that's actually supported by evidence. Let's talk about teaching math in ways that work. Effective math teaching supports new learners with clear, explicit instruction.
There's good research. Charles Hughes and colleagues wrote about this identifying the core components. Breaking complex skills into manageable pieces.
Drawing attention to the important features of content through teacher modeling and think-alouds. Gradually fading prompts and supports as students gain independence. Giving students frequent opportunities to respond and get feedback.
And building in purposeful, structured practice. I'll link to that paper in the show notes. And for an accessible entry point into these ideas, start with Rosenshine's Principles of Instruction, which summarize a lot of this in plain language.
And don't forget worked examples, which are highly effective in mathematics. Check out my episode with John Sweller where we talk about worked examples in detail.
[00:20:06] Specific to math, here's what I'd want every teacher and parent to walk away with from this episode.
Prioritize an efficient, generalizable strategy instead of overwhelming new learners with several at once. Recognize that understanding often develops alongside or after procedural fluency. And make sure students build a genuinely strong foundation in basic procedures and math facts because that foundation is what success in later math depends on.
And practice? That's the secret sauce to getting good at math. A strong math education opens doors.
We owe it to children and their families to get it right. And that means abandoning harmful misconceptions and letting the best available evidence guide instructional decisions. Thank you for listening.
[00:21:06] Thank you so much for listening. If you enjoy this podcast, please consider showing your support by leaving a five-star rating on Spotify or Apple Podcasts. Don't forget to subscribe on your favourite podcast app or on YouTube so you never miss an episode.
You can stay connected with me on Instagram, Facebook, TikTok, X, Blue Sky, or LinkedIn. All links are in the show notes. And check out my website annastokke.com for more information. This podcast is funded by a grant from La Trobe University and from the Trottier Family Foundation through a grant to the University of Winnipeg to fund the Talk & Talk podcast.