Thank you for your passion for math. The importance of math is undeniable, and every post that you write proves your devotion to excellence teaching and learning.
I do have one comment of disagreement. I believe that math competence is measured by the level of math concepts required to solve any particular problem. The testing for proxies generates a record of the measure of understanding. It's that simple. Where a student has learned the basics (as you mention) and can intuit further logical implications that have not been taught, there is competence, understanding and sometimes math genius at work. Those were the bonus questions on any math test in my youth.
In studying theories of knowledge, I have come across the statement that everything perceivable can be measured (even if we don't yet know how). I urge you not to give away such an important weapon to the enemy who says understanding cannot be measured at all. You have the truth on your side.
1 is it easier to discover who doesn’t have a certain level of understanding on a particular topic? And if so is that all you need in practice,
2. Your first point about levels of understanding is key. It makes any definition vague without specific context. Because mathematics is not used in daily life the way written and spoken language is there is no norm for understanding any part of mathematics.
There is an idealistic vision that all students must have 100% understanding of the concepts behind all the procedures one learns. In reality, some students learn the procedure first, and then the concept later; others, it's the other way around. And still others may never fully understand the concept but know what a procedure represents, and how they can operate with it to solve problems.
I discuss levels of understanding in a talk I gave at a researchED conference, and there is a video of the talk for a virtual conference held during Covid. It is available here: https://www.loom.com/share/71a874def3e04a8184821f9f01680f47
I enjoy reading these posts, even if I find the traditional/progressive dichotomy somewhat limiting as an analytical framework. Would love to hear your thoughts on some of these questions:
- What does teaching for understanding look like in a traditional math class? If we all agree that conceptual understanding, fluency, and procedural knowledge are important, how do we find the right balance at every stage of a student's journey? The Singapore Math curriculum seems to balance the three very effectively. What's the key to its success?
-- I agree that it does not make sense to make a fetish of understanding (like asking kids to explain in writing why they arrived at a certain answer or insisting that they solve the problem in multiple ways). But the devil is in the details, no? Yes, it would it's silly to keep students past a certain point from using a standard algorithm to solve a math problem. But it's equally silly to teach the standard algorithm to, say, a kindergartener or a first grader. The problem, it seems, isn't building number sense and conceptual knowledge but that ultra-progressive educators don't seem to know when to move on to standard procedures.
- I also wonder if it might not help to make a distinction between curriculum, assessments, and pedagogy. In other words, I think what you said here is spot on -- the only way to assess understanding is indirectly through assessments of procedural fluency and factual mastery. Ultimately, what matters is what students can do. That seems right to me. But pedagogically, there's a big difference in the teaching styles (the explanations, the activities, the formative assessments, etc.) of someone exclusively focused on procedural knowledge and someone who knows how to expertly balance conceptual, procedural, and factual knowledge.
-- Could all these somewhat abstract arguments about traditional vs. progressive math be avoided if we had a good national curriculum? Rather than standards, like the Common Core, wouldn't we all be better off if we had a nationally used curriculum, akin to the what they have in high-achieving countries like Singapore?
Thanks for your comment and thoughts. Just as measuring understanding is done through the proxies of procedural fluency and reasoning, the framework of traditional vs progressive is a proxy for the cognitive science basis for how people learn.
Also, traditionally taught math is often mischaracterized; i.e., there are probably very few teachers focused exclusively on procedures alone. Yes, Singapore's "Primary Math" series is very effective and I've used it, but if you look at US textbooks from the 60's, 50's, 40's and even earlier, you'll find a similar balance--even some use of bar models though not to the extent Singapore does. (Bar models provide a nice problem-solving heuristic that gives a visual context to what is happening mathematically.) You might wish to read another piece on this site regarding levels of understanding here: https://barrygarelick.substack.com/p/the-fetish-over-conceptual-understanding
You ask "What does teaching for understanding look like in a traditional math class?" I recently wrote a book with J.R. Wilson called "Traditional Math" which provides a description of what it looks like. You might find it useful; it is available here: https://www.amazon.com/Traditional-Math-effective-strategy-teachers/dp/1915261546
Retweeted. I encourage others to do the same while Barry is taking a well-deserved Twitter break.
Thank you for your passion for math. The importance of math is undeniable, and every post that you write proves your devotion to excellence teaching and learning.
I do have one comment of disagreement. I believe that math competence is measured by the level of math concepts required to solve any particular problem. The testing for proxies generates a record of the measure of understanding. It's that simple. Where a student has learned the basics (as you mention) and can intuit further logical implications that have not been taught, there is competence, understanding and sometimes math genius at work. Those were the bonus questions on any math test in my youth.
In studying theories of knowledge, I have come across the statement that everything perceivable can be measured (even if we don't yet know how). I urge you not to give away such an important weapon to the enemy who says understanding cannot be measured at all. You have the truth on your side.
Two thoughts
1 is it easier to discover who doesn’t have a certain level of understanding on a particular topic? And if so is that all you need in practice,
2. Your first point about levels of understanding is key. It makes any definition vague without specific context. Because mathematics is not used in daily life the way written and spoken language is there is no norm for understanding any part of mathematics.
There is an idealistic vision that all students must have 100% understanding of the concepts behind all the procedures one learns. In reality, some students learn the procedure first, and then the concept later; others, it's the other way around. And still others may never fully understand the concept but know what a procedure represents, and how they can operate with it to solve problems.
I discuss levels of understanding in a talk I gave at a researchED conference, and there is a video of the talk for a virtual conference held during Covid. It is available here: https://www.loom.com/share/71a874def3e04a8184821f9f01680f47
I enjoy reading these posts, even if I find the traditional/progressive dichotomy somewhat limiting as an analytical framework. Would love to hear your thoughts on some of these questions:
- What does teaching for understanding look like in a traditional math class? If we all agree that conceptual understanding, fluency, and procedural knowledge are important, how do we find the right balance at every stage of a student's journey? The Singapore Math curriculum seems to balance the three very effectively. What's the key to its success?
-- I agree that it does not make sense to make a fetish of understanding (like asking kids to explain in writing why they arrived at a certain answer or insisting that they solve the problem in multiple ways). But the devil is in the details, no? Yes, it would it's silly to keep students past a certain point from using a standard algorithm to solve a math problem. But it's equally silly to teach the standard algorithm to, say, a kindergartener or a first grader. The problem, it seems, isn't building number sense and conceptual knowledge but that ultra-progressive educators don't seem to know when to move on to standard procedures.
- I also wonder if it might not help to make a distinction between curriculum, assessments, and pedagogy. In other words, I think what you said here is spot on -- the only way to assess understanding is indirectly through assessments of procedural fluency and factual mastery. Ultimately, what matters is what students can do. That seems right to me. But pedagogically, there's a big difference in the teaching styles (the explanations, the activities, the formative assessments, etc.) of someone exclusively focused on procedural knowledge and someone who knows how to expertly balance conceptual, procedural, and factual knowledge.
-- Could all these somewhat abstract arguments about traditional vs. progressive math be avoided if we had a good national curriculum? Rather than standards, like the Common Core, wouldn't we all be better off if we had a nationally used curriculum, akin to the what they have in high-achieving countries like Singapore?
Thanks for your comment and thoughts. Just as measuring understanding is done through the proxies of procedural fluency and reasoning, the framework of traditional vs progressive is a proxy for the cognitive science basis for how people learn.
Also, traditionally taught math is often mischaracterized; i.e., there are probably very few teachers focused exclusively on procedures alone. Yes, Singapore's "Primary Math" series is very effective and I've used it, but if you look at US textbooks from the 60's, 50's, 40's and even earlier, you'll find a similar balance--even some use of bar models though not to the extent Singapore does. (Bar models provide a nice problem-solving heuristic that gives a visual context to what is happening mathematically.) You might wish to read another piece on this site regarding levels of understanding here: https://barrygarelick.substack.com/p/the-fetish-over-conceptual-understanding
You ask "What does teaching for understanding look like in a traditional math class?" I recently wrote a book with J.R. Wilson called "Traditional Math" which provides a description of what it looks like. You might find it useful; it is available here: https://www.amazon.com/Traditional-Math-effective-strategy-teachers/dp/1915261546
If only someone had written a paper on how Singapore maths bar model can be used to get both understanding and fluency.