Is math the only school subject where the existence of concepts are controversial?
I'm curious, do history teachers sit around asking, "Is Democracy even a concept to begin with? Is it possible for a person to have a concept of what democracy is, in the abstract?"
I'm trying to image walking into the English department and saying, "Why do we bother teaching the concept metaphor? The only thing that matters is whether or not students produce provocative, relatable writing, not whether or not they actually know they are doing so. The concept of metaphor is just an illusion, there is no way to measure whether students know it, just if they can use it."
And for all that we argue over the meaning of the word "theory" when we talk about the theory of evolution, I have never heard anyone argue that evolution is not a concept. They might say it should not be taught, but I never heard them say it cannot be taught.
In the past few years, I switched from teaching mostly high achieving secondary students to teaching in under-served communities. I now need to teach concepts that I mostly skipped over in other classes. Here is a smattering: Variables can be replaced with numbers. Decimals and fractions refer to the same set of numbers. Taking half of something is the same as dividing it by two. Multiplication and division are connected.
Let's look at one example. A lot of students at my school double numbers instead of squaring them. Why is this so common? I'm speculating, but I think part the problem is that they were taught about the four operations. They developed the idea that there are four ways to combine numbers: add, subtract, multiply, divide. So they look at 3^2 and assume it's 3*2: what other options are there? Is it that they forgot how to square numbers? Or is it that they could not remember because they never believed that a fifth operation is even possible, so there was no mental framework into which to place this knowledge?
Another option: maybe some students are missing the concept that the superscript matters. Usually if there is a number or a letter floating a little higher, it's because someone wrote it funny by mistake. So, they have not developed the concept that the font size and altitude of a number matters. You can tell them over and over (and we do!) but they won't remember it.
These are not the concepts that come to mind when I think about "conceptual understanding," at least not for high school. But they are the concepts that enable students to take a step forward towards learning procedures. I have to admit, my understanding of "concept" in math class has changed radically in the past few years.
Debbie, I loved your blog post, particularly the point about different forms of equations. If students have surface level knowledge then those will stump them, but if they have a structural knowledge of expressions as representing operations with numbers, and an understanding of what the equals sign means, they will see the relation. That structural knowledge is another word for conceptual understanding.
Several commenters pointed out that it’s often a joyful moment when conceptual understanding is attained, sometimes described as an “aha!” experience. Positive emotional experiences are super important in learning anything! Illustrative Mathematics emphasizes this goal: “… to build a world where all learners know, use, and enjoy mathematics.” Intentionally building pleasure into our lessons should be part of our everyday teaching practice.
My attention was captured by the word "Myth" in Ashman's provocative title. Was he claiming that conceptual understanding is an imagined combination of real entities, like a Hippogriff? Or a story about imagined entities that explains a real phenomenon, like Persephone in the Underworld? I stopped reading when it seemed unlikely to involve heroes and monsters.
Back to Conceptual Understanding: it may be hard to establish its presence, but it's easy to see when it's absent! Any primary school student proficient in the arithmetic of fractions can calculate 1/2 of 3/4. A few can cut shapes from colored paper to explain what 1/2 of 3/4 means, and why it's not the same thing as 3/4 of 1/2 even though the answers are the same.
We would be tempted to say that the latter students have good conceptual understanding of fraction multiplication. Ashman is correct that these children might have memorized the construction, and might have no idea how to explain fraction division using paper and scissors. In his language, transfer has not been demonstrated.
But if a student can't explain with paper, there's no question! Conceptual understanding is absent.
Thanks, Bill—that’s a helpful analysis. “Conceptual understanding is the cognitive structure that supports transfer; transfer is how you measure it.”
A clear counterexample to Ashman’s claim that conceptual and procedural knowledge are the same can be seen in differentiation. Students can often apply standard algorithms mechanically without grasping what a derivative actually represents. However, a question like “find the derivative at x=0 for f(x)=∣x∣" reveals whether they truly understand the concept, since it requires more than using algorithm.
I’m also reminded of Eddie Woo’s short three-minute example on explaining “completing the square”—especially the moment when the student’s reaction shows genuine conceptual understanding clicking into place. - https://www.youtube.com/watch?v=McDdEw_Fb5E
It happens just after minute two in the video. You can't see the students but it's a lovely moment to hear. Anyone teaching this topic should make sure their students see or discover this graphical representation this if they teach this topic.
Another fun example from HS is when students realize the relationship between the equation for the line of symmetry for a parabola / the quadratic equation and the graph of a parabola. I don't have a video I just love the moment when early algebra students realize how one is embedded in the other how "visible" the equations are in the graph. They interact with it all differently afterwards.
There are many moments like these that bring real joy to the classroom. I’m trying to gather as many video explanations of concepts as possible—this one on solving cubic equations is among my favorites. - https://youtu.be/NMiO1iMiAUg?si=WNCD33uDOjkE2EAI
I love posts like this. I think there should be many more discussions around this than we have now, because it is so definitional and semantic that we don't all use it the same.
Before teaching math, I taught English for many years. As a result, I just go to the dictionary to find definitions and etymology.
Concept
1: something conceived in the mind : thought, notion
2: an abstract or generic idea generalized from particular instances
from Medieval Latin conceptum "draft, abstract," in classical Latin stem of concipere "to take in and hold; become pregnant,"
Understanding
1: a mental grasp : comprehension
Comprehend
1: to grasp the nature, significance, or meaning of
2: to contain or hold within a total scope, significance, or amount
3: to include by construction or implication
comprise
1: to be made up of
2: to make up or form
Together, these various definitions would argue that conceptual understanding is simply the grasp of a generalized idea from particular instances. I think of it as knowing the general characteristics and qualities of the thing we are discussing. "Animal" as a concept is just knowing all the characteristics of animals. I can understand that concept without "knowing both what to do and why". The same goes for almost any concept. "Chair" as a concept does not need me to be able to "generate new knowledge and solve new and unfamiliar problems".
I think you have your final statement, "conceptual understanding is the connected cognitive structure" exactly right, but then continue to graft, "that lets you handle problems you haven’t seen before" onto it in order to get around the transfer issue as being separate.
Transfer is incredibly difficult. I have taken time in the past to try to find research that supports transfer happening and I could find very little. Are you aware of research that actually shows transfer happening? I think experts often have this huge base of understanding and knowledge to draw on, they have a deep reserve of organized facts, i.e. conceptual knowledge, and that allows them to quickly find solutions. Very few people can actively transfer those facts to new problems.
I don't have it in front of me, but I believe Kahneman in his famous "Thinking, Fast and Slow" had a section where researchers asked several PhD level statisticians questions outside on the sidewalk in their daily life related to their field and they often missed the answer or jumbled them, showing just how specific learning can be even for the highest level experts.
This is very well understood in sports performance where the SAID principle (Specific Adaptation to Imposed Demands) is a foundational concept in exercise science and physical therapy. This principle essentially captures what any classroom teacher knows already, that if you want to get better at a thing, you need to do that thing repeatedly and often extremely specifically. Everyone is okay teaching to the test when the test is a 400m dash in track and field. We laud athletes for their efficiency due deliberately practiced skills that lead to higher performance.
This also gets at the comment below from Lane that points out standardized tests guide how NC teachers organize their classes for the ACT.
The SAID principle should be much more closely integrated into classrooms because it is really a foundation of how physiology works and brains are physical pathways. Yes, we can broaden the connections via learning, of course, that's what K-12 is about in my view. However, the connections we choose to focus on and broaden shouldn't come with the additional expectation that students can magically transfer their knowledge to new realms or problems. Instead, we'd be much better off focusing on the problems we want them to solve. If we hope advanced mathematics in public schools will transfer to better citizenship via numeracy, then let's just focus on how numeracy allows for more critical citizens in the classroom. Make math class focus on the problems of running a democratic society, rather than hope it will.
If the goal is to serve as a pipeline to college and higher learning, then let's not pretend every student should be able to grasp these abstract concepts at the same level. The majority of people will not graduate with an undergraduate degree (~40% of adults). The PIAAC test results also show roughly 40% of adults in the USA score Level 3 or Above, meaning that the vast majority of adults walking around are Level 1 or 2. Those are described as:
"Adults at Level 1 or below can be considered at risk for difficulties with numeracy. Adults at the upper end of this level can understand how to add, subtract, multiply, and divide and can perform basic one-step mathematical operations with given values or common spatial representations. Adults who are below Level 1 may only be able to count, sort, and do basic arithmetic operations with simple whole numbers or may be functionally innumerate.
Adults at Level 2 can be considered nearing proficiency but still struggling to perform numeracy tasks. Such adults can successfully perform tasks requiring two or three steps, calculations with whole numbers and common decimals, percentages, and fractions. They can interpret relatively simple data and statistics in texts, tables, and graphs."
I think conceptual understanding and transfer need to remain separate. Do you have any literature review or collection of studies on transfer available? Any time I look for something, it shows little to no transfer occurs in real life. Maybe I am overlooking or simply not skilled enough to find where the results are hiding. Thanks for another good post. :)
You got me to think in new ways here, Brian. I love, "Everyone is okay teaching to the test when the test is a 400m dash."
One place things get sticky is in the relevance of the curriculum. I mean, I would love to teach students to analyze news stories and choose the right used car to buy using mathematical tools. But I'm required to teach transformations of functions and the quadratic formula. Those are skills that almost no one uses in life, and I know it.
One thing that I try to teach is the attitude, "If I encounter some math in life, there is a chance that if I look at it closely, I will make sense of it and find the answer." You know, the opposite of the belief that many American adults have, "If there is math somewhere around me, I should not bother to look at it for long because that will be a frustrating waste of time."
I think that helping students develop a conceptual understanding of the math we teach can help them develop this belief in their own efficacy. When they experience transfer in the classroom, they are building a belief in their own ability to do math if they try. This sort of limited transfer matters. Even if they never transfer any of the concepts we teach them outside of the classroom.
But yeah, let's switch to a numeracy and citizenship-enhancing curriculum: sign me up!!
I had a co-worker that when a student said, "I'm just not a math person," would respond with, "Have you ever heard someone say, 'I'm just not a reading person?'". I've adopted that approach myself because it does kind of point out just how silly those sentiments are and usually gets a chuckle when delivered appropriately.
I think it's mainly an American thing. In my eight years teaching across Asia, that was never a thing. If a student wasn't good at math, the parents took it very seriously and addressed it intensely. However, Americans are just as likely to hear from a parent something to the effect, "Yeah they got that from me, I'm just not a math person either."
We just have a culture that adopts a defeatist attitude that sees math as innate, for special people. We never think we aren't history people, English people, etc.
Brian, on the question of what research there is on transfer, it's pretty sparse as you suspected. Daniel Willingham's book Why Don't Students Like School has a nice chapter on it, Why Is It So Hard for Students to Understand Abstract Ideas?, with some teaching recommendations at the end.
I have read his book and several of his other writings. I should revisit it as it has been over ten years now, but I found it to be wonderful at the time. One aspect that has stuck with me over the years is just how much knowing stuff (prior knowledge) helps. Our rate of learning accelerates as we know more stuff and I've always wondered just how much more learning might happen if there was a greater focus on reading extensively just to accumulate knowledge regardless of the content.
More knowledge has more potential connections, but also just more potential comprehension. I've used the literacy trick of deleting 20% of the words on a page and asking people to read it for understanding. It's basically impossible. That really helps to crystallize just how much you're missing when you get 80% of the knowledge at each grade level through school, which obviously compounds over the years so that by high school you are often missing much more than half of the knowledge expected.
This shows up daily as lacking basic vocabulary words to understand the sentences I use to explain a topic. Students can't understand my explanation because they are missing the comprehension of every other word when their lexicon is low and then they also don't have the math skills.
Thanks again for all your time; it is very appreciated and I now requires energy and mental effort. :)
I would say that conceptual understanding is intimately tied with proof (and thus "dock you a couple points" for using conceptual understanding of proof as your personal example). Here's my personal example: as a super-accelerated student, I craved proofs I wasn't getting, becoming excited at any imagined chance to "really know". Sure I could figure out some justification on my own, such as why multi-digit addition works but not really division by fractions - I could see that flip and multiply worked in some cases to my mind, but the curriculum didn't provide anything further so it was on to the next topic. As an eighth grader taking calculus, I thought I would finally get justification, but was generally disappointed, beyond a couple things like being able to set up integrals to compute the area of a circle. Then as a tenth grader taking point-set topology in the context of fractal geometry at the local university, I FINALLY felt like I "understood" mathematics because it had all of the proofs. But what I was missing should have been there all along - for example the use of rectangular arrays to justify commutativity of multiplication.
One of the main reasons I took you up on your invitation to do work related to Common Core implementation is that at a workshop you ran for mathematicians you showed how that the Common Core had proofs implied for all of K-12 mathematics (save a couple topics in high-school that one needs continuity for, such as treating irrational exponents). I saw this as essential to mathematics making sense to students, but grew to appreciate that students can be asked to provide reasoning themselves.
Providing reasoning can provide assessment of conceptual understanding through a different route than transfer. For example, why in performing long division should 823 / 27 have first digit a 3, in the tens place? A good answer is "Because 27 * 30 =810 < 823 < 27 * 40 =1080, we can divide all by 27 and see 30 < 823/27 < 40". (A fun exercise in teacher training I've done, which I could also see done for students: why does sqrt(2) start "1.4..."). Sure, such reasoning could be taught in a rote way, but by demanding a variety of question types such students are more likely to (in some cases "can't help but to") develop a conceptual framework.
To mathematicians, all of these proofs would "just be unraveling definitions". But that's still powerful, and I see it as an important aspect of conceptual understanding.
Addendum 1: in this day and age, having mathematical rules as something a community can all see justification for together (so it doesn't suffice for teachers to briefly say why, as happens even (especially) in college classrooms) rather than something handed down from authority can instill habits of mind with wider implications.
Addendum 2: I see layers of depth to conceptual understanding. This "unraveling definitions" is one I would aspire to see all students engage in, in age-appropriate ways. The conceptual reading of a proof as you described at the beginning of the article is one that graduate students must master. Lately I've been reflecting on the conceptual understanding needed to guide PhD students and postdocs, especially if you want to meet them where their interests and strengths lie.
Thank you for giving a name to something I instinctively understood. I can memorize something, sure, but it doesn’t stick if I don’t understand it. I wonder how much IQ differences are people who stop at procedural understanding versus people who push on to conceptual understanding? I push on because I don’t like procedural understanding, sort of the way I don’t like a bad taste. It triggers an icky, annoyed, uncomfortable feeling that I want to resolve.
As for testing, I remember math books having a page or two of problems based on procedures. All students would practice those. But then there would be one or two “bonus” problems, usually printed in a sidebar at the bottom of the page. Those required applying the procedures in new ways. Those were the questions meant to test conceptual understanding. It would be interesting to track which students can solve the bonus problems and how often. Does that correlate with other cognitive skills? Is it possible — if yes, how? — to improve a student’s ability to solve the bonus problems?
In my experience, ACT has led many teachers in NC to teach formulas like TE-TE-PE-PE-WE-WE and memorize the Angle Bisector Theorm because they hear about or see procedural problems that are related on an ACT test. I am hoping that the ACT revisions help move teachers toward more opportunities to develop conceptual understanding with their students.
Wow. I can see some value in purely procedural knowledge of algebra—it gets you there. But purely procedural knowledge of a geometry proof seems completely without value.
I think the value is that most people are not learning math to learn proofs. The vast majority of people are learning how to get the highest ACT score because it "determines" where they get into college and the belief is that where you go to college "determines" your "success" in life. Most high school students see the "problem" to be solved as how to pass the class, how to graduate high school, or how to get into the best college they can so they can avoid trouble, earn more money, etc. From that perspective, the "value" of procedural knowledge of a geometry proof in their mind is extremely high because they think it will save time, energy, and frustration with the result of higher income in the future. They're simply solving a different problem than that of the proof.
Is math the only school subject where the existence of concepts are controversial?
I'm curious, do history teachers sit around asking, "Is Democracy even a concept to begin with? Is it possible for a person to have a concept of what democracy is, in the abstract?"
I'm trying to image walking into the English department and saying, "Why do we bother teaching the concept metaphor? The only thing that matters is whether or not students produce provocative, relatable writing, not whether or not they actually know they are doing so. The concept of metaphor is just an illusion, there is no way to measure whether students know it, just if they can use it."
And for all that we argue over the meaning of the word "theory" when we talk about the theory of evolution, I have never heard anyone argue that evolution is not a concept. They might say it should not be taught, but I never heard them say it cannot be taught.
A little while back, I wrote a blog post about the concept of concept. Take a look here: https://openset.blog/2025/05/24/concepts/
In the past few years, I switched from teaching mostly high achieving secondary students to teaching in under-served communities. I now need to teach concepts that I mostly skipped over in other classes. Here is a smattering: Variables can be replaced with numbers. Decimals and fractions refer to the same set of numbers. Taking half of something is the same as dividing it by two. Multiplication and division are connected.
Let's look at one example. A lot of students at my school double numbers instead of squaring them. Why is this so common? I'm speculating, but I think part the problem is that they were taught about the four operations. They developed the idea that there are four ways to combine numbers: add, subtract, multiply, divide. So they look at 3^2 and assume it's 3*2: what other options are there? Is it that they forgot how to square numbers? Or is it that they could not remember because they never believed that a fifth operation is even possible, so there was no mental framework into which to place this knowledge?
Another option: maybe some students are missing the concept that the superscript matters. Usually if there is a number or a letter floating a little higher, it's because someone wrote it funny by mistake. So, they have not developed the concept that the font size and altitude of a number matters. You can tell them over and over (and we do!) but they won't remember it.
These are not the concepts that come to mind when I think about "conceptual understanding," at least not for high school. But they are the concepts that enable students to take a step forward towards learning procedures. I have to admit, my understanding of "concept" in math class has changed radically in the past few years.
Debbie, I loved your blog post, particularly the point about different forms of equations. If students have surface level knowledge then those will stump them, but if they have a structural knowledge of expressions as representing operations with numbers, and an understanding of what the equals sign means, they will see the relation. That structural knowledge is another word for conceptual understanding.
Several commenters pointed out that it’s often a joyful moment when conceptual understanding is attained, sometimes described as an “aha!” experience. Positive emotional experiences are super important in learning anything! Illustrative Mathematics emphasizes this goal: “… to build a world where all learners know, use, and enjoy mathematics.” Intentionally building pleasure into our lessons should be part of our everyday teaching practice.
Hi Bill, I just wrote a response to the articles that you and Greg wrote. You can read it here: https://knowledgeforteachers.substack.com/p/why-the-conceptual-understanding
Let me know your thoughts!
My attention was captured by the word "Myth" in Ashman's provocative title. Was he claiming that conceptual understanding is an imagined combination of real entities, like a Hippogriff? Or a story about imagined entities that explains a real phenomenon, like Persephone in the Underworld? I stopped reading when it seemed unlikely to involve heroes and monsters.
Back to Conceptual Understanding: it may be hard to establish its presence, but it's easy to see when it's absent! Any primary school student proficient in the arithmetic of fractions can calculate 1/2 of 3/4. A few can cut shapes from colored paper to explain what 1/2 of 3/4 means, and why it's not the same thing as 3/4 of 1/2 even though the answers are the same.
We would be tempted to say that the latter students have good conceptual understanding of fraction multiplication. Ashman is correct that these children might have memorized the construction, and might have no idea how to explain fraction division using paper and scissors. In his language, transfer has not been demonstrated.
But if a student can't explain with paper, there's no question! Conceptual understanding is absent.
Thanks, Bill—that’s a helpful analysis. “Conceptual understanding is the cognitive structure that supports transfer; transfer is how you measure it.”
A clear counterexample to Ashman’s claim that conceptual and procedural knowledge are the same can be seen in differentiation. Students can often apply standard algorithms mechanically without grasping what a derivative actually represents. However, a question like “find the derivative at x=0 for f(x)=∣x∣" reveals whether they truly understand the concept, since it requires more than using algorithm.
I’m also reminded of Eddie Woo’s short three-minute example on explaining “completing the square”—especially the moment when the student’s reaction shows genuine conceptual understanding clicking into place. - https://www.youtube.com/watch?v=McDdEw_Fb5E
It happens just after minute two in the video. You can't see the students but it's a lovely moment to hear. Anyone teaching this topic should make sure their students see or discover this graphical representation this if they teach this topic.
Another fun example from HS is when students realize the relationship between the equation for the line of symmetry for a parabola / the quadratic equation and the graph of a parabola. I don't have a video I just love the moment when early algebra students realize how one is embedded in the other how "visible" the equations are in the graph. They interact with it all differently afterwards.
There are many moments like these that bring real joy to the classroom. I’m trying to gather as many video explanations of concepts as possible—this one on solving cubic equations is among my favorites. - https://youtu.be/NMiO1iMiAUg?si=WNCD33uDOjkE2EAI
These are great examples, thanks George
I love posts like this. I think there should be many more discussions around this than we have now, because it is so definitional and semantic that we don't all use it the same.
Before teaching math, I taught English for many years. As a result, I just go to the dictionary to find definitions and etymology.
Concept
1: something conceived in the mind : thought, notion
2: an abstract or generic idea generalized from particular instances
from Medieval Latin conceptum "draft, abstract," in classical Latin stem of concipere "to take in and hold; become pregnant,"
Understanding
1: a mental grasp : comprehension
Comprehend
1: to grasp the nature, significance, or meaning of
2: to contain or hold within a total scope, significance, or amount
3: to include by construction or implication
comprise
1: to be made up of
2: to make up or form
Together, these various definitions would argue that conceptual understanding is simply the grasp of a generalized idea from particular instances. I think of it as knowing the general characteristics and qualities of the thing we are discussing. "Animal" as a concept is just knowing all the characteristics of animals. I can understand that concept without "knowing both what to do and why". The same goes for almost any concept. "Chair" as a concept does not need me to be able to "generate new knowledge and solve new and unfamiliar problems".
I think you have your final statement, "conceptual understanding is the connected cognitive structure" exactly right, but then continue to graft, "that lets you handle problems you haven’t seen before" onto it in order to get around the transfer issue as being separate.
Transfer is incredibly difficult. I have taken time in the past to try to find research that supports transfer happening and I could find very little. Are you aware of research that actually shows transfer happening? I think experts often have this huge base of understanding and knowledge to draw on, they have a deep reserve of organized facts, i.e. conceptual knowledge, and that allows them to quickly find solutions. Very few people can actively transfer those facts to new problems.
I don't have it in front of me, but I believe Kahneman in his famous "Thinking, Fast and Slow" had a section where researchers asked several PhD level statisticians questions outside on the sidewalk in their daily life related to their field and they often missed the answer or jumbled them, showing just how specific learning can be even for the highest level experts.
This is very well understood in sports performance where the SAID principle (Specific Adaptation to Imposed Demands) is a foundational concept in exercise science and physical therapy. This principle essentially captures what any classroom teacher knows already, that if you want to get better at a thing, you need to do that thing repeatedly and often extremely specifically. Everyone is okay teaching to the test when the test is a 400m dash in track and field. We laud athletes for their efficiency due deliberately practiced skills that lead to higher performance.
This also gets at the comment below from Lane that points out standardized tests guide how NC teachers organize their classes for the ACT.
The SAID principle should be much more closely integrated into classrooms because it is really a foundation of how physiology works and brains are physical pathways. Yes, we can broaden the connections via learning, of course, that's what K-12 is about in my view. However, the connections we choose to focus on and broaden shouldn't come with the additional expectation that students can magically transfer their knowledge to new realms or problems. Instead, we'd be much better off focusing on the problems we want them to solve. If we hope advanced mathematics in public schools will transfer to better citizenship via numeracy, then let's just focus on how numeracy allows for more critical citizens in the classroom. Make math class focus on the problems of running a democratic society, rather than hope it will.
If the goal is to serve as a pipeline to college and higher learning, then let's not pretend every student should be able to grasp these abstract concepts at the same level. The majority of people will not graduate with an undergraduate degree (~40% of adults). The PIAAC test results also show roughly 40% of adults in the USA score Level 3 or Above, meaning that the vast majority of adults walking around are Level 1 or 2. Those are described as:
"Adults at Level 1 or below can be considered at risk for difficulties with numeracy. Adults at the upper end of this level can understand how to add, subtract, multiply, and divide and can perform basic one-step mathematical operations with given values or common spatial representations. Adults who are below Level 1 may only be able to count, sort, and do basic arithmetic operations with simple whole numbers or may be functionally innumerate.
Adults at Level 2 can be considered nearing proficiency but still struggling to perform numeracy tasks. Such adults can successfully perform tasks requiring two or three steps, calculations with whole numbers and common decimals, percentages, and fractions. They can interpret relatively simple data and statistics in texts, tables, and graphs."
I think conceptual understanding and transfer need to remain separate. Do you have any literature review or collection of studies on transfer available? Any time I look for something, it shows little to no transfer occurs in real life. Maybe I am overlooking or simply not skilled enough to find where the results are hiding. Thanks for another good post. :)
You got me to think in new ways here, Brian. I love, "Everyone is okay teaching to the test when the test is a 400m dash."
One place things get sticky is in the relevance of the curriculum. I mean, I would love to teach students to analyze news stories and choose the right used car to buy using mathematical tools. But I'm required to teach transformations of functions and the quadratic formula. Those are skills that almost no one uses in life, and I know it.
One thing that I try to teach is the attitude, "If I encounter some math in life, there is a chance that if I look at it closely, I will make sense of it and find the answer." You know, the opposite of the belief that many American adults have, "If there is math somewhere around me, I should not bother to look at it for long because that will be a frustrating waste of time."
I think that helping students develop a conceptual understanding of the math we teach can help them develop this belief in their own efficacy. When they experience transfer in the classroom, they are building a belief in their own ability to do math if they try. This sort of limited transfer matters. Even if they never transfer any of the concepts we teach them outside of the classroom.
But yeah, let's switch to a numeracy and citizenship-enhancing curriculum: sign me up!!
I had a co-worker that when a student said, "I'm just not a math person," would respond with, "Have you ever heard someone say, 'I'm just not a reading person?'". I've adopted that approach myself because it does kind of point out just how silly those sentiments are and usually gets a chuckle when delivered appropriately.
I think it's mainly an American thing. In my eight years teaching across Asia, that was never a thing. If a student wasn't good at math, the parents took it very seriously and addressed it intensely. However, Americans are just as likely to hear from a parent something to the effect, "Yeah they got that from me, I'm just not a math person either."
We just have a culture that adopts a defeatist attitude that sees math as innate, for special people. We never think we aren't history people, English people, etc.
Brian, on the question of what research there is on transfer, it's pretty sparse as you suspected. Daniel Willingham's book Why Don't Students Like School has a nice chapter on it, Why Is It So Hard for Students to Understand Abstract Ideas?, with some teaching recommendations at the end.
I have read his book and several of his other writings. I should revisit it as it has been over ten years now, but I found it to be wonderful at the time. One aspect that has stuck with me over the years is just how much knowing stuff (prior knowledge) helps. Our rate of learning accelerates as we know more stuff and I've always wondered just how much more learning might happen if there was a greater focus on reading extensively just to accumulate knowledge regardless of the content.
More knowledge has more potential connections, but also just more potential comprehension. I've used the literacy trick of deleting 20% of the words on a page and asking people to read it for understanding. It's basically impossible. That really helps to crystallize just how much you're missing when you get 80% of the knowledge at each grade level through school, which obviously compounds over the years so that by high school you are often missing much more than half of the knowledge expected.
This shows up daily as lacking basic vocabulary words to understand the sentences I use to explain a topic. Students can't understand my explanation because they are missing the comprehension of every other word when their lexicon is low and then they also don't have the math skills.
Thanks again for all your time; it is very appreciated and I now requires energy and mental effort. :)
You're welcome! I enjoy these discussions.
I would say that conceptual understanding is intimately tied with proof (and thus "dock you a couple points" for using conceptual understanding of proof as your personal example). Here's my personal example: as a super-accelerated student, I craved proofs I wasn't getting, becoming excited at any imagined chance to "really know". Sure I could figure out some justification on my own, such as why multi-digit addition works but not really division by fractions - I could see that flip and multiply worked in some cases to my mind, but the curriculum didn't provide anything further so it was on to the next topic. As an eighth grader taking calculus, I thought I would finally get justification, but was generally disappointed, beyond a couple things like being able to set up integrals to compute the area of a circle. Then as a tenth grader taking point-set topology in the context of fractal geometry at the local university, I FINALLY felt like I "understood" mathematics because it had all of the proofs. But what I was missing should have been there all along - for example the use of rectangular arrays to justify commutativity of multiplication.
One of the main reasons I took you up on your invitation to do work related to Common Core implementation is that at a workshop you ran for mathematicians you showed how that the Common Core had proofs implied for all of K-12 mathematics (save a couple topics in high-school that one needs continuity for, such as treating irrational exponents). I saw this as essential to mathematics making sense to students, but grew to appreciate that students can be asked to provide reasoning themselves.
Providing reasoning can provide assessment of conceptual understanding through a different route than transfer. For example, why in performing long division should 823 / 27 have first digit a 3, in the tens place? A good answer is "Because 27 * 30 =810 < 823 < 27 * 40 =1080, we can divide all by 27 and see 30 < 823/27 < 40". (A fun exercise in teacher training I've done, which I could also see done for students: why does sqrt(2) start "1.4..."). Sure, such reasoning could be taught in a rote way, but by demanding a variety of question types such students are more likely to (in some cases "can't help but to") develop a conceptual framework.
To mathematicians, all of these proofs would "just be unraveling definitions". But that's still powerful, and I see it as an important aspect of conceptual understanding.
Addendum 1: in this day and age, having mathematical rules as something a community can all see justification for together (so it doesn't suffice for teachers to briefly say why, as happens even (especially) in college classrooms) rather than something handed down from authority can instill habits of mind with wider implications.
Addendum 2: I see layers of depth to conceptual understanding. This "unraveling definitions" is one I would aspire to see all students engage in, in age-appropriate ways. The conceptual reading of a proof as you described at the beginning of the article is one that graduate students must master. Lately I've been reflecting on the conceptual understanding needed to guide PhD students and postdocs, especially if you want to meet them where their interests and strengths lie.
Thank you for giving a name to something I instinctively understood. I can memorize something, sure, but it doesn’t stick if I don’t understand it. I wonder how much IQ differences are people who stop at procedural understanding versus people who push on to conceptual understanding? I push on because I don’t like procedural understanding, sort of the way I don’t like a bad taste. It triggers an icky, annoyed, uncomfortable feeling that I want to resolve.
As for testing, I remember math books having a page or two of problems based on procedures. All students would practice those. But then there would be one or two “bonus” problems, usually printed in a sidebar at the bottom of the page. Those required applying the procedures in new ways. Those were the questions meant to test conceptual understanding. It would be interesting to track which students can solve the bonus problems and how often. Does that correlate with other cognitive skills? Is it possible — if yes, how? — to improve a student’s ability to solve the bonus problems?
It sounds like you also possess productive disposition, one of the other strands of mathematical proficiency in Adding It Up!
In my experience, ACT has led many teachers in NC to teach formulas like TE-TE-PE-PE-WE-WE and memorize the Angle Bisector Theorm because they hear about or see procedural problems that are related on an ACT test. I am hoping that the ACT revisions help move teachers toward more opportunities to develop conceptual understanding with their students.
What is TE-TE-PE-PE-WE-WE?
TT: if two tangents are drawn from a common point, external from a circle, the tangent segments are congruent.
PE*PE: If two chords intersect inside a circle, part-part relationship is ab=cd...
Wow. I can see some value in purely procedural knowledge of algebra—it gets you there. But purely procedural knowledge of a geometry proof seems completely without value.
I think the value is that most people are not learning math to learn proofs. The vast majority of people are learning how to get the highest ACT score because it "determines" where they get into college and the belief is that where you go to college "determines" your "success" in life. Most high school students see the "problem" to be solved as how to pass the class, how to graduate high school, or how to get into the best college they can so they can avoid trouble, earn more money, etc. From that perspective, the "value" of procedural knowledge of a geometry proof in their mind is extremely high because they think it will save time, energy, and frustration with the result of higher income in the future. They're simply solving a different problem than that of the proof.