Great article! I love this quote by math prof Steve Wilson: "it's easier to understand something you know how to do than something you don't know how to do"
"Conceptual understanding and procedural fluency often work in tandem — sometimes with understanding coming first, sometimes later. One feeds the other, and usually after a person has more mathematical tools and procedures that make understanding more accessible. (Case in point: many procedures and rules of arithmetic are easier to understand once one has a facility with algebra and symbolic manipulation."
Barry, I agree that holding a student hostage to a verbal explanation before letting them move forward is a bad idea. The kid who can reliably divide fractions is doing something right, and demanding a performance of understanding on top of that can do real harm.
But your principle that "understanding is not tested by words, but by whether the student can do the problems," begs the question of what sort of problems we give students. Carl Hendrick, who as I read him is broadly in favor of explicit instruction, recently wrote a piece called "Correct Answers But No Learning (https://carlhendrick.substack.com/p/correct-answers-but-no-learning)," about his daughter completing a phonics task perfectly by matching colors rather than blending phonemes. She got every answer right and learned nothing.
The interesting question for me isn't "should we require explanations?" but "what kinds of tasks reveal whether the understanding is actually there?" That's a design problem, and I think we need a broader toolbox to solve it that cuts across current debates. I've been writing about that recently on my new Substack, mathematicalmusings.substack.com.
I didn't mean any old sort of surface-knowledge problems, such as the one Carl talks about. I meant if they can solve the sort of problems we expect students to be able to solve, then that serves as a proxy for understanding. Which is how most standardized testing works.
What's funny is how ahistorical this is. We had centuries in progress in mathematics before the 19th century revolution in mathematical rigor, which found and fixed logical gaps in most of the math that came before it. Newton didn't know what a Dedekind cut was, but he still invented calculus. Surely many of these progressive math teachers cannot provide a rigorous definition of the reals, but can still multiply sqrt(2) by itself.
Old fart and traditionalist here - Physcist and Ph.D. Engineer by education. When I didn't like the way my kids were taught math, I taught them myself - and I gave more homework that their teachers did until they learned it. I did not care that the teacher was teaching it a different way. They had to learn it the teacher's way for school and my way for me. Mean dad. And I was hell on principals once the kids of my second family got to Middle School / High School. By 7th grade it was clear that my daughter was heading down the Physical Sciences / Engineering track - a track I know better than the schools do. She did the highest math the middle school had in 6th grade (I raised hell) and had to go to the High School for Math when she was in 7th grade. I had her do Geometry by correspondence over the summer after 7th grade. She skippled 8th grade and went to the high school starting with Honors Algebra 2. I had her do Pre-Calculus by correspondence over the summer after 9th grade and she did Calculus for college credit - taught by one of the community college math professors in 10th grade. She dropped out and did early admissions to the state university after 10th grade. She did her degree in Civil Engineering - Structures and earned her MS by the time she was 21.
But the High School Principal was right - the state had requirements for so many years of math and other subjects. Students who work ahead can not meet the State High School Graduation requirements - other than the trivial GED. My daughter had been planning on the Running Start Program, which her younger brother did do. Running Start / Early Admissions seems to be the only out for the students who are moving faster than expected.
I don't think that educators understand the extent of intervention that educated parents can and will implement when the school instruction program is inadequate. STEM educated parents are probably better educated in math than most of the math teachers. I was quite prepared to carry the full math education load myself. I could afford it, so I used the Gifted Learning Links correspondence classes - to get standard certification. But I certainly could have (and if necessary, would have) done it all myself. As it was, I answered her Geometry questions, and I had not looked at Euclidian Geometry in a good 50 years. All the algebraic areas were straightforward.
I don't think that I am at all unusual in this. My wife could and did provide guidance on the language skills side.
As I said, I am an old fart. I remember the chaos of 'New Math' when I was a young student. I certainly did not see the purpose of doing math in base 3 or base 7 when I was in elementary school - but I certainly had to learn to do math in base 16 and deal with binary IEEE formats for floating point numbers when I was doing some programming and debugging several decades ago. But it is useful to understand that there is a difference between a number and its representation.
I had rather expected to volunteer as a tutor or guide in math at the local high school after I retired, but I find that there is still demand for my computer security skills and am still working full time well into my mid 70's.
This is so true. It also applies to language where people who learned the traditional way and loved it so much that they devoted their professional lives to a foreign language now think that the way to teach kids is to talk at them in the language and never give them a hint as to what is going on.
Thank you for this. Your thoughts on the tensions between conceptual understanding and procedural fluency resonate nicely with an interesting pedagogical approach from Finland that I wrote about here: https://samuelkammin.substack.com/p/the-path-of-abstraction-a-finnish. Would be very curious to hear your take on it. Thank you again
Great article! I love this quote by math prof Steve Wilson: "it's easier to understand something you know how to do than something you don't know how to do"
"Conceptual understanding and procedural fluency often work in tandem — sometimes with understanding coming first, sometimes later. One feeds the other, and usually after a person has more mathematical tools and procedures that make understanding more accessible. (Case in point: many procedures and rules of arithmetic are easier to understand once one has a facility with algebra and symbolic manipulation."
BINGO
Barry, I agree that holding a student hostage to a verbal explanation before letting them move forward is a bad idea. The kid who can reliably divide fractions is doing something right, and demanding a performance of understanding on top of that can do real harm.
But your principle that "understanding is not tested by words, but by whether the student can do the problems," begs the question of what sort of problems we give students. Carl Hendrick, who as I read him is broadly in favor of explicit instruction, recently wrote a piece called "Correct Answers But No Learning (https://carlhendrick.substack.com/p/correct-answers-but-no-learning)," about his daughter completing a phonics task perfectly by matching colors rather than blending phonemes. She got every answer right and learned nothing.
The interesting question for me isn't "should we require explanations?" but "what kinds of tasks reveal whether the understanding is actually there?" That's a design problem, and I think we need a broader toolbox to solve it that cuts across current debates. I've been writing about that recently on my new Substack, mathematicalmusings.substack.com.
I didn't mean any old sort of surface-knowledge problems, such as the one Carl talks about. I meant if they can solve the sort of problems we expect students to be able to solve, then that serves as a proxy for understanding. Which is how most standardized testing works.
What's funny is how ahistorical this is. We had centuries in progress in mathematics before the 19th century revolution in mathematical rigor, which found and fixed logical gaps in most of the math that came before it. Newton didn't know what a Dedekind cut was, but he still invented calculus. Surely many of these progressive math teachers cannot provide a rigorous definition of the reals, but can still multiply sqrt(2) by itself.
Old fart and traditionalist here - Physcist and Ph.D. Engineer by education. When I didn't like the way my kids were taught math, I taught them myself - and I gave more homework that their teachers did until they learned it. I did not care that the teacher was teaching it a different way. They had to learn it the teacher's way for school and my way for me. Mean dad. And I was hell on principals once the kids of my second family got to Middle School / High School. By 7th grade it was clear that my daughter was heading down the Physical Sciences / Engineering track - a track I know better than the schools do. She did the highest math the middle school had in 6th grade (I raised hell) and had to go to the High School for Math when she was in 7th grade. I had her do Geometry by correspondence over the summer after 7th grade. She skippled 8th grade and went to the high school starting with Honors Algebra 2. I had her do Pre-Calculus by correspondence over the summer after 9th grade and she did Calculus for college credit - taught by one of the community college math professors in 10th grade. She dropped out and did early admissions to the state university after 10th grade. She did her degree in Civil Engineering - Structures and earned her MS by the time she was 21.
But the High School Principal was right - the state had requirements for so many years of math and other subjects. Students who work ahead can not meet the State High School Graduation requirements - other than the trivial GED. My daughter had been planning on the Running Start Program, which her younger brother did do. Running Start / Early Admissions seems to be the only out for the students who are moving faster than expected.
I don't think that educators understand the extent of intervention that educated parents can and will implement when the school instruction program is inadequate. STEM educated parents are probably better educated in math than most of the math teachers. I was quite prepared to carry the full math education load myself. I could afford it, so I used the Gifted Learning Links correspondence classes - to get standard certification. But I certainly could have (and if necessary, would have) done it all myself. As it was, I answered her Geometry questions, and I had not looked at Euclidian Geometry in a good 50 years. All the algebraic areas were straightforward.
I don't think that I am at all unusual in this. My wife could and did provide guidance on the language skills side.
As I said, I am an old fart. I remember the chaos of 'New Math' when I was a young student. I certainly did not see the purpose of doing math in base 3 or base 7 when I was in elementary school - but I certainly had to learn to do math in base 16 and deal with binary IEEE formats for floating point numbers when I was doing some programming and debugging several decades ago. But it is useful to understand that there is a difference between a number and its representation.
I had rather expected to volunteer as a tutor or guide in math at the local high school after I retired, but I find that there is still demand for my computer security skills and am still working full time well into my mid 70's.
This is so true. It also applies to language where people who learned the traditional way and loved it so much that they devoted their professional lives to a foreign language now think that the way to teach kids is to talk at them in the language and never give them a hint as to what is going on.
Thank you for this. Your thoughts on the tensions between conceptual understanding and procedural fluency resonate nicely with an interesting pedagogical approach from Finland that I wrote about here: https://samuelkammin.substack.com/p/the-path-of-abstraction-a-finnish. Would be very curious to hear your take on it. Thank you again