Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner, and those who advocate for progressive techniques. Among other things, the progressivists argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself. I am a middle school math teacher and am in the former camp.
I follow my own algebra teacher’s reminder said with a twinkle in his eye. I assumed it was just a bit of doggerel but learned its provenance when reading The Charge of the Light Brigade. Sharing that insight with others, I call it Tennyson’s Rule:
“Ours is not to reason why; just invert and multiply."
I find this a compelling argument for putting practice and competence before "deeper understanding" in K-8 math. However, I wonder if there's a transition point when the reform-style "deeper understanding" becomes more important to teach first. When I was in high school, I found Algebra II a parade of unrelated and uninspiring concepts: e^x's, logarithms, complex numbers, and parabolas. But now that I've spent time tutoring high school students and TAing college physics classes, I've found some basic explanations of Algebra II topics ("Multiplying by an imaginary number causes the number to rotate, just like multiplying by a real number causes it to stretch or shrink," "A logarithm grabs the exponent and pulls it down as a coefficient," etc.) that are big lightbulb-moments for my students. When I tutored a student in AP Calculus AB, he was only able to give me highly technical, textbook-y definitions of what derivatives and integrals do, and the most important thing I had to teach him before he could start solving calculus word problems was the intuitive definitions of what derivatives and integrals do, something he hadn't picked up during his (almost) year of calculus. And teaching him this way really paid off; he passed the AP test when he wasn't at all on track for that a month before the test.
I see similar patterns in college-level physics. I've met too many third- and fourth-year physics majors who don't understand why Taylor series expansions work (and it clearly handicaps them when trying to do their homework), or when I ask my quantum mechanics students "Can anyone explain what a Fourier transform does?" the best they can give me is a formula - and you need a much deeper understanding than that if you are to have any hope of mastering basic quantum mechanics.
I suppose your reply to this might be that I'm teaching them /after/ they've already had the "rote practice" from their prior classes, and now they were ready to learn things more deeply once they could do the basic operations. But I don't see how learning the "deeper understanding" would have gotten in the way of their learning the first time - sure, I can see how asking 3rd graders for too much verbiage with a multiplication word problem is a hindrance, but with things as unnatural as logarithms, complex numbers, and second derivatives, they have to have a deeper understanding in order to use them in word problems (at least once their memorization from class has started to fade); the word problems are too strange and disconnected from their everyday math for just memorized skills to get them very far.
There are levels of understanding. At the foundational level, one has to keep the momentum going and not stress over every step and nuance of a derivation. What is important is the context of the concept. E.g., what does fractional division represent, what types of problems does it solve? There are ways to illustrate it by using whole number divided by a fraction, but the fraction divided by fraction derivation requires some tools of algebra, and I have found it's best to hold off on that derivation until algebra. Even then, my brightest students found the algebraic explanation confusing.
Obviously at upper high school, or at college level, students are further along on the novice/expert scale than are students in K-6 and can accommodate more explanation of the concept. I find that reform math at the K-8 grades obsesses over understanding which serves to obscure the procedural methods and hold students back when they're ready to move on.
Thanks for the insight! I'm just a grad student TA who's trying to read into the pedagogy literature, so I appreciate the response from someone who has a lot of experience at the K-12 level.
I work as an Education Assistant, but my training was in accounting and finance, and I see firsthand how the quest for comprehension affects math proficiency. Students are given all manner of questions to build comprehension, and they are one off questions which are never repeated; hence, they never get the required practice to become proficient in the procedural skills of a content area. For example, they are never encouraged to learn their multiplication tables, and they are taught to skip count and do arrays to understand the concept of multiplication, but never to practice doing multiplication until it becomes routine. The effect - difficulty in learning fraction arithmetic, and learning to become proficient in factorising. I was helping my daughter to upgrade her math to university math evel, and she remarked that the only way that she could become proficient with fact/orization is if she learns her multiplication tables by heart.
I follow my own algebra teacher’s reminder said with a twinkle in his eye. I assumed it was just a bit of doggerel but learned its provenance when reading The Charge of the Light Brigade. Sharing that insight with others, I call it Tennyson’s Rule:
“Ours is not to reason why; just invert and multiply."
I find this a compelling argument for putting practice and competence before "deeper understanding" in K-8 math. However, I wonder if there's a transition point when the reform-style "deeper understanding" becomes more important to teach first. When I was in high school, I found Algebra II a parade of unrelated and uninspiring concepts: e^x's, logarithms, complex numbers, and parabolas. But now that I've spent time tutoring high school students and TAing college physics classes, I've found some basic explanations of Algebra II topics ("Multiplying by an imaginary number causes the number to rotate, just like multiplying by a real number causes it to stretch or shrink," "A logarithm grabs the exponent and pulls it down as a coefficient," etc.) that are big lightbulb-moments for my students. When I tutored a student in AP Calculus AB, he was only able to give me highly technical, textbook-y definitions of what derivatives and integrals do, and the most important thing I had to teach him before he could start solving calculus word problems was the intuitive definitions of what derivatives and integrals do, something he hadn't picked up during his (almost) year of calculus. And teaching him this way really paid off; he passed the AP test when he wasn't at all on track for that a month before the test.
I see similar patterns in college-level physics. I've met too many third- and fourth-year physics majors who don't understand why Taylor series expansions work (and it clearly handicaps them when trying to do their homework), or when I ask my quantum mechanics students "Can anyone explain what a Fourier transform does?" the best they can give me is a formula - and you need a much deeper understanding than that if you are to have any hope of mastering basic quantum mechanics.
I suppose your reply to this might be that I'm teaching them /after/ they've already had the "rote practice" from their prior classes, and now they were ready to learn things more deeply once they could do the basic operations. But I don't see how learning the "deeper understanding" would have gotten in the way of their learning the first time - sure, I can see how asking 3rd graders for too much verbiage with a multiplication word problem is a hindrance, but with things as unnatural as logarithms, complex numbers, and second derivatives, they have to have a deeper understanding in order to use them in word problems (at least once their memorization from class has started to fade); the word problems are too strange and disconnected from their everyday math for just memorized skills to get them very far.
There are levels of understanding. At the foundational level, one has to keep the momentum going and not stress over every step and nuance of a derivation. What is important is the context of the concept. E.g., what does fractional division represent, what types of problems does it solve? There are ways to illustrate it by using whole number divided by a fraction, but the fraction divided by fraction derivation requires some tools of algebra, and I have found it's best to hold off on that derivation until algebra. Even then, my brightest students found the algebraic explanation confusing.
Obviously at upper high school, or at college level, students are further along on the novice/expert scale than are students in K-6 and can accommodate more explanation of the concept. I find that reform math at the K-8 grades obsesses over understanding which serves to obscure the procedural methods and hold students back when they're ready to move on.
Thanks for the insight! I'm just a grad student TA who's trying to read into the pedagogy literature, so I appreciate the response from someone who has a lot of experience at the K-12 level.
I work as an Education Assistant, but my training was in accounting and finance, and I see firsthand how the quest for comprehension affects math proficiency. Students are given all manner of questions to build comprehension, and they are one off questions which are never repeated; hence, they never get the required practice to become proficient in the procedural skills of a content area. For example, they are never encouraged to learn their multiplication tables, and they are taught to skip count and do arrays to understand the concept of multiplication, but never to practice doing multiplication until it becomes routine. The effect - difficulty in learning fraction arithmetic, and learning to become proficient in factorising. I was helping my daughter to upgrade her math to university math evel, and she remarked that the only way that she could become proficient with fact/orization is if she learns her multiplication tables by heart.