If I were in a flippant mood, I would ask those who make light of memorization to trust their lives or those of loved ones to a surgeon who goes to Google at every step of a procedure.
As an academic psychologist all I can say is yes! Yes! Yes! The important thing about education is getting the students to build knowledge structures. Reading, language use and mathematical reasoning are cognitive tools that are not only essential in their own right, but lead to knowledge structures that are integrated with and indeed are foundations of knowledge structures in other domains.
As a high school math teacher, I can tell you that everyone else has it wrong. Memorization is key to success in mathematics, and I don't understand why everyone is against it. In my algebra classes, the kids who have arithmetic of single-digit numbers on automatic are far more successful than the students who have to rely on counting or other "strategies." It is testable, it is quantifiable, yet these clear results are ignored. It boggles my mind.
Great post! As a middle school English teacher, I can tell you that many students do not know their basic math facts. Ask them to computer simple figures--basically anything on a classic multiplication table--and they can't. For years students have lectured *us* because we have analogue clocks. And they can't read them.
This can't be too far apart from knowledge building on the English side of things.
We quit teaching spelling for the same logic: spell checkers and "memorization is bad." I've kept my own database of misspellings for the past four school years, and the average person would be absolutely horrified. Even many future AP students would flunk elementary spelling from the 1990's.
But it's system-wide. Fad after fad have kicked out the ladder from future generations.
(I remember an article years back about a Nobel Prize winner arguing (a) revision of Math standards, and (b) calculators for elementary students. What nonsense! Then again, years of complaining "More teachers should blog" is why I finally started a Substack...)
I agree -- it's quite interesting. As you say, it's fad after fad in which teachers are told that it's bad to ask students memorize multiplication tables or the spelling of words. I don't think people fully appreciate that it's empowers children when they know these things.
I am scared to see what my AP Calc class is going to be like in about 3 years. Our school district just bought the WORST curriculum for math I have ever seen: Illustrative Mathematics (IM). I honestly question whether or not pre-calc and calc will still be around in the high schools because I do NOT see how the kids will be ready for the rigor of the algebraic processes of either class after being prepped by IM. The dumbing down of America rolls on.
Those in charge don’t have the knowledge base to make such decisions. Education is full of idiots with diploma mill doctorates that think they know everything.. they rule with top/down leadership because they have no leadership skills. They weren’t good teachers and quite frankly they are ruining the country.
Don't ask math experts. Ask the neuro developmental scientists. We need kids to learn this so their brain learns how to learn. The brain is plastic & has windows of opportunity when it is open to being changed /structured in certain ways. Rote learning of multiplication tables does this.
Most articles that deemphasize memorization automatically assume that those who memorize do not understand (or try to understand), as if memorization and understanding were divorced from each other. A fallacy.
Along with this assumption, they usually use the term "rote learning" -- which has negative connotations for most people, and is associated with mindless, stultifying repetition.
One can memorize AND understand, and one can understand AND memorize. The automacity this brings -- I sometimes think of it as a "second-naturing" of math facts, that is, making them second nature to one -- often allows memorizers to perform far better and faster than those who don't. A student who sees a right-triangle problem with a given hypotenuse of 29 units is going to immediately consider the possibility of the other two sides being 20 and 21 if s/he has the squares memorized. Indeed, those who do extremely well in tests usually HAVE such facts memorized. Does the anti-memorization squad believe automatically that such students DO NOT understand what they have studied?
A word about repetition. In the realm of physical skills -- be it golf, baseball, cricket or gymnastics -- repetition is taken for granted. A Sachin Tendulkar who scores the required sixers in the last few balls of a match, taking his team to victory, has hundreds of thousands of strokes of various kinds internalized through daily practice on a suspended ball or with a bowling machine. No one makes light of that! Or of a gymnast performing on the tumbling mat, or of a trapeze artiste putting his fate in the reflexes, palms, grip strength and above all -- repetitive practice -- of a partner!
Repetitio est mater studiorum! Repetition is the mother of learning. Also, I'd add, regular revision.
A recounting of a well-known anecdote before I go, about so-called taxicab numbers, so named by mathematician G.H. Hardy, who mentored Srinivas Ramanujan.
(Hardy, I have always thought, was given to dramatizing things a bit, so the story might have been spruced and polished up by him for public consumption!)
Anyway, the story is that, visiting Ramanujan, Hardy mentions that his taxicab had a "boring" number -- 1729. Ramanujan promptly reminds him it's not boring because it's the smallest number that can be expressed as the sum of two cubes in two different ways (one cubed and 12 cubed AND ten cubed and nine cubed).
Thanks for your post! I agree completely. Like you, I am also struck by some people's strange assumption that students can't both understand AND memorize. I think the problem is that most teachers don't appreciate the tremendous value of memorizing certain math facts and procedures. I would go as far as suggesting that elementary teachers who deliberately choose NOT to foster memorization are significantly disadvantaging their students and making it less likely that they will purse a career in the maths and sciences. If you don't have a certain level of automaticity with basic facts by grade 7 or 8, the more advanced math becomes much more difficult. I'm not sure that grade 6 teachers fully appreciate that automaticity with math facts is really about helping their students have enough mental bandwidth to cope with high school algebra and calculus.
Indeed, sir, I agree with you that "teachers who deliberately choose NOT to foster memorization are significantly disadvantaging their students."
I deliberately chose to avoid including "elementary" at the beginning of that quote from you, by the way! :-)
And I think your general point applies to a wide range of subjects. A history or literature student who has the dates/years of important events stored in his memory has a far better perspective than one who doesn't remember that, say, the Big Bang was 13.8 billion years ago, or that Shakespeare lived from 1564-1616.
Thank you (both authors) for your instructive columns. I stumbled upon them a couple of days ago and am very glad I did!
Certain facts must be memorized, however, it must be accompanied with a strong understanding of the fundamentals. For instance, some kids understand the multiplication table better when i approach it from the angle of element groupings (using math tiles) or repeated addition. Is that practical in the real world when they are required to produce fast food solutions?
Hey Edmond, couldn't agree more. Having a strong understanding of foundational concepts is essential before moving on to automatizing them. Ideally, automaticity should precede understanding and being clear about how things work. Thank you for sharing your thoughts and engaging with the post.
learned something similar to this from “Thinking Fast & Slow”. Putting these into system one memory lets you access it from the cheaper part of the brain leaving the expensive system two for more difficult tasks.
Yes, memorize. I have a funny story about this. My son was seated on the love seat going over his multiplication flash cards. Our rabbit Pumpkin sat next to him. Every time my son put a card down, the rabbit picked up the card in his mouth and laid it down on its other side.
When I was in school, memorising maths tables was mandatory and I memorised multiplication tables from 1 to 30. Later on using the calculators and PCs made me forget everything.
I am good at memorizing and I think I have achieved automaticity in the multiplication tables up to 12. I just know what an answer is without thinking.
Our twelve year old son hasn’t reached that point yet. Our nine year old son’s brain seems to be wired more like mine in that he memorizes more easily and then just recalls.
I homeschool our kids and so this intrigues me. I would see how our older son struggles unnecessarily, just because he has to think about the answer of basic multiplication before he can continue with the rest of the problem.
I asked my husband (who is an iOS dev) to write an app for our kids. The kids give their input and requests as well haha.
The kids (even our seven year old) are really enjoying it and they are improving!
It’s going to be a fun journey. I want our oldest to be free to enjoy maths and life.
Memorizing 1-10 times table is useful in the same way as memorizing every single digit combination. Everything beyond that as a math teacher I find pointless.
Let me explain my rationale.
Everything beyond base ten you can teach a student to break into base ten:
16×17=(10+6)(10+7)=(10×10)+(10×7)+(10×6)+(6×7)=100+70+60+42=272. You can display this in a square and fill the insides in. I use this as way to display how FOIL multiplication works with concrete numbers and make it more relatable when we do it with variables. It shows students the value and utility of base ten. Makes them better at mental math which reduces lag time when doing math work and is interesting and relatively simple once you know your times table. Breaking up 16 to 10+6 is a good introduction to more clever algebra strategies as problems get more complicated. I don’t know I hate memorizing a lot but it definitely has a place in education.
If I were in a flippant mood, I would ask those who make light of memorization to trust their lives or those of loved ones to a surgeon who goes to Google at every step of a procedure.
As an academic psychologist all I can say is yes! Yes! Yes! The important thing about education is getting the students to build knowledge structures. Reading, language use and mathematical reasoning are cognitive tools that are not only essential in their own right, but lead to knowledge structures that are integrated with and indeed are foundations of knowledge structures in other domains.
Thank you for sharing this, Graham.
As a high school math teacher, I can tell you that everyone else has it wrong. Memorization is key to success in mathematics, and I don't understand why everyone is against it. In my algebra classes, the kids who have arithmetic of single-digit numbers on automatic are far more successful than the students who have to rely on counting or other "strategies." It is testable, it is quantifiable, yet these clear results are ignored. It boggles my mind.
Great post! As a middle school English teacher, I can tell you that many students do not know their basic math facts. Ask them to computer simple figures--basically anything on a classic multiplication table--and they can't. For years students have lectured *us* because we have analogue clocks. And they can't read them.
This can't be too far apart from knowledge building on the English side of things.
We quit teaching spelling for the same logic: spell checkers and "memorization is bad." I've kept my own database of misspellings for the past four school years, and the average person would be absolutely horrified. Even many future AP students would flunk elementary spelling from the 1990's.
But it's system-wide. Fad after fad have kicked out the ladder from future generations.
(I remember an article years back about a Nobel Prize winner arguing (a) revision of Math standards, and (b) calculators for elementary students. What nonsense! Then again, years of complaining "More teachers should blog" is why I finally started a Substack...)
I agree -- it's quite interesting. As you say, it's fad after fad in which teachers are told that it's bad to ask students memorize multiplication tables or the spelling of words. I don't think people fully appreciate that it's empowers children when they know these things.
I am scared to see what my AP Calc class is going to be like in about 3 years. Our school district just bought the WORST curriculum for math I have ever seen: Illustrative Mathematics (IM). I honestly question whether or not pre-calc and calc will still be around in the high schools because I do NOT see how the kids will be ready for the rigor of the algebraic processes of either class after being prepped by IM. The dumbing down of America rolls on.
Those in charge don’t have the knowledge base to make such decisions. Education is full of idiots with diploma mill doctorates that think they know everything.. they rule with top/down leadership because they have no leadership skills. They weren’t good teachers and quite frankly they are ruining the country.
Don't ask math experts. Ask the neuro developmental scientists. We need kids to learn this so their brain learns how to learn. The brain is plastic & has windows of opportunity when it is open to being changed /structured in certain ways. Rote learning of multiplication tables does this.
Most articles that deemphasize memorization automatically assume that those who memorize do not understand (or try to understand), as if memorization and understanding were divorced from each other. A fallacy.
Along with this assumption, they usually use the term "rote learning" -- which has negative connotations for most people, and is associated with mindless, stultifying repetition.
One can memorize AND understand, and one can understand AND memorize. The automacity this brings -- I sometimes think of it as a "second-naturing" of math facts, that is, making them second nature to one -- often allows memorizers to perform far better and faster than those who don't. A student who sees a right-triangle problem with a given hypotenuse of 29 units is going to immediately consider the possibility of the other two sides being 20 and 21 if s/he has the squares memorized. Indeed, those who do extremely well in tests usually HAVE such facts memorized. Does the anti-memorization squad believe automatically that such students DO NOT understand what they have studied?
A word about repetition. In the realm of physical skills -- be it golf, baseball, cricket or gymnastics -- repetition is taken for granted. A Sachin Tendulkar who scores the required sixers in the last few balls of a match, taking his team to victory, has hundreds of thousands of strokes of various kinds internalized through daily practice on a suspended ball or with a bowling machine. No one makes light of that! Or of a gymnast performing on the tumbling mat, or of a trapeze artiste putting his fate in the reflexes, palms, grip strength and above all -- repetitive practice -- of a partner!
Repetitio est mater studiorum! Repetition is the mother of learning. Also, I'd add, regular revision.
A recounting of a well-known anecdote before I go, about so-called taxicab numbers, so named by mathematician G.H. Hardy, who mentored Srinivas Ramanujan.
(Hardy, I have always thought, was given to dramatizing things a bit, so the story might have been spruced and polished up by him for public consumption!)
Anyway, the story is that, visiting Ramanujan, Hardy mentions that his taxicab had a "boring" number -- 1729. Ramanujan promptly reminds him it's not boring because it's the smallest number that can be expressed as the sum of two cubes in two different ways (one cubed and 12 cubed AND ten cubed and nine cubed).
Did Ramanujan just work it out then and there?
Thanks for your post! I agree completely. Like you, I am also struck by some people's strange assumption that students can't both understand AND memorize. I think the problem is that most teachers don't appreciate the tremendous value of memorizing certain math facts and procedures. I would go as far as suggesting that elementary teachers who deliberately choose NOT to foster memorization are significantly disadvantaging their students and making it less likely that they will purse a career in the maths and sciences. If you don't have a certain level of automaticity with basic facts by grade 7 or 8, the more advanced math becomes much more difficult. I'm not sure that grade 6 teachers fully appreciate that automaticity with math facts is really about helping their students have enough mental bandwidth to cope with high school algebra and calculus.
Indeed, sir, I agree with you that "teachers who deliberately choose NOT to foster memorization are significantly disadvantaging their students."
I deliberately chose to avoid including "elementary" at the beginning of that quote from you, by the way! :-)
And I think your general point applies to a wide range of subjects. A history or literature student who has the dates/years of important events stored in his memory has a far better perspective than one who doesn't remember that, say, the Big Bang was 13.8 billion years ago, or that Shakespeare lived from 1564-1616.
Thank you (both authors) for your instructive columns. I stumbled upon them a couple of days ago and am very glad I did!
Thank you so much for engaging with our posts. We appreciate this so much and welcome healthy and informed discussions.
Certain facts must be memorized, however, it must be accompanied with a strong understanding of the fundamentals. For instance, some kids understand the multiplication table better when i approach it from the angle of element groupings (using math tiles) or repeated addition. Is that practical in the real world when they are required to produce fast food solutions?
Hey Edmond, couldn't agree more. Having a strong understanding of foundational concepts is essential before moving on to automatizing them. Ideally, automaticity should precede understanding and being clear about how things work. Thank you for sharing your thoughts and engaging with the post.
This is a good article from Barbara Oakley that goes in the same direction:
https://lawliberty.org/features/the-promise-of-habit-based-learning/
Thanks Johanne for sharing - great article!
learned something similar to this from “Thinking Fast & Slow”. Putting these into system one memory lets you access it from the cheaper part of the brain leaving the expensive system two for more difficult tasks.
Yes, memorize. I have a funny story about this. My son was seated on the love seat going over his multiplication flash cards. Our rabbit Pumpkin sat next to him. Every time my son put a card down, the rabbit picked up the card in his mouth and laid it down on its other side.
It’s weight lifting for the mind. It’s not the facts or tables it’s the “memorization” that makes a strong mind.
When I was in school, memorising maths tables was mandatory and I memorised multiplication tables from 1 to 30. Later on using the calculators and PCs made me forget everything.
Is’s a shortcut that I'm in favor of.
Hi!
This is very insightful!
I am good at memorizing and I think I have achieved automaticity in the multiplication tables up to 12. I just know what an answer is without thinking.
Our twelve year old son hasn’t reached that point yet. Our nine year old son’s brain seems to be wired more like mine in that he memorizes more easily and then just recalls.
I homeschool our kids and so this intrigues me. I would see how our older son struggles unnecessarily, just because he has to think about the answer of basic multiplication before he can continue with the rest of the problem.
I asked my husband (who is an iOS dev) to write an app for our kids. The kids give their input and requests as well haha.
The kids (even our seven year old) are really enjoying it and they are improving!
It’s going to be a fun journey. I want our oldest to be free to enjoy maths and life.
Wish I had read your piece before I wrote mine. Not sure how I missed it, but I just added it to the list of helpful articles at the end. https://laurenbrownoned.substack.com/p/memorizing-multiplication-tables
Thanks for sharing it Lauren! I am excited to read yours!
Memorizing 1-10 times table is useful in the same way as memorizing every single digit combination. Everything beyond that as a math teacher I find pointless.
Let me explain my rationale.
Everything beyond base ten you can teach a student to break into base ten:
16×17=(10+6)(10+7)=(10×10)+(10×7)+(10×6)+(6×7)=100+70+60+42=272. You can display this in a square and fill the insides in. I use this as way to display how FOIL multiplication works with concrete numbers and make it more relatable when we do it with variables. It shows students the value and utility of base ten. Makes them better at mental math which reduces lag time when doing math work and is interesting and relatively simple once you know your times table. Breaking up 16 to 10+6 is a good introduction to more clever algebra strategies as problems get more complicated. I don’t know I hate memorizing a lot but it definitely has a place in education.