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.
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.
Feb 4·edited Feb 4Liked by Jim Hewitt, Nidhi Sachdeva
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.
This article reminds me of cognitive load theory (which it talks about) and Bloom’s Revised Taxonomy (which it doesn't).
From my experience as a student and teacher, I've seen how much knowing the basics matters while trying to learn or teach something, especially when it comes to understanding, application, analysis, evaluation, or creation. Creativity's undoubtedly important – just like Professor Jo Boaler says in her TED talk (here's the link: https://youtu.be/al6gO9SLqBY?si=jx7tOTLJeOthBMQM). Meanwhile, in my over 20 years in the educational system, I found that most students could perform better by improving their foundation knowledge.
I think front-line teachers do not include enough learning activities to make memorizing basic knowledge easier, and the over-emphasis on creativity may be one of the reasons. If teachers adopt a holistic approach and keep the ultimate goal – helping students obtain knowledge and skills and excel in final assessments – in mind, teaching can become more effective.
Students need to understand the concepts of number facts and times tables otherwise they are limited in their knowledge of how to use them. Seeing these abstract concepts with Numicon shapes and Cuisenaire rods makes a HUGE difference.
Thanks Margi -- great post. I love these kinds of hands-on tools that help students develop a concrete understanding of multiplication and other math concepts. As you say, the conceptual grounding is essential.
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.
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.
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!
This article reminds me of cognitive load theory (which it talks about) and Bloom’s Revised Taxonomy (which it doesn't).
From my experience as a student and teacher, I've seen how much knowing the basics matters while trying to learn or teach something, especially when it comes to understanding, application, analysis, evaluation, or creation. Creativity's undoubtedly important – just like Professor Jo Boaler says in her TED talk (here's the link: https://youtu.be/al6gO9SLqBY?si=jx7tOTLJeOthBMQM). Meanwhile, in my over 20 years in the educational system, I found that most students could perform better by improving their foundation knowledge.
I think front-line teachers do not include enough learning activities to make memorizing basic knowledge easier, and the over-emphasis on creativity may be one of the reasons. If teachers adopt a holistic approach and keep the ultimate goal – helping students obtain knowledge and skills and excel in final assessments – in mind, teaching can become more effective.
Students need to understand the concepts of number facts and times tables otherwise they are limited in their knowledge of how to use them. Seeing these abstract concepts with Numicon shapes and Cuisenaire rods makes a HUGE difference.
Thanks Margi -- great post. I love these kinds of hands-on tools that help students develop a concrete understanding of multiplication and other math concepts. As you say, the conceptual grounding is essential.