Thanks for your post! It was a great read and I love the idea of doing thought experiments!
I see the situations a bit differently. I think mastery of math facts is essential. Students who don't know their multiplication tables will naturally struggle with other math operations. For example, simplifying the fraction 21/56 becomes tricky if you don't know your 7 times table. A student who doesn't know their 7 times table will struggle with this question, even if you give them a calculator. "Trial and error" is the only way they can use the calculator to discover that both numbers are multiples of 7. In general, for any kind of factoring problem, calculators don't buy you much... they aren't a particularly good substitute for knowing your math facts by heart.
I've seen a number of papers that suggest that mastery of math facts is associated with enhanced math learning and problem-solving performance (e.g., Cumming & Elkins, 1999, Lin & Kubina, 2005) and even predicts academic success at the college/university level (e.g., Powell et al., 2020, Hartman & Nelson, 2016). Research also indicates that low-achieving math students experience significant sustained improvement in standardized test scores after developing an automatic recall of math facts (Pegg, Graham & Bellert, 2005, Stickney, Sharp & Kenyon, 2012).
Hi Michael, here's a multiplication intervention involving adolescents. It was for a policy thinktank and not perfect but we demonstrated (among other things) a clear link between pupils having automatic (or even fast) recall of number facts and the ability to answer a range of arithmetic questions. http://www.parliamentstreet.org/wp-content/uploads/2019/09/Maths-Revolution-20-Aug-APPROVED.pdf
I love the post but I would emphasize your final point even more, that knowing multiplication facts helps students to learn lots of other things. It's definitely true for fractions but I think it's true for much more than that. I think that should be the thesis for why math facts are important. My corollary is that you don't have to learn every math fact. I can teach lots of stuff if students just know 2s, 3s, 4s, 5s, and 10s really well.
I think of math facts as a sandbox for students to learn new math in. In 7th grade we solve equations like a(x+b)=c. I want students to eventually be able to solve equations like 3.1(x-0.5)=176.7, and for that they would use a calculator. But if I introduced the topic with examples like that students would get lost and struggle to understand the basic structure of the equation. If I start with 3(x-1)=30 and students know that 3x10=30, they are much more likely to understand the structure of the equation and then generalize to bigger numbers, fractions, decimals, etc. There are plenty of examples using easy numbers (and avoiding stuff like 9x7, 7x8, etc) to get the concept down before moving on to tougher numbers.
I would guess that, looking at my curriculum in 7th grade, about 70% of it relies significantly on math facts. I just glanced at an Algebra I curriculum outline and I would estimate it at 40% though Algebra I teachers I know spend so much time remediating equations skills that it might be more like 50-60% in practice. I can think of lots of other similar topics. I teach a bit of personal finance, and I try to be really thoughtful about picking easy numbers when I'm introducing a topic to help students understand the big idea without getting lost because they lack fact knowledge.
I love how you structured this post. The thought experiment was a great one, and you make some interesting points. I heard some other good ideas in the comments too!
One aspect that was missing from your original post was student perception. A lot of kids get stuck in 3rd and 4th grade when we require them to memorize the multiplication tables. They don't understand why they can't do it, and they start to think of themselves as incapable of doing math. I have found that helping kids learn SOME of the multiplication tables can do a lot of good when it comes to their willingness to engage in math learning, and their opinion of their own efficacy. That alone is a good reason to do it.
One technique I use is to pick a "number of the day" when working with students who don't have multiplication fluency. We start the day reviewing the table for that one number, and we leave the facts up on the board. (I usually pick 2, 3, 4, or 5, since those are the most important ones.) Then, I teach the grade-level topic that I planned to teach, but only using examples with that one number. So, if I'm teaching how to add fractions, and the number of the day is 3, then my examples might be 1/3 + 1/6 or 2/9 + 1/9. Or if I'm teaching how to solve equations, I might give them 3x + 1 = 22 or 6x = 18. Every multiplication fact that comes up that day is already on the board, and has been reviewed earlier that period. It helps kids who lack that confidence to feel much more empowered. And that sort of constant repetition within the period helps with retention too. I did this last year with some high school students in intervention who usually just put their head down and refused to do any math. On the days we did this, they were much more willing to give it a shot. (And they were more capable then they tended to give themselves credit for.)
I also strongly agree with Dylan that learning SOME facts is critical, but knowing ALL of them is unnecessary. I would say that knowing most of the 2, 3, 5, and 10 tables is critical. When you learn the basic concept, you need to work with examples where the numbers make sense to you, so it needs to be true that some numbers make sense to you. After you learn the concepts, then you can move on to bigger numbers using a calculator to help you.
Michael - I appreciate this thought experiment. Where my mind goes right away is the idea of cognitive load. Is it reasonable to think that students who are more comfortable with multiplication, those who don't have to think much about it, can conserve their energy for more challenging tasks as a result?
This is really interesting. Love a good thought experiment.
I struggled with mental arithmetic in primary school and pretty much wrote off the idea I’d ever be good at maths. But in secondary school I found I needed a lot less mental arithmetic (plus we were allowed calculators back then). It made me reconsider maths and I did it right up until A Level at sixth form (I’m not sure what your equivalent is there, sorry) and even into the first year of my combined arts degree. I found I had a genuine love for (and even a small amount of skill in) maths. But my struggles with mental arithmetic had stopped me (and my primary school teachers) seeing it. I’m still not fast with my tables. I have to use little tricks to get to the answer sometimes.
My eldest was very fast with his tables but by secondary school became disillusioned with maths as a subject. I think he’d hoped there would be a LOT more times tables tests (at which he excelled) and a lot less algebra and trig. Still he got a decent GCSE grade. Didn’t want to take it at A Level mind you. I think he’d rather have eaten his own toenail clippings. He is hoping to study psychology at uni so I actually think a little maths could be quite useful, but what do I know; I’m only a mum. 😄
My youngest is currently home educated and I’m trying to strike a balance. It’s harder than one might think (she’s autistic and struggles a tad with what I suspect is her ‘working memory’).
I’m curious what you see in middle school. I find that giving students multiplication tables to reference every single day is more effective than having them sit down with flashcards and learn the multiplication facts in isolation. I see quicker fluency when they practice referring to their table while applying it to a larger, mathematical concept.
Here in Australia we have Tech Free and Tech Active exams: No calculator and Calculator exams. Obviously, students struggling with multiplication facts will struggle without a calculator.
You’re correct that this exhibits in skills with fractions, which ultimately impacts middle and senior school subjects.
I notice a heavy reliance on calculators for what should be mental arithmetic. Possibly because we’ve stopped practicing mental arithmetic: a teacher calling out ten problems, and students calculating in their heads.
As you may know, I've always thought the obsession with math facts was overrated. I teach high school, and the vast majority of the students know most of their math facts, but not all. And it's fine. It's a bit of a struggle when they are learning factoring, but as you say, these are kids who struggle with math anyway.
I wrote once about two kids I taught who I *knew* had IQs less than 90, and one of them had every math fact down cold but couldn't do any abstractions. The other knew math facts to some degree and could solve to some degree, but everything left his head if he didn't do it for two days, and he couldn't build any knowledge.
It's just..not important. Like the whole science of reading mania, the obsession with math facts being The Answer will pass.
Thanks for your post! It was a great read and I love the idea of doing thought experiments!
I see the situations a bit differently. I think mastery of math facts is essential. Students who don't know their multiplication tables will naturally struggle with other math operations. For example, simplifying the fraction 21/56 becomes tricky if you don't know your 7 times table. A student who doesn't know their 7 times table will struggle with this question, even if you give them a calculator. "Trial and error" is the only way they can use the calculator to discover that both numbers are multiples of 7. In general, for any kind of factoring problem, calculators don't buy you much... they aren't a particularly good substitute for knowing your math facts by heart.
I've seen a number of papers that suggest that mastery of math facts is associated with enhanced math learning and problem-solving performance (e.g., Cumming & Elkins, 1999, Lin & Kubina, 2005) and even predicts academic success at the college/university level (e.g., Powell et al., 2020, Hartman & Nelson, 2016). Research also indicates that low-achieving math students experience significant sustained improvement in standardized test scores after developing an automatic recall of math facts (Pegg, Graham & Bellert, 2005, Stickney, Sharp & Kenyon, 2012).
Here are some of the papers:
https://www.tandfonline.com/doi/abs/10.1080/135467999387289
https://link.springer.com/article/10.1007/s10864-005-2703-z
https://pubs.rsc.org/en/content/articlehtml/2020/rp/d0rp00006j
https://arxiv.org/ftp/arxiv/papers/1608/1608.05006.pdf
https://files.eric.ed.gov/fulltext/ED496946.pdf
https://journals.sagepub.com/doi/abs/10.1177/1534508411430321?casa_token=9W8DdHmcJIUAAAAA%3Ap9CKOIRJqEIy_FOeQ8s4WhzPZAkTzq3CJ1KtdltDs5VRfTOxkt6S_wsLINn9_UkMyaTkCC9QAOVDFOA&journalCode=aeib
https://www.iejme.com/download/designing-mathematics-standards-in-agreement-with-science-13179.pdf
Hi Michael, here's a multiplication intervention involving adolescents. It was for a policy thinktank and not perfect but we demonstrated (among other things) a clear link between pupils having automatic (or even fast) recall of number facts and the ability to answer a range of arithmetic questions. http://www.parliamentstreet.org/wp-content/uploads/2019/09/Maths-Revolution-20-Aug-APPROVED.pdf
I love the post but I would emphasize your final point even more, that knowing multiplication facts helps students to learn lots of other things. It's definitely true for fractions but I think it's true for much more than that. I think that should be the thesis for why math facts are important. My corollary is that you don't have to learn every math fact. I can teach lots of stuff if students just know 2s, 3s, 4s, 5s, and 10s really well.
I think of math facts as a sandbox for students to learn new math in. In 7th grade we solve equations like a(x+b)=c. I want students to eventually be able to solve equations like 3.1(x-0.5)=176.7, and for that they would use a calculator. But if I introduced the topic with examples like that students would get lost and struggle to understand the basic structure of the equation. If I start with 3(x-1)=30 and students know that 3x10=30, they are much more likely to understand the structure of the equation and then generalize to bigger numbers, fractions, decimals, etc. There are plenty of examples using easy numbers (and avoiding stuff like 9x7, 7x8, etc) to get the concept down before moving on to tougher numbers.
I would guess that, looking at my curriculum in 7th grade, about 70% of it relies significantly on math facts. I just glanced at an Algebra I curriculum outline and I would estimate it at 40% though Algebra I teachers I know spend so much time remediating equations skills that it might be more like 50-60% in practice. I can think of lots of other similar topics. I teach a bit of personal finance, and I try to be really thoughtful about picking easy numbers when I'm introducing a topic to help students understand the big idea without getting lost because they lack fact knowledge.
I love how you structured this post. The thought experiment was a great one, and you make some interesting points. I heard some other good ideas in the comments too!
One aspect that was missing from your original post was student perception. A lot of kids get stuck in 3rd and 4th grade when we require them to memorize the multiplication tables. They don't understand why they can't do it, and they start to think of themselves as incapable of doing math. I have found that helping kids learn SOME of the multiplication tables can do a lot of good when it comes to their willingness to engage in math learning, and their opinion of their own efficacy. That alone is a good reason to do it.
One technique I use is to pick a "number of the day" when working with students who don't have multiplication fluency. We start the day reviewing the table for that one number, and we leave the facts up on the board. (I usually pick 2, 3, 4, or 5, since those are the most important ones.) Then, I teach the grade-level topic that I planned to teach, but only using examples with that one number. So, if I'm teaching how to add fractions, and the number of the day is 3, then my examples might be 1/3 + 1/6 or 2/9 + 1/9. Or if I'm teaching how to solve equations, I might give them 3x + 1 = 22 or 6x = 18. Every multiplication fact that comes up that day is already on the board, and has been reviewed earlier that period. It helps kids who lack that confidence to feel much more empowered. And that sort of constant repetition within the period helps with retention too. I did this last year with some high school students in intervention who usually just put their head down and refused to do any math. On the days we did this, they were much more willing to give it a shot. (And they were more capable then they tended to give themselves credit for.)
I also strongly agree with Dylan that learning SOME facts is critical, but knowing ALL of them is unnecessary. I would say that knowing most of the 2, 3, 5, and 10 tables is critical. When you learn the basic concept, you need to work with examples where the numbers make sense to you, so it needs to be true that some numbers make sense to you. After you learn the concepts, then you can move on to bigger numbers using a calculator to help you.
Thanks again for an interesting discussion.
Michael - I appreciate this thought experiment. Where my mind goes right away is the idea of cognitive load. Is it reasonable to think that students who are more comfortable with multiplication, those who don't have to think much about it, can conserve their energy for more challenging tasks as a result?
This is really interesting. Love a good thought experiment.
I struggled with mental arithmetic in primary school and pretty much wrote off the idea I’d ever be good at maths. But in secondary school I found I needed a lot less mental arithmetic (plus we were allowed calculators back then). It made me reconsider maths and I did it right up until A Level at sixth form (I’m not sure what your equivalent is there, sorry) and even into the first year of my combined arts degree. I found I had a genuine love for (and even a small amount of skill in) maths. But my struggles with mental arithmetic had stopped me (and my primary school teachers) seeing it. I’m still not fast with my tables. I have to use little tricks to get to the answer sometimes.
My eldest was very fast with his tables but by secondary school became disillusioned with maths as a subject. I think he’d hoped there would be a LOT more times tables tests (at which he excelled) and a lot less algebra and trig. Still he got a decent GCSE grade. Didn’t want to take it at A Level mind you. I think he’d rather have eaten his own toenail clippings. He is hoping to study psychology at uni so I actually think a little maths could be quite useful, but what do I know; I’m only a mum. 😄
My youngest is currently home educated and I’m trying to strike a balance. It’s harder than one might think (she’s autistic and struggles a tad with what I suspect is her ‘working memory’).
I’m curious what you see in middle school. I find that giving students multiplication tables to reference every single day is more effective than having them sit down with flashcards and learn the multiplication facts in isolation. I see quicker fluency when they practice referring to their table while applying it to a larger, mathematical concept.
What do you think?
Here in Australia we have Tech Free and Tech Active exams: No calculator and Calculator exams. Obviously, students struggling with multiplication facts will struggle without a calculator.
You’re correct that this exhibits in skills with fractions, which ultimately impacts middle and senior school subjects.
I notice a heavy reliance on calculators for what should be mental arithmetic. Possibly because we’ve stopped practicing mental arithmetic: a teacher calling out ten problems, and students calculating in their heads.
As you may know, I've always thought the obsession with math facts was overrated. I teach high school, and the vast majority of the students know most of their math facts, but not all. And it's fine. It's a bit of a struggle when they are learning factoring, but as you say, these are kids who struggle with math anyway.
I wrote once about two kids I taught who I *knew* had IQs less than 90, and one of them had every math fact down cold but couldn't do any abstractions. The other knew math facts to some degree and could solve to some degree, but everything left his head if he didn't do it for two days, and he couldn't build any knowledge.
It's just..not important. Like the whole science of reading mania, the obsession with math facts being The Answer will pass.
Great piece.