17 ms·

Why are you advocating memorization? That just gets people to treat mathematics as a set of magical spells you have to receive from the holy scriptures and prie

by jey 8y ago

Why are you advocating memorization? That just gets people to treat mathematics as a set of magical spells you have to receive from the holy scriptures and priests, and that problems just can't be solved without having received the appropriate spell from the masters. (It also encourages strange beliefs like that mathematics is a subjective human construction, like art and music.) Conveying that there's a deeper conceptual "mathematical thinking" seems much more important than memorizing the algorithm for "completing the square" or whatever.

amanaplanacanal 8y ago
My impression is that mathematical thinking isn't really the goal of mathematics education until post-calculus University level. Before that it is mostly learning rules for doing symbol manipulation. Although, this isn't entirely true, I remember doing proofs in Geometry class in Jr High School. So I was at least introduced to it, though we never really used it again.
0815test 8y ago
I'm not seeing any advocacy of memorization? The only thing you might have to memorize, out of what parent listed, is the algorithms for doing written out arithmetic; and that too is in fact an exercise in mathematical thinking. After all, there's a reason we teach these algorithms, as opposed to just telling kids that they can punch numbers into a calculator.
- beaner 8y ago
  Quote: "Even for double digit addition problems they jump right into 'cool tricks' and 'mathematical thinking' while there are just really a few steps to memorize and, guess what, kids are great at memorizing stuff."
  - 0815test 8y ago
    And rdtsc is entirely right about that. The Common Core math curriculum has been taken by many self-described "math teachers" and "educators" as an opportunity to just not teach the effective algorithms for doing arithmetic, and expect students to just "discover the results by themselves" via some mixture of trial-and-error, random guessing, and the rare "trick" or perhaps tiny flash of genuine "mathematical thinking". That was an unmitigated disaster, of course. It's probably part of the reason why Algebra is now being seen as way too difficult for Junior High students, and something that only HS students could approach effectively.
    - hexane360 8y ago
      Holding off algebra to high school predates common core by decades.
    - tzs 8y ago
      Effective arithmetic algorithms are included in Common Core [1]. [1] http://www.corestandards.org/Math/Content/NBT/ http://www.corestandards.org/Math/Content/NBT/
      - 0815test 8y ago
        From your link, knowledge of the standard algorithm for addition/subtraction is only requested by Grade 4, and for multiplication by Grade 5. So the Common Core standard is basically relying on a hidden assumption that students can effectively learn all the other stuff that's expected of them up to Grade 4 and 5, without truly being fluent in addition, subtraction or multiplication! (I'm aware that the standards call for "fluency" slightly earlier, but that's mere wishful thinking until you teach the standard algorithms, or some cosmetic variant thereof such as lattice multiplication.) That's what so bad about the way CC is being applied in elementary education.
bordercases 8y ago
This debate comes up again and again, and any time one view is purported strongly over the other we just get education that is half-assed. This includes people with your point of view that deliberate attempts at committing propositions and procedures to memory should be avoided. The issue is one of cognitive load. You can't get to mathematical thinking and appeal to both a higher-level "intuition" or "key idea" if you aren't fluent in the details. Else it becomes hand-waving that doesn't cash out to proof-ready reasoning. Likewise, you can't internalize the details if you don't have a principle which compresses them and lets you unpack them appropriately for the circumstance of the problem. This is fair, but first there has to be something to compress. The question is, "How much detail can be advanced at this point in the student's growth?" If the key issue is to dispel the notion that mathematics is arcane, you should narrow down the scale of the problem so that you can quickly see how the details contribute to generalizations, and how a procedure can be taken from it. This achieves what you are looking for with less effort and less risk. With that point of view illustrated with a simpler case, it can be developed or built upon with bigger cases. Bigger cases inevitably means more details to internalize. This protracts the phase where memorization would be appropriate. But the student, having seen how mathematics is developed in a simpler case, would understand that these details have a context that will be more adequately realized at a second or third pass of the material. Eschewing the internalization of detail as you are purporting would encourage a sense that math is subjective, because the high-level impressions of mathematics would not have the memories of detail to back them up, allowing for more to be said, with less control.
souprock 8y ago
Memorization is very important. You have to walk before you can run. To perform well at algebra, you can't be stopping to punch numbers into a calculator. You can't be constantly making mistakes. The slowness and errors will make you stumble and lose your train of thought, preventing you from reaching any deeper conceptual "mathematical thinking". Those old-fashioned methods for doing math are reliable and general. If you follow the rules, you get the answer.
- sandworm101 8y ago
  That's what my typing teacher taught me about typing, that by not memorizing all the key+finger combinations I would never type (Owning a computer, I was self-taught and used all the wrong fingers). Twenty years and a law degree later, after authoring literally thousands of pages, I still cannot say which finger is supposed to hit the 'r' key. Somehow I muddled through. The point is that rather than memorize combinations of seemingly random numbers, the same knowledge will come naturally as students learn other aspects of math.
  - WalterBright 8y ago
    I can't recall the commands to my text editor. I just look at my fingers to see which keys they hit.
    - jkingsbery 8y ago
      Yes - When someone asks me what a particular key combination is in emacs or Eclipse, I put my fingers out like they're on a keyboard, think about doing the thing, and, look at what my fingers are doing. To bring it back to the article: It's similar with a lot of things - you sort of need to memorize aspects of it to do it fluently. As part of calculus, you sort of need to know how to do polynomial factorization. Intuition is great and helpful, but you also need to be able to proceed methodically in order to produce or understand proofs. Scales on piano are boring, but you don't just tell kids to feel like they're Bach and they start playing Goldberg Variations. Why on earth we think the intuition is not only necessary but sufficient is unfathomable to me.
p1esk 8y ago
"In mathematics you don't understand things. You just get used to them." -John Von Neumann
rdtsc 8y ago
> Why are you advocating memorization? Because it's something kids do really well and it's something to take advantage of when learning at that age. In other words, it's better to memorize the multiplication table then find insights into how it work later, and do proofs and make connections with calculating an area of a rectangle etc. A lot of the tricks and insights don't mean anything to the children if they didn't already do many examples the rote, repetitive way and in a lot of cases the "insights" are often a distraction as well. Those should come later. Even more interesting is when children see or discover these tricks or rules on their own, then they become really memorable to them. I don't have many samples work with, but I have observed this over the years based on my own experience, my kids and my extended family members and I have noticed the same patterns.
- coretx 8y ago
  Although I tend to mostly agree with your main argument, the example you presented ( memorizing multiplication tables ) is a bad one. My mind was blown when I found out how they teach multiplication to children in Japan. Link: ( Vedic method ) https://www.youtube.com/watch?v=z4X98Mnj0tc https://www.youtube.com/watch?v=z4X98Mnj0tc The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.
  - dhimes 8y ago
    This is similar to the "Lattice Method" that they taught (still teach?) here in the US. I can't stand it. The students simply learn an algorithm, without developing a sense of place value and an intuition for "how numbers work." Sure, they can quickly multiply large numbers and it's a cool "party trick." But they aren't learning math. Might as well use an iPhone.
    - butmath 8y ago
      Fair point that memorizing an algorithm doesn't necessarily advance the student's intuition for how numbers work, but keep in mind that you're replying on a subthread about how learning a general algorithm is an improvement over memorizing multiplication tables. Ideally, we teach children the intuition for mathematics. Just memorizing arithmetic algorithms isn't ideal, but it's surely better than memorizing finite arithmetic tables.
mensetmanusman 8y ago
I recall research showing mathematics being one of the very few subjects where one can get better at through brute memorization. It gets you quite far in that field, as it reduces recall time for proofs etc, and then it becomes like muscle memory. Practice/muscle memory in other fields, like the physical sciences, can not come from a page out of the book, since it requires interacting with nature in a multi-sensory way.
KorematsuFred 8y ago
Sorry that is pure BS. Memorization is a great technique to learn math of course it is not the way to do math. When you learn tables through rot learning you automatically get how they are being computed. At young age it is much easier to rot learn many things and understand the abstract concepts much later. People who despise rot learning are either misunderstanding it or are simply lazy.
rumcajz 8y ago
See Piaget's concrete operational stage [1] of human development: "Abstract, hypothetical thinking is not yet developed in the child, and children can only solve problems that apply to concrete events or objects." In other words, you can only push the mathematical thinking as far as the developmental stage of the child allows you to. Memorization helps though. My son is 6. He doesn't know the multiplication algorithm, but he does know that 2 x 2 = 4 or that 3 x 4 = 12. That's sufficient for him to do discoveries of his own. 3 x 4 = 3 x (2 x 2) ! Picture that! [1] https://en.wikipedia.org/wiki/Piaget's_theory_of_cognitive_development#Concrete_operational_stage https://en.wikipedia.org/wiki/Piaget's_theory_of_cognitive_d...

by jey 8y ago

amanaplanacanal 8y ago

My impression is that mathematical thinking isn't really the goal of mathematics education until post-calculus University level. Before that it is mostly learning rules for doing symbol manipulation. Although, this isn't entirely true, I remember doing proofs in Geometry class in Jr High School. So I was at least introduced to it, though we never really used it again.

0815test 8y ago

I'm not seeing any advocacy of memorization? The only thing you might have to memorize, out of what parent listed, is the algorithms for doing written out arithmetic; and that too is in fact an exercise in mathematical thinking. After all, there's a reason we teach these algorithms, as opposed to just telling kids that they can punch numbers into a calculator.

beaner 8y ago
Quote: "Even for double digit addition problems they jump right into 'cool tricks' and 'mathematical thinking' while there are just really a few steps to memorize and, guess what, kids are great at memorizing stuff."
- 0815test 8y ago
  And rdtsc is entirely right about that. The Common Core math curriculum has been taken by many self-described "math teachers" and "educators" as an opportunity to just not teach the effective algorithms for doing arithmetic, and expect students to just "discover the results by themselves" via some mixture of trial-and-error, random guessing, and the rare "trick" or perhaps tiny flash of genuine "mathematical thinking". That was an unmitigated disaster, of course. It's probably part of the reason why Algebra is now being seen as way too difficult for Junior High students, and something that only HS students could approach effectively.
  - hexane360 8y ago
    Holding off algebra to high school predates common core by decades.
  - tzs 8y ago
    Effective arithmetic algorithms are included in Common Core [1]. [1] http://www.corestandards.org/Math/Content/NBT/ http://www.corestandards.org/Math/Content/NBT/
    - 0815test 8y ago
      From your link, knowledge of the standard algorithm for addition/subtraction is only requested by Grade 4, and for multiplication by Grade 5. So the Common Core standard is basically relying on a hidden assumption that students can effectively learn all the other stuff that's expected of them up to Grade 4 and 5, without truly being fluent in addition, subtraction or multiplication! (I'm aware that the standards call for "fluency" slightly earlier, but that's mere wishful thinking until you teach the standard algorithms, or some cosmetic variant thereof such as lattice multiplication.) That's what so bad about the way CC is being applied in elementary education.

bordercases 8y ago

This debate comes up again and again, and any time one view is purported strongly over the other we just get education that is half-assed. This includes people with your point of view that deliberate attempts at committing propositions and procedures to memory should be avoided. The issue is one of cognitive load. You can't get to mathematical thinking and appeal to both a higher-level "intuition" or "key idea" if you aren't fluent in the details. Else it becomes hand-waving that doesn't cash out to proof-ready reasoning. Likewise, you can't internalize the details if you don't have a principle which compresses them and lets you unpack them appropriately for the circumstance of the problem. This is fair, but first there has to be something to compress. The question is, "How much detail can be advanced at this point in the student's growth?" If the key issue is to dispel the notion that mathematics is arcane, you should narrow down the scale of the problem so that you can quickly see how the details contribute to generalizations, and how a procedure can be taken from it. This achieves what you are looking for with less effort and less risk. With that point of view illustrated with a simpler case, it can be developed or built upon with bigger cases. Bigger cases inevitably means more details to internalize. This protracts the phase where memorization would be appropriate. But the student, having seen how mathematics is developed in a simpler case, would understand that these details have a context that will be more adequately realized at a second or third pass of the material. Eschewing the internalization of detail as you are purporting would encourage a sense that math is subjective, because the high-level impressions of mathematics would not have the memories of detail to back them up, allowing for more to be said, with less control.

souprock 8y ago

Memorization is very important. You have to walk before you can run. To perform well at algebra, you can't be stopping to punch numbers into a calculator. You can't be constantly making mistakes. The slowness and errors will make you stumble and lose your train of thought, preventing you from reaching any deeper conceptual "mathematical thinking". Those old-fashioned methods for doing math are reliable and general. If you follow the rules, you get the answer.

sandworm101 8y ago
That's what my typing teacher taught me about typing, that by not memorizing all the key+finger combinations I would never type (Owning a computer, I was self-taught and used all the wrong fingers). Twenty years and a law degree later, after authoring literally thousands of pages, I still cannot say which finger is supposed to hit the 'r' key. Somehow I muddled through. The point is that rather than memorize combinations of seemingly random numbers, the same knowledge will come naturally as students learn other aspects of math.
- WalterBright 8y ago
  I can't recall the commands to my text editor. I just look at my fingers to see which keys they hit.
  - jkingsbery 8y ago
    Yes - When someone asks me what a particular key combination is in emacs or Eclipse, I put my fingers out like they're on a keyboard, think about doing the thing, and, look at what my fingers are doing. To bring it back to the article: It's similar with a lot of things - you sort of need to memorize aspects of it to do it fluently. As part of calculus, you sort of need to know how to do polynomial factorization. Intuition is great and helpful, but you also need to be able to proceed methodically in order to produce or understand proofs. Scales on piano are boring, but you don't just tell kids to feel like they're Bach and they start playing Goldberg Variations. Why on earth we think the intuition is not only necessary but sufficient is unfathomable to me.

p1esk 8y ago

"In mathematics you don't understand things. You just get used to them." -John Von Neumann

rdtsc 8y ago

> Why are you advocating memorization? Because it's something kids do really well and it's something to take advantage of when learning at that age. In other words, it's better to memorize the multiplication table then find insights into how it work later, and do proofs and make connections with calculating an area of a rectangle etc. A lot of the tricks and insights don't mean anything to the children if they didn't already do many examples the rote, repetitive way and in a lot of cases the "insights" are often a distraction as well. Those should come later. Even more interesting is when children see or discover these tricks or rules on their own, then they become really memorable to them. I don't have many samples work with, but I have observed this over the years based on my own experience, my kids and my extended family members and I have noticed the same patterns.

coretx 8y ago
Although I tend to mostly agree with your main argument, the example you presented ( memorizing multiplication tables ) is a bad one. My mind was blown when I found out how they teach multiplication to children in Japan. Link: ( Vedic method ) https://www.youtube.com/watch?v=z4X98Mnj0tc https://www.youtube.com/watch?v=z4X98Mnj0tc The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.
- dhimes 8y ago
  This is similar to the "Lattice Method" that they taught (still teach?) here in the US. I can't stand it. The students simply learn an algorithm, without developing a sense of place value and an intuition for "how numbers work." Sure, they can quickly multiply large numbers and it's a cool "party trick." But they aren't learning math. Might as well use an iPhone.
  - butmath 8y ago
    Fair point that memorizing an algorithm doesn't necessarily advance the student's intuition for how numbers work, but keep in mind that you're replying on a subthread about how learning a general algorithm is an improvement over memorizing multiplication tables. Ideally, we teach children the intuition for mathematics. Just memorizing arithmetic algorithms isn't ideal, but it's surely better than memorizing finite arithmetic tables.

mensetmanusman 8y ago

I recall research showing mathematics being one of the very few subjects where one can get better at through brute memorization. It gets you quite far in that field, as it reduces recall time for proofs etc, and then it becomes like muscle memory. Practice/muscle memory in other fields, like the physical sciences, can not come from a page out of the book, since it requires interacting with nature in a multi-sensory way.

KorematsuFred 8y ago

Sorry that is pure BS. Memorization is a great technique to learn math of course it is not the way to do math. When you learn tables through rot learning you automatically get how they are being computed. At young age it is much easier to rot learn many things and understand the abstract concepts much later. People who despise rot learning are either misunderstanding it or are simply lazy.

rumcajz 8y ago

See Piaget's concrete operational stage [1] of human development: "Abstract, hypothetical thinking is not yet developed in the child, and children can only solve problems that apply to concrete events or objects." In other words, you can only push the mathematical thinking as far as the developmental stage of the child allows you to. Memorization helps though. My son is 6. He doesn't know the multiplication algorithm, but he does know that 2 x 2 = 4 or that 3 x 4 = 12. That's sufficient for him to do discoveries of his own. 3 x 4 = 3 x (2 x 2) ! Picture that! [1] https://en.wikipedia.org/wiki/Piaget's_theory_of_cognitive_development#Concrete_operational_stage https://en.wikipedia.org/wiki/Piaget's_theory_of_cognitive_d...