Problem solving vs. conceptual understanding?

[regentrude]

 

Or maybe these questions are supposed to provide differentiation at the top? The worst example of this I've seen is a Primary Maths Challenge question which depended on knowing how to determine if a number is divisible by 7. Seriously, the only direct was to solve it was to see how to make a multi-digit number divisible by 7. If you didn't know how to do that, I guess you could plug in each multiple choice answer and divide by 7, in which case the question was a question of your speed with long division. 

 

 

Can you post the question? I am curious now.

 

ETA:

 

 

Or maybe these questions are supposed to provide differentiation at the top?

 

If you are talking about competition question, yes, absolutely - the entire point of competitions is to differentiate at the top.

Edited September 26, 2016 by regentrude

[Guest]

I was talking about elementary arithmetic.

 

Maybe you're talking about working with base-2, cryptography, probability, and stuff like that?

 

We've done some of that, and he thinks they're interesting enough. But he's not super excited about them, not right now anyways. Maybe in the future one of them will pique his interest more.

 

I understand doing these things if a kid doesn't feel ready to jump to abstract concepts. But I seriously question the need to stick to them if it's unnecessary, and the kid isn't interested in them.

I am talking about aritmetic as taught in k to 5. Its not just get the tables memorized, then use the facts to solve word problems. Yes, knowing the place value notational system is part of it, but the rest is seeing part to whole relationships, developing an understanding of the associative, commutative, and distributive properties, developing visualization skills, developing mental math skills, some number theory such as even/odd used to develop number sense as well as check answers, translating to/from words, see and state assumptions, see cases and provide solutions for each, find the trivial solution. Problems as well as exercises, and developing ability to express solutions for problems. When mastered, a word problem such as found in SM CWP will be solved as an adult does, and the student will be declaring them obvious.

 

Base2, cryptography, prob; base 2 in the past here was a fifth grade topic and it cemented place value understanding for many. I am not sure where it is in common core and dont care to look, but if your child will be taking science later its a good thing to know.probability and stats is in every grade level here. Again, a firm foundation needed for future science study and for life. Ex, I was seriously ill when my kid was in 10th grade. He could tell me how to calculate basic prob and stats, but he didnt know it well enough to interpret the info the internet was providing him on my probability of dying. Understanding the data set, understanding and interpreting the language of the reported statistics, etc had not been included.

 

You could ask the school to give the student an achievement test ...the Stanford 10 does have a math subset on problem solving.

Edited September 26, 2016 by Heigh Ho

[SarahW]

Can you post the question? I am curious now.

 

ETA:

 

 

 

If you are talking about competition question, yes, absolutely - the entire point of competitions is to differentiate at the top.

 

 

It's question 10 here.

 

Here's the answer key, and on the second page there's an enrichment suggestion. In neither place does it give any idea about solving without knowing the divisibility of 7. 

 

I suppose a smart test taker would start by putting 6 in first, dividing by 7, and then seeing what the remainder is, and then using the remainder to figure out the answer. But I've always considered using one of the multiple choice answers to solve the problem was simply a test-taking strategy, not really a problem solving one. And the solution doesn't mention this method...so I don't know what they were thinking with this question.

 

I understand trying to prevent too many perfect scores, in England they seem to openly try to prevent them. But this question is in the "mid-level" difficulty. And on this one you either know how to solve directly, or you solve it in a way which the test maker apparently doesn't want to talk about. Which I think is weird, because talking about what information remainders can give you seems like a good thing to talk about, if that was even their idea with this question. 

[Arcadia]

It's question 10 here.

 

Here's the answer key, and on the second page there's an enrichment suggestion. In neither place does it give any idea about solving without knowing the divisibility of 7.

The question is about a 3 digit number with the first two digits given.

 

Given 24* is divisible by 7.

Therefore 24* -210 = 3* is divisible by 7 (7x3=21, 7x30=210)

3* has to be 35 since that is the only multiple of 7 that has 3 in the tens place.

So 5 is the answer for the missing digit

[regentrude]

It's question 10 here.

 

 

I have not looked at the answer key - but you do not need "divisibility rules" at all!

 

With an elementary age student, I would have approached the problem like this:

 

you know the number is between 244 and 248.

So what multiple of 7 do you know that is smaller than 240, but fairly close?

3*7 is 21. So you know 210 is 30*7. (basic knowledge of times tables should get the student there)

 

At 210, we are still 30  away from our region for the numbers.

So what multiple of 7 gets us there, that is larger than 30?

Ah, 5*7=35.

210+35=245 - so that has to be our number.

 

I am fairly sure this is how a young student who just knows times tables is supposed to approach the problem. And then it is neither tricky nor hard and most certainly does not require any knowledge of divisibility rules - just a  simple break down of the number into terms that are multiples of seven.

Edited September 26, 2016 by regentrude

[Guest]

The answer to 'why dont you just tell me' is.......its not about becoming a human calculator. Its about improving your thinking and reasoning skills.

Something about this bothers me and I've seen it from multiple posters, so it isn't just you.

 

Marh isn't strictly about the most difficult or creative problem solving, not every student has that as a goal and that's okay. Some people really enjoy math as a logical, meditative, predictive behavior. They don't necessarily want to expand it out to calculus and beyond. Much everyday math is indeed basic problem solving in steps, or being a capable, quick human calculator. I think there is something to be said for respecting the student's goals in math and not constantly pushing for your own definition of excellence, irrespective of whether they actually share that goal with you.

 

That said, a student calculating in a methodical way IS problem solving with that algorithm, and may just need to be shown in incremental steps how that is occurring. That's more of a discussion of variables and word problems than anything related to the calcs, which are already there. I'd be focusing on translating various real life word problems TO that paper setup. And changing up variables here and there until they had a smooth transition between the words and the operations.

 

And quite frankly a student who lands there who doesn't have a burning desire for a math or mathematical science degree is well educated and has strong foundational math skills for life. Really. Competition math isn't the pinnacle goal that any student who doesn't want to pursue it is somehow failing or inferior. I HATE that attitude and I think it devalues the very real skill and work of the vast majority of students who enjoy math but not necessarily a brain breaking puzzle.

 

That's my .02

[Guest]

This is a false dichotomy. There is no way to do actual problem solving without conceptual understanding.

You can crank out the answers to math "problems" using a memorized algorithm, but that is not "problem solving" - real problem solving is encountering a new, previously unknown,question and applying your toolbox of concepts to find a solution.

 

Conceptual understanding does not exist in vacuum either; if you cannot apply a concept to a problem, you did not really understand it.

 

For learning, it has been my experience as an instructor that practicing problem solving enhances conceptual understanding and conceptual work enhances problem solving abilities. The two go hand in hand.

This. Bingo.

[slackermom]

It's question 10 here.

 

Here's the answer key, and on the second page there's an enrichment suggestion. In neither place does it give any idea about solving without knowing the divisibility of 7. 

 

 

 

I figured that one out using repeated subtraction of multiples, something we saw in second grade math, I think, but I am not sure which book.

 

240-140=100

100-70=30

30-28=2

With 2 left over, add 5 to get to 7

[mathnerd]

The question is about a 3 digit number with the first two digits given.

 

Given 24* is divisible by 7.

Therefore 24* -210 = 3* is divisible by 7 (7x3=21, 7x30=210)

3* has to be 35 since that is the only multiple of 7 that has 3 in the tens place.

So 5 is the answer for the missing digit

 

We did this kind of problem in a similar manner - using "landmarks" on the number line - the easily visible "landmark" on the number line that is closest to 24* and also a multiple of 7 is 210 (which is 7 x 30). From the landmark of 210 on the number line, how many hops of 7 can you make to land in the range of 24* on the number line. Since the difference between 24* and 210 lies in the 30ish range, the first multiple of 7 that we should use to check is 7x5=35. Does it fit the requirements? Yes, it does. Answer: 210+35 

Edited September 26, 2016 by mathnerd

[happybeachbum]

So, here's the thing. I was a gifted child, and I LOVED number stuff. I got a workbook and did multiplication problems on my own, pages of them. I thrilled to the structure of setting up a complicated long division. It tickled me pink when my dad showed me how to set up an algebra problem to solve third grade math. Hearing about your son and the area of a cone takes me right back to my own childhood mindset. ;)

 

And that stuff is great, and loads of fun! But what I was missing as a kid is that it's not MATH. Mathematics IS mathematical thinking -- and using that process to solve problems. Any calculator can plug and chug, but it takes a human being to make connections and draw conclusions.

 

So as I came up through the system I just kept getting heaped more and more praise for quickly understanding more and more advanced algorithmic solving approaches, and getting told I was so good at math because of it. Algebra…Trig…Calculus…all setting up the right equation and then solving it the right way. It took me until sophomore level college math and physics before I hit a point where my gifted-kid skill of matching patterns was no longer enough to get the right answer. Instead I was looking at problems that *had* no one right answer, where there *was* no algorithm in the text book to apply. This blew my mind -- and in the end, crushed my ability to succeed in my chosen field. I limped through and got my bachelor's in physics, but have never returned to math or science.

 

I don't want that for my kids. I want them to have the joys of trying and failing while they're still young enough to appreciate it. But they're public schooled and "math time" for them means just doing lots of plug and chug. So I distinguish very clearly between "school math" (learn the right algorithm for the right problem,) and "real math." They work at school math and often enjoy it, but if sometimes they get bored with the repetition I can say, "It's okay. It's just for school. That's not real math anyway. You have to learn it for school, but don't worry, we can make time for real math too." So at home we can practice discrete math, logic, computations with more than one approach, etc.

 

I highly recommend that you go to Amazon and look up one of the Borac competitive math books. Here's one. https://www.amazon.com/dp/0692244905/

Then use the "Look Inside" feature to read the Foreword. They say it better than I ever could. :) I drive my kids' educations with a mantra I got from the Boracs, "Mathematics is not meant to be easy. It is meant to be interesting."

 

Of course it's okay for your son to just explore around and make his pattern-matching mind feel soothed with mathematical structures. But for the long term, it's really probably good for him to get some frustration tolerance built up, and to find some work that stretches and builds his ability to tackle real math. I bet he'll rock at it. :)

 

That was my experience with math.  Except the higher math I didn't do so well. 

 

 

I think my problem I was a why person and was being taught plug and chug.  The way I did addition, subtraction, Multiplication etc. I always tended to do in probably considered long ways especially algebra.  It finally occurred to me that all of those things were connected, but no one explained that to me.  The problem still was that I was being taught plug and chug.  Had to do developmental/remedial math in CC.  Got into College Math and bombed.  You know why?  Because in those remedial classes they just taught plug and chug and forgot things like mathematical thinking. 

 

​Also after you learn about problem solving then comes problem posing.

Edited September 26, 2016 by happybeachbum

[quark]

NM, not relevant!

Edited September 27, 2016 by quark

[SarahW]

I am talking about aritmetic as taught in k to 5. Its not just get the tables memorized, then use the facts to solve word problems. Yes, knowing the place value notational system is part of it, but the rest is seeing part to whole relationships, developing an understanding of the associative, commutative, and distributive properties, developing visualization skills, developing mental math skills, some number theory such as even/odd used to develop number sense as well as check answers, translating to/from words, see and state assumptions, see cases and provide solutions for each, find the trivial solution. Problems as well as exercises, and developing ability to express solutions for problems. When mastered, a word problem such as found in SM CWP will be solved as an adult does, and the student will be declaring them obvious.

 

Base2, cryptography, prob; base 2 in the past here was a fifth grade topic and it cemented place value understanding for many. I am not sure where it is in common core and dont care to look, but if your child will be taking science later its a good thing to know.probability and stats is in every grade level here. Again, a firm foundation needed for future science study and for life. Ex, I was seriously ill when my kid was in 10th grade. He could tell me how to calculate basic prob and stats, but he didnt know it well enough to interpret the info the internet was providing him on my probability of dying. Understanding the data set, understanding and interpreting the language of the reported statistics, etc had not been included.

 

You could ask the school to give the student an achievement test ...the Stanford 10 does have a math subset on problem solving.

 

 

Your first paragraph are things I consider to be "number sense" which is something I said in my pp.

 

As for the second paragraph, I'm not saying stats or prob are unimportant, I'm just wondering how important they are for elementary kids, who really don't have the math skills to do much more than make a table about possible coin tosses. Does spending time doing this in elementary set kids up to learn these things better later? I mean, if the kid thinks "hey, a fun thing I can do with math!" that's great. If it causes them to start calculating variations, even better. But if not, I think it could just be a waste of time. 

 

Unfortunately, there's no Stanford 10 or similar here. He needs to test out of each grade level of the national standards.

Edited September 27, 2016 by SarahW

[SarahW]

I looked at the question some more when I made the links in my pp, and when I was putting Babypants to bed I realized  that you could start with 210 and then find what in the 30's is divisible by 7. But then I was too tired to come back downstairs to tell ya'll that I figured it out.

 

When we were doing this last year we were doing BA heavily, and with BA I get used to reading through the solutions to see the ways to solve and what the question is meant to teach. I did the same when we were doing these questions. And honestly, part of the problem is that we had been talking about divisibility, so when CP read "divisible by 7" his first reaction was "there's a divisibility rule by 7?!" I had never heard of it either, but, hey, the answer sheet says there is one! What do you know! Now, though, that I'm older and wiser, and haven't seen a curriculum teach the method of divisibility by 7, I realize why it's left alone. But at the time I thought it was just one of those "extras" like divisibility by 6 or 8 which aren't talked about much. And the answer provided being "well, it's 5" made me think that they were suggesting that the student think about the units digits in the 240's to figure it out. Clearly, whoever wrote the answers was not expecting them to be used by idiot moms like me.

 

That being said, I don't think my kid would be impressed by a circuitous method of figuring this out. When we're puzzling over something and I come up with "Aha! This is that, so this must be this!" he gives me a skeptical look which has "So you managed to arrive at the answer this time, young padawan, but will that method serve you well when the numbers become larger and more complicated? Should you not seek to discover why that is the answer, so that you can solve these questions directly in the future? Clever you are, yes. But overconfident you should not become." written all over it. Maybe I cause that attitude a bit, when he's jotting down equations in his math notebook and he exclaims about finding out something new about something or the other, I challenge him to try it out with lots of different numbers and see if it's still true. I think that makes him suspicious of methods of solving which work neatly with this number, without proving that it works with other numbers. Maybe when we're doing competition questions in the future I should first tell him to throw caution to the wind and be more aggressive to find the answer that makes that equation true.

 

 

 

 

Edited September 27, 2016 by SarahW

[SarahW]

Something about this bothers me and I've seen it from multiple posters, so it isn't just you.

 

Marh isn't strictly about the most difficult or creative problem solving, not every student has that as a goal and that's okay. Some people really enjoy math as a logical, meditative, predictive behavior. They don't necessarily want to expand it out to calculus and beyond. Much everyday math is indeed basic problem solving in steps, or being a capable, quick human calculator. I think there is something to be said for respecting the student's goals in math and not constantly pushing for your own definition of excellence, irrespective of whether they actually share that goal with you.

 

That said, a student calculating in a methodical way IS problem solving with that algorithm, and may just need to be shown in incremental steps how that is occurring. That's more of a discussion of variables and word problems than anything related to the calcs, which are already there. I'd be focusing on translating various real life word problems TO that paper setup. And changing up variables here and there until they had a smooth transition between the words and the operations.

 

And quite frankly a student who lands there who doesn't have a burning desire for a math or mathematical science degree is well educated and has strong foundational math skills for life. Really. Competition math isn't the pinnacle goal that any student who doesn't want to pursue it is somehow failing or inferior. I HATE that attitude and I think it devalues the very real skill and work of the vast majority of students who enjoy math but not necessarily a brain breaking puzzle.

 

That's my .02

 

 

I agree with your overall point, but about the bolded in particular: 

 

When I was in grad school I had a friend who was getting a graduate degree in Math, and she told me about how one of her profs showed up to class all excited one day because she found a "new form of math" which was absolutely great because this math didn't actually have a relationship with reality! This exasperated my friend greatly, mostly because her husband was getting a philosophy degree and she seriously questioned the actual existence of something which didn't have a realistic method of existence. It also provided material for late-night snark "So if I number my thesis paper footnotes in your prof's new math system, the bottom half of every page could just be blank, right?"

 

I'm totally burned on Realistic Math Education, and "real life word problems" in general, right now. But I'm wondering if we also can't do a better job of working with numbers just for the sake of working with numbers, because not everything in math exists in reality (at least, not in a way which is immediately obvious to most people, and apparently there's some math out there which doesn't actually exist). There seems like there should be a better way to explore (and test) number sense and number relationships, and their proofs, without talking about birds pooping on cars. At least for some kids. (I suppose some kids like and need RME, sure.)

[regentrude]

When I was in grad school I had a friend who was getting a graduate degree in Math, and she told me about how one of her profs showed up to class all excited one day because she found a "new form of math" which was absolutely great because this math didn't actually have a relationship with reality! This exasperated my friend greatly, mostly because her husband was getting a philosophy degree and she seriously questioned the actual existence of something which didn't have a realistic method of existence. It also provided material for late-night snark "So if I number my thesis paper footnotes in your prof's new math system, the bottom half of every page could just be blank, right?"

 

Actually, the ironic thing is that many mathematical concepts that had been developed without any tangible connection to reality have been found to be immensely useful in physics. Often much much later.

For centuries, nobody thought number theory to be useful for anything, just mathematicians playing with numbers - and now it absolutely vital for cryptography.

[Arcadia]

For a normal kid, having to learn probability and statistics in elementary school isn't that much of a time suck. For an accelerated learner who already have time to spare, it might become a passion or one of many passions.

My kids did the AoPS intro to number theory book and intermediate counting and probability book in 2nd/3rd for fun. They were with a public charter school. If my kid wants to use his playtime for math fun instead of other fun stuff then so be it even though he is very sure from young that he does not want a career in academia.

 

While my kids dislike Zaccaro's books e.g. Real World Algebra, there are kids who enjoy it.

 

As for birds pooping on cars, my kids were entertained by thinking about the chance of our car being pooped on in different parking lots while we were shopping for groceries.

[Guest]

The bird pooping is so funny. My kid, then 11, raised rurally, was walking along a city street when he was the recipient. He wasnt too amused when other people told him his chances would decrease if he didnt walk under a wire. Sure brought home the basic definition of probability in a way that his teachers never could with their busy work up counting m and m colors in halloween sized bags. In the end he decided his odds wouldnt be any different if he had taken the bus rather than walked.

 

Elementary school here doesnt use grades, so students dont get experience calculating averages. Big shocker when they get to middle school and realize what a 10 question test does to their course grade.

Edited September 27, 2016 by Heigh Ho

[Dmmetler]

My DD has found probability and statistics to be useful and interesting-possibly the most useful and interesting part of math for her. Of course, I think she also would greatly enjoy a career in fact checking if she decided to diverge from herpetology. 

[maikon]

I have not looked at the answer key - but you do not need "divisibility rules" at all!

 

With an elementary age student, I would have approached the problem like this:

 

you know the number is between 244 and 248.

So what multiple of 7 do you know that is smaller than 240, but fairly close?

3*7 is 21. So you know 210 is 30*7. (basic knowledge of times tables should get the student there)

 

At 210, we are still 30 away from our region for the numbers.

So what multiple of 7 gets us there, that is larger than 30?

Ah, 5*7=35.

210+35=245 - so that has to be our number.

 

I am fairly sure this is how a young student who just knows times tables is supposed to approach the problem. And then it is neither tricky nor hard and most certainly does not require any knowledge of divisibility rules - just a simple break down of the number into terms that are multiples of seven.

This is exactly how my 6 year old solved the problem in a fraction of a second. Edited September 27, 2016 by maikon

[sunnyday]

 

 

As for the second paragraph, I'm not saying stats or prob are unimportant, I'm just wondering how important they are for elementary kids, who really don't have the math skills to do much more than make a table about possible coin tosses. Does spending time doing this in elementary set kids up to learn these things better later? I mean, if the kid thinks "hey, a fun thing I can do with math!" that's great. If it causes them to start calculating variations, even better. But if not, I think it could just be a waste of time. 

 

All I can say is that when my peers and I took the sophomore course in Discrete Mathematics as required for our physics major, it crushed most of us. Utterly left us staggered and confused. We would turn to each other and whimper, "COUNTING? How is COUNTING the most difficult thing we've encountered after 13 years of math? I thought I knew how to COUNT."

 

So. I personally enjoy introducing a bit of counting and combinatorics and probability to my 9yo. And I get excited when he spends his own time, sitting in the car, figuring out things like how many individual symbols can be represented in Morse code by a maximum of five dots or dashes. (Ah, Benedict Society, thanks for putting that in his head, LOL.) 

 

But anyway, I found the book (and the MOOC) Introduction to Mathematical Thinking several years ago, about the same time I read Knowing and Teaching Elementary Mathematics, and I guess between them they've really shaped my thoughts about mathematical education. I believe there's a place for school math, but there shouldn't be the abrupt distinction between "math as a way to calculate" and "math as a way of thinking and a science in its own right" that currently exists in the transition from school to university. IMO, we need to ease this transition for those who will make it -- and presumably our very precocious young math learners *will* someday encounter *some* sort of math at a college level, as taught by a mathematician -- by introducing mathematical thinking right from the beginning, even if it's just one strand among many, even if it has to ebb and flow with their maturity and frustration tolerance.

And I also really like this quote by Keith Devlin:

 

"Education is not solely about the acquisition of specific tools to use in a subsequent career. As one of the greatest creations of human civilization, mathematics should be taught alongside science, literature, history, and art in order to pass on the jewels of our culture from one generation to the next."

[Have kids -- will travel]

All I can say is that when my peers and I took the sophomore course in Discrete Mathematics as required for our physics major, it crushed most of us. Utterly left us staggered and confused. We would turn to each other and whimper, "COUNTING? How is COUNTING the most difficult thing we've encountered after 13 years of math? I thought I knew how to COUNT."

 

That class, which I took the second semester of freshman year, was the reason I'm not a math major. That and the fact that cutting edge math research involve soap bubbles.

 

But you are totally right about how mathematical thinking thoroughly transcends math-as-calculations. Probability is one of the best tools to teach elementary kids mathematical thinking, in part because you don't need complex calculations to reach the conclusions, and the world needs more people who understand probability -- many, many more people!

 

To OP, I hope you find a good solution for your boy. I've always heard that the best support for a gifted kid to support their strengths and their weaknesses, with an eye to challenging both. Maybe then he'll get to discrete math and say, "Oh, that explains so much!" Rather than, "Why on earth am I here?" followed closely by "Who cares about that damn Chinese postman!"

[Arcadia]

If anyone wants some free discrete maths materials, MEP has some units on it that are for the 12th grade exam but does not need much prior knowledge.

http://www.cimt.org.uk/projects/mepres/alevel/alevel.htm

[daijobu]

This distinction between (1) finding the key insight (say locating similar triangles that are hidden away) that allows you to solve a difficult math problem and then (2) executing the algorithm to actually find the answer.  My math professor would offer up the key insight and then announce, "And the rest you can do with your spine."  Meaning executing the algorithm required only lower brain functions.  

 

Yes, when someone shows you that the two triangles are similar, the rest is easy.  It's finding those damn triangles, or even knowing to look for them, that shows good problem solving skills.  

[SarahW]

I was thinking about this topic some more last night, mostly because something in my dreams made me think again of the recent thread where posters said that Saxon does (eventually) lead to conceptual understanding.

 

I wanted to post with a "Maybe, but..." response, but decided against posting about my childhood again. But maybe it would be somewhat relevant here.

 

My older brother and I got largely the same homeschool education (I got a lot of his old books, lucky me!  :001_rolleyes: ). We did mostly BJU math in elementary, followed by some Saxon. My brother also did a year of Lifepacs in elementary (I didn't get those, obviously, maybe one reason our mom didn't continue it, even though he liked them). Despite the fact that my brother learned math largely from these rote and procedural curriculums, he got math. When he moved into Calculus and physics he would bother me by talking my ear off about how this relates to that, and how that thing works, and so on. He got Gardner books from the library for fun, and then tried to talk to me about those. He took a crazy (to me) amount of advanced math in college and talked about how interesting it was ("Once you see it, it's beautiful!" Umm, okay...?). He was hired by two tech giants right of college (long story) as a technical engineer. So the method worked for him. Me, however, not at all (obviously). My brother maybe could have done well with math competitions, but we didn't know they existed (just another thing in the could've/should've list).

 

So I guess that's what I'm wondering - if maybe even without Singapore bar models / discovery approach / puzzles / complicated word problems method of teaching, some kids just understand math. Just from seeing a procedure or a formula they can see how it works and why. And after they "get" it they can puzzle and play and figure things out to their heart's content. Maybe these sorts of math learners are really rare? But it seems to me that it describes my brother pretty well, and maybe my kid got whatever math gene my brother did?

[regentrude]

So I guess that's what I'm wondering - if maybe even without Singapore bar models / discovery approach / puzzles / complicated word problems method of teaching, some kids just understand math. Just from seeing a procedure or a formula they can see how it works and why. And after they "get" it they can puzzle and play and figure things out to their heart's content. Maybe these sorts of math learners are really rare? But it seems to me that it describes my brother pretty well, and maybe my kid got whatever math gene my brother did?

 

I don't think they are so rare. Otherwise it would be surprising that there are quite a number of strong mathematicians, given the crappy math education in schools that is weak in conceptual teaching.

 

Of course some kids intuit math and get it even with substandard teaching. Just like some kids become excellent writers just by reading, without any formal instruction. 

[Guest]

So I guess that's what I'm wondering - if maybe even without Singapore bar models / discovery approach / puzzles / complicated word problems method of teaching, some kids just understand math. Just from seeing a procedure or a formula they can see how it works and why.

Imho once the student understands part/whole relationships, rather than seeing everything as facts in isolation, things just snowball. My sons' first grade teacher used other domains, especially drawing, to help that realization form. No procedures or formulas in k to 2 here, just mathematical thinking. My son's 3rd gr teacher had a chuckle when he was grade skipped into her class and she realized he hadnt been taught any traditional algorithms. He on the otherhand, was very polite to the sped teacher who was assigned to show him, using lots of colored markers rather than referring to place value, and spent his time as a tutor unmystifying other students. One of the values of SM/etc is that it gets more kids thinking than memorizing tables does. They puzzle and play both before and after the 'aha' or the very quiet 'that's what I suspected'.

 

My mother spent hours playing monopoly with us, as new math had arrived at our ele. school and she was giving my 2nd grade sib the experience with dice to see the relationships and with moving pieces to see make-a-ten. The year before I had spent taking many timed multiplication fact tests...students were give no tools to get faster, so we just figured out how to not let our minds wander, the most efficient way to fill in the sheet, the fastest way to write each numeral, etc. No experience with distributive property at all, so only the ones going on to Alg. would be eventually clued in. It was college before I knew about mental math...I had been taught to make tens if adding columns of figures, how to count back change using the count up method so customers could follow and told that mental math was an inner chalkboard....in college a prof advised us all to learn mental math so we could be conversant in a back-of-the -envelope type of discussion. He left it to the student to figure it out...chisanbop being the pop culture thing at the time. Mental math was a powerful technique...wasnt any faster, but it was less effort and focus for me than the inner chalkboard. Having it earlier would have made my studies easier though.

Edited September 30, 2016 by Heigh Ho

[SarahW]

I don't think they are so rare. Otherwise it would be surprising that there are quite a number of strong mathematicians, given the crappy math education in schools that is weak in conceptual teaching.

 

Of course some kids intuit math and get it even with substandard teaching. Just like some kids become excellent writers just by reading, without any formal instruction. 

 

 

Oh, yeah, that thought occurred to me. Somehow, America puts things in space, usually successfully. This was also exemplified, I think, in the other recent post about Cothran's dad.

 

"Rare" might be a relative term. Rare in terms of the percentage of American population? Or not rare in that they do manage exist in large enough numbers to have an academic society with an annual conference?

 

Maybe the title to this thread should have been conceptual learning vs. conceptual teaching. If a kid picks up the concepts pretty much straight from bare equations, is it necessary to go through all the stuff designed to teach the concepts behind the equations? Crazypants had fun with bar models when I showed it to him, but he doesn't really see the point of them. The same with lots of other "show you understand" things. He'd rather prove it with numbers. He might like writing proofs, when we get there....

[mathwonk]

"where the difference between problem solving and just having conceptual understanding.  Is one without the other okay?"  no.

Edited October 2, 2016 by mathwonk

[regentrude]

. If a kid picks up the concepts pretty much straight from bare equations, is it necessary to go through all the stuff designed to teach the concepts behind the equations? Crazypants had fun with bar models when I showed it to him, but he doesn't really see the point of them. The same with lots of other "show you understand" things. He'd rather prove it with numbers. He might like writing proofs, when we get there....

 

I see no need for visualization tools and manipulatives when the student does not need them. Bar models are one specific teaching tool, but not necessary to understand the concept. At some point, making a student jump through hoops becomes busy work. A student who understands place value and gets arithmetic does not need to be tortured with cuisenaire rods.

Edited October 2, 2016 by regentrude

[Guest]

I would be concerned about visualization skills if a student post 3rd grade couldnt see the reason that concrete or pictorial manipulatives are used in some stages of learning. Its funny, teachers always say that a child peer tutoring learns more about the topic he is teaching, but what my then little guy learned was that many cant visualize...they need those counters. And they have a real tough time with fractions....half of a half of a half is not something they could visualize, not even in terms of clock or pizza or lego, although they could be trained to fold paper or to do the symbol manipulation on paper and get the answer. Knowing and being able to demo the knowing is the goal.

Edited October 3, 2016 by Heigh Ho