Problem solving vs. conceptual understanding?

[SarahW]

Speaking specifically about math here - where the difference between problem solving and just having conceptual understanding. Is one without the other okay? Is is possible? Can you leave the problem solving side alone for a while with a kid, and just pick it up later?

 

Sorry if this is fuzzy. I've got a cold, again (this has been the absolute worst summer ever when it comes to my health, ugh). But while I was laying in bed I got to thinking about what Crazypants likes about math. What in math he likes doing. What really piques his interest. And it occurred to me that maybe one of the reasons why he has historically been so ambivalent about completing the BA practice books is that he (currently) has zero interest in problem solving. What does really interest him is solving equations. He has a graph paper notebook (more than one, actually) where he'll jot down math equations which interest him. Divide a prime number by a prime number to the tenth decimal point, haha look how crazy that is! Oh look, this division results in .1666 at the end, oh yeah, those numbers can be reduced to 1/6th! (Umm, those are examples from my memory, if they're wrong the problem is probably me, not him). And on and on. Base 3 numbers! The area of a cone! I thumb through his notebooks sometimes and try to check his calculations sometimes, but it's hard to for me to keep up with what strikes his fancy each day. 

 

To add another wrench.into this, as y'all may know, he's stuck in school doing "baby" math each day. He's frustrated, and asked me the other day if he already knew "all" of math (I think he was wondering when the school would stop giving him their math workbooks). I explained that he's probably about done with pre-algebra stuff, like division and fractions and negative numbers. But there was still all the math for when you use those things, like Algebra, and Trigonometry, and Calculus. Well, he got it into his head that he wants to "do algebra" now. Like, right now.

 

I'm like, but enrichment! problem solving! going deeper! Nope, after school today he demanded to go to aops.com and watch algebra videos. And that's what he did, grabbing a piece of paper from the printer and a pen and carefully jotting down notes and equations from RR.

 

I don't even know.... Y'all know I'm pretty clueless for this boy, lol. It's frustrating, cause I spent the whole day looking up online math competitions that he could do this year, and try to give him "problem solving" math to do after school (since I know he's pretty fried on doing pages of 2-digit by 2-digit multiplication at school). But no, Tell me what happens when one divides negative fractions, mom!

 

So while I was laying in bed thinking, I remembered something RR said somewhere about how he didn't get into competition math until middle school. And it made me think that maybe it's not really necessary to "slow" a kid down with tons of problem solving and discovery and so on in elementary school. What are these things trying to do anyways? Critical thinking, yes. Number sense, yes. Mathematical thinking, yes. But if a kid has these things already (at least a bit, and is continuing to develop them in his own way), maybe a bunch of "tricky" word problems are superfluous?

 

I feel that the main issue CP has with a lot of "competition" questions is that they ask questions which he doesn't yet know how to write as a pure equation. He's supposed to do guess and check, or draw a chart or something. But his first desire is to write out the problem in an equation with variables, and then he gets frustrated when he doesn't know how to set things up properly to solve it. I tell him to "just think about it" but that makes him mad, because he wants to be able to "solve" it mathematically. 

 

I feel like once he has more math tools, and is grounded in the way to solve equations with different sorts of information, the problem-solving side would make more sense to him. But I don't know. Is it just a characteristic of elementary math competition questions that they rely on guess-and-check methods? Or is this something that he would continue to fact in math competitions throughout high school level?

 

I'm not planning on giving up word problems or problem solving questions with him. But I'm wondering if it's okay to switch focus to "having fun solving interesting equations!" math.

 

Or, if a kid appears to have really good number sense and likes to play with numbers, can I just set him loose to do whatever math he wants?

 

I'm sick, and tired, and this kid is befuddling me again. Help please. At least assure me it will be okay.

[Guest]

Concept....pizza can be split in two equal pieces and we call each piece a half. Two haves equal a whole. Problem solving.....we need enough pizza for 3 children and two parents. How much do we order? How much will be leftover if each kid has 1/4 of a pizza and each parent 1/2?

 

I dont live in your country, but here the elementary problem solving used a lot of number sense and extending the given lesson. They might have to use even/odd knowledge, ball park, and reason. We skipped about half of the packet the textbook mfr provided as 'obvious'.

[SarahW]

Concept....pizza can be split in two equal pieces and we call each piece a half. Two haves equal a whole. Problem solving.....we need enough pizza for 3 children and two parents. How much do we order? How much will be leftover if each kid has 1/4 of a pizza and each parent 1/2?

 

I dont live in your country, but here the elementary problem solving used a lot of number sense and extending the given lesson. They might have to use even/odd knowledge, ball park, and reason. We skipped about half of the packet the textbook mfr provided as 'obvious'.

 

 

Yes, but in that example he would want to write down 3(1/4)+2(1/2). Oh, and look, you can cancel the 2's! That makes 1! But what would happen if the adults only got 1 zero-eth of a pizza? 1 over 0, does not compute. Mommy, you get no pizza, and you made the earth fall into a black hole. Kaboom!

 

Things tend to explode during math time.

 

But if I were to give him a question like that before he understood how to multiply fractions he would have been mad. I would tell him to count it up, and okay, he'd count it up. But the whole time he would have this look on his face like "there is something going on here which would make this a lot easier, so why don't you just tell me!"

 

 

I don't know, I talked to him a bit about what he learned from his Algebra video earlier today and he got all excited about finding the square root and then cubing it. Clearly, I need to go watch that video too, because I'm not sure what that is about. When I ask him why it works, he can give me quite a few coherent sentences, so I think he's understanding it. But...what is this? Math theory? Number theory? Why can't we just spend more time with Grogg flipping coins?? Sigh.

[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.

[sunnyday]

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. :)

 

[Arcadia]

Problem solving and conceptual understanding are so intertwine not just for math.

 

For a silly real life example, I bake a cake and it was too brown on top and too flat. Is the oven temperature off, did I forget baking powder, did I forget something else like preheating my oven? I have to have a basic understanding of baking to start problem solving, which probably lead me to research more and improve my baking and or oven repaid knowledge. Then of course I bake another cake and there would be some other imperfection to solve.

 

So problem solving can lead someone to finding gaps in knowledge which leads to the person reading and thinking more and trying to solve harder and/or different problems and the cycle continues.

[Dmmetler]

This sounds familiar. DD reached a point, at about Singapore 4th/5th grade, when she wanted algebra. Like, right now. She was dying for new concepts, new ideas, and to move ahead, and to her, algebra was that mythical time when math would become "more". She got to the point that she was doing placement tests for fun. Going sideways, IP/CWP, Zacarro, MOEMS released practice...all not what she wanted. She desperately wanted algebra, with it's new notation, new vocabulary, and that just looked different and harder.

 

She didn't seem ready for AOPS. Or maybe, she was, but I wasn't. So, she did LOF PA (two books at the time) and all 10 books of Key to Algebra, and some extra stuff thrown in. Lots of simple, procedural problems, with some problem solving, but definitely not the level of rigor of AOPS PA or Algebra 1. She felt grown up because she was "doing algebra".

 

Then, we went back and did AOPS. She was ready to solve problems once she felt she had the concepts. But that year of lots of new concepts, types of problems, repetition, and vocabulary was what she needed at the time.

 

So, my suggestion is-let him go. If you can get Key to Algebra, it's good for kids who want algebra and equations. There's also a "Complete Book of Algebra and Geometry" which is, again, a good "pre-algebra" and "pre-geometry"-lots of fairly easy problems that introduce the terms and concepts, but definitely don't have the depth and difficulty of a high school level textbook. Key to Algebra has a lot more space to work the problems in the workbook-the Complete book was too cramped for my DD to use as a workbook at the time, but probably would have been fine if she'd been a couple of years older.

 

And both would be easy for him to take to school and do for fun if he has time while waiting, or to do independently.

[Mike in SA]

So, my suggestion is-let him go. If you can get Key to Algebra, it's good for kids who want algebra and equations. There's also a "Complete Book of Algebra and Geometry" which is, again, a good "pre-algebra" and "pre-geometry"-lots of fairly easy problems that introduce the terms and concepts, but definitely don't have the depth and difficulty of a high school level textbook. Key to Algebra has a lot more space to work the problems in the workbook-the Complete book was too cramped for my DD to use as a workbook at the time, but probably would have been fine if she'd been a couple of years older.

 

And both would be easy for him to take to school and do for fun if he has time while waiting, or to do independently.

 

:iagree:   This.

 

Don't block initiative.  It'll create a lose-lose situation.  If you are worried about some basic math skills, then pull out a remedial college math book.  It will be COLLEGE math.  When your kiddo can do everything there, then there is absolutely no reason to hold back.

 

You can add depth in other ways: do every challenger, do alcumus, number theory, counting & probability, et al.  That will allow the knowledge to build at the right pace, and not force anything to be too fast or too slow.

[wapiti]

 Well, he got it into his head that he wants to "do algebra" now. Like, right now.

 

I'm like, but enrichment! problem solving! going deeper! Nope, after school today he demanded to go to aops.com and watch algebra videos. And that's what he did, grabbing a piece of paper from the printer and a pen and carefully jotting down notes and equations from RR.

 

I'm a strike-while-the-iron-is-hot person.  It's not as though you are contemplating acceleration via shallow, procedural resources.  Since you think he's had enough prealgebra, he could start Jacobs algebra.  Or, you could let him pursue the topic of the moment, equations, with whatever (e.g., RR videos, alcumus, ch 5 of Prealgebra, various chapters in Intro to Alg, the section of BA 5A, in any combination that is suitable).  So, I vote to go for it.  Maybe his path will be a zig-zag, going back and forth between concepts more advanced in the sequence and deep but less-advanced stuff, or maybe he'll just take off.  As long as he's learning and having fun, I don't see a downside.  If you hit a hole, you can always pause momentarily if needed.

Edited September 24, 2016 by wapiti

[eternalsummer]

Problem solving and conceptual understanding are so intertwine not just for math.

 

For a silly real life example, I bake a cake and it was too brown on top and too flat. Is the oven temperature off, did I forget baking powder, did I forget something else like preheating my oven? I have to have a basic understanding of baking to start problem solving, which probably lead me to research more and improve my baking and or oven repaid knowledge. Then of course I bake another cake and there would be some other imperfection to solve.

 

So problem solving can lead someone to finding gaps in knowledge which leads to the person reading and thinking more and trying to solve harder and/or different problems and the cycle continues.

 

 

On the other hand, you wouldn't give someone various cake ingredients and say, hey, figure out how to make this a cake!

 

If the person really wanted to learn to make cakes, first you'd give them the tools to figure out all the basics of cake making - some good recipes to try, maybe you'd tell them how each ingredient works and what its purpose is, so they could decide if they might want more or less of it or if they could replace it with something similar.

 

I have a kid who wants to be told how to do things if I know how. She doesn't really care to discover things.  This has not lessened her conceptual understanding, just because I explain it instead of having her discover it, as far as I can tell.  She does like to be challenged to some degree with problem solving exercises (though she prefers to do them in other areas, like art or cooking), but she would be pretty pissed if she encountered a problem, struggled with it for an hour, and then figured out there was a structure to the solution that people had known for centuries/millennia.  

 

I dunno.

[Arcadia]

On the other hand, you wouldn't give someone various cake ingredients and say, hey, figure out how to make this a cake!

I did say it is a silly example :)

My kids did take flour, margarine, baking powder, eggs and make muffins after watching a few YouTube videos.

[quark]

It was for this very reason that we did math in strands. 20 mins of one thing, 15 mins of another and another 30 mins after dinner. It often went beyond the 20-15-30 because he would keep asking why this and why that and why not this and that. A lot of googling and pounding headaches later, he'd go do something else.

 

I still keep the math notebooks. So many graph paper notebooks. To think that at one point I was worried he wasn't dating his entries. Gosh, I was so clueless.

 

So when he asked for algebra (the higher level stuff, not just solving for x for example), I hmm-ed and haw-ed and then DH came home and said yeah why not. We engaged a tutor (because I was so worried I would mess up teaching him and DH was too busy) and bam, kid did algebra. Baulked at the writing so I sat with him and did it with him. Every single page of homework. For our parallel strands, we did LoF Fractions/Decimals, MOEMS, AoPS CP and NT intro books and he kept reading Murderous Maths and all the other books in my siggy and binge watched Singing Banana and later Numberphile. When the Number Theory craze peaked, he would go to sleep with miscellaneous Dover and mass market number theory books (check out this one).

 

No competitions here. Not a single one. But problem solving can continue without competitions. Good luck.

[ebunny]

Its possible that he would have to slowly build up the stamina required for working on multi step problems. Its also possible that some children may be more into the 'theory' than 'application'. And vice versa. I dont know if they can be thought of as 2 distinct and water tight categories,iykwim. I dont know if its possible to focus exclusively on one without impacting the other. I.e. any conceptual learning will involve some procedural and some application.

 

Basically, I would let him continue to lead his own learning and not worry at this point.

 

Edited for clarity

Edited September 24, 2016 by ebunny

[ebunny]

I wonder if this false dichotomy (conceptual vs problem solving) is encouraged by publishers/authors who present their material as filling a special need or gap.

[Mike in SA]

To be honest, I'm not sure I follow many of the analogies, fears, and concerns listed in this thread.  It really isn't all that complicated.

 

If a child thinks of arithmetic as baby math, is it because it isn't "cool," or because he already understands it, and is bored?  If it's the latter, don't confuse adequacy with perfection.  One doesn't need to be perfect with arithmetic to proceed with algebra.  That's a myth.  Over the years, arithmetic will be encountered so frequently that it will become automatic.  By the time you reach calculus - assuming you do everything available before that - algebra will become just as automatic.  It's just how it goes with math.

 

On the other hand, if he has to stop to count on his fingers to do arithmetic, well, algebra is going to be a long slog.

 

At this age, you can spend 2 or even 3 years per course.  Watch his progress, and if he needs additional background, take a pause to secure that background. 

 

That's what a mathematician or scientist does.  We aren't given all the necessary background up front - we get it as we need it.

[SarahW]

Thanks for all the replies. I'm feeling a bit less freaked out today.  :)

 

....

 

I have a kid who wants to be told how to do things if I know how. She doesn't really care to discover things.  This has not lessened her conceptual understanding, just because I explain it instead of having her discover it, as far as I can tell.  She does like to be challenged to some degree with problem solving exercises (though she prefers to do them in other areas, like art or cooking), but she would be pretty pissed if she encountered a problem, struggled with it for an hour, and then figured out there was a structure to the solution that people had known for centuries/millennia.  

 

I dunno.

 

Yeah, that's my kiddo too. There was something in BA where a they were trying to "lead" him into some topic, and he he struggled through the questions until he got to "answer" and he exclaimed, "Well, I already knew that! Why didn't they just say that's what they wanted me know from the beginning!" He was pretty ticked. 

 

I mean, he definitely is not in favor of plug-and-chug. But he's not a "discovery" person either. He just wants to be told what and why. I guess this is why he likes watching RR's videos. he shows him a topic, show why it works, and does an example. Bam, done. And since he appears to have nearly perfect recall and understanding, I guess this method works for him? Maybe he's just sorta unique in that?  :001_huh:

 

 

It was for this very reason that we did math in strands. 20 mins of one thing, 15 mins of another and another 30 mins after dinner. It often went beyond the 20-15-30 because he would keep asking why this and why that and why not this and that. A lot of googling and pounding headaches later, he'd go do something else.

 

I still keep the math notebooks. So many graph paper notebooks. To think that at one point I was worried he wasn't dating his entries. Gosh, I was so clueless.

 

So when he asked for algebra (the higher level stuff, not just solving for x for example), I hmm-ed and haw-ed and then DH came home and said yeah why not. We engaged a tutor (because I was so worried I would mess up teaching him and DH was too busy) and bam, kid did algebra. Baulked at the writing so I sat with him and did it with him. Every single page of homework. For our parallel strands, we did LoF Fractions/Decimals, MOEMS, AoPS CP and NT intro books and he kept reading Murderous Maths and all the other books in my siggy and binge watched Singing Banana and later Numberphile. When the Number Theory craze peaked, he would go to sleep with miscellaneous Dover and mass market number theory books (check out this one).

 

No competitions here. Not a single one. But problem solving can continue without competitions. Good luck.

 

His notebooks! Yes, lol. Base 3 there, multiplying negatives here, oh look, it's One-Punch Man! and...now we're back to pi. There is a serious lack of organization. Right now I just tell him to let me know when he needs a new notebook. 

 

On the one hand, since he is in school (where they are tediously making him show mastery and test out of every grade level) I suppose I shouldn't worry too much about "gaps," at least in the basic skills. But the S&S is so slow and "real-life" based, that I fear it is turning him off of any and all "word problems."  

 

I would so like to get a tutor for him, but I don't see a way to make that happen right now. I don't know what his school will do with him when he tests out of elementary math completely. I don't think they know either.

 

Math competitions seem like a good way for him to engage in math in comparison to other kids. But yeah, he's got other "problem solving" things going, he spends a lot of free time on Scratch and other coding projects, and he knows about the relationship between math and programming. So maybe that's where his interest lies, at least for right now.

 

I've thrown a lot of math "extras" at him, at least a lot of the lower-level ones you've mentioned. It's hard for me to predict what he'll connect with. Some books he finds frustrating because they're too superficial. Others he finds frustrating because they're too dense. But he will read some superficial books, and some dense books. It just depends....on something. Something something, I don't know. 

 

I do know that he finds videos more accessible. We've done nearly all math documentaries. And he likes RR's videos. He's never been a fan of Khan's videos (too slow? too annoying? too something). Somewhere here someone post a link to Holt's Algebra with Borger's(?) videos, I don't know how good they are, but there's a lot there.

 

I guess we'll just keep exploring and filling up notebooks in a random manner. :cool: .

Edited September 24, 2016 by SarahW

[Guest]

.. shows him a topic, show why it works, and does an example. Bam, done. And since he appears to have nearly perfect recall and understanding, I guess this method works for him? Maybe he's just sorta unique in that? .

Lots of bright kids can memorize well. The idea is to develop the ability to come to an understanding and extend that understanding using one's own mind, and solve problems that no one else has solved. Its not a situation of 'teacher shows procedure, explains concept, student recites what was memorized'. There are challenge problems available by concept/grade level, and if they can be solved at a glance the child is misplaced.I would be concerned about a child given a word problem, who ducks answering with distraction. For ex, my then 6 yr old had a little buddy who wanted to multiply, just like big sis. He didnt understand addition, but he was good at memorizing and memorized the times tables, enjoyed telling you answers. Pose a problem that required a solution, and he was

clueless. He didnt fare so well in the future math classes, because he spent elementary jumping to his memorized answers, rather than develop understanding of such concepts as associative, communitive, distributive properties or developing any number sense. Eventually reached a point where memorized concept explanations and procedures could not solve problems. For a lot of kids, that is AP Physics 1, for others, its Calc 2. Taking the time to know a concept, not just remember someone else's explanation, is worthwhile.

[Guest]

I wonder if this false dichotomy (conceptual vs problem solving) is encouraged by publishers/authors who present their material as filling a special need or gap.

There is a problem of low expectations. Putting word problems seperately means the el ed teacher never has to get to them. They can fill the year with memorizing activities. District here has id'd math competent teachers and they teach two classes, while the partner teacher who isnt math competent teaches two classes the SS. They have also hired math coaches, who are consultants.

Edited September 24, 2016 by Heigh Ho

[Arcadia]

I wonder if this false dichotomy (conceptual vs problem solving) is encouraged by publishers/authors who present their material as filling a special need or gap.

A lot of commercial contest prep books for elementary Math Olympiads in Asia are presented as problem solving. Kind of like test prep books.

 

However solving math puzzles that stretch the brain can make a child go and further widen and deepen his/her conceptual understanding.

[regentrude]

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.

Edited September 24, 2016 by regentrude

[ebunny]

There is a problem of low expectations. Putting word problems seperately means the el ed teacher never has to get to them. They can fill the year with memorizing activities. District here has id'd math competent teachers and they teach two classes, while the partner teacher who isnt math competent teaches two classes the SS. They have also hired math coaches, who are consultants.

  

Given that schools have about an hour a day(more or less?) to work on automaticity and number sense; Ive often wondered if early elementary necessitates 'problem solving' as an exclusive activity in math. Maybe its more a time crunch to fit in everything in a school day +low expectations..

 

 

A lot of commercial contest prep books for elementary Math Olympiads in Asia are presented as problem solving. Kind of like test prep books.

However solving math puzzles that stretch the brain can make a child go and further widen and deepen his/her conceptual understanding.

The value of those books is debatable..do kids really benefit? Or do we(parents, educators) assume so? Probably a topic for a different thread..

[SarahW]

Maybe the title to this thread is all wrong. Could be. I'm trying to figure this all out myself.

 

But to go back to my first post, I'm wondering if this is largely an issue with elementary "problem solving" questions. To use another example, yesterday we were doing a math competition question which asked how many paper airplanes kids could make depending on whether they used a whole sheet of paper for each one, or half a paper, or a quarter of a paper. Since it was a "Grade 3-4" question, they expected him to map out the number of airplanes per page and then multiply. He found the way the question was worded, and the solution presented, to be unnecessarily confusing. And, tbh, it was confusing, because it was actually a ratio question, which was "dumbed down" to not look like a ratio question. So that's one of the things I'm wondering - is this "just count up the information because we don't expect you to know the math yet" an issue only with these lower grade questions? Or is "ha! we'll see if you can figure out math you don't know yet!" a characteristic of math competition questions in general?

 

I'll admit that I am nervous about whether he is "done" with Pre-A, though doing all this rote math at school should drill arithmetic into his head quite well. But even getting to the end of "elementary" math doesn't cover all of what I consider to be Pre-A (like I said the S&S here is slow, here is a link to a sample end-of-elementary exit test, math is pp. 5-8). I think he can back up and learn something he's "skipped" okay. It's just the "extra" topics, like finding the median or something, which doesn't come up often (I don't think?) that I worry about. 

 

And I don't think he's "memorizing" from the videos he watches. If the next day he can tell me about the concept and then write down some (his own created) equations as examples and solve them, and tells me how it relates to something else in math, I think he understands it. I mean, it may turn out that he has misunderstood something (which is why I think a tutor would be super nice to have right now) but if talks off and on over the course of a few days about the relationship between a fractional exponent and a square root, I think he's got the conceptual understanding of the topic at some level. 

 

But then, how important is it that he problem solves at this stage??? Hence the title of this thread....

[ebunny]

nm... Tried attaching pdf's that address reading comprehension and problem solving. But formatting issues on the phone.

Edited September 24, 2016 by ebunny

[wapiti]

But then, how important is it that he problem solves at this stage??? Hence the title of this thread....

 

I think there are different levels of problem solving even at a particular point in the sequence.  There's problem solving along the way that can lead to understanding of a concept, as in the lesson problems in aops, and then there's more challenging problem solving with a concept that can lead to even deeper understanding, as in the book challenge problems or years-later math competition problems.  

 

If you are asking whether he needs to have some experience with problem solving using prealgebra concepts, I would say that yes, there's definitely *a ton* of value there.  But, that doesn't mean problem solving with those concepts needs to occur in a linear sequence where there are boxes named prealgebra and algebra 1.  I think that the passage of time may be helpful, that it's actually quite useful to learn more and then go back to do problem solving at earlier math levels.

Edited October 4, 2016 by wapiti

[SarahW]

  

Given that schools have about an hour a day(more or less?) to work on automaticity and number sense; Ive often wondered if early elementary necessitates 'problem solving' as an exclusive activity in math. Maybe its more a time crunch to fit in everything in a school day +low expectations..

 

 

 

The value of those books is debatable..do kids really benefit? Or do we(parents, educators) assume so? Probably a topic for a different thread..

 

No, I'm curious. It seems that elementary arithmetic is largely based on the concepts of place value in the base-10 system and basic number sense. Once a kid already has these concepts down, he'd largely just needs to learn the nomenclature for the conventions of expressing it. (Exponents, like so, express repeated multiplication, and this sign is a square root sign, see? And here's even more fun things we can do with them!). I feel like that where we've been spinning our wheels for the last few years.

 

I've done logic books with him, and problem solving books, and I don't know. "If you know this, what else can you know? What can be eliminated? Using that information, what can be discovered?" is all good. They definitely have a place. Maybe some kids do really, really well with this "mystery" approach to math. But it is ideal/best/optimal to put such an emphasis on it? And is it terribly necessary for elementary students who are still learning how to properly express ideas in numbers?

 

Just thinking out loud here. I feel like "problem solving" and "word problems" in general are used as an antithesis to plug-and-chug, which I suppose they could be (though I've seen plenty of bad word problems which were nothing but plug-and-chug with words around the numbers). But the real goal is to have a deep conceptual understanding of how numbers work, yes? That means being able to use numbers, yes. But maybe at the elementary level there's a tendency to put the cart before the horse?

 

nm... Tried attaching pdf's that address reading comprehension and problem solving. But formatting issues on the phone.

 

Please attach when you can. :) 

 

Reading comp is another thing we're dealing with here. The paper airplane ratio problem was especially hard because it was just so wordy, a half of this, and then another half, and then a half of the half, and each half has two, and so on. Too many words!

 

He's getting better, but even so, half the problem he has sometimes is reading the question. Which is something he needs to work on, obviously, but it's usually better to work with him on sequencing information apart from math.

[SarahW]

I think there are different levels of problem solving even at a particular point in the sequence.  There's problem solving along the way that can lead to understanding of a concept, as in the lesson problems in aops, and then there's more challenging problem solving with a concept that can lead to even deeper understanding, as in the book challenge problems or years-later math competition problems.  

 

If you are asking whether he needs to have some experience with problem solving using prealgebra concepts, I would say that yes, there's definitely *a ton* of value there.  But, that doesn't mean problem solving with those concepts needs to occur in a linear sequence where there are boxes named prealgebra and algebra 1.  I think that the passage of time may be helpful, that it's actually quite useful to learn more and then go back to do problem solving at earlier math levels.

 

I had included an example above but then deleted it later though it might fit better here.  We are somewhat stuck with the boxes because we are not homeschooling.  I have a fifth grader who recently started alg 1 at a new school after completely skipping several levels of math - last year at his old school, he was officially in MM4 although he also participated in some enrichment activities (including a weekly MOEMS group).  Apparently the enrichment activities were enough for him to pick up a lot of concepts involving fractions and ratios such that he scored high on placement, but his prior practice of procedures in these areas is very weak (e.g while he understands a lot intuitively, he had never had instruction on proportions), which obviously presents some risk in a standard school algebra 1 course.

 

So, I told him he could stay in the alg 1 class if we did prealgebra at home and last week we started AoPS Prealgebra.  What's really interesting is that he's so much more aware of why he needs to learn these concepts and where they fit into the context of algebra.  I'm guessing his attention is a little closer than it might have been.  I'm keeping fingers crossed because he hasn't been willing to afterschool math in the past.  He's seen a few of the small, special group of fifth grade classmates drop back to the prealgebra level, which all along I've emphasized might be a great option for him, but he's determined to do this LOL.  (We'll see if that determination can help him through when the going gets tough.)  Anyway, if we continue this way, he'll be working on prealgebra problem solving (not to mention general practice!) at the same time he learns algebra 1.  Just trying to show that you can indeed mix it up.

 

We x-posted.

 

Yes, a lack of practice with procedures makes me twitchy. Though it's hard for me to get him to practice things because he starts complaining "But I already know this!" and so on, and he does write a good number of equations in his math notebook each week, so...

 

I'm wondering what would happen if we "start Algebra" and got to something he didn't Already Know. Maybe he'd see the need to back up? Or maybe he would intuitively grasp the prior knowledge? Might be worth finding out. I did pass College Algebra, and get a kinda decent math score on the GRE, so I just need to psyche myself up for this. Deep breath. 

 

I'm not planning on dumping his elementary problem solving completely or anything. But the other day there seemed to be such a disconnect - do we struggle counting up the possible number of paper airplanes? or do we spend more time discussing the properties of exponents? What?!

[Guest]

No, I'm curious. It seems that elementary arithmetic is largely based on the concepts of place value in the base-10 system and basic number sense.

 

 

Reading comp is another thing we're dealing with here. The paper airplane ratio problem was especially hard because it was just so wordy, a half of this, and then another half, and then a half of the half, and each half has two, and so on. Too many words!

.

There is more to elementary math. Perhaps it would help you to talk to a G&T elementary math teacher.

 

Ah, the problem was worthwhile in developing visualization skills.

[wapiti]

Yes, a lack of practice with procedures makes me twitchy. Though it's hard for me to get him to practice things because he starts complaining "But I already know this!" and so on, and he does write a good number of equations in his math notebook each week, so...

 

I'm wondering what would happen if we "start Algebra" and got to something he didn't Already Know. Maybe he'd see the need to back up? Or maybe he would intuitively grasp the prior knowledge?

 

I'm guessing a fair amount of zig-zag, and I think that's ok.  Example:  the first few weeks of ds's alg class involved assignments that should have been review.  Well he didn't know how to do lots of things (basic percents! proportions!) and I was really concerned (i.e. freaking out) though I didn't let on.  I just gave him a little lesson in the moment, when he was in the middle of working on something and suddenly needed to know.  The first thing he wants to know is *how* to do something so that he can get on with his work.  When he needed to work with proportions, I gave him a little instruction that included "wiping out denominators" a la RR, but he ended up taking away the procedure for cross-multiplication that I think had been mentioned in class.  He didn't absorb everything I tried to tell him the first time around even though he catches on very quickly.  A few days later, he asked why cross-multiplication works.  Doh!  I re-explained the same way I had explained originally, with an example, and he laughed when he understood.  At that point he got it.  It was a bit of zig-zag.  Same thing happened with percents - I went through the concept but, the first time, he took away the procedure and then the second time wanted the concept.

 

FWIW, on your problem solving question on counting airplanes vs exponent rules, I tend to lump all elementary arithmetic as falling under prealgebra.  I suppose that wouldn't quite work for a student who, for whatever reason, needed more depth all the way back at place value, but that's not the case for us thanks to montessori math.  Maybe I should mentally lump all "grade 5-8" math under prealgebra.  I also wonder if some individual kids develop in such a way that depth is more accessible at an older age than it might have been at a younger age.

Edited September 24, 2016 by wapiti

[quark]

 

I'm wondering what would happen if we "start Algebra" and got to something he didn't Already Know. Maybe he'd see the need to back up? Or maybe he would intuitively grasp the prior knowledge? Might be worth finding out. I did pass College Algebra, and get a kinda decent math score on the GRE, so I just need to psyche myself up for this. Deep breath. 

 

 

DS seemed to intuit based on the algebra problem. He probably drew upon some concept he had learned earlier and made connections in his head and voila, he had it. It just came so naturally to him because he wanted it very much. I really think that's what it was. He wanted these algebra tools under his belt and he wanted to solve problems at a more challenging level than the elementary and lower middle school problem solving books. We used the resources we did at the time because there was no BA and no AoPS Pre-Alg available.

 

We skipped prealgebra (we skipped precalculus too). Even when I was afraid of gaps because at that time I had already learned that more repetition meant boredom and losing interest. There is so much more going on than just learning. Especially with a child who is very goodnatured and does not act out, boredom takes a very different form. There was no sullen-ness which I would have mistakenly attributed to only bad attitude. There was just an overall shutting down (which I now have learned is what wilting looks like for my boy). So I wanted to avoid that. Once they lose interest and hope it's not always easy to get it back (lesson learned first with a bad kindy experience and later, with chemistry when he was 7yo and I wouldn't let him go deeper for fear of injury etc.).

 

I agree with Mike now. At that time, I was very worried because I think my own grasp of math isn't strong. But I knew that I wanted my son to love it and keep loving it and so I took that leap and followed his lead. We've only had to dial back at most a couple times. After algebra 1 with Dolciani, we used the first half of AoPS Intro to Algebra to make sure he had the strong problem solving skills. And instead of letting the conceptual and problem solving part stick to just math, we also made sure that he saw how algebra could be used in other ways by having him take honors physics with Derek Owens. That made geometry much more enjoyable and then with algebra 2 and trig more connections came together. By then, if he had needed to go back to algebra 1 to cement some gap, I totally would have done it. I wasn't afraid anymore. But there were no gaps at that level any more. None. They closed up on their own. He had just learned it as he went along. Truth be told, I always felt that we skimped on ratio and probability because these were my own weak areas. But this wasn't the case for DS. He caught on. If he stumbles on more difficult problems now (I think he will stumble on a few of the AoPS Volume 2 and Intermediate AoPS problems), so what? He's going to learn it by doing it. If he doesn't he will leave it aside and come back to it. It's that stick-to-itiveness that I think is very commendable. At this point though, he is applying the stick-to-itiveness on CrowdMath instead.

 

I think he would have lost the stick-to-itiveness the longer I delayed something when he was so ready, wanting it.

 

With natural math kids, there is something going on. I don't know how to put my finger on it. CP sounds a lot like what DS was like as a child. Just intuiting things, going into the "why"s instead of just the "how"s. Maybe many kids are that way and the usual system of following things lock-step robs them of discovering their math potential (sour grapes here for myself too).

[ebunny]

 

 

Please attach when you can. :)

 

Reading comp is another thing we're dealing with here. The paper airplane ratio problem was especially hard because it was just so wordy, a half of this, and then another half, and then a half of the half, and each half has two, and so on. Too many words!

 

He's getting better, but even so, half the problem he has sometimes is reading the question. Which is something he needs to work on, obviously, but it's usually better to work with him on sequencing information apart from math.

Nm...please PM me for those papers. I cant seem to get it linked here Edited September 25, 2016 by ebunny

[wapiti]

But I knew that I wanted my son to love it and keep loving it and so I took that leap and followed his lead. We've only had to dial back at most a couple times. After algebra 1 with Dolciani, we used the first half of AoPS Intro to Algebra to make sure he had the strong problem solving skills. And instead of letting the conceptual and problem solving part stick to just math, we also made sure that he saw how algebra could be used in other ways by having him take honors physics with Derek Owens. That made geometry much more enjoyable and then with algebra 2 and trig more connections came together. By then, if he had needed to go back to algebra 1 to cement some gap, I totally would have done it. I wasn't afraid anymore. But there were no gaps at that level any more. None. They closed up on their own. He had just learned it as he went along. Truth be told, I always felt that we skimped on ratio and probability because these were my own weak areas. But this wasn't the case for DS. He caught on. If he stumbles on more difficult problems now (I think he will stumble on a few of the AoPS Volume 2 and Intermediate AoPS problems), so what? He's going to learn it by doing it. If he doesn't he will leave it aside and come back to it. It's that stick-to-itiveness that I think is very commendable.

 

Thanks for your whole post.  Your experience and perspective are immensely helpful for me.

 

Reading comp is another thing we're dealing with here. 

 

I feel your pain.  My ds10 is pretty much reading on grade level AFAIK.  The Prealgebra text's problems and exercises should be fine for my ds; the Common Core alg 1 text his school uses, I'm not sure, as it's not exactly well-written though I've never looked at it from this angle before - e.g. there was a proportion problem involving high school chemistry that he had difficulty interpreting.

[SarahW]

There is more to elementary math. Perhaps it would help you to talk to a G&T elementary math teacher.

 

Ah, the problem was worthwhile in developing visualization skills.

 

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.

[SarahW]

Here you go..

 

 

The forum's complaining about opening your attachments.

 

Is this one? And the other one?

[chocolate-chip chooky]

I can relate to some of what's being discussed.

 

Maybe 1 or 2 years ago (at around ages 8-9ish) my daughter hated word problems. Loathed them. She just could not stand looking at a problem and not immediately knowing what she was going to have to do. 

 

Like you, SarahW, I was worried about the lack of interest in problem solving. But I was more worried about crushing her spark for maths, so I kind of let it go a bit and let her move on and build up her tool kit of concepts.

 

Now, at age 10, she adores problem solving. She relishes a word problem that is like a puzzle to solve and that include all different areas of maths all rolled into one problem. She doesn't necessarily immediately know all the steps she'll need to take, or which of her 'tools' she'll need to draw on. But *phew* she now loves it. I'll add though that she has zero interest in competitions. She says 'what's the point?'

 

Anyhoo, my point is that problem solving did come for us, with time and with a build up of tools in her tool kit.

I think perfectionism held her back quite a bit and having the tools gave her more confidence to jump in and give multi-layered problems a go. Maturity has also helped, I'm sure.

[sunnyday]

But to go back to my first post, I'm wondering if this is largely an issue with elementary "problem solving" questions. To use another example, yesterday we were doing a math competition question which asked how many paper airplanes kids could make depending on whether they used a whole sheet of paper for each one, or half a paper, or a quarter of a paper. Since it was a "Grade 3-4" question, they expected him to map out the number of airplanes per page and then multiply. He found the way the question was worded, and the solution presented, to be unnecessarily confusing. And, tbh, it was confusing, because it was actually a ratio question, which was "dumbed down" to not look like a ratio question. So that's one of the things I'm wondering - is this "just count up the information because we don't expect you to know the math yet" an issue only with these lower grade questions? Or is "ha! we'll see if you can figure out math you don't know yet!" a characteristic of math competition questions in general?

 

You keep coming back to this idea that the only problem solving being done at the elementary level is effectively a "gotcha", forcing kids to do problems with one hand tied behind their backs. It's just not true. Just because a problem can be solved with both a bar model AND an algebraic equation, doesn't mean that the only value of the problem is in re-inventing the wheel of algebra. Conversely, it doesn't mean that having an algebraic skillset negates the need for a bar model or other strategy that is based more on an understanding of the numbers than of the procedures.

 

I think you're not using the right books, to be honest. I have an idea of what you're talking about, as I've also seen "challenge math" that's just silly. But I find that Beast starred problems and Borac competitive math are much more worthwhile. Where are you getting your "competition" math?

[ebunny]

The forum's complaining about opening your attachments.

 

Is this one? And the other one?

Forum's complaining? Which forum and about what?

 

I can pm the papers too. Less stressful all around. Lol.

[Crimson Wife]

My DS liked BA & AOPS Pre-A, Singapore IP, and MEP but he is absolutely love, love, LOVING Elements of Mathematics. He's blazed through the first 2 courses in less than 3 months. Supposedly a student who does the complete sequence (13 of 18 courses are now available) can go from pre-algebra to calculus in 3 years. I don't know how far he'll get in EMF (his current subscription is through the 4th course) but as long as he's enjoying it, I'm willing to let him do it.

[Arcadia]

The value of those books is debatable..do kids really benefit? Or do we(parents, educators) assume so? Probably a topic for a different thread..

I just saw those books at the bookstores when we went back for a visit. So can't comment on them,

We use the freebies from CEMC U of Waterloo as well as old AMC8/10/12 papers.

Edited September 25, 2016 by Arcadia

[SarahW]

Forum's complaining? Which forum and about what?

 

I can pm the papers too. Less stressful all around. Lol.

 

Was one of them Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training?

 

That's an interesting article. Research was done at VU in Amsterdam actually, so especially relevant to me. Realistic Math Education, yes, ugh, it's the bane of our existence right now. I call it "find the distance to three friends' houses math." Something like 99% of the math book is word problems. It's the reason why we send CP to school with a bilingual dictionary - so he can do the math. The "elementary exit exam" I linked to in a pp is an example of the CITO test referred to in the article as the "norm."

 

That being said, my Dutch isn't great, but the word problems in Dutch books seem to be much more straightforward than many English word problems I've seen. Maybe because another word for these things in English is "story problems" the writers feel free to add in lots of extras like dependent clauses, subordinate clauses, relative clauses, and so on. And Dutch is a more blunt language. 

 

I'm figuring out what a "marked relational term" is, at the very least it could be a good reading exercise to go through the meaning of the different forms of word problems.

[SarahW]

You keep coming back to this idea that the only problem solving being done at the elementary level is effectively a "gotcha", forcing kids to do problems with one hand tied behind their backs. It's just not true. Just because a problem can be solved with both a bar model AND an algebraic equation, doesn't mean that the only value of the problem is in re-inventing the wheel of algebra. Conversely, it doesn't mean that having an algebraic skillset negates the need for a bar model or other strategy that is based more on an understanding of the numbers than of the procedures.

 

I think you're not using the right books, to be honest. I have an idea of what you're talking about, as I've also seen "challenge math" that's just silly. But I find that Beast starred problems and Borac competitive math are much more worthwhile. Where are you getting your "competition" math?

 

But in BA the starred problems are clearly working the topic of the chapter. I haven't seen a BA question which included a concept which they hadn't covered yet.

 

I print off old test papers from lots of different competitions in different countries. Every competition has a different focus, so we get new things with each one. But yes, in lots of old tests there's a question or two which could be easily solved with algebra or something (a dependent variable, for example) but the test makers apparently expect the kids (who aren't expected  to know algebra yet) to go through a series of 12 steps to guess and eliminate possibilities to get to the answer. Or maybe just "see" the solution? 

 

The test makers are trying not to test rote procedures, I understand that. But if the kids can't be expected to have lots of background skills, there seems to be a limited number of the types of questions they can ask, which leads them to throw in these "well, if you think about it long enough....you can do it without algebra" questions.

 

I'm curious if this issue goes away when the test makers have a larger expectation of background knowledge to work with.

[SarahW]

I just saw those books at the bookstores when we went back for a visit. So can't comment on them,

We use the freebies from CEMC U of Waterloo as well as old AMC8/10/12 papers.

 

 

More freebie printable math questions!  :thumbup1:

 

All right, so I looked through last year's 3-4 questions, and the first question sort of goes along with what I'm trying to say in this thread. Last year we actually did a Tower of Hanoi investigation. Like a real one, and we watched some youtube videos of mathematicians doing super complicated Towers and talking about the math for finding the least number of moves. I guess simplifying it down and having kids move the piles around to see the best to make the piles "even" has value and is fun and all. But how superficial that is would drive my kid nuts.

 

So yes, maybe he should skip the 3-4 questions. The 5-6 questions look like something he might like. They are more dependent on doing math.

[SarahW]

DS seemed to intuit based on the algebra problem. He probably drew upon some concept he had learned earlier and made connections in his head and voila, he had it. It just came so naturally to him because he wanted it very much. I really think that's what it was. He wanted these algebra tools under his belt and he wanted to solve problems at a more challenging level than the elementary and lower middle school problem solving books. We used the resources we did at the time because there was no BA and no AoPS Pre-Alg available.

 

We skipped prealgebra (we skipped precalculus too). Even when I was afraid of gaps because at that time I had already learned that more repetition meant boredom and losing interest. There is so much more going on than just learning. Especially with a child who is very goodnatured and does not act out, boredom takes a very different form. There was no sullen-ness which I would have mistakenly attributed to only bad attitude. There was just an overall shutting down (which I now have learned is what wilting looks like for my boy). So I wanted to avoid that. Once they lose interest and hope it's not always easy to get it back (lesson learned first with a bad kindy experience and later, with chemistry when he was 7yo and I wouldn't let him go deeper for fear of injury etc.).

 

I agree with Mike now. At that time, I was very worried because I think my own grasp of math isn't strong. But I knew that I wanted my son to love it and keep loving it and so I took that leap and followed his lead. We've only had to dial back at most a couple times. After algebra 1 with Dolciani, we used the first half of AoPS Intro to Algebra to make sure he had the strong problem solving skills. And instead of letting the conceptual and problem solving part stick to just math, we also made sure that he saw how algebra could be used in other ways by having him take honors physics with Derek Owens. That made geometry much more enjoyable and then with algebra 2 and trig more connections came together. By then, if he had needed to go back to algebra 1 to cement some gap, I totally would have done it. I wasn't afraid anymore. But there were no gaps at that level any more. None. They closed up on their own. He had just learned it as he went along. Truth be told, I always felt that we skimped on ratio and probability because these were my own weak areas. But this wasn't the case for DS. He caught on. If he stumbles on more difficult problems now (I think he will stumble on a few of the AoPS Volume 2 and Intermediate AoPS problems), so what? He's going to learn it by doing it. If he doesn't he will leave it aside and come back to it. It's that stick-to-itiveness that I think is very commendable. At this point though, he is applying the stick-to-itiveness on CrowdMath instead.

 

I think he would have lost the stick-to-itiveness the longer I delayed something when he was so ready, wanting it.

 

With natural math kids, there is something going on. I don't know how to put my finger on it. CP sounds a lot like what DS was like as a child. Just intuiting things, going into the "why"s instead of just the "how"s. Maybe many kids are that way and the usual system of following things lock-step robs them of discovering their math potential (sour grapes here for myself too).

 

 

I've been re-reading this, and it is very encouraging. Thank you.

 

Intuition and asking why, yes. Telling him to do something like "just multiply by the reciprocal" results in his stink face. Mom! you do know that not really how it works, right? Okay, okay!

 

Maybe he does just see math differently. I pulled some problems from chapter 1 of that online Holt Algebra (video solutions for me! Yay!) for him to do, and I noticed that he seemed to understand the equation before he "solved" it. I'm not sure how to explain it exactly, it's a complete mystery to me. But while he could answer what x was, he understood what the equation was saying with the x. I don't know, I think maybe I should start graphing linear equations with him and see what happens.

 

As for "regular" math getting some kids down - yes, I see that. CP has a cousin here who is the same age as him, and this summer while he was sleeping over for the week and we had some rainy days I printed out some old Dutch Kangaroo tests for the two of them, since they're published in both Dutch and English. My nephew actually did really well with it, and had fun with it. My MIL looked at it later and commented how surprised she was. "He's having so many problems with math in school! I don't understand!" Yes, well, clearly wholesale "Realistic Math Education" doesn't make every kid love math, or show his understanding of it!  :glare:

[regentrude]

 But yes, in lots of old tests there's a question or two which could be easily solved with algebra or something (a dependent variable, for example) but the test makers apparently expect the kids (who aren't expected  to know algebra yet) to go through a series of 12 steps to guess and eliminate possibilities to get to the answer. Or maybe just "see" the solution? 

 

The test makers are trying not to test rote procedures, I understand that. But if the kids can't be expected to have lots of background skills, there seems to be a limited number of the types of questions they can ask, which leads them to throw in these "well, if you think about it long enough....you can do it without algebra" questions.

 

I think these questions are very valuable because they encourage a child to think about the problem instead of simply applying a tool.

Many of the problems that are appropriate challenge problems in younger grades for students without the tool set become rather bland and run-of-the-mill once the student has the standard tools to solve the problem efficiently.

I recall competition problems we had to solve before knowing the theory and tools. They separate students who can think creatively about math from students who cannot - assuming that neither had the access to the theoretical knowledge.

Yes, at the highest level competitions, the students who are serious have studied the additional theory to get the advantage over students who have not; you don't win the IMO with just the standard high school math knowledge even if you are creative. But in the lower levels, I think the design is intentional.  It is not about having the students win who accelerated to early algebra - it's about seeing who can puzzle out the question with the available tool set.

Edited September 25, 2016 by regentrude

[Mike in SA]

we had to solve before knowing the theory and tools. They separate students who can think creatively about math from students who cannot - assuming that neither had the access to the theoretical knowledge.

Yes, at the highest level competitions, the students who are serious have studied the additional theory to get the advantage over students who have not; you don't win the IMO with just the standard high school math knowledge even if you are creative. But in the lower levels, I think the design is intentional.  It is not about having the students win who accelerated to early algebra - it's about seeing who can puzzle out the question with the available tool set.

 

I'd say it's also about learning to apply effort habitually.  Younger children don't have a stigma to overcome; if the math they see now requires serious thought and persistence, then as they advance, they won't be put off by the challenges they will encounter.  The math most kids see will be almost trivial.

 

Not too long back, I posted about younger DS actually getting stumped by an easy problem from a regular algebra course - he was just convinced there had to be a hidden trick, because it was too easy to merit placing on a test.  Nope, it was from a real end-of-year algebra test...

 

The key ingredients to his progression were the same, though.  He is doing it because he wants to.  We don't press him at all.  But, he does at least two tracks of math every year.  This year, he has three: AoPS Intermediate Algebra, Kiselev Stereometry (multidimensional and non-Euclidean geometry), and mathematics of relativity (a conceptual introduction with minimal practice, but it does go right through manifold theory).  He loves it.  All of it.  When he started (three years ago), he didn't even have his addition tables down.  He mastered polynomial division before long division.  In fact, polynomial division taught him theoretically how to do long division.  Now, he does it perfectly, because he understands the theory supporting the algorithm.

 

Now, we understand that statistically, he is rather unusual.  But I can say that I have worked with kids from all walks, and when they are ready, they are ready, and most of them know it before their parents do.  Maybe it will take 2-3 years per subject, but so what?  Must we rob them of the joy of mathematics?  Do we have to drill every last ounce out of their systems?

[Dmmetler]

But in BA the starred problems are clearly working the topic of the chapter. I haven't seen a BA question which included a concept which they hadn't covered yet.

 

I print off old test papers from lots of different competitions in different countries. Every competition has a different focus, so we get new things with each one. But yes, in lots of old tests there's a question or two which could be easily solved with algebra or something (a dependent variable, for example) but the test makers apparently expect the kids (who aren't expected to know algebra yet) to go through a series of 12 steps to guess and eliminate possibilities to get to the answer. Or maybe just "see" the solution?

 

The test makers are trying not to test rote procedures, I understand that. But if the kids can't be expected to have lots of background skills, there seems to be a limited number of the types of questions they can ask, which leads them to throw in these "well, if you think about it long enough....you can do it without algebra" questions.

 

I'm curious if this issue goes away when the test makers have a larger expectation of background knowledge to work with.

I think there's also a trickle down effect, because so many kids who do math competitions are probably working ahead in math, if not at school than home or with a tutor. And since every single competition makes practice materials and released tests available, it's going to be likely that kids who are doing a 3rd/4th grade question that can be solved more effectively by setting up an equation than by drawing a chart have had the exposure to both problem solving strategies.

 

I do think that competition math sometimes leads to the "Looking for the trick, because there has to be one". The appropriate antidote is real-world math, where sometimes answers are pretty ugly and problems aren't designed to be able to be done neatly without a calculator. DH brings home such problems from work for DD to play with, and working under snake breeders has shown her that for small values, actual results can deviate dramatically from the statistically predicted results. That's one reason why I'm willing to leave LoF in DD's rotation as well-because I swear sometimes Stan pulls numbers out of a random number generator. Some of those high school LoF problems really involve some ugly values that lead to inelegant looking results (I recall one particularly contentious X coordinate value of 83/13. DD was convinced she'd done something wrong because "you'd never graph an improper fraction". Turned out the whole idea was supposed to be to recognize that ALL those points on the line were included, not just the nice, neat ones, so the suggested value was, indeed something that came out to 83/13. That one ranks with Farmer Fred on the meltdown over math scale).

Edited September 25, 2016 by dmmetler

[regentrude]

Some of those high school LoF problems really involve some ugly values that lead to inelegant looking results (I recall one particularly contentious X coordinate value of 83/13. DD was convinced she'd done something wrong because "you'd never graph an improper fraction". Turned out the whole idea was supposed to be to recognize that ALL those points on the line were included, not just the nice, neat ones, so the suggested value was, indeed something that came out to 83/13. That one ranks with Farmer Fred on the meltdown over math scale).

 

Bahaha. I have vivid memories of Farmer Fred.. for us, it was the most involved problem in the entire book. Certainly the most memorable.

Edited September 25, 2016 by regentrude

[wapiti]

I do think that competition math sometimes leads to the "Looking for the trick, because there has to be one". The appropriate antidote is real-world math, where sometimes answers are pretty ugly and problems aren't designed to be able to be done neatly with a calculator. 

 

I don't have as much exposure to the broader world of competition math and I know there may be "tricks" to some extent especially where extensive contest prep is involved, but I was always under the impression that at least with AoPS problems, the "trick" isn't a procedural sleight of hand but rather the use of a deep understanding of a concept to solve the problem in a more elegant, not to mention easier, way than brute-force calculator, on purpose as a demonstration of that understanding.  Maybe that's why the problems in the CC texts are so messy, they're trying to be real-world?  Could be; I guess always assumed it was because the writers were lazy/weak with the math.

 

Speaking of Farmer Fred, I was thinking of him just last week when ds13 was working on some optimization exercises (Glencoe Common Core alg 2 text from school).  He asked me for help and I said no, I really can't help you LOL (he was fine; after all, he has a calculator).  I was surprised because I thought that optimization wasn't standard but more of an optional topic.

Edited September 25, 2016 by wapiti

[Arcadia]

All right, so I looked through last year's 3-4 questions, and the first question sort of goes along with what I'm trying to say in this thread.

We use their math circles materials, not the problem of the week ones

http://cemc.uwaterloo.ca/events/mathcircle_presentations_gr6.html

 

The CEMC contest questions are here

http://www.cemc.uwaterloo.ca/contests/past_contests.html

[Dmmetler]

It's more the recognition in AOPS that "if you need a calculator, you're doing it wrong"-with the result that if numbers are getting more complex and not cancelling or factoring or whatever neatly, you're going down the wrong path. That is, there is an elegant solution, and it's probably the right one (or it's a bogus solution and there is an equally elegant right one). In the real world, the H.L. Mencken quote is much more likely to apply.

 

They are still excellent teaching problems, and are great for learning how to strategize instead of just applying the algorithm. But some problems are just plain ugly and don't have a nice, elegant solution.

[SarahW]

I think there's also a trickle down effect, because so many kids who do math competitions are probably working ahead in math, if not at school than home or with a tutor. And since every single competition makes practice materials and released tests available, it's going to be likely that kids who are doing a 3rd/4th grade question that can be solved more effectively by setting up an equation than by drawing a chart have had the exposure to both problem solving strategies.

 

I do think that competition math sometimes leads to the "Looking for the trick, because there has to be one". The appropriate antidote is real-world math, where sometimes answers are pretty ugly and problems aren't designed to be able to be done neatly without a calculator. DH brings home such problems from work for DD to play with, and working under snake breeders has shown her that for small values, actual results can deviate dramatically from the statistically predicted results. That's one reason why I'm willing to leave LoF in DD's rotation as well-because I swear sometimes Stan pulls numbers out of a random number generator. Some of those high school LoF problems really involve some ugly values that lead to inelegant looking results (I recall one particularly contentious X coordinate value of 83/13. DD was convinced she'd done something wrong because "you'd never graph an improper fraction". Turned out the whole idea was supposed to be to recognize that ALL those points on the line were included, not just the nice, neat ones, so the suggested value was, indeed something that came out to 83/13. That one ranks with Farmer Fred on the meltdown over math scale).

 

 

Maybe so. I gave CP a "2nd grade" problem the other day, and he went through it and said, "Oh, you can just divide" and that's what he did, and he got the answer. I guess "2nd graders" were expected to count up and then make groups. But CP understands perfectly well that division is making groups, so if you're making groups, just divide.

 

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. We went and looked up the divisibility rules for 7, and we went through it again this summer, but even studied a proof, but it is so unusual and outside what he's doing right now, that I do wonder if our detour with it was particularly efficient or valuable. Maybe there's 4th graders out there for whom divisibility is their favorite things and they work with it all the time for fun, and maybe there's a small number of crazy-gifted kids who love number theory and can look at the question and see the method for determining the divisibility by 7, and there's some kids who have a math enrichment class where they were taught the algorithm and they've remembered it. But for the majority of kids.....I think it was so far beyond anything that could be creatively figured out, that it really was just a fast long division problem.