Math intuition: does it naturally slow down or end?

[Sarah0000]

I'm curious if there's a usual place in the mathematics journey where children who often intuit the next step will stop or considerably slow down? Is there a place where math is simply not as intuitive?

How much do you think that is influenced by whether the child had exposure to higher mathematical concepts while young instead of sticking only to the grade level basics? How much do you think that is influenced by discovery approach or child led learning versus parent led instruction?

 

I feel like DS' math journey is less a steady march forward and more a rocket blast forward and a slow slide back down while filling in gaps then another blast off. I feel like I just figure out what "place" he is in then he surprises me with another leap that I of course have no resources for yet. Then I second guess myself thinking "there's absolutely no way someone his age could possibly be ready for x, y, and z" and then I just throw a bunch of math at him and wait around some more. LOL. Just wondering when its likely, if its likely, we can follow along a math program in a more linear way. I'm kind of wondering if it will be like this until prealgebra when the concepts have to come together more?

[MotherOfBoys]

My son's journey is more of the rocket ship type also. We are at PreAlg (debating when to start) right now. This morning on his math mammoth 5 chapter decimals he figured out the tricks before they were presented in the lesson. He's done this for the whole chapter. I told him decimals might be hard (expecting the wall of learning that people keep telling me about) but he's discovered it all just fine. I'm wondering if we do Aops PreAlg (that I just bought) or MM6 (old version) next fall. We might bounce back and forth as the rocket blasts and refuels.

 

Sorry I'm no help. :)

[Arcadia]

Still not happening at the precalculus/calculus stage for my oldest :lol:

 

There are quite a few boardies here whose kids are doing that beyond the typical high school calculus stage for example Quark's son, Lewelma's son

 

If I have to guess at when it would be more linear, I'll say wait until the brain supposedly mature at 25 :)

[Mike in SA]

Disclaimer going in: math / physics degrees here...

 

I'm not sure it's ever a good thing to stop intuiting solutions.  Intuition in math is part of what makes it an art, not a science.

 

What it sounds like you may be more concerned about is the detailing of the steps taken.  As a professional, it is important to back up your answers in a coherent way.  For a 5-8 year old, not so much.  If they can talk their way to the point of the solution, then there's usually no need to force the issue.  When they reach harder mathematics, they'll learn to write things down just to free their brains for the more important thoughts.

 

[Sarah0000]

I'm not concerned about it at all, just curious. My child is four and the best explanation I've ever gotten for how he figured out some math thing is "I just thought of it in my brain." Usually his explanations are more along the line of "I learned it in all-the-ways-to-multiply-tens school." I'm not expecting him to be able to explain himself in a math way at all.

[imagine.more]

 

 

I feel like DS' math journey is less a steady march forward and more a rocket blast forward and a slow slide back down while filling in gaps then another blast off. I feel like I just figure out what "place" he is in then he surprises me with another leap that I of course have no resources for yet. Then I second guess myself thinking "there's absolutely no way someone his age could possibly be ready for x, y, and z" and then I just throw a bunch of math at him and wait around some more. LOL. Just wondering when its likely, if its likely, we can follow along a math program in a more linear way. I'm kind of wondering if it will be like this until prealgebra when the concepts have to come together more?

 

 

I'm reading other people's answers to your original question too because I've wondered the same thing. But your description of your son's Rocket-like learning progression, especially in Math, describes my son to a T! It can be exhausting sometimes: it almost feels like I'm being dragged along by a rocket, lol! But yes, it's basically Blast off.....Refuel.....Blast off.....Refuel. No rhyme or reason as to when it happens either mostly. 

[Arcadia]

How much do you think that is influenced by whether the child had exposure to higher mathematical concepts while young instead of sticking only to the grade level basics? How much do you think that is influenced by discovery approach or child led learning versus parent led instruction?

A young child's sense of wonder is still restricted by exposure not only for math but everything else.

For example in math, kids see patterns, logic in nature but may not make the connection to Fibonacci spirals if they have not been exposed to the idea.

They may think about what would be the shortest or most efficient path to go from home to point A, B, C, to home without realizing that is graph theory.

They may be looking at lottery and other games of chance and not realize it is game theory at work.

 

For example in terms of languages, we have always stayed in areas with many expats because it is more culturally comfortable for us. Our kids hear many Asian and European languages while out strolling as infants. It pique my DS11's interest in language patterns.

 

I think parent led instruction limits a young child to the parents' ability to educate. However the discovery approach or child led instruction requires more effort on the parents to provide the opportunity until the child is old enough to source for their own opportunities. For example my boys see a poster for an interesting event. We have to register on their behalf if it requires registering, pay up if it is not free, drive and chaperone them. An easy example is MakerFaire, my kids ask to go last year so we make it happen and they learn lots last year by just doing all the hands on and observing those without hands on. A math example would be the Julia Robinson Festival. Kids were curious about the event so we went and they had fun picking up stuff that they don't know yet or had slip their mind. Another math event is the Martin Gardner Festival, which while they didn't pick up as much new stuff, was still lots of fun meeting other kids who enjoy math.

 

As for explaining things, my kid who is stronger on proof is Mr Silent. My kid that is weaker on proofs is actually Mr Chatty and can explain a paragraph worth but not as concise and may leave you bewildered by all the meandering.

[mathwonk]

i thought you meant does intuition slow down with advanced age, as i worry mine is declining in my 70's.  i don't think it slows down much with kids under 30!  that uneven rocket ship analogy followed by slow stuff seems very familiar to me.  probably my intuition is slowing down partly because of my lack of exposure to math activity.  i think intuition follows thoughtful exploration.  some people think intuition slows down when people learn too much.  one of my profs* suggested we try to prove things ourselves "before you have filled your head with too many of other people's ideas".  this is probably not a problem for the first several decades.

 

*here is a nice interview with that prof:  Raoul Bott:

 

http://www.ams.org/publications/journals/all/fea-bott.pdf

Edited April 13, 2016 by mathwonk

[ebunny]

I'm curious if there's a usual place in the mathematics journey where children who often intuit the next step will stop or considerably slow down? Is there a place where math is simply not as intuitive?

 

How much do you think that is influenced by whether the child had exposure to higher mathematical concepts while young instead of sticking only to the grade level basics? How much do you think that is influenced by discovery approach or child led learning versus parent led instruction?

 

 

IMO, intuition possibly has a bigger role during arithmetic. But, as concepts get more complex, maybe some processes need to be explained to make more sense? Off the top of my head, I can think of factoring polynomials as a concept that needs to be explicitly taught rather than intuited because there are so many intricate mini-concepts involved that a child can get lost in rabbit holes.

 

Discovery approach is quite romanticised in math ed. I wonder if it places too high expectations on most children, gifted or not, to discover/unearth relationships between numbers/operations etc? All I know is that I'm not as enthused about discovery based learning when DD is 10 than I was when dd was 6, iykwim. 

[lewelma]

I'm curious if there's a usual place in the mathematics journey where children who often intuit the next step will stop or considerably slow down? Is there a place where math is simply not as intuitive?

How much do you think that is influenced by whether the child had exposure to higher mathematical concepts while young instead of sticking only to the grade level basics? How much do you think that is influenced by discovery approach or child led learning versus parent led instruction?

 

I feel like DS' math journey is less a steady march forward and more a rocket blast forward and a slow slide back down while filling in gaps then another blast off. I feel like I just figure out what "place" he is in then he surprises me with another leap that I of course have no resources for yet. Then I second guess myself thinking "there's absolutely no way someone his age could possibly be ready for x, y, and z" and then I just throw a bunch of math at him and wait around some more. LOL. Just wondering when its likely, if its likely, we can follow along a math program in a more linear way. I'm kind of wondering if it will be like this until prealgebra when the concepts have to come together more?

 

I found that as soon as my ds got to AoPS he slowed down. Less intuitive? Don't think so, but the rocket ship was not quite so zippy. 

 

DS's intuition was definitely not based on exposure to higher maths concepts while young, because I had *no* idea of where he would be 10 years later. We were playing shop at 6.5.  We did puzzles, cooking, estimating, word games, etc. No curriculum to be seen and no introduction into anything abstract. Then one day he asked me about math and what came next, and I told him about algebra, and he asked me to show him.  So I gave him a few easy ones which he solved, and he told me 'well, that was easy.' So I gave him one that he could not solve without studying Algebra, but then he did.  And not by guess and check, he made the intuitive leap to invent algebra, doing to one side what you do to the other as a way to isolate x.  It was that day that I decided that perhaps a curriculum was in order.  :tongue_smilie: But I still only went to primary school maths; I was very stuck in a box about what came first.  So even though he clearly was ready for Algebra, given that he invented it on his own, I bought him a 2nd grade maths book.  oops. That did not end well. I didn't have this board back then so I didn't know that some kids can just do more, earlier than others.  A lot more and a lot earlier. I did figure this out within about 6 weeks and we finally settled on 4th grade Singapore Maths.

 

As for the discovery method, I think it *is* the way you teach mathematical intuition.  From my experience, there is definitely a causation not a correlation.  However, ds refused to do primary school maths as teacher led or child led.  He did it all by discovery method.  He considered the textbook or any parental help to be cheating.  So I don't actually know how he taught himself fractions, decimals, etc, because all he had was the Singapore Intensive Practice which has no explanations in it. It did have answers, so perhaps he tried one thing and if that is not the way you divide fractions then he would try something else.  Back then ds was intensely focused and rigid, so I had to walk a fine line and try not to get involved. The discovery method of AoPS was like a match made in heaven, because they actually designed a discovery approach and integrated it completely into the program, rather than my ds just taking a traditional book and forcing himself to discover all the math by intuition and trial and error. So yes, ds's intuition made direct learning a trial for him which he could not suffer through, but also I am sure that the discovery method was the making of his intuition.  It took him almost 3 full years to get through the AoPS intro Algebra book, and that was because he was not learning algebra really, but rather developing his intuition which is now incredibly strong. 

 

So, my approach has always been to keep him happy with math.  If he did not like a book or it was too easy, we just moved on.  It did leave him with some funny gaps, but gaps are easy to fill later.  Typically, it takes about 10 minutes to fill one a few years down the line. But once the love of math has been lost, I would expect that it would be an uphill battle for years to get it back.  So I say, go with the flow. My younger boy rotated through 6 programs in a year until we found MEP secondary, which he loves. Did he lose time in his progression?  In the short term I think he definitely did, but in the long term he looks forward to math and feels confident in his abilities.  What more could I ask for?

 

Ruth in NZ

[daijobu]

Maybe I'm not as far along as many of the students here on WTM, but I find that as I get farther on in math, things are just plain more abstract and less intuitive.  

 

As an example, there is the chicken mcnuggets theorem.  The problem itself is easy enough to understand:  what is the largest number you can not purchase given they are sold in boxes of 9 or 20.  The formula for the number of integers that can not be expressed as a linear combination of m and n is (m-1)*(n-1)/2.  I have not been able to intuit this formula.  Yes, I get the proof (barely).  But I don't have an intuitive understanding of the formula.  I would go so far as to suggest that one does not exist.  If you disagree, please let me know!    

[EndOfOrdinary]

My son only stopped or walled when I gave him AoPS at 8. Basically the point was to force him to use the book, read the book, write down steps, and stop jumping to the end. I was a math prodigy. I distinctly remember hitting a wall in AP Calc because I never learned long division. Even with polynomials, I could intuit. I just got it and the numbers would come together in my brain. There was no studying. Then WHAM! I had no skills to help myself. So we took care of that one right away with kiddo.

 

He HATED it. Now, he goes with the book for a bit until he can explain things in a way that works for him (he is highly visual/conceptual). Then he just blasts ahead. The intuition is still there, but when asked to find three consecutive odd numbers which add up to 784, he glitches out. The other curriculums were asking for much smaller numbers (under 250) and he could do those in his head like nothing.

 

So I would say the intuition never goes away, but it definitely doesn't help once the complexity increases. There is nothing different about adding up to 784 or 136. The only difference was one my son could pull out of the air without knowing what he was doing, and the other required him to slow down and think it through.

[TerriM]

I think what you're describing is probably fairly healthy.  Even if he seems like he understands something, going back and revisiting a subject may allow him to see nuances and grasp more details that he didn't see at first.

 

I also agree that different kinds of math pose different difficulties.  For me Calculus was the first time I'd ever found math difficult.  In retrospect, I think it's just not my thing, but I do see significant differences in how one thinks about Algebra vs. Geometry vs. Calculus vs.....  So a kid may have intuition in one subject and then feel like another is really hard, and I think that's ok.  I would just say, don't give up, but move to a different kind of math, and then move back to a problematic one later.

[4KookieKids]

Disclaimer going in: math / physics degrees here...

 

I'm not sure it's ever a good thing to stop intuiting solutions.  Intuition in math is part of what makes it an art, not a science.

 

I absolutely agree. I worked on a team of mathematicians / electrical engineers in grad school, and my main contribution to the group was (with relatively high accuracy) being able to intuit when a conjecture someone else gave was false, and then follow my intuition to construct a counterexample. They teased me about it alot, because it became somewhat of a joke that someone would throw something out there and I'd follow it with an "I don't think that's true, because..." Then we'd go our separate ways, each with our own task for the week: they set out to prove it, and I set out to find a counterexample. :) Fun times!