Teaching Math Backwards

[Embassy]

On a previous thread it was mentioned that math cannot be taught out of order. I have been "experimenting" on my kids the past year and a half in our approach to math. My kids do fine with math, but it isn't something that they really love. As the years of teaching them have passed I am coming more and more to the conclusion that they learn some things better when it is approached out of order or even backwards. I still continue with a standard math program, but I reach way out and grab hold of topics and concepts that they will not encounter in their standard math program for quite some time. It is in these more difficult concepts where I find an enjoyment of math or a sparkle in the eye when it is otherwise dull.

 

This past weekend I read a book that resonated with me - with the thoughts I have been having about how my kids learn. The book is called In the Mind's Eye and it is written by Thomas West. It is essentially about visual thinkers, gifted people with dyslexia, other learning difficulties, computer images, and the irony of creativity.

 

He said this in his chapter on math:

Mathematics is essentially sequential and cumulative. You cannot deal with concept or operation 29 until you have mastered operations 1 through 28. That this is the most reasonable way to teach mathematics, there can be little debate,...But what should we do with students for whom the easy things are hard and the hard things are easy - those who naturally jump to the end and skip over the beginning, those who jump to the world of intuitive images without having mastered the basic elementary steps?

Perhaps, for these, one should consider teaching mathematics backwards, that is, teaching the images and patterns first and teaching the conventional symbol system, rules and rigorous process later....

 

Thoughts?

[regentrude]

I have a hard time imagining how this is supposed to work.

Yes, you can discuss general ideas and concepts - but the student will not be able to do math with these concepts if he does not have the prerequisites.

To illustrate what I mean:

I can teach a student who has a rudimentary understanding of functions the two basic ideas that constitute calculus: the derivative is the slope, the integral is the area under the curve. But in order to actually calculate either, the student is limited by her previous knowledge. If the student understands rational functions and limits, I can, with a mathy student, derive what the derivative of a power law function is. But that's where it stops - without trigonometry and algebra 2, exponentials, logarithms and trig functions can not be treated, and realistic problems can not be solved.

 

We actually encountered these limitations when our gifted DD was auditing classes on nonlinear physics and chaos: she got the basic concepts, but was limited in the execution of the actual problems because she had not completed calculus.

So, talking about concepts earlier is great; but the student is not ready to actually apply the concepts until he has acquired the foundation.

 

Or, if you prefer an earlier example: of course a younger child can understand the basic idea of algebra: the balanced equation. Which can be nicely demonstrated using a scale and adding and taking away stuff from each side. But in order to solve any actual algebra problem, the student must have mastered arithmetic - otherwise he will not be able apply the concept to a problem and to perform the necessary calculation.

Edited February 24, 2012 by regentrude

[SunshineMom]

:bigear:

[wapiti]

.

Edited January 12, 2016 by wapiti

[Tanikit]

My twin brother could not do math at school and was almost failing it. He wanted to program a computer when he was 15 and needed some math to do it. He asked for help and was told the math he needed was second year university level so he wouldn't be able to do it. He went on to figure it out by himself without any trouble. I think if something interests you, you can make a plan and the challenge will not be too much.

[regentrude]

My twin brother could not do math at school and was almost failing it. He wanted to program a computer when he was 15 and needed some math to do it. He asked for help and was told the math he needed was second year university level so he wouldn't be able to do it. He went on to figure it out by himself without any trouble. I think if something interests you, you can make a plan and the challenge will not be too much.

 

That would be rare. Extremely rare. Especially the "without any trouble" part.

I see students who are STEM majors struggle with math every day.

[Tanikit]

Yes perhaps that is just what he said after he accomplished it. We are still wondering how he got it right but he constantly passed very difficult computer courses with excellent marks despite never having finished school. He can do what he wants to but if he doesn't want to then he won't.

[Dmmetler]

FWIW, what I've found is that reading about interesting/fun conceptual stuff often provides the motivation for my DD to go back and learn the basics. For example, I suspect one reason why she broke down and learned her multiplication facts and worked on building up fluency with them, instead of just figuring them out every single time she needed them was that last summer she discovered Penrose the Mathematical Cat and a bunch of other books by the same author-and HAD to use multiplication to play with the concepts.

 

I've seen the same thing in gymnastics-for example, she had the hardest time with her cartwheel and her handstand-in both cases, because she struggled with just throwing herself at it. But when she started a cheer tumbling class, where a lot of effort was being spent on walkovers, back and front handsprings, and front tucks (with lots of precursor steps), suddenly, she improved a LOT on the cartwheel and handstand.

 

I think sometimes DD gets in a mindset where "It's too easy, so I'm not going to do it" and stops trying, and when she is allowed to move on to something harder, it gives her a way to "Save face" and a reason to do the "too easy" stuff.

Edited February 24, 2012 by dmmetler

[Temperance]

This describes how my daughters and I learn best. I can't figure out how we would fit into the descriptions provided by the book, however. My verbal scores have always been better than my math scores. Can a visual/spatial learner be better verbally than mathematically?

 

I figure it's easier to explore an island if you've seen a map. Maybe it doesn't show you where every rock and tree are, but you won't get lost.

[Beth in SW WA]

:bigear:

[Embassy]

Can a visual/spatial learner be better verbally than mathematically?

 

I think so. I know an adult VSL who didn't do well at math, but excelled at language arts. My VSL is generally good in math and parts of language arts. Some parts of math (i.e. math facts, CWP) and some parts of language arts (i.e. phonics/spelling/mechanics) are difficult. It is the only way I can explain why my kids struggled with phonics and learned to read with ease. Both of my sons are hard is easy, easy is hard and only one is a VSL.

Edited February 24, 2012 by Wehomeschool

[quark]

Why do all the interesting threads come up when I have the least time? :glare: (LOL).

 

This is essentially what we've been doing with DS. Sequential math would have killed his love for the subject.

 

If you liken my DS to a piece of cloth that needs to be dyed, in the earlier years (much like the thread where you do 3 diff types of math at the same time), we partially dipped him into the "facts practice" and the "grade level concepts" vats but immersed and soaked him for longer hours in the "higher concepts" vat.

 

If I'd soaked him longer in vats 1 and 2 instead of vat 3, he would have tolerated it but I don't think he will love math the way he does now.

 

He isn't working on high school geometry "out of the blue". He does have some background and practice in the 4 operations and a good understanding of pre-algebra concepts and a year-long algebra course. He just did not need it all drummed into him. He spends the most time understanding "patterns". Number behavior is a better word for it. He does this by reading supplementary books on math, math websites (wikipedia included), free pdfs I can find or friends suggest etc. He spends time playing with platonic solids, origami, understanding angles and shapes and such by physically manipulating toys and sticks and paper, by creating contraptions with cardboard and duct tape.

 

But he's not working on calculus yet so I cannot say if our vat experiments have been truly successful. :001_smile: If you look at math enjoyment though, it is obviously a success.

 

I've seen the same in him with everything else. Every book I hand him, he skips pages first. He reads page 200 before page 50, for example. It initially puzzled and frustrated me. But he hasn't lost anything by it yet that I can see. He still eventually reads a book he likes in its entirety, be it Roald Dahl or Asterix or Moby Dick, but he always does it his way first.

 

I really think it's important to reassess how some of these kids are taught. I read about an experiment a few years ago where a college prof, seeing his students dislike biology lectures, changed his method to begin every lecture with a lab first. It was apparently very successful. Even non-science majors began inquiring about joining his class.

 

There is some point where mastery becomes important before moving on, I just don't think sequential learning should be a rigid prerequisite all the time and in every situation.

 

On a previous thread it was mentioned that math cannot be taught out of order. I have been "experimenting" on my kids the past year and a half in our approach to math. My kids do fine with math, but it isn't something that they really love. As the years of teaching them have passed I am coming more and more to the conclusion that they learn some things better when it is approached out of order or even backwards. I still continue with a standard math program, but I reach way out and grab hold of topics and concepts that they will not encounter in their standard math program for quite some time. It is in these more difficult concepts where I find an enjoyment of math or a sparkle in the eye when it is otherwise dull.

 

This past weekend I read a book that resonated with me - with the thoughts I have been having about how my kids learn. The book is called In the Mind's Eye and it is written by Thomas West. It is essentially about visual thinkers, gifted people with dyslexia, other learning difficulties, computer images, and the irony of creativity.

 

He said this in his chapter on math:

Mathematics is essentially sequential and cumulative. You cannot deal with concept or operation 29 until you have mastered operations 1 through 28. That this is the most reasonable way to teach mathematics, there can be little debate,...But what should we do with students for whom the easy things are hard and the hard things are easy - those who naturally jump to the end and skip over the beginning, those who jump to the world of intuitive images without having mastered the basic elementary steps?

Perhaps, for these, one should consider teaching mathematics backwards, that is, teaching the images and patterns first and teaching the conventional symbol system, rules and rigorous process later....

Thoughts?

[wapiti]

thinking about this another way, one person's forward might be another person's backward; e.g., the new BA samples teach area before multiplication...

[Temperance]

I think so. I know an adult VSL who didn't do well at math, but excelled at language arts. My VSL is generally good in math and parts of language arts. Some parts of math (i.e. math facts, CWP) and some parts of language arts (i.e. phonics/spelling/mechanics) are difficult. It is the only way I can explain why my kids struggled with phonics and learned to read with ease. Both of my sons are hard is easy, easy is hard and only one is a VSL.

 

I definitely need to read up on VSL. It sounds like sometimes hard-is-easy/easy-is-hard learners aren't VSL. So perhaps the book doesn't describe non-linear learning as it is needed for all types of learners. Are all auditory learners sequential?

[Beth in SW WA]

Why do all the interesting threads come up when I have the least time? :glare: (LOL).

 

This is essentially what we've been doing with DS. Sequential math would have killed his love for the subject.

 

If you liken my DS to a piece of cloth that needs to be dyed, in the earlier years (much like the thread where you do 3 diff types of math at the same time), we partially dipped him into the "facts practice" and the "grade level concepts" vats but immersed and soaked him for longer hours in the "higher concepts" vat.

 

If I'd soaked him longer in vats 1 and 2 instead of vat 3, he would have tolerated it but I don't think he will love math the way he does now.

 

:bigear: Love this analogy!

 

He spends time playing with platonic solids, origami, understanding angles and shapes and such by physically manipulating toys and sticks and paper, by creating contraptions with cardboard and duct tape.

 

Awesome!

 

There is some point where mastery becomes important before moving on, I just don't think sequential learning should be a rigid prerequisite all the time and in every situation.

 

:iagree: I love the way you think. :)

Graphing is a visual exercise which dd8 enjoys. Did/does your ds enjoy graphing? I expect dd will love geometry once we dig deeper.

[Embassy]

I definitely need to read up on VSL. It sounds like sometimes hard-is-easy/easy-is-hard learners aren't VSL. So perhaps the book doesn't describe non-linear learning as it is needed for all types of learners. Are all auditory learners sequential?

 

The book discusses non-standard learners like visual thinkers, people with dyslexia, and highly creative people. Dyslexia is a main topic. He mostly talks about easy-is-hard and hard-is-easy from the viewpoint of dyslexia. I don't think my kids have dyslexia, but they do have some similarities.

 

The book predicted that my 3 year old would likely fit with the same learning pattern as her siblings because of her strong art/visual skills. She sees patterns and can create art from ordinary objects. As I was typing this she was behind me and told me to look at her whale. She had made a whale on the floor out of a toy necklace, torn bits of tissue paper, and a hair tie. She draws quite detailed pictures on Scratch too. I guess when you take two visual parents, a family history of dyslexia, and a highly spatial dh you get something like what I am seeing. I don't have dyslexia, but I experienced the hard-is-easy and easy-is-hard. College was so much more stimulating and therefore much easier than high school.

 

I was never interested in math. I don't want that for my kids. They are both wanting to go into STEM related fields (one computer, one science) and neither of them really enjoy math. In school math wasn't difficult for me. It was one of my best subjects, but I backed out of the math honors track in high school. I had no interest. In college and graduate school I was only required to take statistics. The only time I have seen a spark with my kids is in geometry with my VSL, learning through videos of people applying math in their lives, engaging living books (with pictures for my VSL), and when they are confronted with math that is over their heads. :confused::bigear:

[2smartones]

When ds1 was tested, we were told he was a VSL, and it was obvious that verbal skills were his strength. When we started the process of testing, we went into it thinking he must have dyscalculia or something, because his conceptual math level was several years higher than his working level. He couldn't remember something tomorrow that he'd learned today, but he could remember it on the third day, and then forget it again on the 4th day. He learned things "out of order" quite often. He learned area before learning multiplication, and learned multiplication before addition. Seems crazy! (Turns out he didn't have dyscalculia, btw.)

 

We were told not to approach math with a step-by-step progression with him like textbooks do. The spiral approach was too confusing, too. We were told to approach math like a bubble, with multiplication being the center of the bubble, because that was his strongest area at the time. Every other math concept could be taught as an extension of multiplication (division is multiplication backwards, addition is broken down multiplication, subtraction (taught last, and in conjunction with long division) is backwards addition, etc.). It worked. He still doesn't turn cartwheels when it's time to do math, but at least he doesn't cry and throw things and call himself stupid.

[quark]

:iagree: I love the way you think. :)

Graphing is a visual exercise which dd8 enjoys. Did/does your ds enjoy graphing? I expect dd will love geometry once we dig deeper.

 

Thanks Beth! He loves graphing. Loves it with a capital L (or G depending on how you look at it), bolded, italicized and all that.

 

I definitely need to read up on VSL. It sounds like sometimes hard-is-easy/easy-is-hard learners aren't VSL. So perhaps the book doesn't describe non-linear learning as it is needed for all types of learners. Are all auditory learners sequential?

 

I don't think hard-is-easy/easy-is-hard learners are always = VSL. I don't think my son is VSL. Perhaps slightly but not very much so from what I have observed. My visual-auditory-kinesthetic learner is not sequential. I do wonder if a purely visual or purely one type of learner will be more sequential or less sequential.

 

Very interesting how each kid works. Amazing stuff.

[catz]

 

I figure it's easier to explore an island if you've seen a map. Maybe it doesn't show you where every rock and tree are, but you won't get lost.

 

:iagree: I think we are a house full of people who tend towards VSL. It helps to have the big picture or we might get lost in a little detail. I've posted this before, but I did not do well in elementary math (very rote, procedural, old style curriculum). But went on to get honors grades through high school and college once I hit a more conceptual curriculum, and went on to get a BS in math and computer science.

 

So with my kids, if they seem stuck on a small thing I just keep moving generally. Math facts in particular I have not held them up for. Often when they get a wider view, things will click for them. I never had my math facts down pat until I got to interesting math because I never found a compelling reason to retain that info.

 

My youngest is self selecting math a couple levels of Singapore in the order she'd like to cover them because going at it sequentially was driving her nuts. And she's not interested in racing through math like my oldest who I had to work hard to stall.

[mom4peace]

I'd disagree that mathematical learning must be linear. My older son, then 10, did a beautiful job with Algebra (Jacobs) without having memorized his times tables or knowing long division (he used a calculator on the division and was just slow with the times tables). He'd finished Singapore 6, including challenging word problems, but really despised arithmetic.

 

By the end of algebra, the times tables were in place. He had to use them for the Algebra, and with that necessity, they wormed into his brain. He figured out long division during Algebra as well -- synthetic division just made sense to him, and from there he learned long division.

 

Without moving along at that point, he'd have increasingly lost interest in math. For any subject, I think if we hold to an absolute order to learn things, we do a disservice to some kids who just aren't linear thinkers. Gifted kids especially are often whole-to-parts thinkers, needing to see the whole picture before the details.

 

As always, there are no absolutes. Watch the kid, not the text.

[SunshineMom]

I have approached math totally wrong but am trying to change that. We have done way too much "drill and kill" here and not enough exploring of math. I have introduced some basic pre-algebraic concepts to my dd8 (x=6, y=x/3) or (x=-12+y,y=5), x=6y, y=3) to coordinate with graphing shapes, in order to change some things up which has helped. I recently read this article http://www.maa.org/devlin/LockhartsLament.pdf which is really making me rethink math and how I am approaching it vs how my children are.

[Greenmama2]

I have approached math totally wrong but am trying to change that. We have done way too much "drill and kill" here and not enough exploring of math. I have introduced some basic pre-algebraic concepts to my dd8 (x=6, y=x/3) or (x=-12+y,y=5), x=6y, y=3) to coordinate with graphing shapes, in order to change some things up which has helped. I recently read this article http://www.maa.org/devlin/LockhartsLament.pdf which is really making me rethink math and how I am approaching it vs how my children are.

 

Thank you for the link. I really enjoyed that article. Now I'm contemplating sending it to my musician friends who are campaigning for higher levels of compulsory music education :001_huh:

[wapiti]

I have approached math totally wrong but am trying to change that. We have done way too much "drill and kill" here and not enough exploring of math. I have introduced some basic pre-algebraic concepts to my dd8 (x=6, y=x/3) or (x=-12+y,y=5), x=6y, y=3) to coordinate with graphing shapes, in order to change some things up which has helped. I recently read this article http://www.maa.org/devlin/LockhartsLament.pdf which is really making me rethink math and how I am approaching it vs how my children are.

 

Thank you for the link. I really enjoyed that article. Now I'm contemplating sending it to my musician friends who are campaigning for higher levels of compulsory music education :001_huh:

 

FWIW, if you search for Lockhart, you can find older discussions of this article.

[Halcyon]

Why do all the interesting threads come up when I have the least time? :glare: (LOL).

 

This is essentially what we've been doing with DS. Sequential math would have killed his love for the subject.

 

If you liken my DS to a piece of cloth that needs to be dyed, in the earlier years (much like the thread where you do 3 diff types of math at the same time), we partially dipped him into the "facts practice" and the "grade level concepts" vats but immersed and soaked him for longer hours in the "higher concepts" vat.

 

If I'd soaked him longer in vats 1 and 2 instead of vat 3, he would have tolerated it but I don't think he will love math the way he does now.

 

He isn't working on high school geometry "out of the blue". He does have some background and practice in the 4 operations and a good understanding of pre-algebra concepts and a year-long algebra course. He just did not need it all drummed into him. He spends the most time understanding "patterns". Number behavior is a better word for it. He does this by reading supplementary books on math, math websites (wikipedia included), free pdfs I can find or friends suggest etc. He spends time playing with platonic solids, origami, understanding angles and shapes and such by physically manipulating toys and sticks and paper, by creating contraptions with cardboard and duct tape.

 

But he's not working on calculus yet so I cannot say if our vat experiments have been truly successful. :001_smile: If you look at math enjoyment though, it is obviously a success.

 

I've seen the same in him with everything else. Every book I hand him, he skips pages first. He reads page 200 before page 50, for example. It initially puzzled and frustrated me. But he hasn't lost anything by it yet that I can see. He still eventually reads a book he likes in its entirety, be it Roald Dahl or Asterix or Moby Dick, but he always does it his way first.

 

I really think it's important to reassess how some of these kids are taught. I read about an experiment a few years ago where a college prof, seeing his students dislike biology lectures, changed his method to begin every lecture with a lab first. It was apparently very successful. Even non-science majors began inquiring about joining his class.

 

There is some point where mastery becomes important before moving on, I just don't think sequential learning should be a rigid prerequisite all the time and in every situation.

 

My younger was sitting on math facts for a while, because of me. I finally moved him on the geometry and he's thrilled. He gets it quickly and interestingly, math facts are coming along nicely now that we're not focusing on them. He is much, much happier with math now. For his style of learning, jumping around topic-wise works better. My personality does not want to embrace this, but that's my problem, not his. He likes doing origami, math puzzles, word problems....and he jumps all over the place, topic-wise.

[quark]

My younger was sitting on math facts for a while, because of me. I finally moved him on the geometry and he's thrilled. He gets it quickly and interestingly, math facts are coming along nicely now that we're not focusing on them. He is much, much happier with math now. For his style of learning, jumping around topic-wise works better. My personality does not want to embrace this, but that's my problem, not his. He likes doing origami, math puzzles, word problems....and he jumps all over the place, topic-wise.

 

:hurray: Good for your younger! And good for you too. I understand the bolded. It isn't easy on me either. But we have to try what we can. Good luck!

[Bluegoat]

:iagree: I think we are a house full of people who tend towards VSL. It helps to have the big picture or we might get lost in a little detail. I've posted this before, but I did not do well in elementary math (very rote, procedural, old style curriculum). But went on to get honors grades through high school and college once I hit a more conceptual curriculum, and went on to get a BS in math and computer science.

 

So with my kids, if they seem stuck on a small thing I just keep moving generally. Math facts in particular I have not held them up for. Often when they get a wider view, things will click for them. I never had my math facts down pat until I got to interesting math because I never found a compelling reason to retain that info.

 

My youngest is self selecting math a couple levels of Singapore in the order she'd like to cover them because going at it sequentially was driving her nuts. And she's not interested in racing through math like my oldest who I had to work hard to stall.

 

I was very much like this as a child. I pretty much gave up on it before the end of elementary school and went on to another field of study (though I did do well in computer science in high school).

 

I'm seeing the same thing in my daughter now though, and i really don't want her to give up on math at such an early stage. Unfortunately because I don't feel like I have a great grasp on how to teach it, I find it hard to know how to tweak what we are doing to keep her interested. It is a bit worrying.