# Conceptual understanding vs. Procedural (alone) understanding in Mathematics....

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### #1 Rosie

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Posted 24 February 2012 - 08:54 PM

Please help me find a way to clearly and concisely explain the difference between math programs that emphasize a conceptual understanding (Singapore, Miquon, RightStart, Math Mammoth, etc.) vs. those that emphasize a procedural understanding (Saxon, Horizons, etc. - not really sure what programs are in this category).

I have a friend who is considering homeschooling and has come to me for advice. I would love for her to truly understand math concepts as she learns along with her dds (just like I have done) through a conceptual program that emphasizes things like number bonds, visualization, and place value, but so far I am doing a terrible job of explaining the difference between the two methods.

So, how would you explain the difference to someone coming from a procedural only understanding? And make it convincing! LOL!

### #2 NittanyJen

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Posted 24 February 2012 - 09:07 PM

Somebody who only has a procedural understanding knows how to knock out the steps, but they don't really understand WHY those steps happen, or more importantly, why the numbers relate to each other the way they do.

Why care?

Because if you run into a novel situation, if you have a conceptual understanding, you can use what you know and apply it to figure it out. Procedural people can generally only do what they have memorized an algorithm for, and get stuck with even small deviation from that situation. There is a limit to how many individual circumstances you can memorize and use effectively.

Procedural people tend to get frustrated with math and declaim, "I'll never use this anyway!" They are memorizing individual circumstance, many of which they may well never use. The example they are memorizing was probably being used to illustrate a particular idea, or to show how to apply an existing idea in a new way, but the P person memorized it as a new case and (correctly) termed knowing that case as useless. If you have a conceptual understanding, your ability to creatively problem solve is unlimited, up to the skill level you obtain.

Math is not really numbers. Math is logic and problem-solving. These skills transfer far beyond the realms of just mathematics. Applying only procedural knowledge completely misses these skills. You end up memorizing the puzzle instead of learning how to solve puzzles and apply strategies. If you get the concepts, you acquire a toolbox of strategies you can apply to a very broad range of problem solving scenarios.

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Posted 24 February 2012 - 09:10 PM

It is the difference between knowing how to put ingredients together vs. having to carefully follow a recipe.

### #4 Crimson Wife

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Posted 24 February 2012 - 10:00 PM

If you only memorize the algorithm and don't understand why the algorithm works, you will be at the mercy of your memory. I actually ran into this recently when I couldn't remember how to calculate the formula of a line running through two points on a coordinate graph to help DD with a Singapore problem. It was so incredibly frustrating because I *KNEW* that I had memorized it at some point in jr. high. Had I been taught the why's and not just the how's of math, maybe I would've either remembered or been able to figure it out again.

I want so much better for my own kids than the mediocre math education I had growing up!

### #5 ElizabethB

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Posted 24 February 2012 - 10:33 PM

What they said.

Two good books:

The Liping Ma book and The Teaching Gap.

### #6 Momling

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Posted 25 February 2012 - 01:20 AM

It's important to have both though... before homeschooling full-time, I had to deal with TERC Investigations. The authors had clearly taken the idea of conceptual understanding and run insanely around in circles with it, tossing procedural understanding out the window with that bath water.

What the school ended up with was kids who had decent number sense and could explain their work, but took a great deal of time to draw pictures or work each problem out in mind-numbing ways that showed conceptual understanding and zero procedural ability.

I recall feeling like I might take my daughters pencil and shove it into my eyeballs if I had to watch her one more time draw 523 objects or write another sentence explaining why 10 + 4 was 14.

Incidentally, I understand they've found the kids test below grade level and they're abandoning the math program... I'm curious what the next trend is that they'll jump on.

### #7 8FillTheHeart

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Posted 25 February 2012 - 07:18 AM

Please help me find a way to clearly and concisely explain the difference between math programs that emphasize a conceptual understanding (Singapore, Miquon, RightStart, Math Mammoth, etc.) vs. those that emphasize a procedural understanding (Saxon, Horizons, etc. - not really sure what programs are in this category).

I think Jen's explanation is great.

I disagree w/you, however, that Horizons emphasizes procedural understanding at the expense of conceptual. My kids have all used Horizons and all of them have excellent conceptual understanding of math. I believe the weakness in Horizons is probably its infrequent word problems. Supplementing w/something like HOE easily fills that gap. Honestly, I think that MM is very similar to Horizons w/the exception of presentation in mastery units vs spiral.

### #8 Rosie

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Posted 25 February 2012 - 10:59 PM

Thank you, everyone!

I'll gladly take more comments if anyone else would like to chime in!

### #9 petepie2

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Posted 25 February 2012 - 11:36 PM

For us, it's a matter of seeing the forest or the trees. We use a combination of Rod & Staff (for the trees) and Singapore (for the forest).

For example, last week in Singapore she had a word problem that required her to subtract 958 from 1000. Instead of seeing "the forest", she set the problem up vertically to do a standard subtraction algorithm. Well, she got all confused borrowing with the multiple zeros because she had never done a problem like that before. I had to remind her of the big picture. Subtract 900. Then subtract 58. It was a matter of not seeing the forest because of the trees. Singapore and other conceptual programs help one to see the big picture. Obviously this is a work in progress for my DD!

### #10 Young Curmudgeon

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Posted 26 February 2012 - 12:01 AM

... when I couldn't remember how to calculate the formula of a line running through two points on a coordinate graph to help DD with a Singapore problem. It was so incredibly frustrating because I *KNEW* that I had memorized it at some point in jr. high.

Have you been able to find a reference to the how and why of finding distance between two point on a coordinate graph yet?

If not:

The basic idea is to form a right triangle that has the two points as end points of the hypotenuse and with one leg parallel to the x axis and the y leg parellel to the y axis.

Then it is possible to find the distance of one leg by taking the absolute number of the difference the the x coordinates of the points. Length leg parrellel to x is |x_1 - x_2|. And similarly for the leg parallel to y axis.

Then break out Pythagorean's formula, a^2 + b^2 = c^2. (Here I can't help with why that is true, I don't know why.)

Once all the pieces are put together to come up with the distance formula, it will still work if the line is parellel to one axis or the other, even though no triangle can be formed as above, because the other leg will have a length of zero, so a^2 + b^2 = c^2 implies a^2 + 0 = c^2.

### #11 LucyStoner

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Posted 26 February 2012 - 02:03 AM

It is the difference between knowing how to put ingredients together vs. having to carefully follow a recipe.

Being a cook or being a chef.

To succeed in science and math degrees later in life (which many bright kids could have done but ended up unable to do because of poor math educations) people need to love math in a way that IMO only comes from conceptual understanding. As Zaccaro says, "the music!"

### #12 Hunter

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Posted 26 February 2012 - 12:50 PM

I often need to practice the procedure before I can understand the concept. Often I will push ahead with a book like Saxon, but then read a bit in several other curricula and real books and whatever I can get my hands on, to cement the concept. Then go back and practice the procedure a bit more.

The breakthrough for me is realizing that I am a whole to parts learner, which isn't exactly the same as a conceptual learner. I can learn almost entirely by procedure, but do best with the problems set up whole to parts. For example learning multiplication tables by all the factors of a certain number, instead of by multiplying one number by a bunch of different numbers. I'm really enjoying reading Waldorf math books right now. Properly taught I can see that would have been the best way I would have learned math. Improperly taught that would be far worse than a procedure only method. Waldorf is big on whole to parts. Complete free grades 1-3 curriculum here. http://www.entwicklu...lfe3.de/?id=786

In general, with untrained teachers trying to cram too much content into too few hours, kids make at least some progress with procedure only. Concept heavy curricula with too little time to practice procedure are a disaster. And if the teacher doesn't even understand the concepts herself and then tries to teach with a concept heavy curriculum, forget it.

So if mom's math is shaky and she hasn't scheduled math as a huge priority, then procedure heavy curricula are best. Then she can plan some quick math unit studies, enrichment, games, etc to fill in the gaps at her leisure, without having the next day's procedure practice being dependent on her having presented a lot of time consuming concept material that maybe she doesn't even understand herself.

I've noticed the Strayer-Upton books are more whole to parts, than some other curricula, while still being heavily procedural. I've taken apart my grade 3-4, and am scanning it, and printing it out. I find working with large flat worksheets easier then wrestling with that cute but small fat book. Strayer-Upton 3-4 teaches all four processes, skip counting and easy fractions for each number at a time, so the child is really able to wrestle with the concept of the whole of each number, and all the ways it is used.

Edited by Hunter, 26 February 2012 - 12:58 PM.

### #13 TracyR

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Posted 26 February 2012 - 01:18 PM

I agree, both are important. But you have to know when to use what when you do. Some children understand conceptual math very early on. Some do not , and this is where you run into the " I hate math syndrome".

I think being able to know when to use which one is important.
For instance my oldest learns well conceptually. She always has. But let me tell you I found that I focused so much on that when we got to doing mutiplication and division that I needed to do a 360 and have her learn her math facts . So memorization is important too.
Then my middle two do well with spiral learning right now. I'm most sure at some point it will change for them and we'll switch over to more conceptual math.
My youngest switches between the two right now.

So it all depends on the child. I know as a child I would have done better with a spiral math program. It makes sense to me. But as an adult conceptual math is starting to make sense. Problem was I was taught math conceptually as a child and therefore learned to hate math because I didn't understand it.

P.S. I'm waiting for that perfect program that really includes the best of both worlds in its curriculum.

Conceptual based math:
Rod and Staff
Math U See
Bob Jones
Math Mammoth
Seton Math
Ace School of Tomorrow
Making Math Meaningful
Miquon
Modern Curriculum Press
K12's Math
Life of Fred
Teaching Textbooks are just to name a few

Spiral math :
Abeka,
Horizons
Saxon
Christian Light

### #14 MeganW

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Posted 26 February 2012 - 02:05 PM

I agree, both are important. But you have to know when to use what when you do. Some children understand conceptual math very early on. Some do not , and this is where you run into the " I hate math syndrome".

I think being able to know when to use which one is important.
For instance my oldest learns well conceptually. She always has. But let me tell you I found that I focused so much on that when we got to doing mutiplication and division that I needed to do a 360 and have her learn her math facts . So memorization is important too.
Then my middle two do well with spiral learning right now. I'm most sure at some point it will change for them and we'll switch over to more conceptual math.
My youngest switches between the two right now.

So it all depends on the child. I know as a child I would have done better with a spiral math program. It makes sense to me. But as an adult conceptual math is starting to make sense. Problem was I was taught math conceptually as a child and therefore learned to hate math because I didn't understand it.

P.S. I'm waiting for that perfect program that really includes the best of both worlds in its curriculum.

Hmmm - I thought I understood, but maybe I don't.

I thought you had a continuum of conceptual vs. procedural, and also of mastery vs. spiral, and so any one program could be any combo so you could have
a conceptual-focused, mastery program,
a conceptual-focused, spiral program,
a procedural-focused mastery program, or
a procedural-focused spiral program.

I thought spiral vs. mastery was really just a matter of how it was organized (all of one topic, then all of the next, vs. continual revisiting each topic and adding a little each time).

Is that right?

### #15 Rosie

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Posted 27 February 2012 - 10:21 PM

Hmmm - I thought I understood, but maybe I don't.

I thought you had a continuum of conceptual vs. procedural, and also of mastery vs. spiral, and so any one program could be any combo so you could have
a conceptual-focused, mastery program,
a conceptual-focused, spiral program,
a procedural-focused mastery program, or
a procedural-focused spiral program.

I thought spiral vs. mastery was really just a matter of how it was organized (all of one topic, then all of the next, vs. continual revisiting each topic and adding a little each time).

Is that right?

Yes, that's how I've always read it. I'm (the OP) asking specifically about teaching concepts AND algorithms vs. teaching only (or mostly) algorithms.

### #16 Targhee

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Posted 28 February 2012 - 02:27 AM

Teaching the algorithm yields answers to equations.

Teaching the concept yields problem solving.

You know the thing I find really wonderful about teaching the concept is that once the concept is understood the child usual finds ways of solving the problems intuitively, and even alternative ways from algorithms taught. If you only teach the algorithm you are asking them to memorize a bunch of algorithms and memorize when to use them (lots to remember).

### #17 Crimson Wife

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Posted 28 February 2012 - 05:14 AM

Conceptual based math:
Rod and Staff
Math U See
Bob Jones
Math Mammoth
Seton Math
Ace School of Tomorrow
Making Math Meaningful
Miquon
Modern Curriculum Press
K12's Math
Life of Fred
Teaching Textbooks are just to name a few

Spiral math :
Abeka,
Horizons
Saxon
Christian Light

I'm going to have to disagree with your characterization of certain programs. Seton and MCP are definitely NOT conceptual programs. They may be mastery/soft spiral but they are heavy on memorization and very light on the explanations of why the algorithms work. Seton & MCP remind me very much of the lousy math books I had growing up.

If a parent is looking for a conceptual program, I would go with one of these:

Singapore Primary Mathematics
Math in Focus
Math Mammoth
Right Start Mathematics
Miquon
MEP
Tokyo Shoseki

There may be others but the above are the ones I have either used or previewed enough of to determine they are conceptual.

### #18 brownie

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Posted 28 February 2012 - 09:13 AM

Understanding math allows you to re-develop the rules for yourself even once you've forgotten them.

Both are critical. We have had to stop at times to make more room for each. DS and I clearly prefer understanding but you can't get through algebraic equations if you can't quickly manipulate fractions.

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### #19 rieshy

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Posted 28 February 2012 - 10:07 AM

How would you characterize McRuffy math- conceptual or procedural?

### #20 ElizaG

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Posted 28 February 2012 - 11:37 AM

I'm going to have to disagree with your characterization of certain programs. Seton and MCP are definitely NOT conceptual programs. They may be mastery/soft spiral but they are heavy on memorization and very light on the explanations of why the algorithms work. Seton & MCP remind me very much of the lousy math books I had growing up.

What levels of Seton have you used? It seems pretty conceptual to me. Lots of different visual representations in K and 1st grade. Diagrams, verbal explanations and definitions in 3rd and 4th grade. It's clear to me that they understand the importance of teaching the meaning of the operations, etc. Their books are fairly new, so it's an open question as to how well the lessons work for most children, but I wouldn't say it's just a "follow this algorithm" program.

That said -- DH and I have talked about the "math wars" extensively, and have come to the decision that we'll present the concepts in various ways, but only require mastery of the procedures. If some of our children have a difficult time with certain concepts now, they can explore them at their leisure, as Hunter puts it. There are a lot of years between basic arithmetic and the point where advanced students "hit the wall" because they don't have deep knowledge. At the elementary level, I think it's enough for us as parents just to know that this wall exists, and to provide the children with ongoing opportunities to figure things out (through games, everyday math, books, word problems, manipulatives etc.). I'm not in a rush to get it all taken care of up front, and I think that might be counterproductive for some students anyway. Children move into abstraction in their own way, and I'm not sure that we can or should force the issue.

I feel sort of the same way about language arts. There was a recent discussion here about teaching grammar concepts in early elementary. Many of the issues raised seem relevant to math, as well.