The importance of practicing math *concepts* to mastery

[forty-two]

Much emphasis was made in the "Cheeseburger" thread about the importance of practice, practice, practice when it comes to mastering procedures and math facts. No argument here. But what about the importance of practicing and reviewing the *concepts* to master them?

 

Lots of people said that they had to abandon "conceptual" curricula because there just wasn't enough practice for their kids to master the facts and procedures. But at the same time "traditional" programs were defended as "teaching the concepts" perfectly adequately so long as they explained the why behind a procedure when it was first introduced. And when kids completed the program without retaining the whys, it was because they just aren't mathy, or because they just weren't developmentally ready, not because they just didn't get enough practice. We don't expect that kids will master math facts or the standard algorithms by just seeing them once - why in the world do we think seeing a math concept once is somehow sufficient?

 

A lot of traditional math defenders made a big point that we often learn by doing, that sometimes it takes doing problem after problem before it suddenly clicks. (In college a common joke was that you don't master the material in class 'x' until you are going through class 'x+1' :tongue_smilie:.) True enough. But you learn what you practice. Are you practicing applying the math concepts? Or are you practicing a rote procedure?

 

Bill's descriptions of how he is teaching his son perfectly illustrates what it means to practice thinking through the concepts. For example, with learning math facts, he is having his son think through combining the axioms with his existing fact base to figure out the ones he doesn't know each and *every* time they work on them. The concepts themselves - not just the facts - are reviewed and practiced in every practice session.

 

That is *vastly* different to the standard "show the concepts, but practice the facts/procedure" approach. There a student is shown how to break down numbers using the axioms a few times, maybe even going through the process with each fact. So far, so good. But then all further practice is straight up memorization! So if it takes 100 times of seeing a given fact to learn it, a student will only practice the underlying concept 1 time while learning the fact. They are spending 99% of their time on math practicing surface facts or procedures with no reference to the underlying concepts.

 

No wonder most of the people in this country think math is nothing more than memorizing facts and formulas - that's what the they spent the vast majority of their time in math *doing*! Only naturally mathy people have a chance to learn the concepts, because they just don't need much repetition to achieve mastery. But that doesn't mean that everyone else is incapable of learning the concepts, just that they need more *practice*.

 

We don't expect kids to master anything else without lots and lots of practice. So why do we persist in thinking that mere exposure to math concepts is somehow sufficient, and any failure is due to a defect in the kid (they're just not mathy) instead of a defect in the instruction?

 

If we want kids to learn math concepts, then they need to *practice* the concepts - just like they do to learn grammar and history and science and art and music and sports and anything else.

[Snickerdoodle]

Are you practicing applying the math concepts? Or are you practicing a rote procedure?

 

For practicalities of time management we do both. We spend a fair amount of our math time daily doing mental math exercises (or numbers and all their tricks) and a small amount of time doing rote work on fact recall.

 

Cruddy mean mommy that I am.

[ladydusk]

Luthermama, I always appreciate your comments on math.

 

I think the biggest issue I have, as a home educator, is that I was always good at math, but didn't like it. So I didn't try and really figure out why things worked, it was more intuitive. Now that I'm teaching, I need to figure out multiple ways to explain a concept that is difficult. It is that "Profound Understanding of Mathematical Concepts" that is missing. I understand, some, but not in a profound way.

 

Anyway, this is really just a [bump] so more people will read and an [applause, I agree]!

[Corraleno]

I think the four curriculum components that specifically help with understanding and retention of concepts are:

 

(1) Clear, explicit, step-by-step explanations when the concept is first introduced, to ensure that the student really truly "gets it," rather than "sort of gets the gist of it."

(2) Explanations of the concept from multiple perspectives, with good visual illustrations.

(3) Continual, explicit references to the relationships and connections between concepts, and links between previously learned concepts and new ones.

(4) Lots of challenging word problems that require the student to understand the concepts in order to set them up properly as well as solve them.

 

Jackie

[Halcyon]

Practice of fact AND practice of conceptual understanding are key. I agree witH Jackie. My son is expected to know how to do a problem operationally, but I will often pause and ask him to explain conceptually what he is doing and why. Challenging word problems can often expose weakenesses in conceptual thinking (or strengths, if you're lucky ;) )

[forty-two]

I'm not trying to say that you don't need to explicitly master facts and procedures. But rote practice isn't the only way in which to master them.

 

In fact, I'm not sure I buy into this whole dichotomy between practicing concepts and practicing procedures/facts, any more than I buy into the false dichotomy between learning and teaching concepts and procedures in the first place. You *can* practice both at the same time ;). It's kind of like how swb has integrated writing skill practice into history/lit/sci studies.

[Snickerdoodle]

You *can* practice both at the same time ;).

 

I spent a minute thinking about this and I agree. I have some long convoluted thoughts on this with regard to my own son as #1) he is the only one that I have taught math to and #2) he is the only person I have seen go through the stages of understanding.

 

It will be interesting to see the younger one progress through math. I can already tell there are differences...

[Carol in Cal.]

In fact, I'm not sure I buy into this whole dichotomy between practicing concepts and practicing procedures/facts, any more than I buy into the false dichotomy between learning and teaching concepts and procedures in the first place. You *can* practice both at the same time ;). It's kind of like how swb has integrated writing skill practice into history/lit/sci studies.

 

I totally agree with you--this is a false dichotomy.

Additionally, there is a TEACHER involved with math teaching, just as with all other homeschooling.

 

As a teacher of my specific DD, I found that Saxon was, quite literally, the only program that worked for her. But, I gave her significant amounts of conceptual practice and review my own self. I talked about the 'general case' until she was ready to hit me. Repeatedly. I reminded her of the links between different ways of writing numbers, all the time. I picked topics where I thought she might be just doing rote learning, and asked her to explain the concepts behind them to me.

 

Now that she is in a brick and mortar high school, taking honors geometry (!) (this is my dd who hates math!), she is really, really grateful that she used Saxon, because she can see how valuable the forced practice was, because she realizes that so far she has already known about 2/3 of the material they have introduced this class, and because I emphasized 3 aspects of algebra so much that they are second nature to her, and 2 of those have been very useful in her geometry class so far--the equation of a line, solving multiple equations in multiple unknowns, and unit analysis (which she has not used in geometry, but which will come in handy in her science classes). All of these were covered in Saxon, and practiced a lot, and I punched them even more as well because I know how important they are going to be later on.

 

Saxon is a fine program if it is right for the child and used properly. It is wrong to say or imply that there is only one way to teach math 'right'. Additionally, the first real math is algebra. Before that you're really just doing arithmetic, which lays the groundwork for math but really isn't math per se.

[Corraleno]

I'm not trying to say that you don't need to explicitly master facts and procedures. But rote practice isn't the only way in which to master them.

 

In fact, I'm not sure I buy into this whole dichotomy between practicing concepts and practicing procedures/facts, any more than I buy into the false dichotomy between learning and teaching concepts and procedures in the first place. You *can* practice both at the same time ;). It's kind of like how swb has integrated writing skill practice into history/lit/sci studies.

Obviously the two aren't mutually exclusive, but I think it's certainly possible to practice mathematical procedures over and over without understanding/reinforcing/practicing the concepts behind the procedures. OTOH, I'm not sure how one would practice the concepts without practicing the procedures, other than doing thought experiments! A math program that provides challenging, multi-step problems (especially word problems) ensures that students do practice both simultaneously.

 

Jackie

Edited October 8, 2010 by Corraleno
terrible typo!

[Spy Car]

I think the four curriculum components that specifically help with understanding and retention of concepts are:

 

(1) Clear, explicit, step-by-step explanations when the concept is first introduced, to ensure that the student really truly "gets it," rather than "sort of gets the gist of it."

(2) Explanations of the concept from multiple perspectives, with good visual illustrations.

(3) Continual, explicit references to the relationships and connections between concepts, and links between previously learned concepts and new ones.

(4) Lots of challenging word problems that require the student to understand the concepts in order to set them up properly as well as solve them.

 

Jackie

 

 

I like this, but I would add one thing: Part "A".

 

Part A being an opportunity (created by the parent/teacher) for the child to learn some of the mathematical relationships through the use of such tools a Cuisenaire Rods on his or her "own."

 

So having the opportunity to use rods and see 2 and 2 make 4, and to be able to "prove" that to themselves. Or see 4 and 6 is the same as 6 and 4.

 

This is the "discovery" sort of learning Miquon promotes. I do think it has a very high value that is different than teacher led explanations of math concepts (which are also valuable). If one can give them the tools to problem solve from the start and allow them (and encourage them) to be able to "explain" how they solved their problems from the very first introduction of math one sets up a dynamic that the child can and will explain what they are doing as the math education progresses.

 

I know we share the idea children ought to be able to explain what they are doing conceptually. Those that doubt it might ask their children "how" they are arriving at the (correct) answers to their math problems. What is the reasoning?

 

If they can't do that, or the answer is they "just know" or they have it memorized, then, for me, I'd take it as a signal something is wrong.

 

Forty-two is quite right in suggesting it takes practice, practice, practice.

 

Bill

Edited October 9, 2010 by Spy Car

[regentrude]

I do not think you can practice isolated concepts. Once a concept is understood, there is not practice needed. It is, however, usually necessary to practice problem solving in order to thoroughly understand the concept.

Those two things really go hand in hand.

 

This is one thing we love about Art of Problem Solving: the practice problems are sufficiently varied that you do not just drill a procedure; for every single problem, you need to think about how the concept is applied in this particular situation. This accomplishes both a thorough understanding of the concept and a practice of the problem solving techniques.

[Spy Car]

I do not think you can practice isolated concepts. Once a concept is understood, there is not practice needed. It is, however, usually necessary to practice problem solving in order to thoroughly understand the concept.

Those two things really go hand in hand.

 

This is one thing we love about Art of Problem Solving: the practice problems are sufficiently varied that you do not just drill a procedure; for every single problem, you need to think about how the concept is applied in this particular situation. This accomplishes both a thorough understanding of the concept and a practice of the problem solving techniques.

 

Another reason I like using more than one conceptual math program, since even the best of them can get into a pattern where the student begins to work "mindlessly." Confronting them with a test of their conceptual understanding using methods that look quite different than the norm helps keep them sharp.

 

Bill

[robsiew]

Obviously the two aren't mutually exclusive, but I think it's certainly possible to practice mathematical procedures over and over without understanding/reinforcing/practicing the concepts behind the procedures. OTOH, I'm not sure how one would practice the concepts without practicing the procedures, other than doing thought experiments! A math program that provides challenging, multi-step problems (especially word problems) ensures that students do practice both simultaneously.

 

Jackie

 

:iagree:Yup.

 

Isn't math like anything else? You can read about cooking, watch TV shows about cooking and know the concepts behind cooking, memorize recipes and procedures for different cooking techniques... but in order to learn to cook you have to do it!

[MeganW]

He seems either uninterested or unable to master the multiplication facts. I have tried everything. He has developed these crazy methods of figuring stuff out in his head (and he is very accurate).

 

 

That's me to a tee!!! I can't recite the times tables to save my life, but can figure a fact out without exerting any effort. I NEVER did learn those stupid facts, but went through advanced math in college, and became a CPA. I have tried to explain the way I figure it to others who are mathmatically-inclined, and they always look at me like I have 2 heads. Then in reading about the Asian methods, I realize that I made an elementary effort to develop this on my own. They have much more efficient ways and have developed much further the way that I thought. But it kind of validates my "cheating" methods.

[nansk]

Can you please share your "way" of calculating? I am interested to know more.

 

I read on the VSL yahoo group about moms and kids who "see different colours" in their mind's eye, each colour associated with a number! Is your way something like that?

[lmrich]

I think if your student can explain how they understand it, then you are headed in the right direction. My youngest dd is learning long division now and it does take tons of practice, but I love to hear her talk to herself. I am reassured that she understands what she is doing. Her conversation is something like this.... I have 43 cookes and six friends. Can 6 go into 4 no, not enough, can 6 go into 43, no not exactly, what is closest, you could give 7 cookies to each friend and one to the dog - your remainder. She knows the steps. She still needs practice on how to show the steps and where exactly to write them down, like the 7 needs to go above the 3 (in the ones place) and not the 4. But her first two weeks of this have been great. I am so grateful that she has the opportunity to talk during math.

[MeganW]

Can you please share your "way" of calculating? I am interested to know more.

 

I read on the VSL yahoo group about moms and kids who "see different colours" in their mind's eye, each colour associated with a number! Is your way something like that?

 

No, nothing that exciting! I just make everything round.

 

Like to add 8, I add 10 and take off 2.

 

Same with multiplying - multiply by 10, and takeoff 2. So to multiply 8 x 12, I think 10x12=120, less 2x12= 24, so 8x12 must be 120-24 = 96. (And of course to do 120-24, I would do 120-20=100 - 4 more =96.) And truly, I do that almost as quickly as most people can spit out the fact, b/c I have been doing it this way my whole life so I'm fast!

 

Everybody else I have ever mentioned it to thinks it is convoluted and harder than memorizing the facts, but I just can't seem to memorize them! And the ones that I do remember, I'm not confident enough to rely on it, so I still go through my quickie calculation to check it.

[rockermom]

No, nothing that exciting! I just make everything round.

 

Like to add 8, I add 10 and take off 2.

 

Same with multiplying - multiply by 10, and takeoff 2. So to multiply 8 x 12, I think 10x12=120, less 2x12= 24, so 8x12 must be 120-24 = 96. (And of course to do 120-24, I would do 120-20=100 - 4 more =96.) And truly, I do that almost as quickly as most people can spit out the fact, b/c I have been doing it this way my whole life so I'm fast!

 

Everybody else I have ever mentioned it to thinks it is convoluted and harder than memorizing the facts, but I just can't seem to memorize them! And the ones that I do remember, I'm not confident enough to rely on it, so I still go through my quickie calculation to check it.

 

I do things in a very similar (and fast) way. Everyone always told me I was weird when I tried to help them with math.

 

My brain would never accept an algorithm unless I could break things down myself and prove to myself that it worked. I always had to know why and it drove the teachers crazy. (They simply wanted me to plug in the numbers and shut up.) Once I broke it down and proved to myself that it actually was a good formula to use, I would use it.

 

I always just thought I was weird.

Edited October 9, 2010 by rockermom

[MeganW]

I do things in a very similar (and fast) way. Everyone always told me I was weird when I tried to help them with math.

 

Had that same experience. My mom is the definition of non-mathy, as is my younger sister. I used to try and help my sister (who is a GREAT memorizer), and it was a huge disaster!

[bbrandonsmom]

No, nothing that exciting! I just make everything round.

 

Like to add 8, I add 10 and take off 2.

 

Same with multiplying - multiply by 10, and takeoff 2. So to multiply 8 x 12, I think 10x12=120, less 2x12= 24, so 8x12 must be 120-24 = 96. (And of course to do 120-24, I would do 120-20=100 - 4 more =96.) And truly, I do that almost as quickly as most people can spit out the fact, b/c I have been doing it this way my whole life so I'm fast!

 

Everybody else I have ever mentioned it to thinks it is convoluted and harder than memorizing the facts, but I just can't seem to memorize them! And the ones that I do remember, I'm not confident enough to rely on it, so I still go through my quickie calculation to check it.

 

I never learned math this way. I wish I did.

[siloam]

Lots of people said that they had to abandon "conceptual" curricula because there just wasn't enough practice for their kids to master the facts and procedures. But at the same time "traditional" programs were defended as "teaching the concepts" perfectly adequately so long as they explained the why behind a procedure when it was first introduced. And when kids completed the program without retaining the whys, it was because they just aren't mathy, or because they just weren't developmentally ready, not because they just didn't get enough practice. We don't expect that kids will master math facts or the standard algorithms by just seeing them once - why in the world do we think seeing a math concept once is somehow sufficient?

All of my kids have required repeated teaching of the base concepts. They don't just get it, and they do forget. Because math concepts build on one another it is possible for a mathy child to get it and not need to be shown again. This just isn't my kids.

 

RS is the best of the programs I have used and bringing up topics again, but even then I still have to re-teach why at times. Part of why I like RS is because I work with my kids enough to know exactly where they are at.

 

A lot of traditional math defenders made a big point that we often learn by doing, that sometimes it takes doing problem after problem before it suddenly clicks. (In college a common joke was that you don't master the material in class 'x' until you are going through class 'x+1' :tongue_smilie:.) True enough. But you learn what you practice. Are you practicing applying the math concepts? Or are you practicing a rote procedure?

 

I personally was considered mathy as a child. I loved numbers and did well in math, but honestly I didn't understand math. I was just happy writing all the numbers out (I know I am weird), even became an accountant. :D

 

My point is neither did I get it by doing. I do think some kids need to do before they will understand the explanation of why. But if you know you have this type of student you can have them do the problems first then explain it.

 

As hsers we are so good about modifying every other topic, but not math.

 

Bill's descriptions of how he is teaching his son perfectly illustrates what it means to practice thinking through the concepts. For example, with learning math facts, he is having his son think through combining the axioms with his existing fact base to figure out the ones he doesn't know each and *every* time they work on them. The concepts themselves - not just the facts - are reviewed and practiced in every practice session.

 

My kids wouldn't learn this way. They in fact despise discovery approach, they need something much more concrete. They need to know exactly what they are doing, why, and what is expected of them. That said most of my kids have test scores that are higher on the concepts than in the computation side.

 

How do they learn? With RS and with anything they run into problems with, I teach it to them, demonstrate it, explain why, tell them what I expect and they rise to the occasion (most of the time). There are whole sections of the Singapore IP and CWP books that they just can't figure out, and I don't push it. I come alongside them and do it with them. You can say this is learning by doing. I don't just have them watch me, I have them do it with me, I ask them questions, but my kids know that if they can't figure it out I will step in and take them to the next step, so they don't worry about me leaving them hanging. I do feel they are always 2 steps behind, because they can't always figure out new stuff on their own, but they do own what they have learned. Not just how to do it buy why. They just are also easily overwhelmed and shut down if they even think it might be too hard.

 

The only other thing I would add, is I only stop moving forward when mastering what we are doing now is required for the next step. Most of the time if there is something that they struggle with I just daily review it with them when we start math till they develop a mastery.

 

Heather

[Snowfall]

No, nothing that exciting! I just make everything round.

 

Like to add 8, I add 10 and take off 2.

 

Same with multiplying - multiply by 10, and takeoff 2. So to multiply 8 x 12, I think 10x12=120, less 2x12= 24, so 8x12 must be 120-24 = 96. (And of course to do 120-24, I would do 120-20=100 - 4 more =96.) And truly, I do that almost as quickly as most people can spit out the fact, b/c I have been doing it this way my whole life so I'm fast!

 

Everybody else I have ever mentioned it to thinks it is convoluted and harder than memorizing the facts, but I just can't seem to memorize them! And the ones that I do remember, I'm not confident enough to rely on it, so I still go through my quickie calculation to check it.

 

Add me to the group that does that! I was never taught to, either. It just seemed easier. A few weeks ago we were out and heard a father grilling his daughter on 8x28. She looked to be about 13. Her mom kept telling her dad to leave the girl alone, and the daughter kept saying, "I could figure it out if I had paper, but stop asking me! It makes me nervous." I was getting really irritated with the father because he was upsetting the girl (he wasn't nice about this) and told DH someone out to teach that poor girl to just multiply 10x28 and subtract the 2x28 (which is NOT 16, as I just typed, but 56 - realized that as soon as I hit send. At the time I got it right. lol) DH thought that was incredibly odd, but I had always thought most adults did math like that, until that conversation with him. I assumed it was something everyone eventually figured out. I'm not mathy or anything.

Edited October 10, 2010 by Snowfall

[OlgaLA]

Add me to the group that does that! I was never taught to, either. It just seemed easier.

 

But doesn't the fact that many people here were not taught this trick yet they use it mean exactly what conceptual understanding is about? If you understand the concept behind, you will find a number of ways to do it. I don't always use this method. If I need to multiply large numbers, I may round and subtract or add depending on whether I round up or down, or in some cases it may be multiply then divide (for 5s), or I may even do a long multiplication in my head, whatever is easier/faster for the particular set of numbers.

 

Honestly, I can't quite imagine "practicing concept". To me concept is an abstract idea of the process that cannot be practiced independently. What I do with my daughter is offer her chances to see the same process from different angles hoping that eventually it will solidify into a thorough understanding of the concept behind. It includes doing different kinds of word problems, without requiring to use any one method. All I ask for is an explanation how she came up with the solution.

 

And it looks like a great excuse to post an example of testing of conceptual understanding in Russian schools. This was used in testing second graders there, so about third graders here.

 

20k + 13 = 22p

k and p are digits from 0 to 9, 20k and 22p are three-digit numbers.

Compare k and p.

 

I do think that this requires conceptual understanding, although I am not sure my daughter (third grader) is there yet.

[MeganW]

Add me to the group that does that! I was never taught to, either. It just seemed easier. A few weeks ago we were out and heard a father grilling his daughter on 8x28. She looked to be about 13. Her mom kept telling her dad to leave the girl alone, and the daughter kept saying, "I could figure it out if I had paper, but stop asking me! It makes me nervous." I was getting really irritated with the father because he was upsetting the girl (he wasn't nice about this) and told DH someone out to teach that poor girl to just multiply 10x28 and subtract the 2x28 (which is NOT 16, as I just typed, but 56 - realized that as soon as I hit send. At the time I got it right. lol)

 

That is so sad. Like she's EVER going to enjoy math with that kind of pressure!!

 

 

 

DH thought that was incredibly odd, but I had always thought most adults did math like that, until that conversation with him. I assumed it was something everyone eventually figured out. I'm not mathy or anything.

 

I always thought (until reading threads here) that I had to do it that way b/c I was too dumb to remember the times tables! It has been such a wonderful thing to realize that other people do it in these crazy ways!

[Chrysalis Academy]

I'm bumping this ancient thread because I just started a thread on mastery on the Logic board, and the tag pulled this one up.  I love the question the OP posed, and I would love to hear more thoughts about it - including those of you who posted here 4 years ago! How have your thoughts about mastery (conceptual vs. procedural, or whether the dichotomy exists) evolved in the past 4 years?