Please define "conceptual math."

[Veritaserum]

It seems to mean different things to different people. Also, I'd like an example of a problem that uses and/or teaches conceptual math. Please explain what you mean by "the whys" behind getting an answer. Does this mean understanding the relationship between numbers (e.g. 9 + 5 = 14 because 9 + 1 = 10 and there are 4 more)? Or is there something deeper?

 

This is a sincere question. Thanks for answering. :)

[Snickerdoodle]

http://www.welltrainedmind.com/forums/showthread.php?t=167422&highlight=conceptual

[Guest Cheryl in SoCal]

You might want to watch MUS's demo. It's what helped me understand what teaching conceptually was and realize the programs we were using weren't teaching conceptually.

[forty-two]

It seems to mean different things to different people. Also, I'd like an example of a problem that uses and/or teaches conceptual math. Please explain what you mean by "the whys" behind getting an answer. Does this mean understanding the relationship between numbers (e.g. 9 + 5 = 14 because 9 + 1 = 10 and there are 4 more)? Or is there something deeper?

 

This is a sincere question. Thanks for answering. :)

To me conceptual math is answering the whys by going back to the basic, fundamental properties, not stopping at an intermediate level.

 

For example, place value is often cited as the fundamental concept in arithmetic. And certainly understanding place value is necessary to do any calculating in our base-10 system - all the standard algorithms - even just writing numbers - are rooted in the concept of place value.

 

But while place value is fundamental to *calculation*, it's not really fundamental to *math*. The Greeks did plenty of real math without having a positional numeration system (i.e. one that depends on place value). Sure, calculations were a lot harder to do (thus the use of counting boards and abacuses), but it didn't prevent them from using axioms and logic to discover and prove all sorts of things.

 

This post does a great job explaining what that sort of conceptual math looks like.

[ChrissySC]

So, if I understand, conceptual teaching is not letting or having the child find that "why", but instead, tells the "why" from the beginning. Additionally, teaching the method to solve the problem with the why it works the way it does.

 

Hmmm, what math program does this exceptionally well?

[Guest Cheryl in SoCal]

So, if I understand, conceptual teaching is not letting or having the child find that "why", but instead, tells the "why" from the beginning. Additionally, teaching the method to solve the problem with the why it works the way it does.

 

Hmmm, what math program does this exceptionally well?

 

Math-U-See is the best I've seen (pun intended:lol:).

[TracyR]

In a nut shell conceptual math teaches the why's of math and what it means.

It gives you the ability to look at the math problem and not just plug in numbers but know what the math problem means.

 

Some children pick up on this quickly and do fine with it. More children then so though do not think this way in the beginning but pick it up later in life when they go through life experiences and gain more maturity and logic skills.

 

Usually children who are good with math and understand it do very well with conceptual math programs. My oldest is like this. She, at times, has more of a conceptual math understanding then I do.

 

 

I was one of those kids who did not understand conceptual math. I just wanted to know how to do the math program. I just wanted to know that 2+2=4 , or how to solve the polynomial algebraic problem. I could of cared less about why the answer was the way it was. I just did not have that type of understanding in math when I was a child Now that I'm older I find that I've known conceptual math for quite some time now. With time and maturity I have picked up on patterns and find myself knowing them at a different level then I did as a child. I really didn't need a curriculum to learn it. I'm just finding a shift in how my brain functions and how I think now verses when I was a child.

 

Children who struggle with math should be using math programs that teach the hows first so they can gain confidence. You can not push a conceptual math program on a child that hasn't developed that type of understanding in math. All it leads to is confusion for them, and hatred for the subject.

 

I unfortunately tried pushing these types of programs on my 2nd daughter believing it was the better way to teach math. Only to know at the age of 10 have to do some major damage control because she just didn't understand math that way. We are using Saxon with her now and there are many more happy faces and " Mom, I understand that now." Then there has been in a very long time.

 

Of course I throw some conceptual math in there when she isn't expecting it but the delievery of math for her with Saxon makes sense to her at this point in her life. Just like with any subject for any of our children I expect it to change as she gets older and we may switch back. Or we may not.

 

I don't think there is any one right way. If you have a mathy child that loves math, loves to play with numbers then I would try math programs like Singapore math, Math Mammoth, Bob Jones math, Calvert math, Right Start Math, Math U See, Life of Fred, Chalk Dust Math , Jacob's Math series , MEP

These math programs do just fine in teaching conceptual math.

[Veritaserum]

I read the long thread posted by Snickerdoodle and the post linked by forty-two and came away with two things:

 

1. Manipulatives are a good way to prove the concepts (a 4 block plus a 3 block is the same length as a 3 block plus a 4 block).

 

2. Knowing the axioms (conceptual math) is to math what phonics rules are to reading. Is that a good comparison?

 

If my child can tell me that x + y = y + x because of the commutative property or because you're not changing anything by switching the order, is that demonstrating a conceptual understanding? Or am I still missing something?

[forty-two]

To simplify my previous (lengthy) comment:

 

Conceptual math is proofy math. Conceptual math wrt arithmetic/algebra involves learning the field axioms.

[Veritaserum]

To simplify my previous (lengthy) comment:

 

Conceptual math is proofy math. Conceptual math wrt arithmetic/algebra involves learning the field axioms.

 

By field axioms, you mean these, right?

[forty-two]

2. Knowing the axioms (conceptual math) is to math what phonics rules are to reading. Is that a good comparison?

I'd say it is more comparable to knowing grammar, to having a conscious understanding of the structure of language.

 

(Will elaborate later - kids need me ;).)

[Veritaserum]

A follow-up question for anyone who has read Knowing and Teaching Elementary Mathematics by Liping Ma: Does she discuss teaching using the aforementioned field axioms?

[mpcTutor]

I'd like an example of a problem that uses and/or teaches conceptual math. Please explain what you mean by "the whys" behind getting an answer. Does this mean understanding the relationship between numbers (e.g. 9 + 5 = 14 because 9 + 1 = 10 and there are 4 more)? Or is there something deeper?

 

This is a sincere question. Thanks for answering. :)

Concept is nothing but certain concrete idea (as against a vague notion). It doesn't matter if the concept is in mathematics or not, the success in identifying "the" concept from all other seemingly similar concepts is needed in understanding that particular concept.

 

Example 1:

 

Explain "All squares are rectangles but all rectangles are not squares."

 

The beginning of above explanation is sometime when a child learns to differentiate a table from a chair and learns to ignore the fact that they both have four legs which in fact are not important in differentiating a chair from a table. The same child will however identify both chair and table as furniture distinctly different from say a car. Use of pictures, verbal description and constant acquaintance with these objects make these non-mathematical concepts pretty clear for most children.

 

Unfortunately, with respect to most mathematical concepts, making the child aware of say, number 3 is not that easy. We can show 3 cars, 3 apples and 3 dogs but identifying the common three-ness in the pictures of cars, apples and dogs is largely left to the child. There is not an easy way to tell the child that the "three" is common to 3 cars, 3 apples and 3 dogs. If the child sees them as "things" then he/she missed the concept of "three". Also, important thing to remember is that even grown ups don't always see everything that is shown. So every child doesn't have to see the idea right after it is explained. There could be several valid reasons for a child not see your point including barriers in communication, child's willingness to be attentive etc.

 

Example 2:

The commutative law of addition a + b = b + a is introduced to the child by dividing 15 marbles in to two groups say 10 marbles and 5 marbles and then adding 1st group in the 2nd group and vice versa and then counting to show that the result is 15 marbles. The concept here is the sum of two numbers is not affected by the order in which they are added. If this "concept of commutation" is understood then the child doesn't need to be shown 20 + 30 = 30 + 20 again with the help of marbles.

 

Example 3:

The commutative law in Example 2 above is normally not questioned with "why" but simply accepted and its reason is known in later grades when field axioms for real numbers are taught only to learn that highly fundamental facts don't have reason. A new concept called "axiom".

 

 

Bottom line:

In earlier years certainly one could learn the basic facts through manipulative and by solving select problems. Particularly after 4th - 5th grades it is through the practice of solving well chosen problems that many hidden concepts can be understood. In my experience, that's the way the mathematics and physics are: not only for reading but largely for solving problems (with paper and pencil). In doing all of these activities a wise goal is to keep the child interested in mathematics and the sought after math mastery will come as consequence in due time.

 

Best regards.

 

MPCTutor

www.mpclasses.com

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AP Calculus, AP Physics, Singapore Math Grades 7-12

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US Central Time: 2:11 PM 5/14/2010

[forty-two]

By field axioms, you mean these, right?

That looks about right; the book I use lists them like this.

 

I read the long thread posted by Snickerdoodle and the post linked by forty-two and came away with two things:

 

1. Manipulatives are a good way to prove the concepts (a 4 block plus a 3 block is the same length as a 3 block plus a 4 block).

 

2. Knowing the axioms (conceptual math) is to math what phonics rules are to reading. Is that a good comparison?

 

If my child can tell me that x + y = y + x because of the commutative property or because you're not changing anything by switching the order, is that demonstrating a conceptual understanding? Or am I still missing something?

 

1) Manipulatives are great for helping kids get an intuitive sense that the axioms are true. I've speculated that judicious use of manipulatives would allow kids to do proofy-type reasoning earlier than normal by giving them something concrete to work with, a la Hands-On Equations - in a few years, we'll see how it works in practice ;).

 

2) Following up on my pp on this, I think that "learning the field axioms" is to math as "learning grammar" is to language. In both cases you are learning the basic building blocks of the subject - both what those fundamental concepts are, and how to consciously manipulate them to achieve the result you want (and to be able to play around with new, legal combinations and see what happens).

 

Being able to state the communtative law and having a feel for what it means, being able to ID it when you see it, is a start. But ultimately you need to be able to *use* it, along with the other properties, to prove things. Going back to the grammar analogy, being able to state the definition of a noun and pick out the nouns in a sentence is a good start. But ultimately you need to be able to use nouns, along with the other parts of speech, to correctly say what you want to say - and be able to explain *why* it is correct (logic). And eventually you want to be able to say what you want to say in the most effective, beautiful way possible (rhetoric).

 

The math equivalent is being able to prove what you want to prove correctly, and eventually being able to prove things *elegantly*.

 

Does that help?

[forty-two]

A follow-up question for anyone who has read Knowing and Teaching Elementary Mathematics by Liping Ma: Does she discuss teaching using the aforementioned field axioms?

My impression is that she does not particularly address teaching them explicitly, with an eye to using them in proofs. But the type of teaching she advocates requires the teacher to have a good *implicit* understanding of them, and a good intuitive sense of math is invaluable - it goes hand-in-hand with a good explicit knowledge. And, just like an intuitive sense of language usage, good math sense comes from being exposed to lots of good math, with ample opportunity to interact with it. Give them the necessary tools and turn 'em loose on lots of good problems.

 

One of my favorite resources that develops basic arithmetic from first principles (for the teacher):

*Professor Wu's chapter drafts for a math-for-math-teachers book (pdfs): Whole Numbers and Fractions. And it's free =). (All of Dr. Wu's material is worth reading.)

[ElizabethB]

My impression is that she does not particularly address teaching them explicitly, with an eye to using them in proofs. But the type of teaching she advocates requires the teacher to have a good *implicit* understanding of them, and a good intuitive sense of math is invaluable - it goes hand-in-hand with a good explicit knowledge. And, just like an intuitive sense of language usage, good math sense comes from being exposed to lots of good math, with ample opportunity to interact with it. Give them the necessary tools and turn 'em loose on lots of good problems.

 

One of my favorite resources that develops basic arithmetic from first principles (for the teacher):

*Professor Wu's chapter drafts for a math-for-math-teachers book (pdfs): Whole Numbers and Fractions. And it's free =). (All of Dr. Wu's material is worth reading.)

:iagree:

 

Read Ma.

Read Wu.

 

They know math, plus their names are syllables, an added bonus, I am a big fan of syllables.

[Corraleno]

Children who struggle with math should be using math programs that teach the hows first so they can gain confidence. You can not push a conceptual math program on a child that hasn't developed that type of understanding in math. All it leads to is confusion for them, and hatred for the subject.

I disagree with this generalization. Some children who struggle with math may need to just learn "how," but for other kids the fact that they haven't learned "why" first may be precisely the reason they're struggling! Kids can struggle in math for a wide variety of reasons, many of which have nothing to do with being unable to understand the concepts.

 

My DS struggled in school and was more than grade level behind in math. I put him in a very conceptual math program (Math Mammoth) and suddenly he understood the point of it all, and he quickly caught up. So for him, a strongly conceptual program was exactly what he needed.

 

Jackie

Edited May 14, 2010 by Corraleno
spelling

[8filltheheart]

It seems to mean different things to different people. Also, I'd like an example of a problem that uses and/or teaches conceptual math. Please explain what you mean by "the whys" behind getting an answer. Does this mean understanding the relationship between numbers (e.g. 9 + 5 = 14 because 9 + 1 = 10 and there are 4 more)? Or is there something deeper?

 

This is a sincere question. Thanks for answering. :)

 

You are correct in it meaning different things to different people. Even mathematicians argue about it actually means.

http://www.math.rochester.edu/people/faculty/rarm/concepts.html (this one is sort of funny but also a sad testament on test writing)

http://www.maa.org/devlin/devlin_09_07.html

 

Based on the discussions on this board, my take is that the vast majority see it as computational and procedural. I think that when I read supporters of Singapore, they are discussing the mental visualization of how operations work.

 

I think it is a combination of both. However, when I am discussing math, those are only part of what I want to see in a math program. My ultimate goal in math is application to problem solving which is again a different category.

 

 

ETA: The varying views made me curious and I spent some time digging around AOPS to see if I could find out how they defined it. Here are some quotes from AoPS describing their texts which indicates a definition of conceptual similar to that of mpcTutor: Concept is nothing but certain concrete idea (as against a vague notion). It doesn't matter if the concept is in mathematics or not, the success in identifying "the" concept from all other seemingly similar concepts is needed in understanding that particular concept.

 

From AoPS website http://www.artofproblemsolving.com/Store/curriculum.php?mode=features

Encourages students to understand rather than memorize:

 

By emphasizing the understanding of underlying mathematical concepts, students will gain a much deeper knowledge of the mathematical principles involved. Also, students won't have to worry about "forgetting" an important formula, because they'll know the formulas, rather than merely having them memorized.

 

Although our books have a number of straightforward exercises necessary to reinforce basic concepts, our larger focus is on teaching the student to solve challenging, multi-step problems. Not only does this provide the student with a challenge more appropriate to his or her ability, but it avoids boring the student with an endless succession of too-easy exercises.

 

Students' curiosity is engaged by discovering mathematics:

 

Students are naturally curious and inquisitive. Rather than giving them information to parrot, we challenge students to discover mathematics on their own by asking them questions before the lesson, in addition to after the lesson. When students figure out the key concepts on their own, they enjoy learning more and internalize the lessons efficiently. The mathematics becomes their mathematics, rather than information that was told to them.

Edited May 15, 2010 by 8FillTheHeart

[Spy Car]

 

Children who struggle with math should be using math programs that teach the hows first so they can gain confidence. You can not push a conceptual math program on a child that hasn't developed that type of understanding in math. All it leads to is confusion for them, and hatred for the subject.

 

 

I disagree with this generalization.

 

 

I also disagree with this generalization, and would suspect the reason children struggle is because they don't understand the concepts, or are not even taught mathematical laws and so lack this "grammar" of math.

 

The key is to be able to teach the concepts in ways that are comprehensible to young minds, not to thrown the baby out with the bathwater. That is where good teaching comes in.

 

I will put in a plug for Miquon math when combined with Singapore, as the laws of math in the four basic operations in elementary math are wonderfully (and simply) demonstrated for parent/teachers in the Lab Annotations book. It does not have to be hard or difficult for either parent or child.

 

Bill

[sagira]

I disagree with this generalization. Some children who struggle with math may need to just learn "how," but for other kids the fact that they haven't learned "why" first may be precisely the reason they're struggling! Kids can struggle in math for a wide variety of reasons, many of which have nothing to do with being unable to understand the concepts.

 

My DS struggled in school and was more than grade level behind in math. I put him in a very conceptual math program (Math Mammoth) and suddenly he understood the point of it all, and he quickly caught up. So for him, a strongly conceptual program was exactly what he needed.

 

Jackie

 

:iagree: I strongly suspect ds is like me - he has to get the big picture. Why am I doing this? Is there a point? From what I've seen, conceptual math is what we've been missing. Ds brightens up when he comes up with the solution through Miquon, so I know there we're on the right track, so I'm keeping this. However, I think we need better explanations too, as in Japanese or Singapore.

 

I believe ds is going to respond well to conceptual math, as it will make him see the whys, which I know he asks all the time. Different strokes for different folks. Some children I bet would love for teachers just to give them the tools and get it over with. "Just the facts, m'am" kind of math :)

[HappyGrace]

Good thread!

 

Love the Dr. Wu pdf's that were linked, thanks. Where can I find more from him?

[forty-two]

Good thread!

 

Love the Dr. Wu pdf's that were linked, thanks. Where can I find more from him?

Here :).