Elementary math & reasoning skills

Someone recently sent me a VERY interesting link:
The Story of an Experiment

Photo by foundphotoslj

This experiment in math teaching was done in the 1930s by by L. P. Benezet, and the main gist of it was that formal arithmetic studies were delayed until the latter half of 6th grade. Instead, the instruction concentrated on "teaching the children to read, to reason, and to recite - my new Three R's."

They also were taught about numbers they encountered in their reading materials, about time, about measuring units, estimation, and coins. Finally in 5th and 6th grade they also learn skip-counting. Formal arithmetic, meaning paper-and pencil work with addition, subtraction, multiplication, and division using a textbook began in latter half of 6th grade and continued till 8th grade.

The experiment was a huge success. Mr. Benezet compared the children's abilities in the experimental classroom to those of the traditional classrooms, and every time the "experimental" children were able to reason out word problems correctly, whereas those taught traditionally just stumbled all around, trying to find some formula to use.

Maria's comments

I find this story very interesting and I've read it before. I can't say that approach wouldn't work better than "traditional math" even in today's world. Maybe it would! Those kids were taught to reason AND they were also taught some basic math skills, just without the use of mechanical formulas. So, they learned to think and reason it in their heads.

My guess is that such an approach would be even better if it was accompanied with some, what you might call, formal, instruction in math, BUT very much avoiding the idea that you use some formula that the teacher gives you "on a platter".

For example, in the experiment the kids were supposed to learn the numbers they saw in books and learn the page numbers such as 76 and 293. But, I wonder if this would have resulted in quicker/better understanding of number system had they been taught explicitly about hundreds, tens, and ones (the place value). Then again, we don't know how exactly the teachers explained those- maybe they did explain them in a very good way.

One of my instructors in university often mentioned the idea of school math being "announcement math" or "announced math". It's announced from a higher authority, without giving kids much in the way of justification, or the opportunity to find the truths themselves and thereby understand them deeper.

I do try to avoid that in the elementary part of math. In fraction studies, I refrain from giving "formulas" for fraction addition or division or simplification until kids have had a lot of experience with "doing" it with the fraction pictures. And even then the formula can be like a "sideline" that is mentioned in passing. I don't want to even deal with LCMs and GCFs when it comes to fraction math in elementary grades.

In general I always try to justify the math and let children experience and understand it on a conceptual level.

As regards to the experiment, they wanted kids to learn reasoning by the usage of good books and reading a lot. Then, when their brains were trained to a certain point, arithmetic was studied from books.

Reading a lot of books and learning reasoning is definitely a good way to go. But, I feel that mathematics, when taught right, can also have a part in this "training the mind" process and learning reasoning. It is very well suited to that if it is taught well during the elementary years - having the emphasis on thinking skills and concepts, not just memorizing formulas.

Elementary math & reasoning skills

Someone recently sent me a VERY interesting link:
The Story of an Experiment

Photo by foundphotoslj

This experiment in math teaching was done in the 1930s by by L. P. Benezet, and the main gist of it was that formal arithmetic studies were delayed until the latter half of 6th grade. Instead, the instruction concentrated on "teaching the children to read, to reason, and to recite - my new Three R's."

They also were taught about numbers they encountered in their reading materials, about time, about measuring units, estimation, and coins. Finally in 5th and 6th grade they also learn skip-counting. Formal arithmetic, meaning paper-and pencil work with addition, subtraction, multiplication, and division using a textbook began in latter half of 6th grade and continued till 8th grade.

The experiment was a huge success. Mr. Benezet compared the children's abilities in the experimental classroom to those of the traditional classrooms, and every time the "experimental" children were able to reason out word problems correctly, whereas those taught traditionally just stumbled all around, trying to find some formula to use.

Maria's comments

I find this story very interesting and I've read it before. I can't say that approach wouldn't work better than "traditional math" even in today's world. Maybe it would! Those kids were taught to reason AND they were also taught some basic math skills, just without the use of mechanical formulas. So, they learned to think and reason it in their heads.

My guess is that such an approach would be even better if it was accompanied with some, what you might call, formal, instruction in math, BUT very much avoiding the idea that you use some formula that the teacher gives you "on a platter".

For example, in the experiment the kids were supposed to learn the numbers they saw in books and learn the page numbers such as 76 and 293. But, I wonder if this would have resulted in quicker/better understanding of number system had they been taught explicitly about hundreds, tens, and ones (the place value). Then again, we don't know how exactly the teachers explained those- maybe they did explain them in a very good way.

One of my instructors in university often mentioned the idea of school math being "announcement math" or "announced math". It's announced from a higher authority, without giving kids much in the way of justification, or the opportunity to find the truths themselves and thereby understand them deeper.

I do try to avoid that in the elementary part of math. In fraction studies, I refrain from giving "formulas" for fraction addition or division or simplification until kids have had a lot of experience with "doing" it with the fraction pictures. And even then the formula can be like a "sideline" that is mentioned in passing. I don't want to even deal with LCMs and GCFs when it comes to fraction math in elementary grades.

In general I always try to justify the math and let children experience and understand it on a conceptual level.

As regards to the experiment, they wanted kids to learn reasoning by the usage of good books and reading a lot. Then, when their brains were trained to a certain point, arithmetic was studied from books.

Reading a lot of books and learning reasoning is definitely a good way to go. But, I feel that mathematics, when taught right, can also have a part in this "training the mind" process and learning reasoning. It is very well suited to that if it is taught well during the elementary years - having the emphasis on thinking skills and concepts, not just memorizing formulas.

Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the impression in children's minds that "multiplication reduces to addition" or, as Denise put it, it is a "subspecies of addition".

So, to try to summarize what I've been mulling over in my mind:

* Multiplication is simply a DIFFERENT operation from addition. It's not a "subspecies" of addition or some special kind of addition. AND, we need to stress that in our teaching.

Denise offers defining multiplication for kids as "counting by groups" - should be same-size groups, of course. In teaching that, we need to emphasize the meanings attached to the two factors: in M × N, M would be the amount of groups, and N would be the number of elements in one group. M is called the multiplier, and N is the multiplicand.

We need to emphasize the difference between additive and multiplicative situations in word problems:

"Mark has two baskets and each basket has five appples." => this is a multiplication situation
"Mark has five apples in one basket and five in another." => this is an addition situation.

Joshua says, "They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different."

It does sound simple and clear, right? I hope it does. The IDEAS are different even though the way we find the answer may be the same. But the IDEAS match these situations:

"Mark has four baskets and each basket has three appples." => this is a multiplication situation
"Mark has five apples in one basket and seven in another." => this is an addition situation.

Let's keep going.

• In multiplication, 1 is the special number so that if you multiply by it, "nothing" happens . Also called the identity element.
  
  In addition, zero has the similar role.

• Multiplication: each (real) number except zero has its multiplicative inverse so that if you multiply the number and its inverse, you get 1.
  Addition: Each (real) number has its additive inverse so that if you add the number and its inverse, you get 0.

• Multiplication has an opposite operation called division.
  Addition has an opposite operation called subtraction.

Pretty nice, eh? As we keep emphasizing these distinctions, hopefully we can develop in the students' minds the idea that they're different animals, not the same. One is not a special case of another. We can strive to define multiplication (initially) as "so many of the same-size groups" or counting by groups. Of course we have to FIND products by adding repeatedly, but we can treat them as different operations.

Then later, students will encounter multiplication of fractions and of decimals, and leave behind the idea that they solve multiplication by repeated addition. Yet, the basid properties of multiplication hold true:

• Its identity element is 1.


• Every number except zero has a multiplicative inverse (a.k.a. a reciprocal number)

I also found an interesting study quoted at Text Savvy. I quote:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . .

Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

Well, I'm trying. Here's the way I changed one page in my Multiplication book to read for now. What do you think?

Click on the link to download the one page (PDF).

Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the impression in children's minds that "multiplication reduces to addition" or, as Denise put it, it is a "subspecies of addition".

So, to try to summarize what I've been mulling over in my mind:

* Multiplication is simply a DIFFERENT operation from addition. It's not a "subspecies" of addition or some special kind of addition. AND, we need to stress that in our teaching.

Denise offers defining multiplication for kids as "counting by groups" - should be same-size groups, of course. In teaching that, we need to emphasize the meanings attached to the two factors: in M × N, M would be the amount of groups, and N would be the number of elements in one group. M is called the multiplier, and N is the multiplicand.

We need to emphasize the difference between additive and multiplicative situations in word problems:

"Mark has two baskets and each basket has five appples." => this is a multiplication situation
"Mark has five apples in one basket and five in another." => this is an addition situation.

Joshua says, "They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different."

It does sound simple and clear, right? I hope it does. The IDEAS are different even though the way we find the answer may be the same. But the IDEAS match these situations:

"Mark has four baskets and each basket has three appples." => this is a multiplication situation
"Mark has five apples in one basket and seven in another." => this is an addition situation.

Let's keep going.

• In multiplication, 1 is the special number so that if you multiply by it, "nothing" happens . Also called the identity element.
  
  In addition, zero has the similar role.

• Multiplication: each (real) number except zero has its multiplicative inverse so that if you multiply the number and its inverse, you get 1.
  Addition: Each (real) number has its additive inverse so that if you add the number and its inverse, you get 0.

• Multiplication has an opposite operation called division.
  Addition has an opposite operation called subtraction.

Pretty nice, eh? As we keep emphasizing these distinctions, hopefully we can develop in the students' minds the idea that they're different animals, not the same. One is not a special case of another. We can strive to define multiplication (initially) as "so many of the same-size groups" or counting by groups. Of course we have to FIND products by adding repeatedly, but we can treat them as different operations.

Then later, students will encounter multiplication of fractions and of decimals, and leave behind the idea that they solve multiplication by repeated addition. Yet, the basid properties of multiplication hold true:

• Its identity element is 1.


• Every number except zero has a multiplicative inverse (a.k.a. a reciprocal number)

I also found an interesting study quoted at Text Savvy. I quote:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . .

Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

Well, I'm trying. Here's the way I changed one page in my Multiplication book to read for now. What do you think?

Click on the link to download the one page (PDF).

Isn't multiplication repeated addition?

I just found out an interesting column by Keith Devlin... he tells elementary teachers to stop telling the students that multiplication is repeated addition.

Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition.

He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process.

Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go with it.

Unfortunately, I immediately ran into problems.

Let me illustrate.

I thought, we need to illustrate the idea of scaling. So I thought, kids might know scaling from computer programs such as scaling images, or scaling maps. I was going to use a picture of a toy car: and scale it by a factor of 2, to let it be TWICE as big.

I scaled its width and length by a factor of 1.414 so as to make the AREA to be twice the original: .

But I realized, kids would feel that that's NOT twice as big... they might feel it needs to be scaled like this, doubling the width and height: . But then, of course, the area is quadrupled (which 2nd or 3rd grades wouldn't automatically know).

Right there I gave up. I SURE don't want to get them confused by doubling the width, height, or area... Later on they need to learn that IF you multiply the two dimensions by some factor r, the area will be multiplied by r².

I did write for my lesson an illustration of scaling a "stick", or a line. In that one-dimensional situation we won't run into this problem. But even so, I would be extra careful of using it a lot, because surely some student will ask about scaling two-dimensional images, and then we have confusion.

I would find it more natural to present the idea of multiplication to 2nd and 3rd graders as "multiple copies", such as 2 ×  =  .

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the same thing over and over, such as "I opened the door three times".

Then, when it comes to multiplication of fractions and of decimals, one has to bring in the idea of taking a part:
1/2 x 7 means 1/2 OF 7. I do not see a problem there.

Later one can tie these "two meanings" of multiplication together with the scaling idea... maybe... somehow... I just do not know myself how to do that without confusing the idea of scaling the width/height by some factor and the area being scaled by the square of that factor.

Or maybe I'm all wrong and it IS possible to use the idea of scaling images?
Maybe someone should TRY it on a few classrooms of kids and see what happens over the years.

Update: Joe Niederberger has left an excellent comment on the issue at Let's Play Math blog. I feel I need to quote him... hope he doesn't mind:

Devlin unfortunately makes the mistake of thinking of multiplication as one "thing". It’s true multiplication of any two real numbers cannot be simply reduced to repeated addition, however, the multiplication of any two integers *can* always be reduced (or thought of, or defined by) repeated addition. Even though we call them both "multiplication" technically they are different functions.

In fact, we learn somewhere along the mathematical way that functions (like multiplication) are only properly defined by specifying their domain (among other things). Two functions that have different domains cannot be the *same* function. One function can be the extension or restriction of another, but they are not the same.

This is the basis of the confusion. Multiplication of integers *is* repeated addition, in some form or other (Peano uses a recursive definition - recursive, repeated; I say to-mae-toe, you say to-mah-toe.) Multiplication of rationals is a different animal (related, but different.) Same for multiplication of reals, complex numbers, etc. All different functions even though they build on one another.

Again, multiplication of rationals is technically a different function, in fact, an extension of multiplication on integers. Defining it requires that multiplication of integers has already been accomplished — and that, yes, means that repeated or recursive addition has already been put in the soup.

Essentially, we can define multiplication of whole numbers (and integers) as repeated addition. We have to define multiplication of fractions in a different way - but that is not a problem. It is extending the idea of multiplication in a way that it will "match" or "work" for integers as well.

Sideline... in other words, multiplication of fractions can be defined as

a/b * c/d = (ac)/(bd) which is the familiar rule. (BTW, definitions vary. That's why I can't say that multiplication of fractions would always necessarily be defined this way.)

If you have integers y and z, they can be written as fractions as x/1 and y/1, and multiplying them using the definition of fraction multiplication we get:

y/1 * z/1 = (yz)/(1*1) which equals yz.

Later on, multiplication of real numbers and that of complex numbers are defined still differently, as "extensions" of the idea of basic multiplication.

Other bloggers have their take, too:

If it ain't repeated addition... by Let's Play Math, Devlin on Multiplication by Rational Math Education, and Devlin's Right Angle at Text Savvy.

Isn't multiplication repeated addition?

I just found out an interesting column by Keith Devlin... he tells elementary teachers to stop telling the students that multiplication is repeated addition.

Why? His point is, this idea does not carry through. As soon as the student encounters multiplication of fractions (or of decimals), it won't work. You can't think of 3/4 x 6/11 as repeated addition.

He feels it's better to portray multiplication as a scaling process: say 5 x 9 means 9 is scaled by a factor of 5. Then, students can have a true "aha" moment as they discover for themselves that you CAN use addition to find the answer to 5 x 9. But, Devlin says, they should be taught and shown the multiplication idea as a scaling process.

Now, I feel that Devlin has a point here... so since I'm constantly in the process of writing math materials for my Math Mammoth series of books, and right now I'm writing lessons on multiplying decimals for 5th grade, I took this idea just yesterday and tried to go with it.

Unfortunately, I immediately ran into problems.

Let me illustrate.

I thought, we need to illustrate the idea of scaling. So I thought, kids might know scaling from computer programs such as scaling images, or scaling maps. I was going to use a picture of a toy car: and scale it by a factor of 2, to let it be TWICE as big.

I scaled its width and length by a factor of 1.414 so as to make the AREA to be twice the original: .

But I realized, kids would feel that that's NOT twice as big... they might feel it needs to be scaled like this, doubling the width and height: . But then, of course, the area is quadrupled (which 2nd or 3rd grades wouldn't automatically know).

Right there I gave up. I SURE don't want to get them confused by doubling the width, height, or area... Later on they need to learn that IF you multiply the two dimensions by some factor r, the area will be multiplied by r².

I did write for my lesson an illustration of scaling a "stick", or a line. In that one-dimensional situation we won't run into this problem. But even so, I would be extra careful of using it a lot, because surely some student will ask about scaling two-dimensional images, and then we have confusion.

I would find it more natural to present the idea of multiplication to 2nd and 3rd graders as "multiple copies", such as 2 ×  =  .

Even our word "multiply" refers to multiple copies of the same... people and animals "multiply", we talk about multiples, etc. We use the word "times" referring to doing the same thing over and over, such as "I opened the door three times".

Then, when it comes to multiplication of fractions and of decimals, one has to bring in the idea of taking a part:
1/2 x 7 means 1/2 OF 7. I do not see a problem there.

Later one can tie these "two meanings" of multiplication together with the scaling idea... maybe... somehow... I just do not know myself how to do that without confusing the idea of scaling the width/height by some factor and the area being scaled by the square of that factor.

Or maybe I'm all wrong and it IS possible to use the idea of scaling images?
Maybe someone should TRY it on a few classrooms of kids and see what happens over the years.

Update: Joe Niederberger has left an excellent comment on the issue at Let's Play Math blog. I feel I need to quote him... hope he doesn't mind:

Devlin unfortunately makes the mistake of thinking of multiplication as one "thing". It’s true multiplication of any two real numbers cannot be simply reduced to repeated addition, however, the multiplication of any two integers *can* always be reduced (or thought of, or defined by) repeated addition. Even though we call them both "multiplication" technically they are different functions.

In fact, we learn somewhere along the mathematical way that functions (like multiplication) are only properly defined by specifying their domain (among other things). Two functions that have different domains cannot be the *same* function. One function can be the extension or restriction of another, but they are not the same.

This is the basis of the confusion. Multiplication of integers *is* repeated addition, in some form or other (Peano uses a recursive definition - recursive, repeated; I say to-mae-toe, you say to-mah-toe.) Multiplication of rationals is a different animal (related, but different.) Same for multiplication of reals, complex numbers, etc. All different functions even though they build on one another.

Again, multiplication of rationals is technically a different function, in fact, an extension of multiplication on integers. Defining it requires that multiplication of integers has already been accomplished — and that, yes, means that repeated or recursive addition has already been put in the soup.

Essentially, we can define multiplication of whole numbers (and integers) as repeated addition. We have to define multiplication of fractions in a different way - but that is not a problem. It is extending the idea of multiplication in a way that it will "match" or "work" for integers as well.

Sideline... in other words, multiplication of fractions can be defined as

a/b * c/d = (ac)/(bd) which is the familiar rule. (BTW, definitions vary. That's why I can't say that multiplication of fractions would always necessarily be defined this way.)

If you have integers y and z, they can be written as fractions as x/1 and y/1, and multiplying them using the definition of fraction multiplication we get:

y/1 * z/1 = (yz)/(1*1) which equals yz.

Later on, multiplication of real numbers and that of complex numbers are defined still differently, as "extensions" of the idea of basic multiplication.

Other bloggers have their take, too:

If it ain't repeated addition... by Let's Play Math, Devlin on Multiplication by Rational Math Education, and Devlin's Right Angle at Text Savvy.

I'm bad at math... and fine with that!

I just received a nice article from Jim Stone, a math teacher at Global Institute of Mathematics.

I enjoyed it; I thought you might too. I've seen this issue raised and talked about elsewhere as well.

Namely, that it seems to be socially acceptable to admit how bad you're at math... while no one would comfortably admit that they can't read.

So here goes:

---

I'm bad at math and I’m okay with it!
By Jim Stone

For the past eighteen years I’ve been reading articles and editorials lamenting the mathematical performance of America’s school children. It has become an annual ritual for politicians and educators alike to bemoan the results of the latest tests showing American kids falling behind their international counterparts.

What is the problem? More importantly, what is the solution?

The answer to the first question is almost always laid at the doorstep of our system of education. Blaming the system is safe. No one person represents the system. No one person is held accountable. Everyone can look outside themselves for accountability.

Attempts at fixing the system have largely focused on three areas: overhauling mathematical curriculum, changing the methodology of teaching that curriculum and the implementation of standardized tests to measure what kids know.

But here is the problem. The performance of American students has remained stagnant. Despite the implementation of integrated curricula, the establishment of benchmarks and a seemingly never ending battery of standardized tests, American students still lag behind students beyond our borders.

Perhaps we are missing the boat. Perhaps the problem isn’t really the system with its traditional curriculum. Perhaps the problem isn’t with a traditional lesson beginning with going over homework from the previous day, an introduction to a new topic followed by time working on a new homework assignment. Perhaps the problem is more societal. Perhaps the problem lies with us all.

Since becoming a high school mathematics teacher I have been struck by the seemingly countless times I have heard parents and other adults, well educated or not, utter statements such as, “math was my worse subject” or “I can’t do math.”

I have never once heard a person say, “I can’t read.” Why, then, is it so acceptable to publicly declare, without the slightest bit of embarrassment, one’s inability to do mathematics? In our society one would be shamed and embarrassed to admit their inability to read. The point is it is perfectly acceptable to be poor at mathematics in this country. We are an anomaly compared to most of the rest of the world when it comes to our comfortable attitude that it is acceptable to be poor at math. Is there any doubt that our children haven’t learned this lesson?

Or have they? While American adults casually admit to their mathematical ignorance, American children appear to overestimate their mathematical ability. In a 2003 study, 84% of American 8th graders agreed with the statement, “I usually do well in mathematics.” Of their counterparts in Singapore, 64% agreed with the same statement. Confidence often, especially in America, is considered as a positive personality trait. However, it is ironic that of the same group of 8th graders, the least confident students in Singapore, on average, outscored the most confident American students on an international math test.

If the results of the 2003 study are accurate, then our young students are overconfident. If my informal observations over the years are accurate, they ultimately learn of their shortcomings and then shrug them off with a cavalier attitude of acceptable incompetence. If so, then new curriculum, changes in pedagogy and an implementation of standardized tests are not addressing the real problem and, therefore, doom us to perpetual mediocrity.

Perhaps we all share the responsibility for the dismal mathematical performance of our children. If so, it begs a difficult and frightening question: What can be done about it?



Jim Stone, a math and physics teacher for more than 18 years, teaches at Global Institute of Mathematics.

I'm bad at math... and fine with that!

I just received a nice article from Jim Stone, a math teacher at Global Institute of Mathematics.

I enjoyed it; I thought you might too. I've seen this issue raised and talked about elsewhere as well.

Namely, that it seems to be socially acceptable to admit how bad you're at math... while no one would comfortably admit that they can't read.

So here goes:

---

I'm bad at math and I’m okay with it!
By Jim Stone

For the past eighteen years I’ve been reading articles and editorials lamenting the mathematical performance of America’s school children. It has become an annual ritual for politicians and educators alike to bemoan the results of the latest tests showing American kids falling behind their international counterparts.

What is the problem? More importantly, what is the solution?

The answer to the first question is almost always laid at the doorstep of our system of education. Blaming the system is safe. No one person represents the system. No one person is held accountable. Everyone can look outside themselves for accountability.

Attempts at fixing the system have largely focused on three areas: overhauling mathematical curriculum, changing the methodology of teaching that curriculum and the implementation of standardized tests to measure what kids know.

But here is the problem. The performance of American students has remained stagnant. Despite the implementation of integrated curricula, the establishment of benchmarks and a seemingly never ending battery of standardized tests, American students still lag behind students beyond our borders.

Perhaps we are missing the boat. Perhaps the problem isn’t really the system with its traditional curriculum. Perhaps the problem isn’t with a traditional lesson beginning with going over homework from the previous day, an introduction to a new topic followed by time working on a new homework assignment. Perhaps the problem is more societal. Perhaps the problem lies with us all.

Since becoming a high school mathematics teacher I have been struck by the seemingly countless times I have heard parents and other adults, well educated or not, utter statements such as, “math was my worse subject” or “I can’t do math.”

I have never once heard a person say, “I can’t read.” Why, then, is it so acceptable to publicly declare, without the slightest bit of embarrassment, one’s inability to do mathematics? In our society one would be shamed and embarrassed to admit their inability to read. The point is it is perfectly acceptable to be poor at mathematics in this country. We are an anomaly compared to most of the rest of the world when it comes to our comfortable attitude that it is acceptable to be poor at math. Is there any doubt that our children haven’t learned this lesson?

Or have they? While American adults casually admit to their mathematical ignorance, American children appear to overestimate their mathematical ability. In a 2003 study, 84% of American 8th graders agreed with the statement, “I usually do well in mathematics.” Of their counterparts in Singapore, 64% agreed with the same statement. Confidence often, especially in America, is considered as a positive personality trait. However, it is ironic that of the same group of 8th graders, the least confident students in Singapore, on average, outscored the most confident American students on an international math test.

If the results of the 2003 study are accurate, then our young students are overconfident. If my informal observations over the years are accurate, they ultimately learn of their shortcomings and then shrug them off with a cavalier attitude of acceptable incompetence. If so, then new curriculum, changes in pedagogy and an implementation of standardized tests are not addressing the real problem and, therefore, doom us to perpetual mediocrity.

Perhaps we all share the responsibility for the dismal mathematical performance of our children. If so, it begs a difficult and frightening question: What can be done about it?



Jim Stone, a math and physics teacher for more than 18 years, teaches at Global Institute of Mathematics.

Developing positive attitude

What are the incentives needed in order to develop a positive attitude in children and other students towards mathematics?

I am not sure if (special) incentives is the main factor in developing a positive attitude towards mathematics.

I feel it is probably sufficient to get a few of the basics right, and then that alone will take care of most of it, and students can like math just fine.

Disliking math is not something that is inherent in us or in our kids. Little kids don't dislike math or numbers. They're just fine with math. As we know, this "I hate math" or "I don't like math" attitude seems to develop during school years.

Now, I also don't think that children are disliking reasoning, because they're happy to do puzzles and go through games where you have to think.

And, students' negative attitude towards math also is NOT due to (school) math being difficult. The math we learn in school is not difficult. You don't have to be a math whiz to understand it.

If you can learn to read and to use computer software, surely you can learn basic math. It's not that complex.

So... here are the two main factors that I feel contribute most to what attitude children develop towards math:

1) The teacher's attitude.

If you love math and are enthusiastic about it, it is seen in your teaching, and your attitude will be somewhat contagious.

It's true the other way around as well: if you don't like math and are teaching it, students will sense it. (I've written about the teacher's attitude in this article.)


2) How the math is being taught.

Children can end up not liking math when it is taught in such a way that they don't end up understanding it.

When they don't understand it, then they don't like studying more of it.

The teacher is obviously hugely influental in how the math is taught, but curriculum or the math book also plays a role.

So if we as teachers can get these two things straightened out and thus have the basics covered, then some special incentives and other extra "math goodies" won't hurt either.

---

See also:
Four habits of highly effective math teaching

Is your math curriculum coherent?

How to motivate and prevent math anxiety

Developing positive attitude

What are the incentives needed in order to develop a positive attitude in children and other students towards mathematics?

I am not sure if (special) incentives is the main factor in developing a positive attitude towards mathematics.

I feel it is probably sufficient to get a few of the basics right, and then that alone will take care of most of it, and students can like math just fine.

Disliking math is not something that is inherent in us or in our kids. Little kids don't dislike math or numbers. They're just fine with math. As we know, this "I hate math" or "I don't like math" attitude seems to develop during school years.

Now, I also don't think that children are disliking reasoning, because they're happy to do puzzles and go through games where you have to think.

And, students' negative attitude towards math also is NOT due to (school) math being difficult. The math we learn in school is not difficult. You don't have to be a math whiz to understand it.

If you can learn to read and to use computer software, surely you can learn basic math. It's not that complex.

So... here are the two main factors that I feel contribute most to what attitude children develop towards math:

1) The teacher's attitude.

If you love math and are enthusiastic about it, it is seen in your teaching, and your attitude will be somewhat contagious.

It's true the other way around as well: if you don't like math and are teaching it, students will sense it. (I've written about the teacher's attitude in this article.)


2) How the math is being taught.

Children can end up not liking math when it is taught in such a way that they don't end up understanding it.

When they don't understand it, then they don't like studying more of it.

The teacher is obviously hugely influental in how the math is taught, but curriculum or the math book also plays a role.

So if we as teachers can get these two things straightened out and thus have the basics covered, then some special incentives and other extra "math goodies" won't hurt either.

---

See also:
Four habits of highly effective math teaching

Is your math curriculum coherent?

How to motivate and prevent math anxiety

Math learning and unhappiness

Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.

The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.

The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.

American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!

Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.



The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.

I can almost hear my homeschooling readers' anger raising...!

But wait a minute.


Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."

"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'


This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.

Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.

In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.

In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.

But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.

Is there a solution?

Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.

But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.

The truth, as always, must be somewhere in the middle.

When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.

Maybe group work could be used with those, as well.

It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.

I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.


What are your thoughts?

Several other bloggers have gotten on to this too:

Gooseania

Luboš Motl's reference frame

Mathematics and (un)happiness by Alexandre Borovik

Mathematics & (un)happiness at NeverEndingBooks

See also:

CNN news: Confident students do worse in math; bad news for U.S.


The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores

Math learning and unhappiness

Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.

The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.

The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.

American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!

Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.



The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.

I can almost hear my homeschooling readers' anger raising...!

But wait a minute.


Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."

"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'


This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.

Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.

In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.

In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.

But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.

Is there a solution?

Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.

But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.

The truth, as always, must be somewhere in the middle.

When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.

Maybe group work could be used with those, as well.

It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.

I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.


What are your thoughts?

Several other bloggers have gotten on to this too:

Gooseania

Luboš Motl's reference frame

Mathematics and (un)happiness by Alexandre Borovik

Mathematics & (un)happiness at NeverEndingBooks

See also:

CNN news: Confident students do worse in math; bad news for U.S.


The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores

Living and Loving Math

X (however many) Habits of Highly Effective Math Teaching:

Part 4: Living and Loving Math

You are the teacher. You show the way - also with your attitudes, your way of life.

Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?

And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?

Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.

Math is not a drudgery, nor something just confined to math lessons.

Some ideas:

• Let it make sense. This alone can usually make math quite a difference and kids will stay interested.


• Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.


• Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.


• When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.

I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.

Living and Loving Math

X (however many) Habits of Highly Effective Math Teaching:

Part 4: Living and Loving Math

You are the teacher. You show the way - also with your attitudes, your way of life.

Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?

And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?

Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.

Math is not a drudgery, nor something just confined to math lessons.

Some ideas:

• Let it make sense. This alone can usually make math quite a difference and kids will stay interested.


• Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list here.


• Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. Julie at LivingMath.net has suggestions for math history books to buy.


• When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.

I've talked about all this before, so I really don't want to go into repeating myself too much. But I did want to include this principle in my "mini-series" of effective habits of math teaching.

Spiraling or mastery in a mathematics curriculum

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better?

Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on.

Mastery approach simply aims to teach to mastery any single topic, before going on to the next.

There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula.

So is the mastery principle then better?

Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles.

For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers.

At each grade, mastery is required - but in reality it is only partial mastery since there is more to learn about adding numbers.

Or, take fractions. You can visit the topics of fractions on 1st, 2nd, and 3rd grades just passing, noting what is a fraction, maybe having a few addition problems. Then 4th grade some more. Then on 5th grade, you study them a lot, require mastery of certain topics. On 6th, require mastery of all fraction topics. And you're done.

In essence, there is nothing wrong with spiraling. Certainly it is GOOD to present some fraction topics on lower grades, and some later on, revisit the topics next year; or study some long division on 4th and more on 5th. But we need to require mastery of these arithmetic subtopics each year. That way we can at some point easily move into ALGEBRA and leave arithmetic behind.

Yet when it comes to the teaching of math, the divide between the two approaches may be somewhat artificial, as a spiral curriculum could be taught with a judicious eye toward mastery at every level, and any good mastery curriculum will build on concepts in a spiral fashion.
Teens and Tweens Blog, a post on Spiral Curricula (taken offline)

The approach to beware is when a mathematics topic is studied so briefly that kids don't get to master it at all. This can happen with the 'inch-deep-mile-wide' curricula that is so shock-full of topics that all a teacher can do is race thru the book...

See also:
Is your math curriculum coherent and logical?

Scope and sequence chart suggestion

Spiraling or mastery in a mathematics curriculum

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better?

Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on.

Mastery approach simply aims to teach to mastery any single topic, before going on to the next.

There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula.

So is the mastery principle then better?

Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles.

For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers.

At each grade, mastery is required - but in reality it is only partial mastery since there is more to learn about adding numbers.

Or, take fractions. You can visit the topics of fractions on 1st, 2nd, and 3rd grades just passing, noting what is a fraction, maybe having a few addition problems. Then 4th grade some more. Then on 5th grade, you study them a lot, require mastery of certain topics. On 6th, require mastery of all fraction topics. And you're done.

In essence, there is nothing wrong with spiraling. Certainly it is GOOD to present some fraction topics on lower grades, and some later on, revisit the topics next year; or study some long division on 4th and more on 5th. But we need to require mastery of these arithmetic subtopics each year. That way we can at some point easily move into ALGEBRA and leave arithmetic behind.

Yet when it comes to the teaching of math, the divide between the two approaches may be somewhat artificial, as a spiral curriculum could be taught with a judicious eye toward mastery at every level, and any good mastery curriculum will build on concepts in a spiral fashion.
Teens and Tweens Blog, a post on Spiral Curricula (taken offline)

The approach to beware is when a mathematics topic is studied so briefly that kids don't get to master it at all. This can happen with the 'inch-deep-mile-wide' curricula that is so shock-full of topics that all a teacher can do is race thru the book...

See also:
Is your math curriculum coherent and logical?

Scope and sequence chart suggestion

The daily grind of math - making connections

I wanted to briefly touch some more on this topic of "grind" or daily grind that learning math might sometimes become that I mentioned briefly last time.

I didn't mean to imply that learning to add, subtract, multiply, and divide whole numbers, decimals, fractions, percents, numbers with exponents, and integers has to be such a boring task. You can avoid that type of feeling.

One way:
Make connections between the concepts. Don't make math appear as fairly separate compartments of "fractions" and "decimals" and "percents" and "geometry".

I sometimes wonder how students feel when every year (on 4th, 5th, 6th, 7th, and 8th grade) they have a chapter on fractions, a chapter on decimals, a chapter on whole numbers, a chapter on geometry... Almost the same thing over and over.

We need to make sure they can see how these things connect.

An example:
A jogging track is 3.5 km long. If you jog 2/5 of it, how far was that? How many percent of the track is that?

|--------------------------------------------|
0                                           3.5 km

|--------|--------|--------|--------|--------|
0                                            1

|--------------------------------------------|
0                                           100%

Can your student solve it? What do you think, when are they ready for this sort of problem?

Tags: math, mathematics

The daily grind of math - making connections

I wanted to briefly touch some more on this topic of "grind" or daily grind that learning math might sometimes become that I mentioned briefly last time.

I didn't mean to imply that learning to add, subtract, multiply, and divide whole numbers, decimals, fractions, percents, numbers with exponents, and integers has to be such a boring task. You can avoid that type of feeling.

One way:
Make connections between the concepts. Don't make math appear as fairly separate compartments of "fractions" and "decimals" and "percents" and "geometry".

I sometimes wonder how students feel when every year (on 4th, 5th, 6th, 7th, and 8th grade) they have a chapter on fractions, a chapter on decimals, a chapter on whole numbers, a chapter on geometry... Almost the same thing over and over.

We need to make sure they can see how these things connect.

An example:
A jogging track is 3.5 km long. If you jog 2/5 of it, how far was that? How many percent of the track is that?

|--------------------------------------------|
0                                           3.5 km

|--------|--------|--------|--------|--------|
0                                            1

|--------------------------------------------|
0                                           100%

Can your student solve it? What do you think, when are they ready for this sort of problem?

Tags: math, mathematics

Reform math - we need a BALANCE

Recently Spunky wrote about reform math, and I feel I want to say something about it, too.

She was critizing the movement heavily. Some people call reform math 'new new math' or 'fuzzy math' (in a degrading way) - and like you can read in Spunky's post, sometimes when new ideas are implemented, it can cause math to become 'fuzzy' to the children.

But that is not necessarily a fault of the reform on a whole - maybe the teacher in question didn't understand how to implement the methods, or for some reason (unwisely) totally abandoned 'traditional' math fact drills.

I would say not all of the IDEAS that have come from this mathematics education reform are bad.

For example, encouraging kids to explore and investigate mathematical concepts, or discovering their own rules, can be a good thing. Group work CAN be a good thing.

The reformists promote understanding of concepts, problem solving, critical thinking. They try to distance themselves from the "drill ' n' kill" and rote memorization parts of traditional math education.

If kids "learn" math as a set of disconnected rules, they soon forget those since they never learned WHY they worked. (Well that can easily happen. I just witnessed it this past week; a youngster didn't know how to multiply two two-digit numbers since it had been so long that he had done it.)

The key is a BALANCE. Truth is somewhere in the middle. Kids absolutely need to learn their addition facts and memorize their multiplication facts. Otherwise, they're going to go about learning math without a FOUNDATION.

But, I feel it is very important to teach them also WHY the rules and procedures work.

For example, let kids learn how to multiply 2-digit numbers by 2-digit numbers. Let them practice. But also show them what it is based on (distributive property). They need to know that anyway in algebra class!
(For example: 34 x 58 is based on doing it in parts: 4 x 8, 4 x 50, 30 x 8, and 30 x 50, and adding all those. Compare to (x + 1) (2x - 5) which is done using distributive property.)

Learning to estimate and critically look at your final answer is very important, I feel. (For example, 34 x 58 is about 35 x 60, which is same as 70 x 30 = 2100. Exact answer is: 1972. OK.)

Let them SOMETIMES explore and investigate, find real-life connections. Occasional "math labs" can do much good in motivating and showing where math is useful in real life, letting them gain confidence, etc.

But if all the class time is spent on letting the kids find their own division methods and forgetting all about memorization, the pendulum has swung WAY too far to the other direction.

Memorization is needful, very needful. But let us not forget the 'reform' aspects of math altogether.

And like was pointed out in the comments to this post, it is the TEACHER that decides what is done in the math class and how it is done.

I feel the NAME of this reformist website summarizes it all: Mathematically Sane.

P.S. Here's a nice piece Newer Math by Jack Lee about what is going on at Seattle.

Tags: math, mathematics, reform, education

Reform math - we need a BALANCE

Recently Spunky wrote about reform math, and I feel I want to say something about it, too.

She was critizing the movement heavily. Some people call reform math 'new new math' or 'fuzzy math' (in a degrading way) - and like you can read in Spunky's post, sometimes when new ideas are implemented, it can cause math to become 'fuzzy' to the children.

But that is not necessarily a fault of the reform on a whole - maybe the teacher in question didn't understand how to implement the methods, or for some reason (unwisely) totally abandoned 'traditional' math fact drills.

I would say not all of the IDEAS that have come from this mathematics education reform are bad.

For example, encouraging kids to explore and investigate mathematical concepts, or discovering their own rules, can be a good thing. Group work CAN be a good thing.

The reformists promote understanding of concepts, problem solving, critical thinking. They try to distance themselves from the "drill ' n' kill" and rote memorization parts of traditional math education.

If kids "learn" math as a set of disconnected rules, they soon forget those since they never learned WHY they worked. (Well that can easily happen. I just witnessed it this past week; a youngster didn't know how to multiply two two-digit numbers since it had been so long that he had done it.)

The key is a BALANCE. Truth is somewhere in the middle. Kids absolutely need to learn their addition facts and memorize their multiplication facts. Otherwise, they're going to go about learning math without a FOUNDATION.

But, I feel it is very important to teach them also WHY the rules and procedures work.

For example, let kids learn how to multiply 2-digit numbers by 2-digit numbers. Let them practice. But also show them what it is based on (distributive property). They need to know that anyway in algebra class!
(For example: 34 x 58 is based on doing it in parts: 4 x 8, 4 x 50, 30 x 8, and 30 x 50, and adding all those. Compare to (x + 1) (2x - 5) which is done using distributive property.)

Learning to estimate and critically look at your final answer is very important, I feel. (For example, 34 x 58 is about 35 x 60, which is same as 70 x 30 = 2100. Exact answer is: 1972. OK.)

Let them SOMETIMES explore and investigate, find real-life connections. Occasional "math labs" can do much good in motivating and showing where math is useful in real life, letting them gain confidence, etc.

But if all the class time is spent on letting the kids find their own division methods and forgetting all about memorization, the pendulum has swung WAY too far to the other direction.

Memorization is needful, very needful. But let us not forget the 'reform' aspects of math altogether.

And like was pointed out in the comments to this post, it is the TEACHER that decides what is done in the math class and how it is done.

I feel the NAME of this reformist website summarizes it all: Mathematically Sane.

P.S. Here's a nice piece Newer Math by Jack Lee about what is going on at Seattle.

Tags: math, mathematics, reform, education