uses analogies to football (numerous quotes from Blanchard and Shula's The Little Book of Coaching.)

Three ideas which are relevant to mathematics: (a) capacity limits performance - mathematical performance will eventually break down as task difficulty is increased; (b) capacity is used in performance - adding more components to a problem will use more capacity and increase task difficulty; and (c) demands on capacity are less for individuals who have higher levels of relevant skills. (29)

Over-learned skills can be directly retrieved from memory rather than constructed (Logan, 1988) (68). Cognitive pychology clearly states that you cannot problem solve effectively if you have to simultaneously use working memory to process basic skills (87)

If a learner experiences difficulty learning some bit of mathematics explicitly, he or she has to imitate the pattern of the technique as many times as it takes to be able to successfully duplicate it without error. (118)

Modern textbooks that emphasize problem solving over skills do not contain enough problems of a similar type to allow the implicit learner to develop the sense of rhythm of the technique that comes with repeated practice. (119)

The Math Plague slams NCTM's Curriculum and Evaluation Standards for School Mathematics. It is easy to see that a curriculum that is guided by these standards is unlikely to produce students who will be able to compute without the aid of a calculator. (84)

Cognitive pychology clearly states that you cannot problem solve effectively if you have to simultaneously use working memory to process basic skills.

The fuzzies should write this on their forehead. Where is Skippy when he needs to hear this crucial message?

The fuzzies talk endlessly about problem-solving and don't realize that their own prescriptions are undermining the means to achieving their purported goals.

As a high school math teacher for ten years in Eastern Canada, and now a doctoral mathematics education student in Western Canada, I find the answer to allow students to think numerately is to cut our curriculum, allowing teachers the time to do what the curriculum intends. Look at what Western canada (4 provinces and 3 territories) are doing with their curriculum changes being implemented in 2008 (www.wncp.ca). They are cutting the content to allow teachers to have time to "teach for understanding".

The Math Plague is NOT an answer for teachers. It is an answer to the traditional curriculum where students did too many concepts and did not have time to grasp them.

This is from the K-9 Math Common Curriculum Framework at www.wncp.ca.

"The learning environment should value and respect all students’ experiences and ways of thinking, so that learners are comfortable taking intellectual risks, asking questions and posing conjectures. Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. learners must realize that it is acceptable to solve problems in different ways and that solutions may vary."

"... explore problem-solving situations in order to develop personal strategies and become mathematically literate."

"personal strategies"

Yep. That's what math is all about. They are in charge and they get to define math however they want.

"solutions may vary"

"They are cutting the content to allow teachers to have time to 'teach for understanding'."

OK, I'll bite. What is "teach for understanding"? Please give an exact example of a content for understanding tradeoff. While you're at it, please define "understanding" with an example.

I would love to let others define math any which way they would like. Just give me (and everyone else) the money to go somewhere else.

It is not about defining mathematics, but on defining how to teach mathematics.

Mantyka's book mentions lots of problems, but does not give solutions for teachers in their classrooms.

At Memorial (where she works), all students need Grade 12 Academic or Advanced mathematics to be accepted into the university. Also, all Arts students are to have a numeracy/science requirement upon graduating, where they must take 6 credit hours (2 courses). How many students end up taking mathematics for this requirement, and never end up finishing their Arts degree, just because they did not pass mathematics?

Mantyka is trying to help students overcome the deficits of a school system where the basics of numeracy is not second nature to them. Some students who fail the Mathematics Placement Test before entering Memorial go to the Math Help Centre to better their skills so that they can take the test later and eventually pass it. She claims that they then go on to succeed in first year mathematics. Students need mathematics to get their degree, and she is providing an opportunity to help students pass a course and essentially pass a test.

Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test? Is our curriculum too progressive, and is she addressing a shortcoming in our current teaching? Or have the changes the NCTM proposed funneled through our children’s curriculum but they have not yet reached university? Secondary school is about preparing them for the world. NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills. Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test? We have to question what the outcome for secondary mathematics is. Is it for students to prepare themselves to go to university, to become part of the workforce, to love mathematics for the beauty of the language itself and how it works in our world?

"Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test?"

You could look at this two ways. First, all teachers test. Presumably, they know how to test for important knowledge and skills. I assume that you are not arguing against all testing.

Second, one could complain about gateway or high stakes tests. These tests shouldn't be necessary, but schools allow many students to slide through grades without making sure that they know the required knowledge. These tests are usually very easy with very low cutoff points. Since high stakes tests tend to punish the students, some states, like ours, use the results of standardized testing to rate the different schools. The downside to this approach is that the proficiency levels are so low that the schools look good when all they did was to meet very minimal cutoff standards. Our local newspaper had a big headline that talked about the "high rigor" at our schools because our scores were higher than many other schools in the state. This is for a simple test where the goal is only to see if a school gets most all of its students over a very minimal cutoff.

There has to be some way to verify that students master basic knowledge and skills. For very simple standardized tests, what other knowledge or skills are so important that it would make flunking these tests acceptable?

This is what you have to show. You have to go through the test and explain why it is OK for the student to flunk the test.

NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills.

The brain is hardwire to learn new material in concrete terms and then knowledge gradually becomes organized around abstract principles.

The NCTM has it exactly backwards. You can't teach abstract understanding directly. There is no magic shortcut.

"NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills."

What, exactly, does "numerate" mean? I'm really tired of these code words. It sounds like you are changing the definition of math. Does it mean that the school teaches kids the required knowledge and skills to get into the top math track in high school? Does the school have a carefully-defined curriculum that leads to a proper course of algebra in 8th grade?

Our schools think they are doing just fine even though they only offer CMP + a few topics in algebra in 8th grade. There is a curriculum gap between 8th grade math and the high school college prep math track. Most kids who make this jump do so ONLY because of outside help from parents and tutors. The rich get richer, so to speak. The poor get low expectations because it's better than really low expectations.

" ...teaching for understanding before they teach for memorization and fluency ..."

Once again, you have to explain this carefully. You can't just toss it out. Is it even possible to understand something properly if you have little content knowledge and mastery? Is there no linkage between mastery and understanding. Some ed school type once told me that experience only adds speed, not understanding. I surely want the CHOICE to have my son taught by somebody else.

"Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test?"

They want students who have the basic knowledge and skills to do college work. These are called prerequisites.

"We have to question what the outcome for secondary mathematics is. Is it for students to prepare themselves to go to university, to become part of the workforce, to love mathematics for the beauty of the language itself and how it works in our world?"

Wow. My outcome would be to teach them the knowledge and skills of math. ONLY THEN, can they truly love mathematics for the beauty of the language itself. Anything else is clueless dreaming.

Is this the modern ed school mantra, understanding without hard work?

"...to prepare themselves to go to university, to become part of the workforce, ..."

If this is NOT your outcome, then you have the responsibility to tell all parents up front and allow them the choice (and money) to go elsewhere.

The brain is hardwire to learn new material in concrete terms and then knowledge gradually becomes organized around abstract principles.

The NCTM has it exactly backwards. You can't teach abstract understanding directly. There is no magic shortcut.

This is our situation with the Earth Science course.

The teacher is teaching "conceptually."

Therefore only kids scoring in the top 90th percentile of the country can handle it.

Conceptually means they have to be able to generalize knowledge to novel situations.

The example the Science Chair gave was a question on the CTBS in which the student has to say why a candle won't burn on the moon (answer: no oxygen)

"Conceptual" simply means flexible knowledge the student can generalize.

The school sees this as a matter of "maturity."

The reason kids can't take the course in 8th grade but CAN take it in 10th is that they're "more mature."

I assume "more mature" means more able & willing to teach the course to themselves, but I don't know. The school may also be confusing greater reading comprehension and background knowledge with maturity.

"It is not about defining mathematics, but on defining how to teach mathematics."

That's not true. Take a look at state tests and you'll see that numeration and computation are increasingly being displaced by visuals and estimation.

Also see David Klein http://www.csun.edu/~vcmth00m/AHistory.html on the topic:

It would be a mistake to think of the major conflicts in education as disagreements over the most effective ways to teach. Broadly speaking, the education wars of the past century are best understood as a protracted struggle between content and pedagogy. At first glance, such a dichotomy seems unthinkable. There should no more be conflict between content and pedagogy than between one's right foot and left foot. They should work in tandem toward the same end, and avoid tripping each other. Content is the answer to the question of what to teach, while pedagogy answers the question of how to teach.

The trouble comes with the first step. Do we lead with the right foot or the left? If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow. In the same way, the choice of a pedagogy can naturally limit the amount of content that can be presented to students. Therein lies the source of the conflict.

"Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test?...Secondary school is about preparing them for the world."

I haven't reviewed what research shows, but my sense is that students who are proficient in mathematics will be better prepared for the world. But I leave open the possibility that research shows otherwise.

"NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills."

You can't treat these abilities as separate. They are intertwined.

For example, at the upper middle grades level, when I teach the conversion between fractions, decimals and percents, I'd like to get across the concept that these are different forms of the same thing, namely the relationship between part and whole. I can't do this if the students haven't become adept at working out fraction and decimal problems. Usually they lack that skill because they have spent the previous years on a fruitless, non-instruction discovery journey with Trailblazers and CMP, two of the worst fuzzy math programs.

"Second, one could complain about gateway or high stakes tests. These tests shouldn't be necessary, but schools allow many students to slide through grades without making sure that they know the required knowledge."

In the large urban school system in which I teach a high proportion of students "slide through grades without making sure that they know the required knowledge."

You would think it should be obvious that these students should be identified and that there should be some sort of intervention but that appears to be an alien thought around here. It would be easy enough to have students take the Stanford Diagnostic test each year for an hour and to provide intensive math support as is done with reading. But educationists are only concerned with reading and can't relate to math.

David Klein said: "If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow."

The problem with Everyday Math is that they have way too much content. Like they couldn't decide what was important and so they threw everything in there. But one does preclude the other. So, in EM, you don't get a lot of discovery learning and you don't get a lot of practice. I'm not quite sure what you are getting a lot of.

Games. In EM, you get a lot of games. The games drive the content.

If you can't come up with a game to play connected to the math concept, the concept is gone. And they have such cute names for those games, too.

EM includes a "Games Correlation Chart" in the back of the Teacher Reference Manual, p. 358 & 359. The games are all mapped out for the skill and concept area they address. I count 65 games in all. (Although, EM encourages teachers to supplement this exhaustive list with their own creations as well).

How about a little "musical name collection boxes" anyone? "Pocket Billiards," "Greedy," "Baseball Multiplication Advanced Version"?

NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills. Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test?

Well yes.

May I ask what do you think the merit of understanding maths is, if not to be able to competently solve problems?

If a student can't solve the mathematical problems on a test then what does it mean to say they have understanding?

If a student can't solve the mathematical problems on a test then what does it mean to say they have understanding?

Ed and I were going 'round about this one this weekend.

I argue that it's possible to have some pretty decent understanding without being able actually to do a multi-step problem.

I was thinking about....I'm trying to remember the name for the procedure (that's another issue: I have some reasonably-OK understanding of this procedure, but nothing is firm in memory at this point including the name)....

The problem is to reverse a polar coordinate into rectangular coordinates, I think.

I think it's probably correct to say that I have an NCTM-like understanding of how and why it's possible to do the procedure. (In this case I use "NCTM" neutrally.)

But I can just barely do the procedure independently; often I have to go back and re-read the lesson when I get stuck.

Ed says that if I could do the procedure easily my understanding would also be better.

He's probably right about that (although he may only be right that I'd be able to generalize my understanding, not that my understanding would be appreciably different....)

Still, my experience so far is that it's possible to have a kind of...."journalistic" understanding of mathematical procedures and concepts without being able to do the procedure.

"Ed says that if I could do the procedure easily my understanding would also be better."

I agree with Ed.

For polar coordinates, one could understand it graphically without knowing trig. Take a piece of graph paper. Pick an origin. Use a protractor to define the angle. Draw a straight line for that angle. Measure the distance to the point and mark it. Now measure the [X,Y] coordinates off of the graph paper. No trig involved.

Understanding? Yes, but mathematically limited. If someone doesn't understand what I described (basically the definition of polar coordinates) then I would question their understanding at any level.

This level of understanding is fine (along with examples of why polar coordinates are useful) for middle school (and journalists). It's obviously not fine for high school trig class, where one would have to write:

X = R*cos(theta)

Y = R*sin(theta)

off the top of their heads. This is the required understanding at this level.

"Still, my experience so far is that it's possible to have a kind of...."journalistic" understanding of mathematical procedures and concepts without being able to do the procedure."

This is what NCTM math does, but schools don't know which kids are going to end up as journalists when the are in K-8. They apparently assume that all of the future engineers will figure it out on their own.

## 25 comments:

wow!

Fantastic find!

amazing

I'm ordering this book today

wow

check out her bio

I've ordered the book.

This post is exactly what Ed and I were talking about last night.

Back later --- (must go engage in some deliberate practice!)

Cognitive pychology clearly states that you cannot problem solve effectively if you have to simultaneously use working memory to process basic skills.The fuzzies should write this on their forehead. Where is Skippy when he needs to hear this crucial message?

The fuzzies talk endlessly about problem-solving and don't realize that their own prescriptions are undermining the means to achieving their purported goals.

I'm right here. Sorry, can't concentrate on your question right now. I'm trying to add 2 + 5.

Wow!

Mr. NCTM is here and hopefully processing after he is done with the addition problem!

Your line gave me a good chuckle.

I do want to apologize for the somewhat irrevent form of address I used.

As a high school math teacher for ten years in Eastern Canada, and now a doctoral mathematics education student in Western Canada, I find the answer to allow students to think numerately is to cut our curriculum, allowing teachers the time to do what the curriculum intends. Look at what Western canada (4 provinces and 3 territories) are doing with their curriculum changes being implemented in 2008 (www.wncp.ca). They are cutting the content to allow teachers to have time to "teach for understanding".

The Math Plague is NOT an answer for teachers. It is an answer to the traditional curriculum where students did too many concepts and did not have time to grasp them.

This is from the K-9 Math Common Curriculum Framework at www.wncp.ca.

"The learning environment should value and respect all students’ experiences and ways of thinking, so that learners are comfortable taking intellectual risks, asking questions and posing conjectures. Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. learners must realize that it is acceptable to solve problems in different ways and that solutions may vary."

"... explore problem-solving situations in order to develop personal strategies and become mathematically literate."

"personal strategies"

Yep. That's what math is all about. They are in charge and they get to define math however they want.

"solutions may vary"

"They are cutting the content to allow teachers to have time to 'teach for understanding'."

OK, I'll bite. What is "teach for understanding"? Please give an exact example of a content for understanding tradeoff. While you're at it, please define "understanding" with an example.

I would love to let others define math any which way they would like. Just give me (and everyone else) the money to go somewhere else.

It is not about defining mathematics, but on defining how to teach mathematics.

Mantyka's book mentions lots of problems, but does not give solutions for teachers in their classrooms.

At Memorial (where she works), all students need Grade 12 Academic or Advanced mathematics to be accepted into the university. Also, all Arts students are to have a numeracy/science requirement upon graduating, where they must take 6 credit hours (2 courses). How many students end up taking mathematics for this requirement, and never end up finishing their Arts degree, just because they did not pass mathematics?

Mantyka is trying to help students overcome the deficits of a school system where the basics of numeracy is not second nature to them. Some students who fail the Mathematics Placement Test before entering Memorial go to the Math Help Centre to better their skills so that they can take the test later and eventually pass it. She claims that they then go on to succeed in first year mathematics. Students need mathematics to get their degree, and she is providing an opportunity to help students pass a course and essentially pass a test.

Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test? Is our curriculum too progressive, and is she addressing a shortcoming in our current teaching? Or have the changes the NCTM proposed funneled through our children’s curriculum but they have not yet reached university? Secondary school is about preparing them for the world. NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills. Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test? We have to question what the outcome for secondary mathematics is. Is it for students to prepare themselves to go to university, to become part of the workforce, to love mathematics for the beauty of the language itself and how it works in our world?

"Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test?"

You could look at this two ways. First, all teachers test. Presumably, they know how to test for important knowledge and skills. I assume that you are not arguing against all testing.

Second, one could complain about gateway or high stakes tests. These tests shouldn't be necessary, but schools allow many students to slide through grades without making sure that they know the required knowledge. These tests are usually very easy with very low cutoff points. Since high stakes tests tend to punish the students, some states, like ours, use the results of standardized testing to rate the different schools. The downside to this approach is that the proficiency levels are so low that the schools look good when all they did was to meet very minimal cutoff standards. Our local newspaper had a big headline that talked about the "high rigor" at our schools because our scores were higher than many other schools in the state. This is for a simple test where the goal is only to see if a school gets most all of its students over a very minimal cutoff.

There has to be some way to verify that students master basic knowledge and skills. For very simple standardized tests, what other knowledge or skills are so important that it would make flunking these tests acceptable?

This is what you have to show. You have to go through the test and explain why it is OK for the student to flunk the test.

NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills.The brain is hardwire to learn new material in concrete terms and then knowledge gradually becomes organized around abstract principles.

The NCTM has it exactly backwards. You can't teach abstract understanding directly. There is no magic shortcut.

"NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills."

What, exactly, does "numerate" mean? I'm really tired of these code words. It sounds like you are changing the definition of math. Does it mean that the school teaches kids the required knowledge and skills to get into the top math track in high school? Does the school have a carefully-defined curriculum that leads to a proper course of algebra in 8th grade?

Our schools think they are doing just fine even though they only offer CMP + a few topics in algebra in 8th grade. There is a curriculum gap between 8th grade math and the high school college prep math track. Most kids who make this jump do so ONLY because of outside help from parents and tutors. The rich get richer, so to speak. The poor get low expectations because it's better than really low expectations.

" ...teaching for understanding before they teach for memorization and fluency ..."

Once again, you have to explain this carefully. You can't just toss it out. Is it even possible to understand something properly if you have little content knowledge and mastery? Is there no linkage between mastery and understanding. Some ed school type once told me that experience only adds speed, not understanding. I surely want the CHOICE to have my son taught by somebody else.

"Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test?"

They want students who have the basic knowledge and skills to do college work. These are called prerequisites.

"We have to question what the outcome for secondary mathematics is. Is it for students to prepare themselves to go to university, to become part of the workforce, to love mathematics for the beauty of the language itself and how it works in our world?"

Wow. My outcome would be to teach them the knowledge and skills of math. ONLY THEN, can they truly love mathematics for the beauty of the language itself. Anything else is clueless dreaming.

Is this the modern ed school mantra, understanding without hard work?

"...to prepare themselves to go to university, to become part of the workforce, ..."

If this is NOT your outcome, then you have the responsibility to tell all parents up front and allow them the choice (and money) to go elsewhere.

Or have the changes the NCTM proposed funneled through our children’s curriculum but they have not yet reached university?God help us...

The brain is hardwire to learn new material in concrete terms and then knowledge gradually becomes organized around abstract principles.

The NCTM has it exactly backwards. You can't teach abstract understanding directly. There is no magic shortcut.

This is our situation with the Earth Science course.

The teacher is teaching "conceptually."

Therefore only kids scoring in the top 90th percentile of the country can handle it.

Conceptually means they have to be able to generalize knowledge to novel situations.

The example the Science Chair gave was a question on the CTBS in which the student has to say why a candle won't burn on the moon (answer: no oxygen)

"Conceptual" simply means flexible knowledge the student can generalize.

The school sees this as a matter of "maturity."

The reason kids can't take the course in 8th grade but CAN take it in 10th is that they're "more mature."

I assume "more mature" means more able & willing to teach the course to themselves, but I don't know. The school may also be confusing greater reading comprehension and background knowledge with maturity.

"It is not about defining mathematics, but on defining how to teach mathematics."

That's not true. Take a look at state tests and you'll see that numeration and computation are increasingly being displaced by visuals and estimation.

Also see David Klein http://www.csun.edu/~vcmth00m/AHistory.html on the topic:

It would be a mistake to think of the major conflicts in education as disagreements over the most effective ways to teach. Broadly speaking, the education wars of the past century are best understood as a protracted struggle between content and pedagogy. At first glance, such a dichotomy seems unthinkable. There should no more be conflict between content and pedagogy than between one's right foot and left foot. They should work in tandem toward the same end, and avoid tripping each other. Content is the answer to the question of what to teach, while pedagogy answers the question of how to teach.

The trouble comes with the first step. Do we lead with the right foot or the left? If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow. In the same way, the choice of a pedagogy can naturally limit the amount of content that can be presented to students. Therein lies the source of the conflict.

"Is this what our elementary and secondary curricula are all about, for students to be proficient in mathematics to pass a test?...Secondary school is about preparing them for the world."

I haven't reviewed what research shows, but my sense is that students who are proficient in mathematics will be better prepared for the world. But I leave open the possibility that research shows otherwise.

"NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills."

You can't treat these abilities as separate. They are intertwined.

For example, at the upper middle grades level, when I teach the conversion between fractions, decimals and percents, I'd like to get across the concept that these are different forms of the same thing, namely the relationship between part and whole. I can't do this if the students haven't become adept at working out fraction and decimal problems. Usually they lack that skill because they have spent the previous years on a fruitless, non-instruction discovery journey with Trailblazers and CMP, two of the worst fuzzy math programs.

"Second, one could complain about gateway or high stakes tests. These tests shouldn't be necessary, but schools allow many students to slide through grades without making sure that they know the required knowledge."

In the large urban school system in which I teach a high proportion of students "slide through grades without making sure that they know the required knowledge."

You would think it should be obvious that these students should be identified and that there should be some sort of intervention but that appears to be an alien thought around here. It would be easy enough to have students take the Stanford Diagnostic test each year for an hour and to provide intensive math support as is done with reading. But educationists are only concerned with reading and can't relate to math.

Help!

rightwingprof posted some links to math help in comments recently that looked interesting, and now I can't find them.

I do miss the search capability of the old wiki.

David Klein said: "If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow."

The problem with Everyday Math is that they have way too much content. Like they couldn't decide what was important and so they threw everything in there. But one does preclude the other. So, in EM, you don't get a lot of discovery learning and you don't get a lot of practice. I'm not quite sure what you are getting a lot of.

Anne Dwyer

Games. In EM, you get a lot of games. The games drive the content.

If you can't come up with a game to play connected to the math concept, the concept is gone. And they have such cute names for those games, too.

EM includes a "Games Correlation Chart" in the back of the Teacher Reference Manual, p. 358 & 359. The games are all mapped out for the skill and concept area they address. I count 65 games in all.

(Although, EM encourages teachers to supplement this exhaustive list with their own creations as well).

How about a little "musical name collection boxes" anyone? "Pocket Billiards," "Greedy," "Baseball Multiplication Advanced Version"?

Yes, this is where EM excels.

NCTM is helping teachers develop the minds of pupils to become numerate by teaching for understanding before they teach for memorization and fluency of mathematics skills. Do Memorial and many other universities seem to want students to be competent computing machines who can pass a test?Well yes.

May I ask what do you think the merit of understanding maths is, if not to be able to competently solve problems?

If a student can't solve the mathematical problems on a test then what does it mean to say they have understanding?

If a student can't solve the mathematical problems on a test then what does it mean to say they have understanding?Ed and I were going 'round about this one this weekend.

I argue that it's possible to have some pretty decent understanding without being able actually to do a multi-step problem.

I was thinking about....I'm trying to remember the name for the procedure (that's another issue: I have some reasonably-OK understanding of this procedure, but nothing is firm in memory at this point including the name)....

The problem is to reverse a polar coordinate into rectangular coordinates, I think.

I think it's probably correct to say that I have an NCTM-like understanding of how and why it's possible to do the procedure. (In this case I use "NCTM" neutrally.)

But I can just

barelydo the procedure independently; often I have to go back and re-read the lesson when I get stuck.Ed says that if I could do the procedure easily my understanding would also be better.

He's probably right about that (although he may only be right that I'd be able to generalize my understanding, not that my understanding would be appreciably different....)

Still, my experience so far is that it's possible to have a kind of...."journalistic" understanding of mathematical procedures and concepts without being able to do the procedure.

Which doesn't suit my purposes at all.

"Ed says that if I could do the procedure easily my understanding would also be better."

I agree with Ed.

For polar coordinates, one could understand it graphically without knowing trig. Take a piece of graph paper. Pick an origin. Use a protractor to define the angle. Draw a straight line for that angle. Measure the distance to the point and mark it. Now measure the [X,Y] coordinates off of the graph paper. No trig involved.

Understanding? Yes, but mathematically limited. If someone doesn't understand what I described (basically the definition of polar coordinates) then I would question their understanding at any level.

This level of understanding is fine (along with examples of why polar coordinates are useful) for middle school (and journalists). It's obviously not fine for high school trig class, where one would have to write:

X = R*cos(theta)

Y = R*sin(theta)

off the top of their heads. This is the required understanding at this level.

"Still, my experience so far is that it's possible to have a kind of...."journalistic" understanding of mathematical procedures and concepts without being able to do the procedure."

This is what NCTM math does, but schools don't know which kids are going to end up as journalists when the are in K-8. They apparently assume that all of the future engineers will figure it out on their own.

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