# Let math be math

Connected Math is controversial in Madison, Wisconsin — especially with parents who are mathematicians. The Capital Times compares a traditional and a Connected Math problem for seventh graders.

A gas station sells soda in three sizes. A 20-ounce cup costs 80 cents, a 32-ounce cup is 90 cents and a 64-ouncer goes for \$1.25.

Traditional: What size offers the most soda for the money?

Connected: If the gas station were to offer an 84-ounce Mega Swig, what would you expect to pay for it?

A student, for instance, could argue that the 84-ouncer would cost what the 20-ounce and 64-ounce cups cost together. Another student could say that soda gets cheaper with volume, and then choose an answer based on some per-ounce price slightly less than what was given for the 64-ounce drink.

Or a student could skip calculating the per-ounce prices and just pick a number: \$1.50 seems about right, based on my knowledge of real-life pricing strategies.

On one side, those who support Connected Math say that engaging students by presenting problems as real-life scenarios, often with no absolute solution or single path to arrive at an answer, fosters innovation and forces students to explain and defend their reasoning as they discover mathematical concepts.

The other side says the approach trades the clear, fundamental concepts of math, distilled through thousands of years of logical reasoning, for verbiage and vagary that may help students learn to debate but will not give them the foundation they need for more advanced mathematical study.

. . . University of Wisconsin-Madison math and computer science Professor Jin-Yi Cai began to become concerned with Connected Math when he saw the questions on his seventh-grade son’s test.

He went to the UW Math Library to investigate the textbooks, and he said that he was dismayed to find them “thicker than the collected works of Tolstoy.” Where the Chinese math books he remembers fondly were thin and contained concise explanations of math’s fundamentals, these books were cumbersome and full of long story problems and written passages.

“It goes around and around and things never really get down to the really crisp, elegant, basic fundamental principles,” Cai said of the Connected Math texts.

“It takes away the elegance, it takes away the beauty, it takes away the most basic logic structure. And the students are left with a vague, touchy-feely idea,” he said.

Cai and UW-Madison mathematician Melania Alvarez, who is running for school board, also object to all the essay writing required by Connected Math. Writing isn’t math, they say.

My algebra teacher, Miss Diedrick, said that math is a language of its own.

Update: Alvarez lost her school board race.

1. Bill R says:

My wife teaches Math Connections. They adopted the program because the students lacked basic math skills and couldn’t complete a traditional course.

Now they’re dropping it because it turns out the kids lack basic reading skills as well.

This is not an inner city school. It’s in a small Maine city where the students are mostly middle class.

2. Mike McKeown says:

Nine years ago, when I first started fighting fuzzy math, one of the first things I realized was that the programs deliberately denigrated computation.

This used to be explicitly stated by fuzzy advocates. Indeed, they presented this as a social justice issue. Just because kids failed to learn basic arithmetic, and computations with fractions, decimals and percents, was no reason to keep them from expressing and utilizing their higher order algebraic thinking skills. After all, this would just be the tyranny of computation being used to keep the disadvantaged from succeeding at all the gatekeeper courses, like algebra. If it weren’t for the minor matter of computation, a whole cohort of under represented math geniuses would be moving through the system and into mathematics based fields.

This idea may be comforting, but it does not fit with reality. Algebra is generalize arithmetic. If you can’t do numbers, how can you do the same things with symbols? If you are not well practiced in rates (speed, slope, cost per unit, etc.) you don’t have the tool to think deeply about the problem in the example.

3. Bob Diethrich says:

No, you are all missing the major problem. The question itself is invalid.

The use of the out-dated term “gas station” will lead students to associate math with convenience stores. In the United States most of these ventures are owned by persons of Middle Eastern and/or Indic descent.

While answering the math problem little Johnnie or Janie cannot but think of the dark skinned person serving them their soda which will lead to feelings of racial superiority and cause them to re-enforce sterotypical views of white supremacy.

Or they may associate soda at a gas station with the current war on terror and they will be too concerned with their safety in the world made unsafe by the Bush administration, that they will be unable to concentrate on their math and hence will not pass the test, and thus their school will fall victim to the “No Chiild Left Behind Act.”

They should not be performing math problems anyway. They should be writing essays about how Bush’s policies will lead to the destruction of all life and make the need for math skills a moot point.

This question is invalid on its face and should be removed no matter how students seek to solve it.

4. People so easily forget that word problems exist to enrich a real-life setting with pure, sweet mathematics, not to enrich mathematics by injecting a real-life context into it.

5. Roger Sweeny says:

Paul Samuelson’s “Foundations of Economic Analysis,” which probably did more to mathematize economics than any other book, has this right after the dedication:

“Mathematics is a language.”
–J. Willard Gibbs

6. Steve says:

You cannot assume that a “traditional” approach and the “Connected” approach are doing the same level of problems. Even if you disregard the child-centered, group, fuzzy approach, a look at the problems that the students have to solve year by year will show a steady, but ever widening gap, starting in first grade.

Our school uses MathLand for grades 1-4 and CMP for grades 5-8. They are still trying to get third graders to know their adds and subtracts to 20 in the middle of third grade. These fuzzy math programs meet the needs of schools that try to educate all ability students in common groups. This requires lower, fuzzier expectations that show up as “spiraling”. The idea is that students do not have to master the material when it is taught because they will see the same material year after year. Some math programs, like EverdayMath carry this to an extreme and have the kids jumping from topic to topic on a daily basis. These programs assume that all students will learn at their own rate and eventually master the material. In practice, this doesn’t work unless the program is carefully supplemented by lots of practice and testing of basic skills. Unfortunately, many think that this is unnessary because of the calculator.

Even with supplementation, by eighth grade, there is a wide gap between these new new math programs and a rigorous traditional approach. They can talk all they want about problem solving and critical thinking, but all you have to do is look at the problems and skills that the students have to master year by year. That will tell you everything you need to know about students’ problem solving abilities.

7. I think the kids should answer both questions.

8. Steve says:

“… math is a language of its own.”

I prefer to think of math as a toolbox. I mix and match techniques to solve a variety of geometric and engineering analysis problems. For example, I have used parametric curve-fitting followed by line integrals to find properties of complex shapes. A specific example of this toolbox idea is represented in a series of books called Graphic Gems I – V, which give 2/3 page write-ups on hundreds of small, individual techniques for solving all sorts of geometric and computer graphics problems.

I have said before that you have to know what is in the box before you can think outside of the box. I could say that you have to know what is inside of the toolbox before you can think outside of the toolbox. New math programs like CMP don’t think you need a toolbox and the skills to use the tools. Just use the “Think Method”.

9. Chris says:

On one hand I want to say it’s sickening that students aren’t getting the proper math education (IMHO). The other question to ask is how were students doing before Connected Math (or pick your fav new math)?

I’m feel fairly confident that for students with decent math aptitudes and abilities Connected Math is a disservice. I don’t know about students with lesser abilities.

I do know at the school my 7th grade daughter attends, where Connected Math is enthusiastically used, the CSAP (Colorado) math scores decrease every year starting at 7th grade. They have bought the whole program hook, line and sinker so I’m sure it hasn’t even entered their stream of consciousness that their is a cause and effect between Connected Math and declining test scores.

Oh well. Also, good math teachers go a long way as well. Haven’t encountered on of those yet either. 😉

10. DBL says:

This is so frustrating. Math opens doors. The ability to do real math, real algebra and real calculus, opens many doors for high school students. The inability to do real math leaves those doors shut. How any educator that claims to have a genuine interest in his/her students could foist this phony math on them is beyond me.

I thought my son’s Chicago Math was bad until I read your Connected Math problem. Whoever brought that into the schools ought to be horse whipped, tar-and-feathered, and run out of town on a rail.

11. Doug Purdie says:

It’s a good problem for a business theory class. It would be a good problem for a math class if students were required to calculate the cost per ounce, or if they were required to identify the pattern of price increase as the volume increases, but it doesn’t, so it’s not.

12. linden says:

Why doesn’t Professor Jin-Yi Cai just get copies of his old math books and translate them into English?

Indeed, they presented this as a social justice issue. Just because kids failed to learn basic arithmetic, and computations with fractions, decimals and percents, was no reason to keep them from expressing and utilizing their higher order algebraic thinking skills. After all, this would just be the tyranny of computation being used to keep the disadvantaged from succeeding at all the gatekeeper courses, like algebra. If it weren’t for the minor matter of computation, a whole cohort of under represented math geniuses would be moving through the system and into mathematics based fields.

What was that about left-leaning bias in curricula that was a recent thread?

Ha-ha. It is to laugh.

14. boo says:

Studying physics long ago, I found the concept of math as a language to be apt, since I was using math to express ideas that could not be expressed in English. Of course, engineers don’t use math in the same way, so to them it’s a toolkit.

In either case, it is disturbing to see math books full of banal discussions of low-level business problems. Does anyone really think a discussion of soft-drink pricing is going to excite kids? More likely it’ll turn off the natural mathematicians and bore everyone else.

15. theAmericanist says:

FWIW, my objection is the lousy psychology involved. Somebody needs to take an acting class: “What’s my Motivation?” They take for granted that kids will want to learn math with a concept that is essentially abstract (what “should” it cost?), despite being cast as a ‘practical’ problem. They don’t even use the excellent motivational tool of “if you can’t do math, you will be ripped off”.

Consider a different motivational approach: If it costs more per ounce to buy smaller amounts of soda, and costs progressively less per ounce as you spend MORE, common sense would suggest that two people should split one soda — so long as they get another cup. If an empty cup costs a dime, how much does an 84 ounce soda have to cost to make it worthwhile for two people to buy one and an empty cup, to split it evenly?

Then add: What if the 84 ounce cup is wider as well as bigger than the cup that costs a dime? If the empty dime cup holds 64 ounces when it is full, what is the best way to know when you’re sharing the soda 50-50? (Extra credit: why do we say 50-50? It’s EIGHTY-FOUR ounces, right?)

LOL — all a matter of motivation.

I forget if it was JJ who posted an 19th century 8th grade final exam from Kansas, but look at the PRACTICAL character of the problems. This was the math section:

“1. Name and define the Fundamental Rules of Arithmetic.

2. A wagon box is 2 ft. deep, 10 feet long, and 3 ft. wide. How many bushels of wheat will it hold?

3. If a load of wheat weighs 3942 lbs., what is it worth at 50cts/bushel, deducting 1050 lbs. for tare?

4. District No. 33 has a valuation of
\$35,000. What is the necessary levy to carry on a school seven months at \$50
per month, and have \$104 for incidentals?

5. Find cost of 6720 lbs. coal at \$6.00 per
ton.

6. Find the interest of \$512.60 for 8 months and 18 days at 7 percent.

7. What is the cost of 40 boards 12 inches wide and 16 ft. long at \$20 per metre?

8. Find bank discount on \$300 for 90 days (no grace) at 10 percent.

9. What is the cost of a square farm at \$15 per acre, the distance of which is 640 rods.

16. JuliaK says:

“Mary Ramberg, the director of teaching and learning for the Madison Metropolitan School District,…”I think that it does indeed work to keep the kids really engaged in mathematics, but I don’t think that it’s at the expense of a certain group of kids,” she said. “I wish I would have had this curriculum. I did not like math. I learned the procedures without learning how I could apply them and why they worked.””

So, in other words, the math curriculum is at the mercy of people who _do_not_like_math_, who _expect_it_to_be_difficult_? This isn’t a trivial point. If you don’t like math, and see it as a necessary evil to be endured, in order that you can later do the family shopping, you will not choose a math program which a mathematician would choose.

17. Walter Wallis says:

Sodas are no longer allowed in California schools, so the question is moot.

18. JuliaK says:

The answer to the second question should be, “Don’t be silly, that’s an obscene amount of soda to sell in one cup. Don’t you know there’s a campaign against juvenile obesity?”

19. Erin says:

Right on, Americanist. The thing that struck me about that question was how astonishingly UN-realworldly it is, especially compared to the denigrated “traditional” question. As a soda-buying kid with limited cash, the most for the money might indeed be something I’m interested in. But why should I care what I *might* be charged for a hypothetical drink that’s not actually available?

Incidentally, the “connected math” suggestions are both impossible without some basic grasp of calculation fundamentals. What good is this unless you can first recognize that 20+64=84, calculate .80+1.25, divide all the costs by the volumes to find a cost per ounce… It’s almost a vindication of the idea that computation must be taught *before* these so-called higher-order thinking skills come into play.

20. Independent George says:

One of the things I love most about math is the fact that there are definite, absolute right & wrong answers. An infinite number of methods, but the answers are known with absolute certainty (or uncertainty, but I’m getting ahead of myself). If you get something wrong, you can examine your work, and identify the precise spot where you made the error, what you did wrong, and how to correct yourself. And if you get it right, you can still improve by finding more elegant ways of doing it. Good math is a thing of beauty.

21. Steve says:

I agree with JuliaK. Many of the problems start in grades K-6. These teachers do not have to know much of anything about math and many don’t like it. These are the teachers who are selecting the curricula. Perhaps they suffered through very poor math classes when they were growing up. Their problem may not have been a poor curriculum, but poor teachers. All it takes is one or two. The problem could also be that math requires precision and hard work. Some people have difficulty with that. You can’t simply redefine math to eliminate the two. Our state tries to do that. Here is what they think math is.

“For our purposes in this framework, readers should consider the definition of (math sic) to be the study of patterns and relations, with people interacting with each other and the physical world as they explore the process of thought, solve problems, make connections, reason, and communicate ideas.”

Chris said:
“I’m feel fairly confident that for students with decent math aptitudes and abilities Connected Math is a disservice. I don’t know about students with lesser abilities.”

Perhaps, but how do you know in grades 1-6. How do you tell the difference between those who have aptitude problems and those who can’t or don’t know how to buckle down and do the work. If someone doesn’t like math, does that mean that they don’t have an aptitude for it and they don’t have to meet better standards? Do only students that find a subject easy (have an aptitude?) get the rigorous material? Even the people who developed Everyday Math say that it is not for the “elite”. I wondered how they defined that term, what they think is missing or different, and how they know this difference between the kids in the early grades. Traditionally, it is in seventh or eighth grade where different tracks show up to accomodate the needs of different students. However, I find that many lower schools are making big assumptions and expecting less from the kids. This forces many parents to create their own early track and pull their kids out of public schools.

22. Margaret says:

Someone asked why Professor Cai didn’t just translate over his Chinese books. Actually, he doesn’t have to– Singapore Math is already available, and since it’s taught in English in the Singapore schools, there is no real translation necessary. The one thing my kids found funny about it (in a good way) was that the word problems involved rambutans and mangoes instead of apples and oranges. I think that’s been changed, though, in a new “American” edition which simply replaces some of the vocabulary and probably teaches English instead of metric measurement…

23. Steve LaBonne says:

‘ “I wish I would have had this curriculum. I did not like math.”‘
“So, in other words, the math curriculum is at the mercy of people who _do_not_like_math_, who _expect_it_to_be_difficult_?”
OK, I have to get this off my chest:
AAAAAAAARRRRRRRGGGGGGHHHHHH!!!!
Thanks, I feel better now.
No wonder _good_ (and cheap!) math texts like Singapore don’t get selected…

24. EcoDude says:

Nice to know that the professors in the mathematics departments are objecting to a mathematics curriculum developed in education departments.

25. The math from a 19th Century 8th Grade exam is something I’ve been hoping will return to schools some day. It would be nice if students had an idea of the costs associated with credit and interest some time before they get their third or fourth credit card.

26. Bob M says:

I am not arguing here for the particular programs that are being discussed. But I am arguing against a false dichotomy of learning exact reasoning with mathematics and the ways we figure out how and when to use math to reason about real situations.

In most of the projects I work on an applied statistician, the challenge is determining models, not doing calculations. Anybody with the formulas and a computer could do them, once the model is determined; and yes, there is one right answer under the model, and yes, it might be arrived at through different calculations. But doing calculations better isn’t what distinguishes better statisticians. Every real situation is more complicated than a model. In building a model, you need to figure out which patterns are the important ones to parameterize and which are better modeled as noise for the particular job you need to do—and the right model for one job can be the wrong model for another job. Reasoning within the mathematical model does need to precise, accurate, and correct. Determining, comparing, testing, and revising models has standards too—like efficiency, robustness, and fit—but often involves tradeoffs and choices. Isn’t this true in accounting, economics, carpentry, etc. as well? If both are necessary to using math in practice, shouldn’t students learn both?

27. Steve says:

Bob, this is not about a dichotomy. It is a matter of sequence. Do you believe that anyone can create mathematical models (economic, engineering, etc.) before or without a solid understanding and mastery of the basics of calculation? That is what these new new math programs try to do. I have seen cases where they expect kids to solve what are simple linear 2 equation and 2 unknown problems using trial and error. They don’t even want the students to write down explicit equations.

The lower grades should be all about mastery of the basics. This doesn’t mean there can be no interesting problem solving, but it does mean that schools have to first teach basic skills that can be then applied to increasingly difficult problems. Working top down from problem solving to the basics is a ticket to failure.

All you have to do is look at any of the new top down math programs and compare them with a math program like Singapore Math. It is not an either/or or dichotomy issue. It is a sequence issue and a low expectation versus high expectation issue.

28. Yes, Bob, but I think that learning how to do calculations should precede learning how to construct models. One can’t make workable “tradeoffs and choices” if one can’t manipulate the numbers involved. Introducing model construction at a point when most students may not be ready for it sounds like an invitation for disaster.

29. Wow, Steve, you made the same point simultaneously, but better.

Sometimes I wonder if advocates of top-down programs are mathematicians who have forgotten what it’s like not to have the skills at the bottom. If one takes one’s foundations for granted, top-down programs don’t seem so bad. The appeal of these programs is further enhanced by their paradoxical nature: they enable educators to claim that they have taught students higher-level theory without the sweat needed to master low-level basics.

30. Math requires discipline, and it builds upon itself. You cannot progress without knowing the previous steps.

31. Algebra is generalize arithmetic. If you can’t do numbers, how can you do the same things with symbols?

Our tenant’s daughter had done fairly well in her arithmetic classes, but was failing algebra, so I was asked to tutor her. I basicly just gave her the converse of the above statement, “Arithmetic is just algebra with numbers.” Things fell into place for her and she started getting A’s.

32. Even if they computed the per-ounce price for each instance, and saw how the per-ounce price changed with the number of ounces, you could make multiple conclusions as to what was a reasonable price for a MegaSwig. But this idea requires a lot of thinking. I think this idea is asking children to jump from the bottom to the top of Bloom’s taxonomy in one leap. Bad idea.

33. T. in FL says:

Margaret,
Yummy! Rambutans! They are one of the best fruits I have ever eaten. If your kids would like to try their juice you can order Leechee (not sure of the spelling) from an upscale grocery store. I just might have to switch my kids to Singapore Math for the memories. 🙂

34. Richard Brandshaft says:

Major teaching fads in math and reading hit at 20-odd year intervals (sometimes less) which means that anyone old enough to be in a decision-making position is old enough to remember the last fiasco. The cultural memory of educators is less than an adult lifetime.

35. Claire says:

“Anyone who cannot understand mathematics is not fully human. At best, he is a tolerable sub-human who has learned to bathe, wear shoes, and not make messes in the house.”

–‘Lazarus Long’ in Time Enough for Love by Robert Heinlein

Although the character’s sentiments are somewhat extreme (the character of Lazarus Long is a feisty and exasperating old curmudgeon), I would have to agree that mathematics is a uniquely human endeavor.

36. Tim from Texas says:

Well, here we go again, teachers et al are at it again-new math,fuzzy math vs traditional math and all the blame-storming. Blame-storming doesn’t accomplish anything other than,possibly, high blood pleasure.

Maybe a mixture of both would accomplish the goal of teaching math to the greater majority of our youth. After all, finding methods to reach a greater majority or all the youth must be the goal.

With regards to this goal, traditional math’s track record,at best, could be called an “also ran” and perhaps fuzzy math by itself would accomplish same.

Traditional math and its teachers have allowed too many youth to fall out and evolve into a state of disinterest. Let’s face it,traditional math,and moreover,traditional algebra and the way it’s been taught by most is stilted. Too many traditional math teachers are,in my opionion, “legends in their own minds”. I say this because they approach the students and their lessons with the following attitude. “I made it with lessons just like this,so dammit, so are you.”, hence the name, “traditional”.

Some traditions, I suppose, are just wonderful and should be left intact just as they are. Traditional math-left completely intact-I think has shown itself to be a non-winner. A winning math program is what we want isn’t it-or is it?

37. Steve says:

Tim – I remember a few years back when my son was in preschool, I began to think about what I wanted for my son’s education. I thought about many things I didn’t like about my traditional math schooling. Then I read a glowing article about TERC and was dumfounded.

The new new math assumption is that all of the problems of the past are due to the curriculum only. There are actually lots of reasons why many people have trouble with math – poor teachers, lack of student effort, no role models, peer pressure, curriculum and the fact that math is just plain difficult. Why pick on just the curriculum?

My opinion is that blaming the “traditional” curriculum is just an excuse to apply a modern progressive approach to teaching – real world, thematic, child centered, wide mixed ability group learning and lowered expectations. Rather than ensuring that lower school teachers know much more about math (not just pedagogy), these new math programs completely change the curriculum to a lower expectation, top down, spiraling approach to teaching. You may get more students up to a slightly higher mediocre level of math understanding, but these programs will also increase the number of students who will not be able to meet the demands of rigorous high school and college math courses. Anyone can get kids further along by lowering expectations. The trick is to get kids further along without lowering expectations.

“After all, finding methods to reach a greater majority or all the youth must be the goal.”

No. The real goal is to get more kids prepared for a rigorous course in algebra by eighth or ninth grade. I call this a “winning math program”. This is a specific, target goal and not some sort of vague idea of getting more kids to “like” math or do math a little bit better. The goal is not to offer a slightly better education for everyone. It is to provide the best, most rigorous education for each individual. I really detest the assumption that kids in grades 1 – 6 cannot handle an approach to math that emphasizes the fundamentals. Are public schools all about the best lowest common denominator or are they about providing the best opportunities for all? The best opportunities should not be limited to those affluent enough to go to private schools.

I also want to say that this argument is not about blaming teachers and schools. (This is a cop-out.) It is criticism about schools’ assumptions and curriculum. The criticisms are specific and include critical comments that come from practicing mathematicians, engineers and scientists. Just because some people like to lay vague blame on teachers does not mean that schools do not have to deal with specific criticisms. For example, our school did not restart a Citizen’s Curriculum Committee, as the superintendent promised, and went ahead and kept MathLand (long after even the publisher dropped it) and extended CMP back from 8th grade to 5th grade. Obviously, the schools felt that they owed no responsibility to answer specific criticisms. I am ready to give constructive and professional help to the school and they don’t want it. I don’t expect that the school will do exactly what I want, but it didn’t even get that far.

38. Cousin Dave says:

Tim: do you have some data to back that contention up? I’m willing to bet that Joanne or Kimberly Swygert can produce quite a bit of data that contradicts your assertion.

39. Steve LaBonne says:

The main reason why too middle school and high school kids can’t handle a real math curriculum is that they didn’t get the essential foundations in elementary school. And in turn, the reason for that is that few elementary teachers are competent to teach the subject (yet again I have to put in my obligatory plug for Liping Ma’s _Knowing and Teaching Elementary Mathematics_). I don’t see many people talking about making a serious attack on that problem, yet I don’t see how real progress can be made otherwise. (Yes, the fuzzy-math curricula compound the problem, but I think they are a symptom rather than the disease itself. They exist _because_ so many elementary teachers are so uncomfortable with real math.)

40. Steve LaBonne says:

Joanne must have read that last message. 😉 See her new post, “Content makes a comeback”.

41. JuliaK says:

Let’s posit that different children think differently, and approach learning with different capabilities. Math is an abstract topic with concrete applications. Many children may need the abstract concept of multiplication to be presented in a concrete context. While instruction may proceed at a slower pace, many children will learn more math than they would otherwise.

This does not mean that all children need every step illustrated with “real world” examples. Those children who grow up to be mathematicians or physicists, for example, have the potential to grasp the field’s language and grammar without intermediary steps. With all the talk of different learning styles, what of the instruction these children need? It seems to me that denigrating “traditional” math instruction willfully ignores the fact that a significant number of children had no problem at all with such instruction. If there is any truth to the theory of different learning styles, then it must also be true that such children may be ill-served by programs designed to reach those children who have problems with traditional math instruction.

Of course, if schools allowed parents the option of transferring their children into “traditional” math courses, they would be instituting de-facto tracking.

On a separate note, I am troubled by the fact that progams such as Connected Math render scoring of student work much more subjective. I was fortunate enough to have excellent math teachers, and they had no problem pinpointing the point at which a student’s reasoning went off the rails, without the student justifying their work with a paragraph of prose.

42. Richard Brandshaft says:

“Anyone who cannot understand mathematics is not fully human…” When I read the Heinlein book I thought that was funny. Then I started thinking about where you draw the line. Next to a world class physicist, I’m just a smart monkey.

43. Steve says:

The goal in math is to get kids from point A (counting numbers in kindergarten) to point B (algebra by eighth or ninth grade). In high school there are different tracks to meet the needs or abilities of each student. Up until then, schools should not make any assumptions about the abilities or aptitudes of the kids. If a school doesn’t get to point B, then many kids will be effectively locked out of a technical career. Schools should not make assumptions and close doors. This is about content and skills, not learning styles. If there was a program that taught students math using hand puppets, but got the students to point B, then I would be all for it.

As for learning styles, I assume that teachers learn all about this in Ed school. Back in my college math teaching days, it was always clear who was “getting it” or not by the blank stares and the poor or missing homework. I would try different approaches. This applies to any curriculum. (This reminds me of a satire I read about “nasal learners’ who demanded scratch-and-sniff books.)

Math programs like CMP are not strictly about appealing to different learning styles. They are about redefining math and lowering expectations. We have CMP at our middle school and some eighth graders’ parents state that they have to help their (good at math) kids (or get them tutors) to prepare them for college prep math courses in high school. There is a specific gap in the topics and skill level between eighth and ninth grades. Our school talked about this gap a few years ago, but the best they could do is a half year of pre-algebra in eighth grade for a select few students. (… who were select because they were getting help at home.)

Scoring in these math courses is more subjective because of their redefinition of math, where in real life there is usually no one exactly correct answer. That is why they go out of their way to find problems that don’t have a single answer, like the one in the article. Math without calculation.

However, this approach shows a complete lack of understanding of what goes on in the real world of engineering, math and science. There are a lot of unknowns and variables in real problems, but the process is neither vague or unexact. Just because engineers sometimes use large factors of safety doesn’t mean that the supporting calculations are unexact.

To approach the soda problem mathematically, one would plot the data points and fit (or regress) a curve through the points. Then one would extrapolate to find the expected answer. Regression analysis and extrapolation. These are specific skills or tools that students can use for many real life problems. Do you think that CMP talks about these things and discusses the problems and potential errors in their use? No. They expect kids, with few background skills, to come up with some kind of zen-like pure math solution. (The “Think Method”) Teaching them specific mathematical skills and tools is too much like rote learning and drill and kill.

I also want to quibble with Steve L. about his comment that the curriculum is a symptom and not the main cause. I wouldn’t downplay the curriculum too much. Well prepared teachers can make up for bad curriculum, but what if the curriculum demands that kids from a very wide range of abilities be taught in child centered, mixed ability discovery groups with the teacher as a facilitator. What if spiraling is mandated where the teacher cannot enforce mastery of basic skills when the material is taught. What if the curriculum states specifically that advanced students can only get enrichment and not acceleration of material? What if the curriculum states that kids do not have to know their adds and subtracts to 20 until the middle of third grade? What if it’s your child that doesn’t get the well-prepared teacher? We have to live in the real world and expect average teachers, but demand superior curricula. I don’t buy the argument that better prepared teachers will make all of the difference. I hear that too much lately from those who support these new curricula. However, this usually involves in-service training that is more about pedagogy than learning math. (How much math can one learn in a seminar or a week’s course?) For better prepared teachers, the work starts in college. We can and should expect that too.

44. Tim from Texas says:

If the traditional math curriculum had been perceived as doing as well as needed,I doubt the new math curricula would have evolved. The new approachs to teaching math used in the new math curricula were written by mathematicians, scientists, engineers, teachers and the like, and were backed by the NSF, which would make one assume that at least the methods therein were not conceptualized by blithering idiots.

As I wrote in my previous post, perhaps a blend of the traditional and new curricula could be the answer.

The math traditionalists and the mathematically correct proponents et al want to be the sole gate keepers as they have in the past. That is history now. It seems it would be prudent for them not to be so contentious.

What appears to be underlying all of the posted statements is an incorrect assumption that skills aquisition, problem-solving and conceptual understanding somehow exist in some well-defined hierarchy, when in fact the mastery of skills is not necessarily a prerequisite for conceptual understanding or problem solving (yes, some skills must be there), nor are problem solving and conceptual understanding somehow the secret to skills aquisition. They are complementary and must be part of a total mathematics education.

45. Steve says:

The new math curricula were written by “teachers and the like”, not by practicing engineers, mathematicians and scientists. “blithering idiots” – those are your words, not mine, and a poor rhetorical ploy. Besides, this curriculum change is all about ideology and pedagogy.

The new curricula do not just try to fix whatever problems existed before, they redefine what math is and how it’s taught. They also lower expectations. As I said before, the new curricula blame all of the problems on “traditional” curricula to justify a wholesale change in what is taught and how it’s taught.

“…perhaps a blend of the traditional and new curricula could be the answer.”

Balanced Math! Just like the balance between whole language and phonics (phonics as a last resort). Who do you think will define that balance? Who gets to define what is meant by “problem solving”? Does this mean problems like the fuzzy CMP “soda” problem? As for balance, I like the balance of skills and problem solving in Singapore Math.

“The math traditionalists and the mathematically correct proponents et al want to be the sole gate keepers as they have in the past.”

You’ve got this wrong. The K-12 teaching establishment (along with the NCTM) is and has always been the sole gatekeeper.

“…when in fact the mastery of skills is not necessarily a prerequisite for conceptual understanding or problem solving…”

… Gee, I always thought that it is better to solve problems using an existing base of skills and knowledge. It may not be “necessary”, but it surely is much more effective. Your approach is why CMP has it’s fuzzy “soda” problem. It is possible to teach badly and still have students learn something. Regression and extrapolation versus the “Think Method”.

46. Rita C. says:

Actually, in practice, I’ve always found that a blend of traditional and progressive works best. When I get the balance right (note that *I* determine the balance based on the children I am teaching), I can keep the bright kids challenged and the slacker kids making progress. It’s a tricky thing to pull off and certainly puts more responsiblity on me than if I were blindly following a set pedagogy. I’m lucky that I get to teach this way; it’s a little messier than many are comfortable with. But I get good results.