View Full Version : American math students ...

whiteflags

05-20-2008, 01:42 AM

It's now boiled down to "use a calculator or do it a lazy way."

I caught a news blurb on television and meant to share it with you all here. The way that mathematics is being taught in American elementary schools has drastically changed. We were all probably taught the foundations of arithmetic in similar ways, but I saw a method of solving a double-digit multiplication problem I didnt recognize, and wanted to know more. I found this youtube video (http://youtube.com/watch?v=Tr1qee-bTZI).

In it, you'll get to see the methods in the text books that school districts are using to teach basic arithmetic. The lady makes a pretty good argument why this is a bad idea.

But mostly, I'm stunned at what the textbooks lead teachers to spend class time on, and how at the end of the video, parents are encouraged to import textbooks used in another country and work through that if their child is struggling. Im concerned. There should be no need to outsource basic math education.

What do you think this all means? Assuaging my worries is also appreciated.

I tutor algebra I students at my high school four days a week and the way they are teaching students now is complete rubbish. A lot of important details are hidden through abstractions (they call them "methods") that don't really teach the students much of anything. The students end up having little to no understanding of how and why things works the way they do because they simply aren't being taught everything they should know.

Of course the new math programs are imposed by the school and the No Child Left Behind act, and all of the math teachers agree that it doesn't really help the students at all.

laserlight

05-20-2008, 02:52 AM

Yes, go buy Singapore textbooks! We learn sigma notation for summation, abachler didn't (http://cboard.cprogramming.com/showpost.php?p=751551&postcount=30)! :D

jEssYcAt

05-20-2008, 04:03 AM

This is (scary) news to me as the parent of a child who will be starting kindergarden this year with a little brother 4 years later. I've heard humor at the expense of "new math" but never understood what could be new about 2+2=4.

Apparently this isn't only in Washington State either. This place (http://wheresthemath.com/blog/) has links to sites for other states that are having issues like this.

Seeing the video that citizen linked frightened me actually. The examples she gives to show how multiplication and addition are taught seem bizarre. Putting emphasis on keeping track of the digit places isn't so bad when learning the basics, but I don't think problem clusters are a way to learn math. And I protest with all my being that a school, any school, should ever EVER promote using a calculator over learning the principles and concepts yourself. That is ludicrous! And one of the pictures in the math book showed a child confused about f(x) and h(n) (or something similar) floating by his head. GOOD GRIEF!!

I have a couple conspiracy theories about it all, but I won't air those just now. :devil:

I believe this is a natural consequence of not thinking (or not correctly anticipating) far enough ahead. It is a commendable goal, this "No child left behind". But rather than putting the burden on the students and their families to learn at least the minimum, the burden was placed on the school system, which reacted in the only way it could; make it easier to pass so children are more likely to pass. This had the effect of lowering the bar for everyone as this guy (http://youtube.com/watch?v=ymvSFunUjx0&feature=related) points out, and now we are starting to recognize the flaw in this situation.

I think in the grand scheme of things though, this will work itself out and we as a people won't be crippled by the less than stellar education some received. I hope so anyway, and I intend to make sure my sons learn properly. There will be a few victims, like the kids who were taught this way and never mastered the basics, but I have no idea how to fix that.

laserlight

05-20-2008, 04:45 AM

After watching the video again and thinking about my own Singapore brand of mathematics education, I posit that McDermott (McDermitt?) only has one main argument: the students in question lack practice because their syllabus is too concerned with appreciation of mathematics with respect to other fields.

To address her dismay that "most parents go: what's a cluster?", I would provide a counter example from Singaporean primary school mathematics: my uncle, an accountant, asked "what's a model?" when asked to help me. Guess what? The model method has been acclaimed as an excellent teaching method, after "Singapore Math received media attention and worldwide recognition in 1995 when Singaporean students were ranked first in an international study of mathematics and science education" (Teach Kids Math With Model Method (http://www.teach-kids-math-by-model-method.com)). Clearly, the fact that parents are not familiar with a method should have no bearing on the validity of its use in pedagogy.

She scorns the TERC method of problem clusters, but that method is taught in primary schools in Singapore as a mental shortcut for specific calculations. In fact, I would argue that it is a closer precursor to algebraic manipulation than the other methods.

I agree that stopping at the partial products method is a Bad Thing. However, it seems to me as a useful intermediate step between a theoretical exposition of place-values and practical calculation with long multiplication.

The lattice method is perfectly fine, except that it can be tedious to draw up the lattice. The algorithm is essentially the same as long multiplication. (As a side note, it would help the students survive as book keepers if they are transported back in time and have to use Napier's bones. Then again, if the time machine brings them to Asia, they would be better off learning to use an abacus instead.)

Long multiplication provides a convenient form to express the calculation, and in this it has value by making practice less tedious. With more practice, students become more familiar with (numeric) symbol manipulation, and symbol manipulation is the essence of algebra, which is necessary to understand higher mathematics.

Therefore, if McDermott really wants it her way, she should not just be advocating the purchase of Singapore textbooks. Rather, she should advocate the purchase of mountains of Singapore assignment books and Ten Year Series past exam papers (model answers provided), just like what the students in Singapore are force-fed everyday.

zacs7

05-20-2008, 05:13 AM

Long multiplication provides a convenient form to express the calculation, and in this it has value by making practice less tedious. With more practice, students become more familiar with (numeric) symbol manipulation, and symbol manipulation is the essence of algebra, which is necessary to understand higher mathematics.

Not to forget that it has direct applications to calculus... *shivers*

whiteflags

05-20-2008, 06:02 AM

Thanks for contributing so much Laserlight. I'm very interested in Sigapore's program now. :D

It's not that I'm afraid to revamp the mathematics curricula, or that I have anything against Singaporean textbooks. I don't want to pretend that you didn't get the argument she put forth either. There must be better ways of teaching math than rote memorization of multiplication tables and so forth.

The point that I got seems to be that under reform math:

- There is little structure and guidance in order to learn whole concepts or formulae, in the worst cases avoiding these all together.

- Conventional algorithms are written off by proponents of reform math and are consequently avoided.

- Some of the alternative algorithms fall back on multiplication or addition skills that I have little confidence are being developed because of the above, plus calculator dependency. To solve a reform math problem you have to breakdown until you have a long string of subproblems you can solve.

Basically it lives and dies on the powers of factorization, but who knows when they're made aware of this, almost entirely because of how the books are written. I get the distinct feeling that reform math is the same type of shortcut stuff your math teacher would give you in grade school as nuggets of wisdom. I really don't see a reason to model a whole curricula after this, though.

medievalelks

05-20-2008, 06:12 AM

It's even worse when it comes to English and spelling. If you're in the US, do all you can to send your kids to private schools.

medievalelks

05-20-2008, 06:26 AM

After watching the video again and thinking about my own Singapore brand of mathematics education, I posit that McDermott (McDermitt?) only has one main argument: the students in question lack practice because their syllabus is too concerned with appreciation of mathematics with respect to other fields.

I think her main argument is their argument that methods really aren't necessary since we have calculators. The anecdote of the college students not knowing how to multiply 4 and 6 is entirely believable. I can't tell you the number of times I stand in a grocery line facing a flummoxed young adult cashier who was asked for $20 cash back.

hk_mp5kpdw

05-20-2008, 06:31 AM

How about a lesson in Country Bumpkin Math (http://youtube.com/watch?v=z6TrGn5Xr3Y).

laserlight

05-20-2008, 06:38 AM

I think her main argument is their argument that methods really aren't necessary since we have calculators.

That's what she's pointing out by calling out the lack of practice. Since calculators are perceived as a substitute for the ability to perform numeric symbol manipulation, the syllabus stresses appreciation of mathematics instead.

indigo0086

05-20-2008, 06:58 AM

It also depends on the school. There are inner city and low funded schools who probably fill up math course time teaching for the reading portion of a standardized test.

Mario F.

05-20-2008, 08:29 AM

You mind if I play the devil's advocate and spur things a little? ;)

This will be somewhat long. Hope you enjoy reading it.

...

The problem these text books are trying to solve is one and one only; current math skills of the student population in generally is poor, comparatively to early years. And this has been happening long before these new (radical) methods were introduced.

One can argue standard algorithms aren't making a case of conveying basic math skills and promoting math as a "easy" discipline. I'm don't agree with this statement at all. But we'll get back to it shortly. First a critic of the proposed algorithms on both books...

On the books

Of the two, I think at least Everyday Math's is more coherent in the teaching method it proposes. The so called cluster problems are nothing more than putting on paper how we do mental math. Take a close look at their Partial Product algorithm for solving multiplication, or their version of the Long Division algorithm. Both algos use factorization, which is basically how we do it on our heads. Incidentally, the traditional and ancient Long Division method also uses mental math for the calculation of every digit. We can argue it's not a good algorithm, even though its the one I and my parents and their parents were taught.

The Terc Investigations based book is on the contrary completely laughable. There's no coherence in the proposed algorithms. Multiplication is taught by means of a cumbersome graphical Lattice which, while I was watching the video made me curious I must say as to what method they proposed for division. However disappointment came down like a rock when it was revealed to be nothing more than a skewed version of the cluster method proposed by Everyday Math's meshed with the Long Division graffiti. The Lattice Method for multiplication becomes a nightmare once you try operations like 353534*1256. Have fun drawing a 60 cell matrix and the diagonal lines to go along. Have fun adding all the diagonals. The division method, which I forget the name, is on the other hand dangerous. Very dangerous! Because it introduces the same graphical composition of the Long Division method, but then goes about solving the division in a completely different way (by means of a form of factorization).

Both books have another problem... space. Problem solving becomes curiously a large mantra of operations with the potential to fill in entire sheets.

On Calculators

Of the two books, if anything, I would at least want my daughter to be learning from Everyday Math's. At least the mental stimulation is there and it even seems to have the power to nurture her ability in mental maths, which will be what she will be doing most of her life if she later decides to not follow a scientific career. The only no-no I see is the fostering of calculator usage at such an early stage.

Calculators are no strangers to European schools. We have been using them since shortly after I finished my studies (late 80s). However the traditional practice is to only allow them past 7th grade, and past 5th grade only allow them during exams. Before that, students are taught basic math skills without the use of calculators. Schools are allowed some level of freedom, but the usual practice is this. Of course not all calculators are allowed, depending on the teaching phase. All in all, they are not that bad as long as they serve the purpose of helping in the exactness of the end result and not as a replacement for the teaching method; something that I think goes more or less in the line of what Everyday Math's was suggesting. It however fails by wanting to introduce them below the 7th grade.

On Math Teaching

But the fact is that even Everyday Math's is not a book I would like my daughter to learn from. I would be in the first line of angry parents if such occurred.

I said earlier, and immediately rebuked it, that traditional algorithms don't help math teaching in our schools. At least in Europe (speaking from what I know) there's a growing concern with math results for the past decades. In Portugal, for instance, we've had years in which the national average for mid-school was... get ready... 80 in a 1-200 scale! That's national average, ladies and gentleman. Shocking at it may be, it's no more than the fact that universities had years in which they had to accept students that flunked on their math admission exams!

The problem has been discussed all over Europe extensively for the past years. At least two things are today common knowledge:

Teaching Methods have to change to accommodate a more easily distracted student community and with a more active social life.

Their parents (the group I belong) were the first generation of poor math students and are today completely incapable of following and helping their offspring in their studies.

This is, agreed by many, the major problem with math, but also general student results. So, I almost feel as natural the appearance of new teaching methods like the ones presented in that YouTube video as new solutions that are being tried out. Of course, the guinea pigs are real life students and I have to wonder if this is not the time for the federal government to regain control of education. It is as such in most countries in Europe, but I have a feeling in US, each state owns its own education policies, which can have devastating results if these type of books are actually being used in some states.

Anyway politics aside, and because my aged laptop is starting to page this web control and writing is becoming slow, it is my feeling these books are not addressing the problem from the correct angle. It's not the algorithms that need to change. Any of the solutions proposed is ridiculous because it is nothing more than Yet Another Algorithm. The real problem with Math as I've perceived it during my days and as I seem to perceive in my daughter's progress (thankfully a terrific student. I'm 7 feet tall with pride) is Environment. More than ever before math needs to be taught in a context and in a less abstract manner. I was taught math outside the real world. Never, even once, I was given a real life example where I could apply the skills being taught. This is not a good method in these days of neon lights, flashy movies, cellphone messages and a more self-aware, and I risk shallow (proportionately), youth.

It is my belief this is the real problem with math. Not so much in early school where results have been historically consistent, but starting with middle-school.

Thantos

05-20-2008, 08:53 AM

I can't agree with all of her arguements.

The cluster method may not be effecient for pen and paper usage but it is great for mental solving. For the 26 x 31 problem I would definately break that up into something like ((25 *3)+3)*10 + 20 + 6 so that it would end up being:

25 * 3 = 75

75 + 3 = 78

78 * 10 = 780

780 + 20 = 800

800 + 6 = 806

While parents might not be familiar with "clusters" (I know I've never heard the term before) a quick look at the book (oh no we can't do that can we?) should make it apparent what is going on.

I do agree that the reliance on calculators is troubling. I remember I had a professor for a math class (might have been linear or DE can't remember) that gave a problem that our big bad TI-89s would choke on but if you knew how to do it would only take you a minute or so.

As to the homework question she pointed out about 36 / 6. Well the problem wasn't to find the solution but to describe how you could go about solving the problem. Making sure students can logically get to the solution is much more important then the solution itself.

If anything my experiences tutoring everything from pre-algebra to Intro DiffEq showed the opposite of her point about math language. Most of the students I helped could only understand the symbolic language and couldn't do a word problem if their life depended on it.

I would actually encourage teaching multiple different methods to solve the same problem. People think differently and what works for one person may not work for another. Additionally, the type of problem can make one method more favorable then another.

abachler

05-20-2008, 09:14 AM

Yes, go buy Singapore textbooks! We learn sigma notation for summation, abachler didn't (http://cboard.cprogramming.com/showpost.php?p=751551&postcount=30)! :D

Yes, this is the unfortunate truth, and I took great interst in math above and beyond what was taught in school. We simply don't have access to the resources. Part of the problem is the mistaken notion that all children are of equal capacity. While I don't think we shoudl abandon those that are not as academicalyl gifted, equally horrible is holding back those that are. We need a comprehensive multi-track educational system with a primary focus on core subjects like math science, art, and music with less focus on things like sports. The problem is that when you start talking about educational reform, people get emotional over it. Parents equate grades with success, not knowledge gained or skilsl acquired. So the kids don;t learn to think critically and grow up and fail to think critically about how ot improve the education of their kids.

medievalelks

05-20-2008, 09:19 AM

As to the homework question she pointed out about 36 / 6. Well the problem wasn't to find the solution but to describe how you could go about solving the problem. Making sure students can logically get to the solution is much more important then the solution itself.

Use a harder problem, then. If someone can't look at 36/6 or 20/4 and tell me the answer immediately is troubling. When I ask for change for a dollar, I don't want someone pulling out a pencil and paper and "clustering".

medievalelks

05-20-2008, 09:23 AM

Yes, this is the unfortunate truth, and I took great interst in math above and beyond what was taught in school. We simply don't have access to the resources. Part of the problem is the mistaken notion that all children are of equal capacity. While I don't think we shoudl abandon those that are not as academicalyl gifted, equally horrible is holding back those that are. We need a comprehensive multi-track educational system with a primary focus on core subjects like math science, art, and music with less focus on things like sports.

Teachers and schools don't have the bandwidth to teach everyone at their own pace. The simple fact is that some people are good in some subjects and not in others. I couldn't grok chemistry if my life depended on it, but I don't think it would fair for the teacher to let the better students suffer by changing teaching methods for me.

So I'll never be a pharmacist or chemical engineer, which is good for society. ;)

Thantos

05-20-2008, 09:24 AM

Remember who the target is. It isn't adults it is 4th and 5th graders.

abachler

05-20-2008, 09:50 AM

Teachers and schools don't have the bandwidth to teach everyone at their own pace. The simple fact is that some people are good in some subjects and not in others. I couldn't grok chemistry if my life depended on it, but I don't think it would fair for the teacher to let the better students suffer by changing teaching methods for me.

So I'll never be a pharmacist or chemical engineer, which is good for society. ;)

I agree that schools dont have the execute capacity to tailor a class to each student, I'm not talkign abotu individual tutoring, but smaller class sizes, 7-15 students, MAXIMUM. Combine this with a mtultitrack class structure (easy medium hard for lack of better terms) where kids that just want a general knowledge of a subject can take Easy math, and kids that excell at math can take hard math. This way the individual class doesnt have to tailor itself to the students, btu the individual students curriculum can eb tailored to meet theri needs.

brewbuck

05-20-2008, 09:56 AM

I can't agree with all of her arguements.

The cluster method may not be effecient for pen and paper usage but it is great for mental solving. For the 26 x 31 problem I would definately break that up into something like ((25 *3)+3)*10 + 20 + 6 so that it would end up being:

When I see that, my approach is... 31 is close to 30. Divide by ten to get 3. So what's 3 * 26? You can do that in your head: it's 78. Now multiply that ten back in to get 780. Now you're left with 1, so add another 26 to get 806.

Thantos

05-20-2008, 10:08 AM

Yeah that was how I started to it in my head also but when I got to 3 * 26 I solved it as (3 * 25) + 3. When it came to 780 + 26 I knew that 780 + 20 was 800 immediately so then I just had to add the 6. Luckily with mental math you can take a few more steps without it costing any more time.

laserlight

05-20-2008, 10:11 AM

When I see that, my approach is... 31 is close to 30. Divide by ten to get 3. So what's 3 * 26? You can do that in your head: it's 78. Now multiply that ten back in to get 780. Now you're left with 1, so add another 26 to get 806.

Yes, that is what I mean by "that method is taught in primary schools in Singapore as a mental shortcut for specific calculations". My memory is hazy now, but I think the students are around 8 or 9 years old (primary 2 or primary 3) when they are introduced to that, so it should be within the 4th and 5th grade thing. In other words, the very syllabus McDermott (McDermitt?) recommends teaches what she condemns.

indigo0086

05-20-2008, 10:15 AM

I suck at numbers, I have to see it visually in my mind, which is why I have a hard time when doing things like division of things that don't involve the "grouping" concept with regular integers. Like dividing the length of one side of a triangle to that of another or some such ratio based shenanigans.

Thantos

05-20-2008, 10:20 AM

I have a feeling that clustering would also help later with higher level algebra and beyond because there are times when stopping and thinking out the problem, recongizing the "clusters", grouping the clusters together, and then solving those, is a lot easier and less error prone then trying to tackle it all in one go or using some predefined algorithm. For example problems when you need to insert two terms that are equal to 0 (ex: x - x) but help you factor out the problem and reduce it down.

lindy

05-20-2008, 10:24 AM

The real problem with Math as I've perceived it during my days and as I seem to perceive in my daughter's progress (thankfully a terrific student. I'm 7 feet tall with pride) is Environment. More than ever before math needs to be taught in a context and in a less abstract manner.

I couldn't agree more. I was never one for math (I flunked H.S. algebra)... That is until I joined the Navy. It's amazing how fast you can learn the calculus of of a torpedo shot when the bad guys are hot on your, ah, backside. And yes I used a calculator. I'd have pulled off my boondockers and used my toes if would have gotten me a firing solution any faster.

Maybe if I'd been taught algebra in a less abstract manner I wouldn't have flunked it. Of course being dyslexic didn't help but that's a whole different can of worms. :p

medievalelks

05-21-2008, 05:14 PM

BTW, did anyone else catch the bit at the end about the "inability to work alone" of her colleagues in math classes when she went back to school in the late 90s? Anyone else kind of notice that here in the overwhelming majority of newbie questions that could be solved with a book and a little bit of self-study?

Mario F.

05-21-2008, 06:11 PM

I dismissed that argument since I'm not sure of what she meant. Certainly not basic math (the context was her university student colleagues), and since basic math was the issue at hand, I can't see how it fits in her reasoning.

You make a good point. But then, again, the problem is not the actual algorithms. If we followed that reasoning, the problem with her colleagues was instead laziness.

I generally agree with her arguments. I see nothing wrong with the traditional algorithms. They have been used for centuries(?) and they still produced good math students, scientists, geniuses, moderate students and bad students. And, speaking of laziness, they give nothing to it, whereas all the others are highly questionable.

On the other hand, the fact the proposed algorithms are easily recognized as a mirror of how we do mental math is yet another testimony of the traditional algorithms abilities. After all the latter is the ones we learned.

Simplifying math teaching to the level of those proposed algorithms is a mistake. The problem students have with math is not one of basics. No student has more trouble understanding more complicated issues later in college because they can't do a division. They can! Naturally there are exceptions, but that's what they are; exceptions. Instead students find certain concepts simply hard to understand because math is taught to them in a completely abstract manner and becomes essentially a memorization discipline. This is what kills it (and even more so in these rushing days).

medievalelks

05-22-2008, 06:28 AM

That argument was based on one of books stressing "group problem solving". I think a lot of the newbies that post here are too quick to throw up their hands after trivial effort (or none at all) and ask to be guided through the solution. Even after a nudge in the right direction, they stop after minimal gain and come back for another spoonful.

mike_g

05-22-2008, 07:13 AM

I think it depends the way in which the question is asked. When you're around someone that knows what they are doing then usually quickest way to get an answer is to ask. Often you can get the same answer in a couple of seconds compared to a couple of minutes (or more) if you had to look it up somewhere. On a forum its different tho as its usually quicker to look something up than wait for a reply, and, lots of trivial questions get spammy.

Mario F.

05-22-2008, 07:14 AM

Well, lets not forget those books are meant for children. Basic school, and even 5th - and 6th grade to an extent - are characterized by group activities. And it's an highly successful way of teaching children. Math is no stranger to that. Stressing group activities is the meat of basic school.

Despite generally agreeing with her, I still don't see the validity of that argument. Besides I highly doubt her university colleegues where taught from those books she so well criticizes. So, again, that argument makes little sense to me and could even turn against her. Apparently she also goofed when proposing those Singapore text books, from what I read on a post by laserlight. I think, she makes a good point when exposing the somewhat ridicule of the proposed algorithms, but lost it in the conclusions.

I can however generally agree on the dependency on others. Although I'm not sure how much of that is due to basic school teaching methods. I'd risk... very little.

Thantos

05-22-2008, 07:33 AM

I think the primary reason for group learning is because we often learn the material even better when we have to teach it to someone else.

I find it kinda humorous that the complaint is that they don't work well alone and before it used to be that they didn't work well in groups.

Perspective

05-22-2008, 08:35 AM

Her college peers had a dependence on calculators? Man, I wasn't allowed to use a calculator in *any* university level math class.

and teaching these "cluster" methods as short cuts is great, but they should definitely master the basics first.

twomers

05-22-2008, 10:10 AM

>> Man, I wasn't allowed to use a calculator in *any* university level math class.

That would be very awkward for me... I do elec eng. Even with log tables there wouldn't be enough time to do anything worthwhile. Sure you should learn mental arithmitic, but there comes a point where doing sums isn't important anymore. Most things, you find, are nonlinear, and with that comes computer territory. And a lot of things you can't do even with a calculator. Try solving "x + cos(x) = 0.53" by hand. You can't. You have to do it graphically or numerically.

laserlight

05-22-2008, 10:14 AM

I wasn't allowed to use a calculator in *any* university level math class.

Well, if you are supposed to prove a universal statement, a calculator would not help all that much anyway.

abachler

05-22-2008, 10:15 AM

I find it kinda humorous that the complaint is that they don't work well alone and before it used to be that they didn't work well in groups.

I don't. Both are important. As an engineer it is vital that I be able to communicate and work with others as part of a large project. It is also essential that I be able to work for extended periods by myself on my part of the project. It's not an either or issue.

Thantos

05-22-2008, 10:24 AM

The humor is more about the large swing in "what the problem is"

Sang-drax

05-26-2008, 12:47 PM

Those books seem alright. I think clustering is better than a standard algorithm that the student doesn't really understand. I think clustering helps when the math gets more advanced.

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