Disqus

Wednesday, February 24, 2010

Math Q&A in Times Article

For entertainment value read the Discovering Math Q&A in this article in the Seattle Times. The Discovering Math guy (1) doesn't always answer the question asked, (2) answers but doesn't address the topic properly - see the question on if Discovering Math is "mathematically unsound" and (3) sounds like he works for the district.

Here's one example:

The Discovering books have been criticized by parents, but they've been the top pick of a couple of districts in our area, including Seattle and Issaquah. Any thoughts on why the textbooks seem to be more popular with educators than with parents?

Ryan: I think because (parents) lack familiarity — this doesn't look like what I was taught. I don't know how you get students to a place where more is required of them by repeating things that have been done in the past. That's not how we move forward in life.

What?

I thought the Holt person was able to answer the questions in a more straight-forward manner. And when given the chance to pounce on the math lawsuit, she didn't.

In Seattle, the fight over textbooks ended up in court. Is that happening elsewhere in the country?

Blakely: I haven't seen any other lawsuits like this ... It's really not surprising to hear this happening at all. Everybody is really taking math, in particular, very seriously right now. There's always been a controversy about the best way to teach math.

14 comments:

dan dempsey said...

"In Seattle, the fight over textbooks ended up in court. Is that happening elsewhere in the country?"

Could it be that in the rest of the country districts do not attempt to adopt books that are rated mathematically unsound by the state board of education?

Or perhaps districts realize that after more that ten years of widening achievement gaps they should try something other than inquiry?

Or perhaps when an NSF project is producing horrible results with extensive NSF funding and UW coaching.... that the Superintendent does not attempt to get IMP adopted in 2008.

Or perahps after a failed attempt in 2008 ... they don't come back with the same failing pedigogical approach from the same publisher this time with more numbers but still lacking practice and enough examples.

Or is just that they do not have two wing-nuts like Dempsey and McLaren, who after the May 6, 2009 adoption decision gave a six minute tag team testimoney on May 20, 2009 essentially saying for years we have asked you to be rational decision makers using evidence and relevant data... well we give up .. we are going to take legal action.

dan dempsey said...

Or is it that the community will not fund well thought out and successful efforts? In Seattle we have raised close to $8,000 of a $13,000 legal bill...

Check it out HERE

and thanks to the 52 for $7,600 plus

HOORAY HOORAY and thanks again..

T-Shirts for sale at the next board meeting. Perhaps off the SPS grounds ....

dan dempsey said...

Perhaps elsewhere they might be looking for a record of positive academic performance unlike us.

Look here for results that SPS is apparently unable to understand.

KSG said...

Are there any good comparisons of the various math texts? The response by Ryan to the first question may have been odd, but unfortunately I do think that much of our math education is stuck in the 19th century. Teaching things like long-division is a joke in MHO. And I do think learning to use calculators and do rapid approximation/estimation are more important skills. I really do hope we're preparing students for the 21st century, and not what we've seen in ages past.

Dorothy Neville said...

"Teaching things like long-division is a joke in MHO. And I do think learning to use calculators and do rapid approximation/estimation are more important skills."

Sure, you can skip long division. But don't expect to really understand polynomials, rational expressions, integral calculus without a solid foundation that includes long division.

Rapid approximation skills are very important. I cannot understand how anyone could be successful at this skill without a solid number sense that includes mastery of addition and multiplication facts and factoring.

The Discovering fellow gives an inaccurate portrayal of "traditional" math and of his company's books. Good math teaching always includes an "aha" element, but this is much more effective and efficient in a guided fashion. Inquiry based lessons can be great. But the material I have seen in the Discovering book and what I have seen tutoring is that too much of what is called "inquiry based math" really leads to hit or miss random guessing.

There is a big difference between teaching approximation and estimation in a framework of understanding how math works as an integrated system and what I see in practice, which is teaching kids to guess with very little overall feel for mathematical reasoning.

hschinske said...

*scratch head* How do you teach rapid estimation skills, if you leave out all the parts of the curriculum (such as mental math and long division) that *require* good estimation?

Helen Schinske

KSG said...

@Dorothy, I'm unclear about the connection with the long division algorithm and the various math concepts you listed. I'm not doubting it exists, but what are they? I feel like I could teach all of those concepts, yet long division would never come up in any way.

@Helen, I'm not sure what you mean by mental math, but it sounds like fun. The long division algorithm on the other hand is not useful for rapid estimation, at least not for me, nor other people I have asked. We tend to use tricks like rounding numbers up and down to nearest tens/fives, where division/multiplication is much easier -- and various other tricks you can do to easily manipulate and maintain tracking of calculation. The actual long division algorithm isn't of much use.

Ask most students how long division works and you'll get blank stares. Give them a better feel of numbers with some basic number theory, like for the naturals (a*b) < (a+1)*(b+1), and you'll see that long division falls out for free from their understanding of the numbers. No more sitting through an algorithm that for many students just seems like magic.

One of my favorite courses, which you wouldn't teach to elementary school kids (obviously) or even most high school children is Street Fighting Math: http://ocw.mit.edu/OcwWeb/Mathematics/18-098January--IAP--2008/CourseHome/index.htm

But it shows you were you go with this line of reasoning, and how it can be fun, even at higher levels of mathematics.

Dorothy Neville said...

KSG, sure, you could probably teach yourself all the concepts without ever doing a long division problem. But would you have mastered them without also mastering the underlying concepts of division? If you've mastered division, would it be a horrible burden to practice the algorithm once in a while? Probably not. It's more a burden as you say to memorize the algorithm without a clue of what's going on.

You say that too much kids are taught long division as if it were magic. Well, good teaching of long division would not make it look like magic. Good teaching would incorporate it into understanding of number sense, factoring, multiples, remainders AND how to use arithmetic skills to estimate and avoid having to slog through the algorithm when appropriate to do so.

As for exactly where and how long division is used in those topics? Understanding the concept is more fundamental, but the actual algorithm pops up explicitly from time to time as well. My son is currently in UW Math 125, second quarter calculus and sure enough long division of polynomials is one of the things they expect students to know and be able to do (a tool for integrating rational expressions). Practically speaking, after finishing Math 125, students might never solve a real life integral using those techniques, but will use numerical analysis instead. But it still pays to have gone through the tools one solid time, to help build the conceptual understanding of integration and to know those tools exist and can be referenced when needed.

Couple years ago, my son and a friend were doing Integrated 2 homework at our house. They were simplifying rational expressions and disagreed about how to divide one rational expression by another. All I had to say was "Remember that division is simply the inverse operation of multiplication, so dividing is the same as multiplying by the reciprocal." Aha! his friend said. She'd never heard it explained like that, but my explanation made sense. And here she was, an extremely smart girl in advanced math class, with all those years in SPS of "inquiry" math and she had never really gotten the relationship between multiplication and division so succinctly. My son credits his mathematical success to my ongoing confidence that mathematics is a unified system that flows logically.

Inquiry lessons in math are WONDERFUL. That's not really the divide between traditional and reform. Conceptual understanding MUST develop along with mastery of the tools and algorithms. In either "traditional" or "reform" situations, the tools can seem like magic or they can seem like building blocks to a logical structure.

Dorothy Neville said...

My problem with what I have seen of TERC and CMP over the years is that there isn't a unified grand plan structure behind the way lessons are taught and skills are supposed to develop. The kids are not guided to find patterns in a unified way, draw conclusions and build on the previous concepts. No, way too often, the games and lessons that are supposed to build conceptual understanding in actuality lead to confusion. This is partly (imnsho) due to the texts and partly due to the teachers themselves not fully seeing math as unified and gloriously logical.

Your examples of estimating involve having strong mental arithmetic skills, of knowing how to multiply and divide multiples of five. That's exactly the sort of number sense needed and many parents are appalled that they don't see their kids learning. If the current curricula were teaching strong estimation skills, parents would be much happier. Instead, I see a LOT of guessing, random guessing. That comes from kids not seeing that the patterns they are (supposed to be) learning all really build on each other. That comes from books not using standard terminology. That comes from jumping from one topic to another in what seems haphazard.

So, back to the specific topic of this particular high school textbook series. What I have seen of Discovering Math (see for yourself in the pdf link on the Times article) is that it doesn't use standard terminology, it speaks in a condescending tone and reinforces the idea of guessing, without discriminating between random guessing and thoughtful reasoning. And that MIT on-line course you mentioned. Yes, those are exactly the same sorts of tools that can be taught to kids. The course uses calculus examples, but you don't have to. This is exactly the sort of mathematical discussions I have with my son and my tutoring kids. Do the units make sense? Did the answer give you the type of units you expect? Does your reasoning work in special cases, like 0 or 1 or what about a million? Exactly the sort of discussions around number sense that kids don't get enough of in the classroom.

There you go. "traditional" math without conceptual understanding makes math seem arbitrary. "reform math" without enough skill building to guide the conceptual learning makes math seem capricious.

KSG said...

Dorothy, I agree with your general sentiment, although I'd err more away from mechanics and algorithms (make kids program long division for arbitrary length numbers -- that will teach them more about it than manually doing the algorithm ever will), and toward concepts and examples.

And I do want to state that I have looked at the texts and they do seem bad. My concern is that if they were actually good texts, the response would be the same from many parents. Math should not be taught the way it was taught -- at least not how it was taught when I grew up.

Dorothy Neville said...

What would a good "inquiry" based textbook look like? I have some ideas, but am not sure others would agree. Fundamentally, the extreme proponents of inquiry based math do NOT want to expose the kids to standard algorithms or any formalism, thinking that it will taint them. So they go out of their way to hide the summary or objectives. I think that's something parents are reacting to. Especially after having to deal with their kids' confusion with the CMP booklets.

Dan shared a link to a powerpoint someone made comparing a lesson in Discovering and Holt.


http://www.mathunderground.net/PHvsDA/PHvsDA.html

Look particularly at page 12 of the powerpoint which has page 97 of the Discovering book. See how it treats guessing the solution to a proportion. Awful.

hschinske said...

@Helen, I'm not sure what you mean by mental math, but it sounds like fun. The long division algorithm on the other hand is not useful for rapid estimation, at least not for me, nor other people I have asked.

By mental math I just meant doing math in your head. And re long division I meant almost the opposite of what you said there: it's not that long division helps you estimate, it's that you hone your estimating skills by doing long division (which is basically a series of estimations).

As I recall, long division was the first arithmetical task I was ever taught where I had to work with numbers from left to right, rather than right to left, from most significant to least significant figure (skills that are obviously key to estimation and hence to doing math in your head). Once I could *really* do it (and it took a long time -- I was either badly taught or didn't pay proper attention, or both, the first time around, and had to re-learn it the next year), a whole lot of stuff made way more sense to me.

Helen Schinske

KSG said...

@Dorothy, I'm not sure I'm a proponent for inquiry textbooks, as I'm not 100% clear what that means, but I do know what kind of books I think are effective. I need to know how to do something, I need to know why. This bugged me when I took geometry as a child, and my teacher couldn't explain why 2pi*r was the formula we were using. My view is if I don't understand it, I'm just playing Mad Libs.

The other thing I want to know is why other techniques/ideas don't apply, common mistakes, and why people make those mistakes. Also I'm a bit of a math historian, and I loved the little sections on the history of famous mathematicians or the story of an equation.

One of my favorite texts is "Concrete Mathematics". From an Amazon review: "The smart-aleck marginal notes notwithstanding, this is a serious math book for those who are willing to dot every i and cross every t. Unlike most math texts (esp. graduate math texts), nothing is omitted along the way. Notation is explained (=very= important), common pitfalls are pointed out (as opposed to the usual way students come across them -- by getting back bleeding exams), and what is important and what is =not= as important are indicated."

Although, I should also state that I do think that different people have different learning styles. I think different people may prefer different types of texts. It would be nice to give students a choice.

For example, I was a big fan of Strang's "Linear Algebra and Applications" whereas my sister much preferred Sheldon's "Linear Algebra Done Right". A lot of this is more about the individual than any one text.

Dorothy Neville said...

"This bugged me when I took geometry as a child, and my teacher couldn't explain why 2pi*r was the formula we were using."

That's the problem right there. The teacher should have been able to find some resource to help you understand this. After all, people figured out pi a looooong time ago. While discussing that might have gone over the heads of other kids, probably bored some and made others anxious, the teacher should have been able to answer it for you. All too often math teachers -- especially elementary but some higher levels as well -- really don't understand the math well enough to think flexibly about it.

The reform math materials are supposed to be idiot proof, to be able to guide inquiry based lessons even if the teacher doesn't have a strong understanding. In reality, that just does not work.

All too often when I've tutored, the reason for the kid's confusion is that the teacher couldn't do exactly what you want. To take the child's student created algorithm and either approve of it or demonstrate the errors. So too often the teacher would just rigidly not accept any method of solving a problem but the one they taught. I hate that.

Often when I taught, a student would come up with their own method for solving something. Every time I felt it was my duty to really understand what they did, convince myself that it was equivalent to an algorithm I knew, or come up with an explanation of why it might work in a special case for some reason, but could lead to incorrect thinking so needed tweaking.

Math isn't dogma nor is it magic. Not only does the book need to address this, the teacher needs to understand it as well.