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With respect to "patterns", someone got the idea in his/her head that algebra was about patterns. Maybe they said math was about patterns. It doesn't matter. In any case, it all reminds me of a twelve-year-old's version of Paradise. They seem to think that giving students exercises in which they are given a pattern like x=1, y=2; x=2,y=4; x=3, y=6. and asked to find the "next term" lays the groundwork for future understanding of functions. Well, it doesn't, really. It may lead students to believe that any pattern they can see must be it. So in the above pattern, the student comes up with "Oh, the rule is y = 2x, so the next term is 8." Well, what if I told you the next term is 14? It could be if y = 2x + (x-1)(x-2)(x-3). Actually, some people who advocate this approach have recognized this, and instruct teachers to watch for students who give "one" answer to the problem, illustrating "convergent thinking" and watch for students who give many answers to the problem, illustrating "divergent thinking". What this has to do with teaching about functions is another matter. But why teach about functions when life doesn't lend itself to functions, bringing us to our next topic: Data anlysis and probability.
Everyone knows that life is messy and that life's problems frequently don't lend themselves to "nice" solutions that are in algebra texts like Dolciani. Besides, students are bored learning the algebra that will be necessary to succeed in college science and engineering courses. So let's turn math into an empirical science. That way, expressing ideas in mathematically precise fashion need not ever be a concern.
The issue of why so much emphasis on data analysis and probability came up at the November meeting of the National Math Advisory Panel. Vern Williams questioned a member of the College Board why there was so much emphasis. Here is the exchange:
"WILLIAMS: Maybe one reason why students need more advanced courses to become successful in college is because so many things have been taken out of the basic courses because of the addition of topics like data analysis. I can't understand why data analysis would be a part of a geometry course. American students are extremely weak in geometry. In many cases, that is the only proof-based course, or at least it used to be a proof-based course, that students get. So, of all places, why would data analysis be included in geometry?
"DR. MANASTER: That's a very hard question to answer. We share your concern about mathematical reasoning and proofs being more apparent in geometry than in any other subject. This is a major concern of mine and has been for at least 10 years. I think that the only answer I can give here is that what we put into the geometry course was largely probability, for which there are geometric models. It still isn't traditional geometry by any means. But we tried to have a progression of treatments of data analysis throughout the curriculum."In other words, "we found a way to make it fit, and whenever we can make data analysis and probability fit into any aspect of math, we make sure that is done." Which doesn't answer Vern Williams initial question but these guys apparently get paid big bucks to not answer the public questions, just like our teachers in "constructivist math" classrooms (pardon the expression for those of you who insist on maintaining that constructivism doesn't exist) do not answer students' questions.
Any questions? If so, please direct them to Tyrrell Flawn, Executive Director National Math Advisory Panel: Tyrrell.Flawn@ed.gov
17 comments:
In the last two weeks my 3rd grader has rolled dice, drawn colored beads out of a bag, done worksheets on odds of selecting vowels from a bag of letters, etc...
I suppose they will be teaching him how to count cards and play Texas Holdem next year.
He also does a whole lot of bar graphs as well... for what purpose I don't know.
I think that pattern recognition is stressed because it is thought to involve higher-level thinking skills. Some people deprecate so-called lower-level skills, such as memorization and rote procedures, because they don't understand that they are the foundation for higher-level skills.
The reasons for emphasizing data analysis and probability should be clear. We are living in the computer age. Our society is collecting and analyzing ever-larger data sets, and drawing conclusions from these data. This means that data analysis and probability are more important than ever before.
However, this does not mean that data analysis and probability are important topics at the K-6 level. Probability and statistics are founded on algebra and calculus, which in turn are founded on basic arithmetic. It is a mistake to teach these advanced topics before the basics have been mastered.
This means that data analysis and probability are more important than ever before.
And it doesn't make sense to me that a kid who can't or doesn't get basic arithmetic is seen as some sort of hopeful candidate for success in this area. Johnnie can't do arithmetic but can do data analaysis well? Is that the idea? It's worth speculating that folks that design courses with everything but the kitchen sink thrown in perhaps don't believe that there is any sort of foundation in math...maybe they think math is just a hodgepodge of applications.
I think that pattern recognition is stressed because it is thought to involve higher-level thinking skills. Some people deprecate so-called lower-level skills, such as memorization and rote procedures, because they don't understand that they are the foundation for higher-level skills.
They also believe that the "find the rule" type problems ready students for algebra. I find that Singapore's bar modeling device is far more of a stalking-horse for algebra than the so-called "find the rule" problems.
"The reasons for emphasizing data analysis and probability should be clear. We are living in the computer age. Our society is collecting and analyzing ever-larger data sets, and drawing conclusions from these data. This means that data analysis and probability are more important than ever before."
Yet the quality of data analysis and statistical analysis across the spectrum has been sharply declining over the last two decades.
From what I've seen, they don't cover data analysis or stats or probability to any depth, so I fail to see the point of addressing them at all.
I forgot the other big thing in math instruction is estimating.
Today my 6th graders homeword was on estimating sums and differences of unlike fractions.
rightwingprof,
If I was king... I would add a statistics class to the HS curriculum.
"I forgot the other big thing in math instruction is estimating."
Estimating can be good practice since it develops number sense and reasoning.
Estimating tips is an example. Or take the square root of 10 and position it approximately on a number line. Some reasoning should tell you that it is close to 3 on the right side and not 4 or what have you without resorting to a calculator.
So let's turn math into an empirical science. That way, expressing ideas in mathematically precise fashion need not ever be a concern.
Um, but applied maths is just as rigorous as pure maths. The whole point of applying maths to empirical topics is to get some mathematical precision in there - or to know very exactly what you're being imprecise about.
From the MAA's PMET report that rightwingprof linked to recently,
Finally, we say a word about data analysis and probability. In keeping with our concern for in-depth learning, we would prefer to see a solid development of data analysis for, say, six weeks during one year in high school, rather than a two-week chapter of data analysis every year starting somewhere in the elementary grades. A senior-level AP statistics is, of course, an even better way to learn statistics well. In elementary grades, data analysis can be presented informally in the context of data collection for applied arithmetic problems. The same arguments apply to probability, which first arises informally in applications of fractions.
Score one point for the MAA.
"empirical science"
You should see some of the complicated empirical equations I use.
"The question has come up as to why many math texts, curricula , state standards etc emphasize patterns, probability and data analysis."
Patterns? Pretend math.
Probability? Trivial examples.
Data analysis? Data collection, mode, median, range. That's it.
Hard work is a filter and they don't like filters.
My son brought home some notes he took in EM math class today about critical thinking. He actually has written in his notes: "Think outside of the box."
I can just imagine a mathematician giving a detailed analysis and comparison of mathematical curricula to a school only to be told: "Well, that's nice, but how are our kids supposed to learn to think outside of the box?"
Everyone knows that life is messy and that life's problems frequently don't lend themselves to "nice" solutions that are in algebra texts like Dolciani. Besides, students are bored learning the algebra that will be necessary to succeed in college science and engineering courses. So let's turn math into an empirical science. That way, expressing ideas in mathematically precise fashion need not ever be a concern.
I'm going to remember this way of putting it.
Which doesn't answer Vern Williams initial question but these guys apparently get paid big bucks to not answer the public questions, just like our teachers in "constructivist math" classrooms (pardon the expression for those of you who insist on maintaining that constructivism doesn't exist) do not answer students' questions.
For some reason, this reminds me of a conversation I had years ago with a good friend who'd moved to Chicago and bought a beautiful townhouse.
They'd done a huge renovation job, and she and her husband were regaling us with tales of contracting woe.
She said that every time she brought up a piece of defective work to the contractor he'd respond by giving her a reason why the defect had come to be.
She'd point to some huge gouge in the wood flooring and he'd say, "The floor guy dropped his hammer there."
And that would be it.
He'd explained why there was a huge gouge in the floor.
So case closed.
Presumably if he hadn't been able to explain why there was a huge gouge in a brand-new as yet unpaid for wood floor, that would have been bad.
I suppose they will be teaching him how to count cards and play Texas Holdem next year.
He also does a whole lot of bar graphs as well... for what purpose I don't know.
These are the ONLY subjects being taught to mastery.
Seriously.
I find that Singapore's bar modeling device is far more of a stalking-horse for algebra than the so-called "find the rule" problems.
Absolutely.
My son brought home some notes he took in EM math class today about critical thinking. He actually has written in his notes: "Think outside of the box."
oh my gosh!!
I have a good friend whose son spent 4th grade being told to "think outside the box."
Seriously.
They would give the kids Math Olympiad problems, then tell them to think outside the box.
Um, but applied maths is just as rigorous as pure maths. The whole point of applying maths to empirical topics is to get some mathematical precision in there - or to know very exactly what you're being imprecise about.
Right. Which means mastering the basics, the stuff that lends itself to precise answers so that we know, as you say, what we're being imprecise about. Sorry I didn't say it properly.
Introducing "line of best fit" type problems in an algebra 1 class (as I have seen talked about) can take away from the material that needs mastery. For kids going through algebra, it turns it into a three ring circus of ideas, some of which are extraneous to mastering the basics. To a novice--particularly a student struggling with algebra-- it may not appear to be applying math to empirical topics but that math itself is empirical.
Speaking of "patterns"
Jacob's Algebra refers to the distributive property as a "pattern."
Quilt patterns, traffic patterns, patterns in math...all indistinguishible.
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