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I bought a copy of the freshman algebra book we used in high school (Pearson and Allen, Modern Algebra: A Logical Approach, 1961). Here's a problem example dealing with fractions (and before you complain about the quality of the scan, you try scanning a book bound in 1961–they don't make 'em like that anymore):
and here are some of the problems we did:
Look at 13, which says:
Why is it that you cannot find the cost of one ball and one bat in this case although you could find the cost of one uniform and one hat in Exercise 12?
This is excellent. You're not asked to solve the problem here. Instead, you're asked to explain why you cannot solve this problem, but could solve a similar problem. In other words, you're being asked to think analytically about the problem–about the mathematics behind the problem. If "higher-level thinking" meant anything that had any pedagogical usefulness, this is what it would mean.
Also note that the problems are literate–that is, they're written in educated Standard English:
Were he to reverse the amounts, the yield would be only $330 per year.
That speaks less to the math and more to the place of literacy in education and the way educators treated students (then) seriously, rather than condescending to them with "hip" nonsense and pathetically trying to be "relevant."
Now, contrast with these current examples of 8th grade math:
In a couple of paragraphs, explain how you would estimate the square root of 170.
x2 + 2y = 10
What do you notice about the expression?
In the first, the student is asked to do no math. Instead, he's asked to talk about it. Note that he's not asked to think about it–just write a narrative about how he would approach the problem. Also note that the embedded problem he doesn't have to solve is to estimate, and not find, the square root of 170. The second problem isn't a problem, just as the first isn't a problem, and it's ambiguous. What is the answer supposed to be? That the expression contains an x and a y? That it's a quadratic equation?
And you wonder why we geezers can make change and these kids can't?
9 comments:
Wow... actual math!
Prof,
I've heard my husband rant about Morris Kline's attack of the SMSG. It's been said that the fifth chapter of Why Johnny Can't Add was really directed at Edwin Moise and it would seem obvious that the 4th chapter is against Allen.
I play the devil's advocate: One can't just poo poo a letter signed by 75 mathematicans against these methods of teaching (new) math:
http://michel.delord.free.fr/kline62.html
David Robert's deconstruction of this letter published in the AMS:
http://www.ams.org/notices/200409/comm-roberts.pdf
Kline's blistering attacks on rigor in math:
http://www.marco-learningsystems.com/pages/kline/johnny/johnny-chapt5-6.html
Thanks, Myrtle.
What I find amusing about those letters is that my colleagues look at the SMSG books (e.g. Modern Algebra and Trig by Dolciani et al) as the good old days!
Sadly, the weakest point today seems to be the linkage with language. Story problems, like the ones shown above, or in my book or in my dad's college algebra book, are a mystery to my physics students.
Ralph Raimi has written extensively on the history of the 60's New Math and his website has a section devoted to it, with his writings on various aspects of its history. I have talked at length with him about the 60's New Math in general and SMSG in particular. He has mentioned that Allen's algebra text was a "bust" because of its formality. Dolciani's text is also formal but not to that extent. In fact, such formality in algebra was introduced by the pioneer of the New Math before the term was even invented: Max Beberman. He sought to introduce the axioms and logic of mathematics into algebra because prior to that it was utterly missing. Algebra texts prior to Beberman were a hodge podge of terms, some used correctly, some not.
The algebra book I used (by Pingry, Henderson and Daikan) was written in 1957 but definitely was influenced by Beberman, but wasn't quite so formal as Allen's or Dolciani's.
The letter signed by the 75 mathematicians was brought about in part because of Morris Kline's attack. Ed Begle, the head of SMSG, viewed the New Math as a "work in progress" and knew of its shortcomings and sought to correct them.
Unfortunately, the letter sought to criticize the high school texts, but the mathematicians were also, and perhaps more, critical of the early grade texts, for which the formality of mathematics was not appropriate. A glance at those SMSG texts finds much belaborment of logical distinctions between "numeral" and "number" (5 is a "numeral" as is V and IIIII, that represents a number that has the name of "five"). It was thought that such distinctions would then cement in the fact that x, y and z were yet other names for "numbers" and so by the time the student got to algebra, x + 2 = 6 would be understood as "x+2" being a "numeral" as is the numeral "6", both representing the same number. A bit abstract for most kids; and in fact, Beberman introduced this concept in his algebra books to find that it got adopted into the lower grade texts.
Ironically, the early grade SMSG texts introduced alternative algorithms such as those used in EM and Investigations (like the partial quotients method for division, partial products of multiplication, etc) and relied to some extent on the "discovery" method, promoted by Bruner, though the form of discovery at that time was more of the "guided discovery" variety than the "throw the students in a pot with a problem and let them discover what they need to know to solve it" variety that we see today. Still, some of the early books required a bit of stretching and withholding of info.
Begle was the first to admit things needed to change and listened to the debate. This is not the case w/ NCTM now. Also, the math although perhaps too formal and the approach inappropriate for early grades was sequenced properly, was correct and logical and the writing was absolutely first rate.
There were problems with New Math. I plan to get there eventually. Right now, I'm having too much fun looking through this book.
I plan to get there eventually. Right now, I'm having too much fun looking through this book.
I know what you mean. For all their faults, the books that came out of that era had some great stuff in them. I liked the SMSG "Geometry" text which became the commercial text by Moise and Downs.
In a couple of paragraphs, explain how you would estimate the square root of 170
I'm struggling to figure out how to get a couple of paragraphs out of "It's a tiny bit more than 13."
--DaleA
rightwingprof says:
"Also note that the embedded problem he doesn't have to solve is to estimate, and not find, the square root of 170."
Can you tell me the exact value of the square root of 170? I don't want any of those fuzzy estimated numbers.
Left Wing (Math) Professor
Dear Ghu, I'd love to get hold of those SMSG books! (I got put in New Math, along with about a third of my fellow students, when I got to junior high. Looking back, it had some of the most useful math learning I've met, not all of it from the textbooks.) I keep hearing arguments about it being bad, but so many of the peopel I knew from those classes are in jobs requiring math that it doesn't seem really likely. (YMMV.)
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