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?
Robert says:
That question in Exercise 13 is actually exactly what I mean by “critical thinking” in math — the student is doing more than just number- or symbol-crunching. (Although the crunching is important too.)
I happen to think that the problem about estimating sqrt(170) is pretty important, however. Having a sense of how big non-obvious numbers (like sqrt(170)) are is important in reality-checking answers to other problems. I would have phrased it differently though — something like “By hand, estimate sqrt(170) and explain you did it.” I don’t like math problems that have explanations and no results.
May 2, 2007, 2:30 pmCasting Out Nines / Looking back at an algebra text says:
[…] I love old books, especially textbooks and particularly math textbooks. RightWingProf’s post here explains why, as he walks us through a page out of his old high school algebra textbook. […]
May 2, 2007, 2:57 pmBogusia says:
I am so impressed with this blog post! I hate the math textbooks nowadays (I wrote about this in one of my recent posts also), but you hit the nail on the head! I teach in Canada, and the question you sited from your recent grade 8 textbook is exactly the types of questions I have to deal with when I teach my classes (I usually ignore these questions anyway, but sometimes I have to pretend I’m doing the new math - what a waste of time). Although, I know the reason behind the whole “explain this” and “estimate that”, I don’t necessarily agree with it, or believe this trend is correct. You gave the perfect example for this nagging annoyance of mine. Thanks!
May 2, 2007, 9:48 pmMyrtle says:
I’m still ignorant on how to do a trackback but I linked to you.
We are working out of this text right now with our son. The question he came across today that was the zinger was, “What is the contradiction of the contradiction of the contradiction of” x is a natural number.”
I am embarrassed to say that my kid nailed it, but I didn’t.
You should post this entry to KTM. I’ve been wanting to discusss Morris Klein’s screed against the SMSG.
May 2, 2007, 11:30 pmEllen K says:
I miss the old reading and phonics books as well. I recall fondly making my way up to the top reading group, having phonics tests and actually learning to enjoy reading and writing. Instead of helping kids to decipher language, we resort to all kinds of bells and buzzers which are often no better, and sometimes a great deal worse, than the conventional teaching of forty years ago. Furthermore, I think if we would concentrate on teaching kids to read in the first two years of school, almost to the exclusion of anything else, we would end up with kids that could understand much more of the world, because they would READ THE INSTRUCTIONS FIRST.
May 3, 2007, 12:04 amPeggy U says:
For anyone who has a child taking algebra, I have found Paul Foerster’s books to be really useful. They used to use his texts for Algebra II and PreCalc at our high school, but updated to books I don’t like as well. Part of the reason was that the Foerster books didn’t contain enough statistics content. However, I think they would have been better off to supplement the old texts, rather than abandon them entirely. I noticed that the teachers kept their copies!
Ellen: I miss SRA reading cards. Those were fun! Also, if anyone is interested, my son is using a really excellent grammar book, the BK English Language Handbook (Senn and Skinner). I highly recommend it, and I am going to find a copy for our family after we return this book to the school.
May 3, 2007, 2:43 amRight Wing Nation says:
[…] Looking through my high school algebra text is really quite fascinating. The last time I saw this book, I was a high school freshman, and I don’t recall having any reaction to the textbook, other than reading it and working through the problems. My perspective now is very different, after having taught for over two decades and reviewed God knows how many textbooks. […]
May 3, 2007, 7:48 amrightwingprof says:
I remember SRA! Something about color-coded cards, readings and pencils, but for the life of me, I don’t remember how the system worked.
May 3, 2007, 4:13 pmPeggy U says:
I remember SRA! Something about color-coded cards, readings and pencils, but for the life of me, I don’t remember how the system worked.
The SRA cards were color-coded, and I think the ones at the highest level were brown (the reading instruction equivalent of karate belts?). Each card would contain a short story, a nonfiction passage, or an excerpt from a decent piece of literature. At the bottom of the card would be reading comprehension and vocabulary questions. Our class got pretty competitive moving through the box of cards. But it wasn’t hard to do, as the reading material was interesting. I loved them! I looked at purchasing SRA cards for my kids when I decided to home school. However, they are quite expensive! I have read that SRA also had a system for teaching mathematics that mirrored the reading one. However, I am unfamiliar with it.
May 3, 2007, 11:35 pmWheels within Wheels » Blog Archive » I remember this sort of thing, too says:
[…] I don’t know if I used this algebra book when I was in high school, but it certainly looks familiar. Part 2 of the discussion of the book can be found here. […]
May 4, 2007, 9:11 amwheels says:
I remember the SRA cards from 2nd or 3rd grade. My biggest memory is that the teacher wouldn’t allow me to read to my level; I wasn’t allowed to read cards beyond the grade level for the class.
Just my luck that I was an avid reader from early on - I’d whip through the SRA cards I was permitted to read quickly enough that I’d be bored for the remaining “reading time.”
May 4, 2007, 9:16 amPeggy U says:
Wheels: What a rotten teacher! I mean that! Mrs. Schuler let us have at it, and we pretty much burned through the box. I would bet that by the end of the year nearly all of the students in our class were reading “ahead” of grade level. What on earth do you suppose teachers get out of holding kids back, especially on self-guided programs like SRA?
It is exactly the reason we are home schooling the youngest boy. He could read well before kindergarten (which is fairly common, I think), but was still required to sit and “learn” the alphabet with the rest of the class. We requested that he bring a book from home so that he could sit aside and read during this activity. However, we were told that they couldn’t accommodate him this way because it would show favoritism! We asked to place him in first grade (which he could have adapted to), but were again told “no”. It really is too bad, because the first grade teacher at the time was excellent, and I wish he could have gotten the opportunity to learn from her. She retired at the end of that year.
May 4, 2007, 2:36 pmwheels says:
Looking back from the perspective of 45 or so years later, she could have been afraid to let the class separate into groups by capability. She could have had lesson plans based on the class as a whole reading particular cards at the same time. I really don’t remember much from that long ago.
At least it didn’t hurt my feelings the way having a teacher with a pronounced regional accent read the words for the spelling bee did - I presumed it was a word that I hadn’t come across yet, so I spelled it the way she pronounced it. I can remember being upset about it … “Why did she say ’strinth’ if she wanted me to spell ’strength’?”
Funny what you remember from childhood.
May 4, 2007, 4:56 pmPeggy U says:
Wheels: Here’s something you might enjoy.
May 4, 2007, 10:57 pmwheels says:
Oh, that’s hilarious! Thank you.
May 5, 2007, 9:04 pmRight Wing Nation says:
[…] As you may or may not know, I ordered a copy of the book we used in high school Algebra I class many years ago (see here and here). Now indulge me, please. Imagine for a moment that you have been out of high school over three decades, that you have been teaching a substantial amount of time since, and you got a copy of the textbook you used in your favorite class. What would you do? What would you say? […]
May 22, 2007, 3:48 pm