I’m writing on Christmas Eve day, and I’m sure by the time you read this we are long into 2012 and possibly past it. But this moment is so important for math instruction in our 40+ states that have adopted Common Core standards; we are also on the eve of significant ramping up of Common Core implementation.

I’m looking forward to the Standards for Mathematical Practice. I have a lot to say about these eight mandates, which are repeated on each page of the Common Core content standards in each grade. They appear as a floating reminder that math instruction is not (only) about memorization and regurgitation, but about deep understanding, proof and argumentation, focused exploration and interpretation.

I’m convinced that the Standards for Mathematical Practice are doomed to fail in most schools.

Why? Because it seems that most teachers and principals don’t understand a simple fact: to teach elementary school math well, you have to know elementary school math really well. And most people (be they teachers, principals or otherwise) simply don’t understand much elementary school math.

I don’t know what teacher preparation programs are doing out there when it comes to math instruction, but from my experience in hiring teachers and my stint as an adjunct in one program, my guess is that if there is a math course in most of them it consists of something like, “Here’s the Harcourt Brace textbook. Here’s the Saxon textbook. Here’s the Scott Foresman textbook. Here are some tricks for teaching long division.”

One of the beautiful babies in the bathwater of teacher preparation is the program I went through at Bank Street College. At Bank Street, where theories of learning were developed by watching and learning from children rather than from following a bureaucrat’s mandate, my math mentor taught me that children need to struggle with mathematical concepts, and teachers need to guide them through that struggle with strategic questioning that builds understanding, always with the next math concept in mind. Children also should know why they are learning math concepts and facts, and have an authentic contextual basis for their study.

Linda Metnetsky said, “When the teacher gives the answer, all math thinking stops.”

And of course, the easiest thing to do is to give the answer, and demand that the kids memorize it. After all, that’s what Scott Foresman tells you to do. Teaching math progressively is far from the fluffy, no-facts, fuzzy math of popular culture. If done correctly, it’s a far more rigorous and intellectually demanding exercise than traditional math instruction on the part of the teacher.

Common Core. Lack of math knowledge among teachers. No concern in graduate programs for this problem. What are we to do?

As always, in times of crisis, I turn to books for advice. (Real books, written by authors, not textbooks written by committees, that is.) I’m not talking about how-to books, manuals of how to teach mathematics. I’ll take plenty of time to explore those in a future post, including books by Marilyn Burns and Cathy Fosnot among others. I’m talking about books that inspire or make clear the importance of loving and learning more about math.

Luckily there are a few friendly books out there that do a good job of either laying bare the crisis of math deficits or of explicating just why it’s so beneficial to understand math.

I’m going to describe a few of these books here, but I’d love to hear some reader comments recommending more. And please, don’t say, “The McGraw Hill series has some great looking times tables in it.”

** Innumeracy by John Allen Paulos**: Paulos wrote this tract around 25 years ago but its message is still relevant. While there is tremendous shame associated with illiteracy, society still finds it acceptable to be innumerate. And the consequences for that portion of our society that can’t read a stock table or tell an increasing rate of oil production in a foreign power from a drop in GDP from one quarter to the next extend far beyond the realm of whether 2 + 2 is always equal to 4.

** How Mathematics Happened: The First 50,000 Years by Peter Rudman**: Rudman is not quite a feminist, and you have to avert your eyes at some of the turns of phrase, but he brilliantly catalogues the timeline of the use of mathematical concepts beginning with our hunter gatherer days. Two powerful ideas I took away from this book are that (a) the development of mathematical knowledge in our concept mirrors the development of these concepts in individual children (that’s self-similar like a fractal, although he doesn’t use those words; you will if you love math as much as I do) and (b) there really is a reason why we should explore our base-10 system and other bases with children as we study math. I hadn’t understood it before, but after reading this book every time I look at a clock I think about it.

** Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter**: As I’m kind of slogging through it right now (because it’s dense, not because it isn’t interesting), this tome is not nearly as accessible a read as the two books above. It didn’t win the Pulitzer Prize for nothin’—the author calls it a “metaphorical fugue” inspired by Lewis Carroll, and that’s pretty much what it is, tracing the history of mathematical thinking about patterns and puzzles, their relation to paradoxes, music and computers. Imagine Willy Wonka wrote an autobiography but his obsession was puzzles, not chocolate.

What do you think about the math thinking behind math teaching? Any great books to recommend?

I just want your permission to share this quote widely. “Teaching math progressively is far from the fluffy, no-facts, fuzzy math of popular culture. If done correctly, it’s a far more rigorous and intellectually demanding exercise than traditional math instruction on the part of the teacher.” Ain’t that the truth!

Permission granted!

Found this post through the Twittersphere. Not a lot of folks out there working on questions of the relationship between Common Core and the mathematical preparation of elementary teachers (at least not that many doing so in a spirit of inquiry and hope and in a public way). As a result, you may appreciate some of my thoughts on related matters.

I appreciate what you’ve written here and look forward to reading more. I’ll take your recommendation on “How mathematics happened”. Especially the place value stuff-a topic that’s near and dear to my heart.

I’m with you that Paulos’s main thesis is strong. His examples get a bit esoteric, I think. (I’m supposed to be alarmed that the populace cannot calculate the rate of hair growth in miles per hour?) And GEB is great. Let us all know if you’re able to pull any pedagogical lessons from it.

If you’re able to make it through GEB, you may also appreciate David Foster Wallace’s adventures in mathematics, “Everything and More”. Also dense and it’s crazy to read the lit guy and have him be so, so smart on the math. Though it’s not really on topic w/r/t crisis in math learning or necessity of knowing math.

Thanks for commenting. I do appreciate your thoughts (and your “overthinking”) and I’m adding you to my blogroll. And thanks for the tip on Everything and More; he sounds like a polymath!

I can not agree more with your statement about needing to know elementary math to teach it. Elementary teachers need to know content well in all subjects they are responsible for teaching. I would add to your list of books, Knowing and Teaching Elementary Mathematics by Liping Ma. I also had the privilege to learn from Roger E. Howe from Yale University and he is very passionate about elementary math and instruction. You can find more about his work here http://teachers.yale.edu/pdfs/ocg/ocg14.pdf starting on page 15, he discusses the complexities of first grade mathematics. Great post!

Roger Howe’s review of Ma’s book is here.

Concern about the mathematical preparation of elementary teachers exists and is increasing, but perhaps not in venues familiar to all readers of this blog. See recommendations from 2001 from the Conference Board of the Mathematical Sciences here. These are in the process of getting updated for the current CCSS context.

Specific examples of the mathematical practices in the context of the CCSS are in the Progressions for the CCSS. You can access these and other CCSS-related things here. If you search on “MP” in a Progression, you can get to examples of the mathematical practices.

I haven’t read How Mathematics Happened, but I’m adding it to my list. I do have some suggestions, first, the alternates:

The Drunkard’s Walk by Leonard Mlodinow. It covers everything Innumeracy does and a great deal it doesn’t, including math/stats history, and common statistical misunderstandings. I think Paulos probably was forced to shorten his book to a point where it started doing damage because a long book about math might have seemed risky at the time, but Mlodinow has no such difficulty.

Number by Tobias Dantzig. An oldie but a goodie (Einstein actually blurbed it). Super interesting, covers all the basics, but in depth. Perfect for informing elementary math instruction. Sometimes we aren’t explicit enough about what we’re teaching. I agree that we need to know something about math history and also how kids learn math, and it’s definitely true that we see a recapitulation of historical progress in each individual as they learn these concepts for the first time. I can’t directly compare them, but it might be a good alternative or a companion to How Mathematics Happened.

Other suggestions:

How to Lie with Statistics by Darrell Huff. It’s short but I feel it’s stood the test of time better than Innumeracy. So for quick and easy summer book lists for faculty, it’s a good choice.

Where Mathematics Comes From by Lakoff and Nunez. I see this as the other side to a math history book like Number or How Mathematics Happened. It covers the same basic math in the same order — counting, arithmetic, geometry, algebra — by explicitly looking at the cognitive processes involved. It can be a difficult read at times (for non-psychologists) but also includes some fascinating insights for math teachers. It lays out the metaphors we all use to understand math, including combined metaphors like the number line, which are sometimes thrown at kids without any real explanation. The difference between ordinal and cardinal numbers and when we use one way of thinking versus another. Inborn abilities we all have, like subitizing with small numbers.

A bonus: How about this short essay on big numbers? A great read, and it’s cheaper and faster than a book. I had it posted up in my classroom on 23 printed pages for awhile, and once in awhile a kid would stand there and read all the way through it: http://www.scottaaronson.com/writings/bignumbers.html.

I am recently retired after teaching high school math and physics (35 years, whew!). I always left books by Martin Gardner around the class, and kids would always start checking them out. One book that Gardner recommends is The Number Devil. It is a humorous and understandable set of tales.

I liked GEB so much I took it on my honeymoon in 1980. After rereading it a million times I totally destroyed the binding and had to get a newer edition. I never let that book leave the room.

Several of my better students became elementary teachers, and I made sure they knew I expected that they would be as accomplished in math in college and as a teacher as they were as my students. We discuss pedagogy at football games in the “old fart” section of the stadium, where they know I can be found.

Most years I would have a section of the weakest science freshmen as well as the smarty-pants seniors. When we studied motion with velocity/time graphs I would give both groups the same classwork and homework. It made the freshmen feel important. The seniors didn’t say too much.

Students can tell right away when you don’t have a good grasp of a concept. You have to let them know that you are learning as you go along. Some of the “innovative” things I did were just ways I had learned “the hard way”.

Thanks for the post. I was directed here by another blog. I’ll check in. I like your core values. My class had “Work hard, Have fun, Grow up.”

I like

Gödel, Escher, Bach, but I wouldn’t say it has much to offer as far as understanding math.See my next post for my response!

I recommend the book

The Myth of Abilityby John Mighton. If you read it, I’d be interested in knowing what you think.I saw John Mighton speak about his book once, and was sufficiently amazed by his enthusiasm and conviction that I bought it. After reading a few chapters the gloss wore off a bit. That was about seven years ago and I haven’t revisited it. So I can’t give a real opinion of the book.

But having taught high school maths for six years, I can say I don’t believe ability is a myth.