A friend asked for tips on getting started understanding some new domains in mathematics teaching the other day. An experienced high school teacher, he wants to know more about elementary and middle school topics, especially **fractions**, **place value** and **multiplication and division algorithms**.

For obvious reasons (mainly that I won’t shut up about these topics), I was on his short list to ask for recommendations.

It occurred to me that others might be interested in this particular brain dump. So here it is, lightly edited. Enjoy.

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**Fractions. **Entry level stuff on this is *Connected Mathematics.* In particular, *Bits and Pieces 1*, *Bits and Pieces 2*, and *Comparing and Scaling. *Any version of these units is fine. Work the problems from the student edition; have the teacher edition there for guidance.

I made major progress on understanding student thinking when I constrained myself to using *only ideas that must have come earlier (i.e. in elementary school) and to those that had been previously developed*. When I tried to appreciate the problems on their mathematical merit, or to build connections to my undergraduate mathematics knowledge, I didn’t make much progress that was useful to working with kids.

Then turn to *Extending Children’s Mathematics *(written by the Cognitively Guided Instruction team—CGI—and published by Heinemann). There is a lovely research perspective that should give you new ways to think about the CMP stuff.

More advanced perspectives are to be found in the work of the Rational Number Project (RNP), and there’s Susan Lamon’s book, *Teaching Fractions and Ratios for Understanding*. For contrast, read Hung Hsi Wu’s Math for Teachers curriculum. For extra credit, write a comparative analysis paper reconciling Wu’s work with CGI and with RNP; argue which has the greater influence on the Common Core fractions development.

Conspicuously absent from these recommendations is the “Essential Understandings” series from NCTM, published relatively recently. I find the writing style of these texts hard to process. Others may recommend them, and if so, perhaps you ought to take them more seriously than I have been able to.

**Place value. **There is an oldish JRME piece by Karen Fuson, the CGI folks and another research team about place value. It’s a seminal piece and totally worth your time. There is no one book I can recommend; my exploration of the conceptual landscape of place value has been idiosyncratic and informed more by small pieces of others’ research work combined with my own classroom experience and experiments. Most of that is documented on this blog.

The “Orpda” number system that Cady and Hopkins wrote about (and which I bastardized as “Ordpa”) was foundational to these explorations. Short, short article but the ideas opened a whole new space for me in thinking about what it means to learn place value.

The *Young Mathematicians at Work* book on number sense, addition and subtraction is pretty good. But those articles and the blog are better starting points.

**Multiplication and division algorithms. **I am trying to recall how I came to know the algorithms I know. I have to say that these steps I cannot really retrace. I am loathe to recommend digging through Everyday Math for them, because things are so diffuse; it’s hard to get the right book in your hand in that curriculum to learn any one particular thing.

The Kamii piece I recommended a while back is good. It was published in the 1998 NCTM Yearbook on algorithms. Sybilla Beckmann’s *Mathematics for Elementary Teachers* book is good, too.

But looking back at my *standard algorithm* diatribe last week and trying to think about what small set of resources would prep someone else to build a similar case (or to counter it), I am less clear than I am about fractions or place value. I do not know what this says about my knowledge, nor about the topic.

Particularly for a secondary teacher, I’d also recommend Pam Harris’s

Building Powerful Numeracy for Middle and High School Students.It looks at how building students’ flexibility with the basic operations relates to later mathematical learning. It’s particularly good for thinking about algorithms, as it’s focused on building students’ mental math skills and ability to use a variety of approaches beyond the formal “standard algorithm.” There’s also discussion about fractions (including fraction operations).(Full disclosure: I edited this book. But I can also honestly say that working on it changed the way I do mental math!)

I’m like what you said about understanding where the student has come from and attempting problems with only that knowledge – interesting!

Going back to Marilyn Burns’ original publications in this series is very helpful. http://store.mathsolutions.com/product-info.php?Teaching-Arithmetic-Series-pid290.html

Also, when in doubt, I always go back to Van de Walle when I don’t want to venture into the research handbooks. It’s a good source, with references to all kinds of articles at the end of each chapter. http://www.amazon.com/Elementary-Middle-School-Mathematics-Developmentally/dp/0133006468/ref=la_B001H6MFCC_1_3?ie=UTF8&qid=1369769738&sr=1-3

Many other, earlier, editions make this book accessible, cost-wise, but Drs. Karp and Bay-Williams are worthy educators to add their piece to the book.

I would be eternally grateful for a similar brain dump for high-school topics in Algebra 1 or Geometry. I know the Van Hiele levels but am curious about content like variables, the distributive property, equivalence, trigonometry, congruence and similarity, using formulas, etc.