### Changing my Mind

As I'm reading chapters and articles about math education, I keep changing my mind. This isn't helping my course planning for the fall — I want to do a good job with my teaching, and I want to learn from the successes of others. I keep mulling over questions about what defines good teaching or a successful outcome.

Even looking back over my own elementary education I keep changing my mind. At first I thought that my teachers used "new math" types of techniques. I have few, if any, memories of ever being taught any specific skill in an elementary school math class. Mostly I remember using base 10 manipulatives and playing chess. Aside from that I recall one fifth grade word problem which had me stumped. (Were the floor tiles in the problem squares 12 inches on a side? Or were they squares with a total area of 12 square inches? I couldn't tell.) I seemed to already know how to do base

But now I'm looking critically at some of my own mathematical weaknesses. Am I as familiar with the base 10 system and its advantages for mental arithmetic as I should be? When I'm doing examples of estimation problems using numbers provided by my students, I'm very slow to get the order of magnitude correct. I can't multiply or divide decimals in my head. I always get the decimal point in the wrong place — even when dividing by 0.001 or a similarly "easy" number. I usually write it down and count the decimal places. Only this summer did I realize that I could think of it as dividing by 1/1000 which is multiplying by 1000! I'm finding other areas where my understanding of elementary mathematics is more superficial than I would have expected it to be.

How important are these perceived weaknesses? You can argue that since I've gone on to get a Ph.D. in math, avoided being ripped off in my financial dealings, and can (usually) buy the correct amount of supplies for my home improvement projects that it probably doesn't matter. Or you can argue that my innate draw to mathematics plus my extensive training have allowed me to compensate. What should be our goals for the mathematical education of elementary and middle school students? I don't know. However, I think that it is important for elementary and middle school

As I've been crafting my syllabi for the fall I've been struggling with this idea of what students should learn and how they should be taught. I've torn down most of my assumptions about teaching and now I only have one principle which remains: Mathematics should make sense.

Even looking back over my own elementary education I keep changing my mind. At first I thought that my teachers used "new math" types of techniques. I have few, if any, memories of ever being taught any specific skill in an elementary school math class. Mostly I remember using base 10 manipulatives and playing chess. Aside from that I recall one fifth grade word problem which had me stumped. (Were the floor tiles in the problem squares 12 inches on a side? Or were they squares with a total area of 12 square inches? I couldn't tell.) I seemed to already know how to do base

*n*arithmetic when it came up in a college course. When I do the standard algorithm for multiplication of multi-digit numbers I put down the zeros instead of indenting from the right. I do arithmetic from left-to-right. These small clues were all I had to go on, as I'm not going to call up my teachers 20+ years later and start interrogating them about how they taught.But now I'm looking critically at some of my own mathematical weaknesses. Am I as familiar with the base 10 system and its advantages for mental arithmetic as I should be? When I'm doing examples of estimation problems using numbers provided by my students, I'm very slow to get the order of magnitude correct. I can't multiply or divide decimals in my head. I always get the decimal point in the wrong place — even when dividing by 0.001 or a similarly "easy" number. I usually write it down and count the decimal places. Only this summer did I realize that I could think of it as dividing by 1/1000 which is multiplying by 1000! I'm finding other areas where my understanding of elementary mathematics is more superficial than I would have expected it to be.

How important are these perceived weaknesses? You can argue that since I've gone on to get a Ph.D. in math, avoided being ripped off in my financial dealings, and can (usually) buy the correct amount of supplies for my home improvement projects that it probably doesn't matter. Or you can argue that my innate draw to mathematics plus my extensive training have allowed me to compensate. What should be our goals for the mathematical education of elementary and middle school students? I don't know. However, I think that it is important for elementary and middle school

*teachers*to have what Liping Ma calls a profound understanding of fundamental mathematics.As I've been crafting my syllabi for the fall I've been struggling with this idea of what students should learn and how they should be taught. I've torn down most of my assumptions about teaching and now I only have one principle which remains: Mathematics should make sense.