### Teaching elementary mathematics - don't skip stages

I received recently the book Arithmetic for Parents. A book for Grownups about Children's Mathematics by Ron Aharoni.

I think this book is exceptionally good, and very worthwhile to read if you're a teacher OR a parent.

Ron Aharoni was like many: he thought he could easily teach elementary school mathematics because he knew college level math and beyond (he was teaching math in a university).

But... he was in for a big surprise when he entered the fourth, fifth, and first grade classes in a backward town in northern Israel, in 2000.

One surprise he had was that...

• ...he did NOT know how to teach elementary mathematics, in spite of his knowledge of university level math.

For example, during his first lessons, he took children outside to measure shadows of trees and buildings, and then also to measure the children and their own shadows. The idea was to use the ratio of a child's shadow to his height in order to find the height of a tree or a building

He also took children outside to draw circles on the pavement, and to measure their diameters and compare them.

But as he admits in the introduction to his book, his first lessons were just a mess and confusion, with little meaningful teaching going on. He wanted to do "hands-on experimentations" with children, but he didn't know how to integrate the hands-on experiences into a SYSTEMATIC learning process.

He lacked the idea of building concepts one upon another, and the fact that you cannot skip any of these steps.

But... he found out all that, and wrote an excellent book about how to teach elementary math.

The other big surprise he had was that
• during his teaching career in the elementary school he actually learned a lot of mathematics... not new facts, but subtleties: how that concepts we adults think are easy, are actually built upon other simpler concepts and notions, and how children need to be explained ALL those little steps.

For example, Aharoni learned that he couldn't just present first graders a question such as
"Donna has 4 pencils more than Joseph. How many pencils does Donna have, if Joseph has 5 pencils?"

He had skipped stages. Children couldn't do it. They needed first to learn the concept of there being a certain number more than another number.

You can read in the book how he handled that lesson from that point on, and many other things.

The last part of the book goes through all the important concepts in elementary mathematics from grade 1 to grade 6, and explains these subtleties and little steps that you should teach.

I highly recommend this book for each and every teacher and parent who's teaching elementary math. It costs about \$20 new, you can find it at Amazon at this link. (Kindle version is also available.)

For more advanced mathematics, I highly recommend another book
from www.sumizdat.com : Kiselev's Planimetry. I can claim that no geometry textbook in history of Western civilisation was printed in more copies than Kiselev.
Anonymous said…
Many researchers have commented on how Japanese elementary mathematics curriculum is cohesive and coherent. You can now actually look at an elementary math textbook series in English. The books are available through Global Education Resources (http://www.globaledresources.com/).
Mama Squirrel said…
This sounds like a great book for our homeschool library! Thanks--I'm going to link to your post as well.
Anonymous said…
Good blog and great points. I also want to share Simple Kid's Math website with ya'll. It offers free math exercises. Sometimes, I just feel too bored to create math exercises by myself.
Mark said…
You make a great point about students needing to have all the steps explained in math--especially the lower grades. Often enough we forget as adults, we already have those skills!!

Cheers

Mark--Teacher Education Candidate at Wilfrid Laurier Univeristy
Anonymous said…
I teach third grade (public school) and use a curricular supplement called Picture-It-Math. It helps students 'see' and understand word problems as they are often overwhelmed by the 'words' and don't know where to begin. This program gives a structured step by step strategy to solving any and all word problems. When they don't 'skip steps' comprehension of how to solve the problem increases dramatically.
Anonymous said…
I agree. You can assume nothing with elementary math students! HAving the knowledge is vital, the art of presenting it is of equal importance.
Great article. I think I'll be reading this book.
Chloe Andrews said…
I think it is extremely important not to skip stages or steps in the child's learning. If they miss one detail or that one important fact, they will continually miss it as they grow up and continue in their education careers. So, in the long run, it is just hurting the child if a step is skipped over.