Wednesday With Words: Elementary Math

I have heard about the book Knowing and Teaching Elementary Mathematics by Liping Ma and recently I saw it Half Price Books so I couldn't resist.  Last week I spoke with a homeschooling mom who is a little bit further down the road and she really encouraged me to focus on my children's strengths and let them run as fast as they could in topics that interest them.  Well, my son loves math, so I figured I should try to figure out more about how to help him see the bigger connections in math. So, while I was at the Tae Kwon Do tourney this weekend I decided to give it a read. Just using the math curriculum we do (Miquon) has helped me better understand some concepts.  I was a plug and chug girl and did well enough - but I don't "get" math.  Honestly, my oldest probably does well because he had Montessori math as a 4 yo.  I need to use some of that with my current 5 yo.

Anyhow, back to the book.  The premise is that she posed basic math problems to American teachers and Chinese teachers and asked them how they would teach the concept.  IT IS FASCINATING! Honestly, it shows how little depth we have in our math understanding.  The few concepts she addressed have really helped me to better teach already.

Here are some key thoughts:

Here is a very interesting difference in understanding between the countries.  In the United States, problems like "5+7=12" and "12-7=5" are considered "basic arithmetic facts" for students simply to memorize.  In China, however, they are considered problems of "addition with composing and subtraction with decomposing within 20".  The learning of "addition with composing and subtraction with decomposing within 20" is the first occasion when students must draw on previous learning, in this case their skill of composing and decomposing a 10 is significantly embedded.

The question posed was about teaching borrowing/carrying, regrouping or as they say decomposing in subtraction.  When you teach the math facts up to 20 in this manner, students really GET what it means to "borrow" a ten to help out your units that are lacking.

Next they move on to multiplication and helping students to multiply multidigit numbers.  Here the Chinese teachers use the distributive law.  WHAT?  Yes, that's what you are doing because really in 436x812 you are multiplying each digit - 812 x (400 +30 +6).which become 812 x 400 + 812 x 30 + 812 x 6).  Why didn't I ever really get that?

They go further and say

Besides the distributive law (above) there is another argument that should be included in the explanation. That is the multiplication of a number by 10 or a power of 10.  

They go on to explain why in this special case we can just add 0 at the end of the number.  When done the traditional way, you may remember that you have columns and move over for each column.  This also speaks to learning place value as well.

Finally, fractions.  So, most of the American teachers didn't even get the fraction problem right, forget explaining it to a student.  I just remember "When dividing don't be shy, flip the second number and multiply".  I never really knew why - but the cheerleaders in middle school came in so we would remember to "flip" the second number.

Instead of "invert and multiply,"  most of the Chinese teachers used the phrase "dividing a number is equivalent to multiplying by its reciprocal".

So

Dividing by 5 is equivalent to multiplying by 1/5.  

The light bulb went on - of course it is. That makes it SO MUCH easier.  Of course we remember that we always multiply the top times the top and the bottom times the bottom when it comes to fractions.
So, it's a great read.  It helped me to better understand math and see how to teach it to my kids.  I don't know that you need to own it - but checking it out is probably worthwhile.

See what others are reading over at Ladydusk Wednesday with Words.