Traditional mathematical education starts with the abstract and moves towards the concrete.
"Here's the formula. Now let's use it."
More progressive education moves in the opposite direction.
"Here's a pattern or characteristic of physical objects. Let's figure out what's going on here."
The former of these approaches is often much easier on the teacher, but rarely is helpful for the student.
During one of the conversations last night, I realized that I needed to do more with the physical aspects of multiplication and division and less with the abstract (for now). This is doubly true if I want my students talking about these operations using a specific language of those physical objects.
@getting_sharper they didn't. Any suggestions on how I could frame those questions?— Justin (@JustinAion) October 5, 2016
@JustinAion in other words keep the whole the same so less attention is on finding "the answer" and more about communicating actions— Charlotte D Sharpe (@getting_sharper) October 5, 2016
@JustinAion like, "if we have 48 apples, how could they be arranged on 2 tables?"— Charlotte D Sharpe (@getting_sharper) October 5, 2016
@JustinAion revoicing student responses as addition (40 and 8 more) and multiplication (2 groups of 24) and division— Charlotte D Sharpe (@getting_sharper) October 5, 2016
It is by will alone I set my mind in motion...
Or, more accurately, it is by incredibly helpful suggestions from brilliant educators alone that I set my mind in motion.
Since I wanted them to work on these either individually or in pairs, I had a logistics problem. I don't have 48 apples for each of my students. I also decided against stopping to buy a giant bag of M&Ms on my way to work for obvious reasons.
Thankfully, I happen to have a giant tub of interlocking hexagonal building blocks and my OCD has been nudging at me to sort them anyway!
I spent the morning putting them into stacks of 10 and then groups of 50.
Each student got 50 hexagons and was told to put 2 aside so they had 48 (because there are tons of great factors for 48!) I emphasized that since we all had the same number of blocks, the answer was always going to be 48, but what I wanted to see was the representation.
"Show me 40 + 8."
"...Yes. I know. Nicely done. Now show me how you know that."
Second Student: "48!"
**slow blink** "Good. Show me."
Then we got the hang of it and they were really into it!
We moved from addition to subtraction to multiplication. Each time, I reminded them to talk about it in the language of groups.
"Show me 6*8."
"Tell us what you have there."
"We needed to make 6 groups of 8, so I made 6 stacks of 8 blocks."
"Nice. What about you?"
"I made 6 rows with 8 blocks in each row."
We got a slower start than I would have liked, so we didn't get to division today, but that will be tomorrow.
When my 5-minutes-left alarm went off, and they were cleaning up, I asked them what they thought of the activity. The majority expressed that they thought it really helped them to understand what was going on and, particularly, the language that we're using.
I have a student who is an English Language Learner and I watched her thrive more today than in any other activity we've done so far.
I am indebted to Charlotte for pushing my thinking in this direction!