This has really gotten me thinking lately. I gave my students the following picture:
Yesterday my students and I did a problem adding and subtracting a string of numbers. We looked at patterns that could be found in it, and even looked at why PEMDAS is misleading for order of operations. They were finding patterns, multiple strategies and stretching their thinking. I had hoped that this would translate to this problem- I was wrong.
#1 answer (98% of my students): I counted all of the squares.
I had tick marks, numbers, dots, colors, everything on their sheets showing how they made a 1-1 relationship with each square to find the total. I was very disheartened, but I didn’t show it. I asked if anyone had an alternate method, and only received a small handful (of which I will also share). So I asked those who counted all of the squares: “How many of you still find yourself using your fingers when computing?” All of them raised their hand. This is a strong indication of where my students are developmentally with math, and how I need to provide them with opportunities to explore and stretching their thoughts about problems.
So, I then reminded them about yesterday and asked what we looked at with that problem. They talked about the patterns and multiple representations- they remembered the whole segment! So I asked, “How can that apply here?”
4+3+4+3+4+3+4 or 4×4 + 3×3
1+3+5+7+5+3+1 or 2×1 + 2×3 + 2×5 +7
15+4+4+1+1 or 15+ 2×1 + 2×4
And then the bomb dropped
A student came up with this and blew the minds of every one of his peers. I let it sit there for 3 minutes before I even uttered a word. I could see each of them processing what just happened. Finally one student’s face lit up and he said “Oh, I totally get it!”
It took a while for them to realize that yesterday was not a “one and done” day for math, that I will expect them to do this every day they are with me. Tomorrow we will have another pattern and see how they do with that one.
Go out and drop the bomb on your class.