Bruce finished off his major chapter on subtraction last month, and he wasn’t too keen on it at first. I teach math from a Constructivist perspective, which means enabling children to develop their own meaningful strategies for solving problems, instead of just blindly teaching traditional algorithms. Here’s a nice website that explains more about the Constructivist philosophy: http://mathforum.org/mathed/constructivism.html
Bruce, myself, his classroom teacher, and my husband were all in the trenches there for a few weeks as Bruce explored different strategies for solving triple digit, minus double digit problems with and without regrouping. In normal classroom situations, Bruce would have been exposed to his classmate’s strategies and thinking, which would spur his own ideas, creativity (and showmanship). But since he is working independently through the second grade book, I had to introduce some ideas to get him started.
The first strategy I introduced was solving equations on two abacuses, flipped over on side A, so that they represented 200. (We were starting with problems no greater than 200.) This went on for about a week, and Bruce was not impressed. He does not like using an abacus at all, and so I never got to teach him the side B methods. I was really bummed about this, because my husband worked with an engineer from China, who was just brilliant at math, and Chang said that when he solved problems in his head he could picture the little beads on the abacus sliding. I wanted this for Bruce, but maybe I didn’t introduce the abacus early enough. I’ll know better next time for Jenna.
The second strategy I introduced to Bruce was thinking about the problems in terms of money, and then counting out the equations in pretend change. 236-89 would become $2.36- .89. Bruce was not too thrilled with this strategy either. He could easily solve problems this way, but under protest.
Finally I broke down and taught him one of the methods the book demonstrated, ungrouping, which uses pictorial representations of hundreds tens and ones. I had resisted showing him this method, because Houghton Mifflin uses it as a precursor to borrowing, which I do not want Bruce to learn until later. But I’m eating a big slice of humble pie right now because this is the method Bruce absolutely loved.
All of a sudden Bruce started tearing through his subtraction work at top speed. He called this strategy “Busting open the hundreds”. I’d usually draw out the squares, lines and dots for him on the white board, and he would erase them and then redraw things as he started working. Pretty soon, he was solving problems in his head, by just looking a the picture and not even erasing anything. Then he didn’t even need the pictures at all.
The way Bruce would solve the above problem is that he would think: “100-89 = 11. 11+36=47 100+47=147” He doesn’t need to do any carrying or traditional algorithms at all, which is a testament to the Constructivist method. Constructivism encourages children to think and understand what numbers really mean, instead of just blinding computing an algorithm, which can stunt their development of number sense. Bruce’s accuracy isn’t 100% yet, but it is on par with your typical second grader. I’ll teach him the traditional methods of borrowing and carrying someday, but not until Bruce can do even harder problems intuitively, in his head.