One of the best things about teaching math from a Constructivist perspective is the emphasis on metacognition, or “thinking about thinking”.
In a good Constructivist classroom the teacher should know, and be constantly assessing, the thinking strategies her students are using to solve problems. One of the ways I accomplished this when I was a teacher was through small group white-board activities. I would group three to five students according to ability level, and let them create and solve word problems on the white-board for me, while I reordered their thoughts and strategies.
From a Constructivist perspective, listening to how children solve problems is where the true art in teaching comes.
You need to know when to ask probing questions, when to lead them on to more efficient ideas, and when to pick apart what they are saying. The idea behind Constructivism is not to let kids wildly flounder, but to gently encourage them towards efficiency and accuracy.
My son Bruce(6.5) attends a school where the district uses Houghton Mifflin Math Expressions, which I like, but would not describe as a Constructivist program. Although there are occasionally times built into the curriculum where Bruce is asked to write out his own math problem as part of his homework, or explain his thinking in words and pictures, there doesn’t seem to be the same heavy focus on metacognition that there was in my own classroom.
Last night after dinner, I decided to do an experiment.
How would Bruce handle a third grade Constructivist classroom? How would he do if he was in one of my white-board groups, from long ago? It turns out, not so well, at least not without more practice. And let me tell you, after realizing this type of activity was a struggle for him, we will be working on it at least once a week from now on!
Here is our first example. I asked Bruce to tell me a subtraction story problem about the number 263. That was the first bit of difficulty as you can see! Then I asked him to solve the problem with numbers on the left side of the paper, and to keep his work neat, and organized within the box. Second bit of difficulty! I also wanted him to write down his final answer and circle it, but I forgot to articulate this very well, so that was “my bad”. Lastly, I asked Bruce to explain his thinking to me while I reordered his strategies. This is when he really started to flounder.
Okay, let’s try again! If you look at this next example, you can see a slight improvement. The story follows the word problem, his number work stays to the left, and he circled his final answer…but his explanation? On paper it looks okay, but I really had to tease this out of him. It did not come clearly or easily even with me doing the writing for him!
In a Constructivist classroom, by the end of fourth grade I would expect a learner to be able to see a number problem, write a story to go with it, and then write out their proof of how to solve it both in numbers and words. I would expect to see clear thinking, organized work, and neatness. None of that, is as easy as it might sound.
Sometimes, these types of Constructivist activities make certain parents turn vinaceous with annoyance and anger that comes from misunderstanding. I would never expect a child to have to write about every problem they solved. Nor do I think it is fair to hold children back mathematically because they might struggle with reading, writing or language. But doing proof work like this once a week in a small group setting is really helpful for everyone. Teachers get insight into how children are thinking, kids learn from the strategies of each other, and mathematical communication skills get practiced.
As an at-home activity, this has the added bonus of being completely free. I am going to continue working on this type of proof work with Bruce in an Afterschooling setting. I like Houghton Mifflin and I am extremely happy with Bruce’s teachers, but I want him to be successful at work like this too.
My six and a half year old son Bruce has been doing Math Without Worksheets to earn screen-time recently. I write down a number problem, and he writes down a word problem to match. Then he solves the problems numerically on the left-hand side of the page, and in words on the right. This type of Constuctivist activity is the nidus of someday being able to write out proofs in higher level math.
Bruce has been solving problems this way once a week for three weeks in a row now, and you can already see a really big difference in his work. For one thing, he’s writing down his thinking himself now, whereas before I had to write down his answers for him. But there is still a lot of room for improvement. As a former teacher, these are the things I am looking for (rubric style):
- The word problem should match the number problem. (+3)
- The word problem should be neatly written and in its own space. (+3)
- Numerical work should be neat, tidy, and in its own space. (+3)
- The numerical work should clearly show the strategy being used. (+3)
- The answer should be circled. (+1)
- The written explanation should be neat, tidy, and in its own space. (+3)
- The written explanation should clearly show the strategy being used. (+3)
If I was a third grade teacher scoring Bruce’s work in the above example, I would give it 12 out of a possible 19 points. The word problem, as well as the written explanation, both need a lot of work. Numerically, Bruce is showing that he knows that since 20 divided by 5 = 4, then 20 divided by 4 is going to equal 5. That’s a solid strategy for solving the problem.
Here’s another example from the same day that shows a lot of improvement. In this next problem, Bruce’s numerical explanation is even stronger, but the written parts are still a bit confusing. I’d give this example 15 out of a possible 19 points.
For the twenty minutes Bruce spent working on these two pages I let him earn half an hour of Lego Ninjago on the computer. Am I a “Tiger Mom”? No. Am I looking for extra ways to boost my son’s academic potential? Yes! The best part of all of this is that it is free and rooted in solid Constructivist pedagogy. No matter what math curriculum your child is using at his or her school, this type of practice can help.