The last post about enVisionMATH and how I, as a math person and dad, go about trying to make sense of what my 6-year old brings home from first grade seems to have struck a chord among parents. The comments have been outstanding and there seems to be a real need for this kind of conversation. So I have a few more such posts coming up soon, starting with this one.
The 6-year old brought this home on Monday. Click to enlarge:
It’s about adding “near doubles”, like 3 + 4 or 2 + 3. In case you can’t read the top part or can’t enlarge the photo, here are the steps — yes, there are steps, and that’s kind of the point of this post — for adding near doubles:
- “You can use a double to add a near double.” It gives: 4 + 5 and shows four blue balls and five green balls.
- “First double the 4”. It shows 4 + 4 = 8, and the four blue balls, and four of the green balls with the extra green ball sort of falling to the ground.
- Then it says: “4 + 5 is 4 + 4 and 1 more.” At this point you really have to look at the worksheet itself, because it’s hard to put into words what is going on:
And from there, in the fourth frame, one of the girls in the earlier frame concludes that 4 + 5 must be 9 because 8 and 1 more is 9.
The Guided Practice section has the kids doing four near-double sums. Clearly, the way the worksheet wants kids to learn how to do this is not simply to add 2 + 3, but (1) to recognize that 3 is 2 plus 1 more, (2) add 2 + 2, and (3) then add 1 to the result of 2 + 2:
There’s a thing at the bottom asking kids to explain the process and then a bunch of near-double sums to practice — presumably kids are supposed to use the method described above, but there’s nothing forcing them to do so — and some “algebra” questions with blanks in the place of variables.
I’m not sure exactly how my brain goes about adding near doubles — whether it just somehow does the addition in ways that are almost automatic thanks to 35 years of repetition, or whether there are little tricks it employs — but I am absolutely certain that I don’t do it the way enVisionMATH is telling kids how to do it. I tried it. When I read the worksheet, I thought about near-doubles that aren’t so easy, like 121 and 122. Quick! Add those together. Did you think, “122 is 121 plus one more; 121 + 121 = 242; 242 and 1 more is 243” ? I didn’t — not by a long shot. I just added the numbers together. No methods, no tricks; just old-school addition. There may be some tricks that my brain invokes to “just add the numbers” — for example, I tend to visualize the two terms of the sum stacked atop each other in the classic vertical arrangement for adding, and then visually add the digits — but I am most definitely not going through the four-step process on this worksheet.
In fact, the four-step process complicates matters so much that it’s inexplicable why they are even bringing it up. Most kids at this stage can add 2 + 3 or 5 + 6 in one step. But by introducing this method, there are four operations: comparison (find the larger of the two near-doubles), subtraction (take 1 from the larger number), addition (add the two duplicates), and another addition (add 1 to the result). Technically there is a fifth operation kids have to perform, namely recognize that the two numbers they are adding are near-doubles in the first place.
One might argue that doubling a number (in the third step) is easier than adding it to itself — kids just recognize that doubling 5 gives 10, for instance — and subtracting 1 is a very easy special case of subtraction in general that nearly everybody at this age can do without thinking, similarly for adding 1 at the end. That may be so, but it can’t be so much easier that adding in steps 1, 2 and 4 results in a net reduction in complexity or a net gain in conceptual understanding.
But what about kids who can’t add two one-digit numbers together in one step? There are some of those out there, including probably a few in my daughter’s class. This method doesn’t help those kids. Again, we may argue that adding 4 + 5 is considerably harder than the combined process of comparison, subtracting 1 from 5, doubling 4, then adding 1. But I don’t think so. A four-step process is no less cognitively demanding than a single-step process, even if the four steps are easy. And besides, life does not throw near-doubles at you to add. How is a kid going to learn to add 2 + 5, or 2/5 + 7/8, or 123.38 and 99.99 this way?
If there is some research that suggests that people really do add near doubles this way, I would love to see it. Otherwise it’s hard for me to believe that any more than a tiny fraction of the human population actually does it this way. Is there going to be some mind-blowingly cool way to do complicated arithmetic in one’s head farther down the road that uses this idea, like multiplying numbers that are near-squares or something? Perhaps I should be more patient. But for the time being, I told the 6-year old just to add the numbers together like she already knows how to do, the old-fashioned way.