Moreover, algorithms unteach thinking. Do we believe this student would say that 15 x 2 = 10? Or that 15 + 15 <= 20?
What seems to have happened, however, is that the student multiplied 5 x 2, got 10, put the 0 down under the 5 and 2, "carried the 1" at least to the extent of putting it at the top of the next column, but left it there and multiplies "1 x 1," got 1 and wrote that under the column with all those 1s. Makes "perfect" sense. Kind of. Particularly if there's no thought whatsoever about what the answer should be just by estimation.
That was the scary part…I asked them what 15 + 12 would be and they got it immediately. I asked if it’s possible to add and get a bigger number than when you multiply. They said no and understood why but their number sense is pretty concerning.
I’m a high school math tutor and will have to read that link more in depth later, and perhaps you’ve already answered this question in other posts, but isn’t there some value in having algorithms so that everybody is working on the problem in at least one common way? Of course estimation helps too and having the confidence to be able to develop one’s own shortcuts and check them.
The issue with algorithms is not IF but WHEN? Traditionally, we teach kids algorithms immediately in arithmetic and never let up thereafter. Researchers like Constance Kamii (who worked with Piaget as a young woman) have argued for decades that we kill mathematical independence, confidence, self-reliance, etc., by doing this. Young children are perfectly capable of inventing their own algorithms (which likely include already known ones) and then over time sorting through their own and those of peers (and, if need be, those taught later by the teacher) and selecting what best suits them. Since adults have ZERO faith in the natural intelligence or judgment of children, we ensure they never get to do anything of the kind.
In this regard, I am solidly in her camp. She has frequently been mischaracterized as arguing against EVER teaching algorithms. This claim is simply not true. Nor would it make sense to support that position. But very few teachers or parents have what it takes to support actual investigative and discovery learning. And the propaganda against it is extensive and a beautiful example of educational fear-mongering at its worst.
Yes, I’ve also seen kids doing this “vertical multiplication” algorithm by analogy to the addition algorithm. If you can add ones and ones, and tens and tens, why not multiply the same way?
I find the estimation suggestion here useful. Also reframing the problem as one about money or something else that’s tangible and important to the kids can be helpful. I’m a huge fan of the area model, too, and lots of sheets of graph paper.
5 replies on “15 x 12 = 20”
Moreover, algorithms unteach thinking. Do we believe this student would say that 15 x 2 = 10? Or that 15 + 15 <= 20?
What seems to have happened, however, is that the student multiplied 5 x 2, got 10, put the 0 down under the 5 and 2, "carried the 1" at least to the extent of putting it at the top of the next column, but left it there and multiplies "1 x 1," got 1 and wrote that under the column with all those 1s. Makes "perfect" sense. Kind of. Particularly if there's no thought whatsoever about what the answer should be just by estimation.
That was the scary part…I asked them what 15 + 12 would be and they got it immediately. I asked if it’s possible to add and get a bigger number than when you multiply. They said no and understood why but their number sense is pretty concerning.
I’m a high school math tutor and will have to read that link more in depth later, and perhaps you’ve already answered this question in other posts, but isn’t there some value in having algorithms so that everybody is working on the problem in at least one common way? Of course estimation helps too and having the confidence to be able to develop one’s own shortcuts and check them.
The issue with algorithms is not IF but WHEN? Traditionally, we teach kids algorithms immediately in arithmetic and never let up thereafter. Researchers like Constance Kamii (who worked with Piaget as a young woman) have argued for decades that we kill mathematical independence, confidence, self-reliance, etc., by doing this. Young children are perfectly capable of inventing their own algorithms (which likely include already known ones) and then over time sorting through their own and those of peers (and, if need be, those taught later by the teacher) and selecting what best suits them. Since adults have ZERO faith in the natural intelligence or judgment of children, we ensure they never get to do anything of the kind.
In this regard, I am solidly in her camp. She has frequently been mischaracterized as arguing against EVER teaching algorithms. This claim is simply not true. Nor would it make sense to support that position. But very few teachers or parents have what it takes to support actual investigative and discovery learning. And the propaganda against it is extensive and a beautiful example of educational fear-mongering at its worst.
Yes, I’ve also seen kids doing this “vertical multiplication” algorithm by analogy to the addition algorithm. If you can add ones and ones, and tens and tens, why not multiply the same way?
I find the estimation suggestion here useful. Also reframing the problem as one about money or something else that’s tangible and important to the kids can be helpful. I’m a huge fan of the area model, too, and lots of sheets of graph paper.