My sister’s friend is a first grade teacher in a ‘challenging’ school in New York. People who know about education know that poor students often enter kindergarten, already a few years behind their more affluent peers. Good teachers, when given the freedom to teach their students at an appropriate level for their incoming skills, are able to help students progress.
But when teachers are forced, from a top down mandate, to teach an untested curriculum based on the concept that by adding more ‘rigor,’ students will rise to meet the new expectations, well, that can be a problem.
I have a vivid memory of being in first grade and having to answer some questions like : 5 + blank = 7. I would do these by counting up from 5 until I got to 7 and see how many I needed to count. My teacher, seeing that I was pretty good at this suggested that I could get the answer even more quickly by just doing 7 minus 5. I remember ignoring this suggestion. Even now, as a math teacher and enthusiast, I have to think twice to be convinced that this method should work. (I’m pretty comfortable with it now, but I still see it as a subtle thing.)
Something that ‘math people’ sometimes have is ‘number sense.’ There are actually some great mathematicians who truly have trouble with basic arithmetic since the math they study is in a completely different realm, but most, I’d say, have the number sense too. So when I need to add 98 + 147, I might, instead add 100 + 145. How and when I learned to do this, I’m not sure, but it certainly wasn’t in first grade.
The new first grade common core standards, apparently, were written by people who thought something like “Math people have this number sense so if we ‘teach’ kids to have it at an early age then they will become math people.”
So my sister’s friend has to teach students who would struggle to add 9 + 7 by just counting seven numbers beyond 9, to take one from the 7, give it to the 9, turning the problem into 10 + 6 which then is a ‘trivial’ problem. My first issue with this is what if 10 + 6 is not trivial to the students? Then this conversion is a complicated process that leads them to a question that is just as hard for them. Also, this idea of getting students to ask: “How far is 9 from 10? Take that away from the other number and then add the result to 10″ is extremely complicated.
The students are also expected to do a question like 12 – 7, by noticing that 12 is 2 away from 10, so if you could do 12 – 2 to get 10, you’d just have to take away 5 more (since 7 – 2 = 5) so the problem becomes 10 – 5 which is ‘obviously’ 5.
Finally, students are taught the doubles of each number so they can add two number that are very close together, like 7 + 9 by doing 2*7 + 2 = 16 (since 9 is 2 more than 7).
Now I am not a specialist in early childhood education. I certainly can’t be expected to know as much as the new Teach For America Early Childhood Education director Brittany Toll, who I believe taught lower elementary for two years. But I have some experience as the father of a precocious five year old.
When my daughter Sarah was about 7 months old, I got the very stupid idea that I would train her to be the youngest ever solver of the Rubik’s cube. So I got a cube and removed all the stickers except for two white stickers. My plan was to have her ‘discover’ how to get the two white stickers together and then I’d slowly add more of the stickers until she could do the whole cube. I didn’t force her to practice, but I would conveniently leave this custom cube around the apartment to see if it would interest her. I learned pretty quickly that you can’t force someone to understand something that they are not ready for yet. I gave up on the training and Sarah still has no interest in the cube (and she’s way past the world record age anyway.)
A private joke I have with my wife is “Seven … beven” since about six months ago I thought I’d work with Sarah on her number sense. I said “Let’s play a game where I say a number and you say what number comes next.” I found that when I’d say 5, she’d have to start counting from one until she got to five and then went one more, rather than just start at 5. So I was quizzing her and she was getting a bit better at this. I said 3 and she said “Three, four.” I said 9 and she said “Nine, ten.” Then I said 7, and she said “Seven … beven” indicating that the game was over.
Here are some videos illustrating what happens when you try to teach something developmentally inappropriate. I hope nobody sees these as me exploiting my children. I think of my blog readers as my friends and my kids are so cute I can’t resist. But if people feel strongly that I should remove the videos, please say something in the comments, and if enough people do, I’ll take down the videos.
Here’s my daughter trying to teach my almost two year old son a complicated color matching game before he’s developmentally ready.
Here’s Sarah when she was about 2 1/2 trying to teach me how to properly drink from a sippy cup and properly say “ahhhh” afterwards, before I was developmentally ready.
meghank
test
If a child is struggling with addition or subtraction conceptually, counting methods are a better fit than combination methods, and certainly no less rigorous than doubling or recombining. It is far more important that the student is convinced that the algorithm works and gives a consistent and true answer than to memorize a specific method.
Take the 9+7 example. To make a ten is cognitively complex, involving a memorized concept, some understanding of betweenness, and remembering and manipulating several intermediate values. I’d have had to repeat these steps hundreds of times to be convinced that this algorithm is true (without the advantage of having variables).
Given x,y in Naturals, where x+y>10, and y<x<10. Find the sum by "making a ten:"
1. Recognize the sum is greater than ten.
2. Recall the rule: the sum of 10 and some digit, d, is 1d.
3. Apply an understanding of betweenness to determine that y<x<10.
4. Subtract and hold a value in memory: 10-x.
5. Subtract and hold a second value in memory: d=y-(10-x).
6. Add the first intermediate value to x to make 10: x+(10-x)=10.
7. Apply the rule: 10+d=1d.
The "make a ten" algorithm for x+y is six or seven steps to obtain "x+(10-x)+y-(10-x)," with no guarantee that the student understands where the result comes from (because step 6 can be skipped and the result replaced with 10 every time, without affecting the answer). It does not sound like a strategy for a student struggling with addition concepts. It may be okay for an advanced first- or second-grader (and even then, it is perilous if she doesn't convince herself of it's truth on her own).