I have come to believe that children develop their own strategies. Sometimes, it happens as a result of confusion. A student could be taught one strategy from one teacher and another strategy from another teacher. This could cause one child to incorporate two ways into one, resulting in confusion and the wrong answer. This is with a student at my intern school.We were working on regrouping with subtraction. I showed them a way of subtracting that includes showing the number of 10's in the tens place. For example, with the the number 43, there are 4 ten's. These tens would be wrote on the bottom left side of the math problem. I asked him to cross out a ten when he is adding it to the one's place. When he got the chance to use this method, with no help, he would take the ten away but would add it to the number that is being subtracted. He would put that in the place of the regrouped tens place (I hope that makes sense). I taught him how to demonstrate the adding of 10 to his tens place and another teacher taught him to add that ten to the ones place ( I realized this when asking him to subtract and regroup before I tried to teach him any strategies). I acknowledge the fact that I taught him this strategy poorly. But I wrote about this to show that students create strategies from ones that they've been taught. He also incorporated using blocks for his problems to help him maintain the placement of the numbers.
An example: 21 - 7 = 14. In this picture, he got it wrong at first. It showed another mistake that he was making, but I will discuss this later in the blog. He first added 10 to the 7 down at the bottom of the paper. You can't really tell because he erased his first response. He eventually added the 10 to the 1 in the tens place and gotten the problem correct.When subtracting, one of the students would use tally marks. This was a strategy used by another student. With 2 of the 4 problems given using tally marks, he had gotten wrong (picture below). Noticing this, I introduced him to a new strategy. I called it the 'Count Up' strategy. When subtracting smaller numbers, take the subtracted number and count up until you get the bigger number (I can't remember the proper name for the subracting the number and the 'original number'). In other words, if c - b = d, I told him he could take 'b' and count up (not including b) until he got to 'c'. The number it took to get to 'c' was the answer or 'd'. This was also a problem that the other student was making in the first picture. I showed him the 'Count Up' strategy as well. (Below) His tally marks caused him to get the problem wrong. I tried to help him perfect this way, but it was much easier to explain the 'Count Up' method to both of them.
I have a student in my internship, who just transferred here from a school in North Texas, and she was taught to solve word problems two ways: add or subtract. And on top of that she was taught to solve them by, "showing your work," and that work was taught to her as the traditional addition and subtraction ALGORITHM. So her background knowledge with word problems is centered primarily on reading the word problem and solving it using either a subtraction or addition algorithm. A basic strategy in it's simplistic, ideal form. However, my student also has English language difficulties (even though she is not qualified as either ELL or ESL), which is preventing her from reading the word problem with accuracy and interpreting what digits she needs, the order, and the correction "operation." Upon asking her why she chose to solve every word problem using an addition algorithm, she responded that is what Ms. _________ told her to do. Not that I think her previous teacher told her to exactly do that, but it is an example of a student being taught one strategy, without establishing or understanding why or how it works, and then applying it in a confusing and incorrect way (just as you stated). To make a long story short (cutting all the boring details), just two week of instruction with me and her peers, she is taking the direct modeling strategy we are working with within our word problem group and using it concurrently with her own algorithm strategy! She is combining two strategies that worked for her, and finally she is experiencing success!
ReplyDeleteGabe, I think it is really interesting how students develop their own strategies despite being taught a certain way to solve the problem. It makes me think about how effective teaching math strategies to students really is. Every student is unique and the way the do math is no different so why should they be taught a certain way to solve a problem. Even though I think helping a student discover their own method of doing math is beneficial I think being effective at it is another question.
ReplyDeleteI like how you said you might of not of explained the strategy to the student well enough. The first step of to fixing something is being aware that it can be fixed. I we are all improving on our teaching and that is going to make us into better teachers in the future.