My student at Houston Elementary used several strategies when attempting to solve math problems. She set up the problems originally using the traditional algorithm that she had seen so many times in the classroom. This is to say that when presented with a subtraction word problem orally, the student would first write on the paper: 7-5=. She recognized that this is the standard form of what I would see on homework or a test. She did not seem to have her math facts memorized, but that did not stop her from using student-invented strategies alongside those taught to her in class to creatively solve multiple problems.
One problem is as follows: John had 13 pennies. He lost 4 pennies. How many pennies does he have now? My student would identify that lost meant that the pennies were removed and that must be subtraction. She wrote 13-4= on her paper. Then she grabbed some of the cube manipulatives and counted out the total number of pennies he started with which were 13. She then removed 4 of the cubes to represent the pennies being lost. Then she counted up the remaining cubes and determined the answer to be nine. She then wrote 9 on her paper to complete the equation.
The student was familiar with the standard form of how we expect children to set up a subtraction problem. For the actual process of finding an answer she used direct modeling with the cubes and counted to find the answer needed. This was just one of several strategies she could have used to solve the problem.
She could have used mental computation. Visual representation is not necessary for all students to solve the problem. She could have counted down in her head to solve the problem. I notice that many students use this strategy with varying results. This is a fast way to solve a problem and is useful because manipulatives may not always be present. The major drawback is the process is invisible to the teacher. If a mistake is made there is nothing that can be referenced to see where an error occurred.
Another strategy that she could have used is subtracting by counting up. She could have started with 4 and counted up to 13 noting how many units were between the two numbers. This is helpful in identifying the relationship between addition and subtraction. This strategy does have limitations when dealing with bigger problems such as 99-23. The thing to take away from this is that there is no one way to solve a problem and it is important to encourage students to use traditional and student-invented strategies to gain a better understanding of mathematics. Counting Up Subtration Model
I enjoyed reading about your description of math strategies that your student used, and I agree with you about the pro's and con's you stated about various strategies.
ReplyDeleteI personally feel like the algorithm method and the counting up method are the most common strategy that students refer back to. (I know I even still use it!!). I don't know about you, but I have found it hard to continue questioning students about "other ways you might solve the problem" when the answer the problem correctly with one of those strategies.
Any thoughts?
Eric,
ReplyDeleteI enjoyed reading about your student's strategy in solving the math problem. What I found interesting about it is that she used direct modeling along with the traditional algorithm. This seems like an excellent way for her to check her work.
Jennifer
I'm glad you discussed the student you worked with for the Interviews we conducted at Houston elementary, especially because I was able to see your video of this student working on a problem and got to discuss with you what it was like working with this student. It's interesting that your student began working the problem by setting up the traditional algorithm we have all been taught, and then resorted to using manipulatives for concrete representations. I'm glad that she used what she was comfortable with and the strategy (direct modeling) that she understood.
ReplyDeleteIf I were to think of another way for the student to work this problem I would use our example of counting up. We could go as far as to use the "add tens to overshoot, then come back," (a student-invented strategy) to find the answer. This helps with students that rely on counting by tens, which is one of the counting strategies that students learn early and understand early.
My student also used direct modeling for various problems to demonstrate her thought process. I agree that while this strategy may be easier for the teacher to "see the work," it is not necessarily a better or faster way to reach the answer. Nevertheless, students must understand what they are doing through addition, subtraction, multiplication, and division, and manipulatives are one way to physically display their steps. When doing work on paper, it is also important that students get in the habit of showing their work. I remember having to even write written explanations of my work, in addition to the computation/algorithms used, beginning in middle school. At the time, I hated this requirement but now understand how useful this information might be to my teacher who is assessing my process, not just the final answer.
ReplyDeleteLike you, I am curious to know whether the students we worked with actually didn't know their math facts or did they feel a pressure to use the manipulatives we had set before them?