“But I learned it that way”: The case for student-invented strategies

Recently, I have been receiving a lot of push back from parents and teaches who don’t think it is worthwhile to expose students to methods other than traditional algorithms. They tell me how they memorized the procedures in math class and they don’t understand why their students or children can’t do the same. They complain about all these diagrams I encourage students to use and don’t understand why students should invent their own strategies to solve problems.

I held this belief when I first started teaching as well. However, I soon realized that my students weren’t learning math by memorizing procedures. Maybe they could do the problems in class that day, but ask them to do a mixed set of problems a couple weeks later and they were lost. When I began to encourage students to invent their own strategies, two things happened:  1. They got the problems right more often. 2. They started to like math better and become more confident.

Last week, I was working with 4th graders on the following problem:

Ms. Placa bought 15 packets of pencils.  Each packet had 12 pencils in it.  How many pencils did she buy?

Students were able to represent the problem with a diagram or explain why they knew they needed to multiply 15 times 12.  Awesome, right? Yes. Until they tried to multiply 15 times 12 using the traditional algorithm. Over half the class got the wrong answer.

I asked them whether 30 was a reasonable answer. They knew it wasn’t, but kept going back to the algorithm. Then, one student pointed to the picture she drew–15 boxes with the number 12 written in each one. She suggested counting by 2’s 15 times and then counting by 10’s 15 times and then adding them together. She could even explain to the class why that worked. Later, I learned she was one of the “weaker” math students in the class.

I tell the story because it is amazing what students can do when we allow them to invent strategies that make sense to them.  

The research supports my experience.  The benefits of student-invented strategies include:

1. Students make fewer errors than when using standard algorithms that they do not understand.

2. There is less of a need for remediation later on if students understand what they are doing and make connections on their own.

3. In many cases, using an invented strategy can be faster than using a standard algorithm.

Standard algorithms that are taught with meaning can have their place as well, but I think the case for allowing students to make sense of math needs to be made as well.

Want to know more? Check out:

Fuson, K.C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 68-94). Reston, VA: NCTM

Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products of students’ own
mathematical activity. In J. Kilpatrick, WG Martin, & D. Schifter (Eds.), A research companion to
principles and standards for school mathematics (pp. 114-122). Reston, VA: NCTM