Does efficiency = comprehension in mathematics?

Last year, I co-taught in a grade 6 classroom and gave the students the following problem as a part of an initial lesson to begin a unit on division,

“165 parents will be attending a family math night in your school gym. They will sit at tables in groups of 6. How many tables need to be set up?”

We asked the students to work collaboratively to solve the problem in more than one way and to show their math thinking. Below are two solutions from two different groups.

Which group has a better understanding of division? I showed these two solutions to many teachers and at first glance, many chose the group that solved the problem using long division. Why? The main reasons were efficiency and a percieved higher level of thinking. Many teachers viewed long division as a faster strategy and the latter, a time consuming low level strategy. However, is efficiency the ultimate goal in mathematics and does it mean that students that use very efficient standard algorithms to solve problems have a good comprehension of the mathematics?

The standard algorithm for division was invented long before the calculator and was viewed as the most efficient way to divide. However, I don’t think it was viewed as the most efficient way to teach division. Yet, the standard algorithm is frequently the first (and sometimes the only) strategy that is introduced to students when teaching operational sense. The standard algorithm, like many other division strategies can be a great strategy but only if it is understood. For many students, the standard algorithm for division is difficult to understand which forces them to rely on the memorization of the steps without any conceptual understanding of division. If given the opportunity, students are capable of coming up with their own invented strategies for division. More importantly, these are invented strategies that make sense to them which would lead to a better conceptual understanding. Below are the same solutions to the division problem mentioned above but I have included the students’ final statements that answer the question.

The group of students that used a longer and “less efficient” strategy had a better understanding of division than the group that solved using long division. It is crucial that students be given an opportunity to solve problems and on their own in as many ways as they can and allow them to share and dicuss their strategies with their classmates. By doing so, they may be more likely to make connections between their own strategy with others and move towards a conceptual understanding of a more efficient strategy. When you take a step back as a teacher and let the students loose with math and allow them struggle with guidance, you could be pleasantly surprised with their results and the discussion of their results is where you will find that the most teaching and learning occurs.

8 thoughts on “Does efficiency = comprehension in mathematics?”

1. Great activity. Hope you don’t mind if I copy.

I tell my students that there are 2 reasons why maths was invented:

1. To save time

2. For fairness (so everyone gets an equal share)

Both reasons appeal to the students. The first example you show may be efficient, but the second is concerned that all audience members have a seat to sit on. Both methods are equally sound. The answer is important, but not as important as the reasoning underpinning it.

1. Not at all. The problem comes from a great Ontario, Ministry of Education resource titled “Guide to Effective Instruction”. There is an individual junior guide for different math strands. Electronic copies of the documents can be found at the following website: http://www.eworkshop.on.ca

2. Research shows that being able to use all methods (aka multiple representations) deepens understanding. That is why my physics students must use charts, vector diagrams, and graphs in addition to writing equations.

Great post!

1. I definitely agree. I always tell the students I work with in math to solve problems in more than one way and then to make connections between their solutions to really develop their conceptual understanding. Thanks for commenting!

3. I found this blog post through a comment you made today on another blog. It is exciting to find examples of children being given the time to think and reason their way to understanding. In the 1960s, I was lucky to have teachers who taught math this way, and I have been drawing my way through math problems ever since. I am an older new elementary teacher, hoping to have my own class soon. I look forward to guiding my students to true understanding of math concepts, rather than memorization of the processes. I hear a lot of talk about the need for year-round school to avoid summer brain drain. I believe that kids forget what they memorize, but not what they learn and understand.

1. Thanks for comment Sue! I was a product of procedural teaching and as a result, never really understood the math concepts in school. I developed into a very good memorizer and basically got me through high school math. However, in University I was basically a fish out of water and went from a good memorizer to a great memorizer and relied on my study groups and many sleepless nights. During my first few years of teaching, I taught my students how I was taught and witnessed my students go through the same struggles that I did, forgetting math concepts that I covered a week ago, disengaged from learning. I’m so thankful that I discovered teaching through problem solving and rediscovered a passion for math. Becoming a math facilitator was the best thing that could have happened to my teaching career.

4. I absolutely LOVE your message here. I taught 3rd and 4th grade in a district that used Marilyn Burns, as well as TERC Investigations. A lot of teachers were wary of teaching math in this new vein, but I was shocked at the results. By the time my students got to 4th grade, many of them could multiply 3-digit x 3-digit numbers in their head faster than I could because their understanding was so deep. They didn’t rely on an algorithm that required them to write, carry, etc. When we got transfer students, they would often get the answer right using an algorithm, but when asked to explain WHY it worked, they couldn’t. After a couple of months with our curriculum, their understanding truly deepened.

1. That’s a great story! It funny that so many people think the standard algorithm is the most efficient way to multiply. However, in certain situations that’s not the case. It’s all about knowing when to use appropriate mental strategies for certain problems.

Thanks for commenting!