At a workshop last week, the following task caused a bit of confusion:
If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?
Several participants were convinced it was 144.
As we discussed the problem, it was clear that those who were getting 144 were setting up the proportion below and cross-multiplying.
Small Gear 8 = 96
Big Gear 12 x
They weren’t trying to reason about the situation. They weren’t thinking about whether it made sense that the big gear turned more times than the small gear. Instead, they identified this as a proportion problem, set up a proportion and solved for x because it was the method they were “supposed to use” to solve proportion problems.
It was a great example of why you should hold off on teaching students the cross-multiply algorithm until after they have had experience reasoning about ratios using their own strategies.
It also points to the fact that students need to develop meaning for the algorithms they use or they won’t know how or when to apply them.
Lobato and Ellis (2010) describe a student who set up a proportion and used cross-multiplication to correctly solve the following problem:
How much water would drip from a leaky faucet in 4 minutes, given that it dripped at a steady rate of 6 ounces in 8 minutes?
Although the student could provide a correct numerical answer to the problem, she could not explain why it was correct or why it made sense.
Her limited ability to reason proportionally was even more apparent when she could not make sense of the following problem:
One faucet was dripping 6 ounces in 20 minutes. A second was dripping 3 ounces in 10 minutes. Was the first one dripping slower, faster, or at the same pace as the second?
Her inability to solve the second task, as well as her inability to explain her answer in the first task, indicate that she was using a procedure that held no meaning for her. When presented with a new task that she couldn’t solve in the same way, she didn’t know how to reason about it.
The research also shows that student have a hard time making sense of the cross-multiplication algorithm. It doesn’t necessarily connect to their intuitive strategies. If the algorithm is introduced before they have developed meaning for ratios and proportions, it can interfere with and even hinder their ability to develop proportional reasoning.
So how do we help ratios make sense to students so that they can begin to reason proportionally? I’ll talk more about that next time, but if you want to know know, take a look at the resources below.
Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 629–67). Charlotte, NC: Information Age Publishing.
Lobato, J., & Ellis, A.B. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics, Grades 6–8. National Council of Teachers of Mathematics.
Teaching Ratio and Proportion in the Middle Grades: NCTM research brief
Smith, J. (2002). The development of students’ knowledge of fractions and ratios. In B. Litwiller & G. Bright (Eds.), Making Sense of Fractions, Ratios, and Proportions (pp. 3–17). Reston, VA: National Council of Teachers of Mathematics.
Thanks for posting. This is more justification for thinking before applying algorithms and/or procedures. Nothing wrong with solving stuff intuitively.
Thanks! I agree. Letting students solve things intuitively is also great for figuring out how to build on what they know so that algorithms and procedure do make sense to them.
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Cross multiply is hardly an algorithm, it is a method for dealing with equality between fractions (or proportions, same thing). The error in the gear problem solution described above is entirely due to the FACT that the two proportions (fractions) 8/96 and 12/x are NOT equal, so “cross multiply” cannot possibly give the correct answer. Besides, were the people who got 144 as the “answer” happy with it? Algorithms and procedures are there to get the results less painfully, but are no use if one does not understand the problem and cannot use common sense (Is this now called intuition, they used to be different things).
Interesting comment. I think you are right about cross-multiplying being an efficient method to help students get the solution less painlessly and the method can be useful when students understand the problem and have meaning for the procedure they are using. The issue I noted (which is supported in the research) is with teaching the procedure before students have had a chance to develop an understanding of what ratios are or how to maintain invariance between different ratios. Also, what is common sense to someone who already understands ratio and has developed proportional reasoning does not necessarily common sense to a student who has never been formally taught ratio. However, students do bring intuitions about situations with them when they are beginning to learn a new concept and it can help to build upon that when teaching.
Here’s a great document that a handful of teachers worked on a while ago which includes ideas like this. http://goo.gl/qQ6k01
Just the other day a teacher talked about how she was teaching “Cross-multiplication now and the kids just weren’t getting it.” I wanted to go into all the bad things associated with just teaching an algorithm, but, not wanting to offend her, I just kept my mouth shut and thought about ways of showing her that document without turning her off to it.
How would you recommend helping your co-workers “see the light” about teaching the way you explained above?
Very nice document. I love the idea of “nixing the tricks.” I think you raise a really good point about how to share information with co-workers. In my experience, when someone brings up something that the kids aren’t getting, I ask them why they think the students are struggling. I think when they have to think about why the students are struggling, they may be open to other suggestions of what to do to help the struggling students. Then I might offer a suggestion of how I might teach the concept. When I was teaching, I also offered to have my co-workers come visit my class. In your case, you might share the document you created with everyone and ask what people think in a very open way. It might lead to a very interesting discussion.
Oh, I want to clarify that I did not create that document! That was made by many other teachers who were much more thoughtful in their explanations than I.
I like your approach to handling the delicate issue of helping your co-workers improve their teaching. I hope that I am likewise open to criticism and suggestions in my own teaching!