I found this blog entitled, "Why We Don't Cross Multiply," The idea of cross multiplying seems to pop up whenever I introduce Multiplication of Fractions. But, why? Where did these students learn about cross multiplying? In most cases, students seem to be confused with reducing before multiplying. Does anyone else struggle with learners like mind?
It comes up for the same reason we teach it: it's relatively easily learned as a motor-memory thing. Lots of math has a "look and feel" to it that IMHO should be incorporated into our instruction.
Since multiplying fractions generally is devoid of actual meaning, it's tough to remember the rather arbitrary rituals for reducing before multiplying. I'm still working on a consistent way to address that... maybe working on reducing simple fractions 'til that's a nice "canceling" motor memory and then incorporating it into multiplying fractions.
What about using visual models to help create meaning? I know that when I was in school we were taught to just do and not really understand the why; I find myself teaching the why now but it is sometimes a struggle.
Proportional thinking is a huge big deal and worth spending time building w/ visuals and "real life" examples. (Add that to my list...) The bulleted list in the blog post is good w/ describing a path to that...
In my experience, though, because these things aren't linear, students need extra practice with the "visuals" of the abstract symbols, too, so that the 'canceling' (dividing out 1) becomes as easy to access in the mind as "cross multiplying." Yesterday I had a student practicing for the TEAS test and three or four times she decided a problem with percents was a straight proportion problem because they sort of look the same (things like 13 percent of 89 is 17 % of what?) ... now she knows to look for that % sign in between and use what she knows about percent problems...
I believe that this incorrect idea stems from how students were taught to solve proportions. Sometimes instructors jump to this shortcut of cross-multiplying to obtain equal expressions in a proportion without first demonstrating why that procedure works. I have adopted the habit of staying away from this strategy and teaching proportions as equivalent fractions and clearing fractions on both sides of the proportion equation instead. Eventually, we can show that in a proportion equation, clearing fractions equates to "cross-multiplying" but we should never start there!
Proportional Reasoning is a "big idea" that is very practical in many domains both in careers and in every day life. I strongly believe it needs a lot time to be developed.
Hi Sarah! I so agree with your post and others. Building proportional relationships first is so important as many proportions can be solved easily by looking at relationships either within the ratio or across the two ratios. It can also be used to estimate even with numbers that are not nice. I think we also need to focus on what the equals sign means as well. Your suggestion of equivalent fractions is a great example of seeing one ratio "is the same as" the next ratio by looking at the relationships. I think if we keep emphasizing this maybe students wouldn't even think about cross multiplying when multiplying fractions or using cross multiplying incorrectly in other situations. As we learned in ANI, there are many approaches to looking at proportional relationships before jumping to the cross products.
My background is reading, writing, and social studies, yet, I found myself teaching adult education math. I have taught the cross-multiply method of proportions - because it was an easy method to demonstrate. (So, there's my confession....sad as it is.) As I read through the conversations in this group and other forums about math instruction (Shout out to the Adult Numeracy Network), I am buildng a better background for mathematical undederstanding, BUT, I think we are tapping into a bigger concern about how many educators are afraid of the math that is now expected to be taught, and students need to master. I'll never forget a PD workshop with educators as the 2014 test was emmerging. The teacher said, "I better get good books because I don't know this math."
Are we teaching the cross-multiply method (or other shortcuts in math instruction) becasue that is how the curriculum explains it?
Interested in your thoughts.
In adult ed, I was frustrated *forever* at how consistently *everything* was procedural instruction, compressed and accelerated because, after all, students were so far behind. Even in those cases where materials actually broke things down further, it was still... breaking procedures into smaller steps.
That's shifting in places and I rejoice that the Internet can make it happen so much faster... but most curricula (*especially* "personalized learning") breaks math into procedural slivers.
To really do the concept thing right, you need some common themes ("number bonds" and "parts and wholes" would be examples) that you can revisit with new concepts. I love the "Instructional Routines" that are spreading, too.
And (because not enough students are thoroughly confused yet) we get instructors who use the term "cross cancelling" when they teach multiplication of fractions and we have students "cross cancelling" in proportions. This topic speaks to the need for modelling and for allowing learners to discover sweet shortcuts like cancelling to simplify fraction multiplication - too often we "spill the beans" before learners have had a chance to develop their own thinking. Regarding cross-multiplying in proportions, failure to teach students to look for relationships both across the equal sign and within the numerator and the denominator of either ratio puts them at a disadvantage and precludes many estimation and "common sense" possibilities. Another failure in our teaching of solving proportions by cross-multiplying involves the question, "Why does that work?" Not relating that answer to getting a common denominator for the two ratios smacks of teaching rote procedure rather than building understanding (not that I have an opinion ).