The English language has many confusing phrases but "as many as" IMO has blighted the youth of many an algebra student. Perhaps you think I'm exaggerating this? At the beginning of the school year, write the phrase in the title of this post on the board and have your PreAlgebra/Algebra I (or higher) students write one of the two equations on their paper. Give them only a few seconds, then compile the results. Let us know if the vast majority choose the correct equation. Of course, the outcome depends on the group and many other factors but if we have enough data it might prove interesting. I'm basing this on many years of questioning students. Perhaps I am the only one who has experienced this phenomenon!
The abstraction of algebra is difficult enough for some youngsters. Students who are new to our language have particular difficulty with idiomatic phrases but those born here also seem to struggle with the verbal parts of word problems - that's completely obvious to any algebra teacher of course. If only we could remove the words from a word problem!
Certainly teaching vocabulary and math terminology is an essential part of what we do as instructors. We should also hold students accountable for this vocabulary by assessing it directly.
In this post, I'm inviting readers to share some of the coping mechanisms and pedagogical strategies they use in the classroom to help students survive phrases like "as many as." What phrases seem to cause the most confusion among your students? How about "x is four less than y?"
Here is my initial offering. Let me know if you do something similar or if you feel this might be helpful (or if you vehemently disagree!).
KEY STEP: First decide from the wording of the problem if there are more girls or more boys. In fact, this should have been my original question -- not the equations! It is critical for students to be able to translate the verbal expression into a comparative relationship: Which is the larger quantity? Number of boys or number of girls? Hopefully, most youngsters would interpret the original problem to imply that there are more girls than boys. Hopefully! Ask this question first (metacognitively, students need to learn to ask themselves questions like this when they are reading).
NEXT STEP: Now the issue is where to place the "2" in the equation. Based on the key step above, we know that the number of girls is the larger quantity. Ask them why 2G = B would be incorrect.
Better alternative for some:
We all know that those who have difficulty handling abstraction benefit from concretization, i.e., using numerical values:
Have them write both possibilities:
B = 2G and G = 2B
Now have them substitute values for G and B that make sense for the original problem, say
G = 12, B = 6. Some struggle with this!
By substituting (students like the phrase "plug in") these values into both equations, they should see that 6 = 2⋅12 does not make sense. The correct equation should become apparent. Should...
Of course, most youngsters need to practice many of these before they reach comfort level.
Your thoughts, suggestions, anecdotal evidence???