Friday, September 12, 2014

Math problems of the week: a Common Core-inspired multi-step word problem

A continuing series...

Today's problem comes from ixl, which promotes itself as "the Web's most comprehensive K-12 practice site" and which, on its math page, explains that

IXL's math skills are aligned to the Common Core State Standards, providing comprehensive coverage of math concepts and applications. With IXL's state standards alignments, you can easily find unlimited practice problems specifically tailored to each required standard.
Here's of its multi-step word problems for 8th grade:


This page provides a randomized series of problems of similar difficulty. What they all have in common is that you can pretty much set things up as you read along. In this problem, for example, each sentence corresponds to a clear-cut mathematical step. Sentence 1: do nothing. Sentence two: we clearly have a bunch of costs to add up. The only complication is doubling that of the lemonade--no big deal for 8th grade. 4.40 plus 4.25--you can do that in your head; add 1.40, also in your head. Total: 10.05 Add the tax (already calculated for us)--easy: 11.05. Then we have the $15 given to the waiter: it's clear before you even get to the last sentence that you're going to be subtracting the 11.05 from the 15.00--again, a "friendly number" computation easily done in your head.

Compare this to one of the challenging word problems for 6th grade Singapore Math:
Each pencil cost $0.30 less than each ruler and each ruler cost $0.40 less than each pen. Weimin bought 2 pencils, 2 rulers and 5 pens and paid $5.45 altogether. How much did each ruler cost?
Though the numbers here are no less "friendly" than those in the Common Core-inspired problem, this problem does not reduce to a series of simple arithmetic computations that you can do as you read it through sentence by sentence. "Each pencil cost $0.30 less than each ruler"--what do you do with that? You could either do it algebraically, defining variables, expressing quantities, and setting up equations, or you could do what Singaporeans do, and draw and label some bar diagrams--once again deciding how best to represent things. In either case, it's helpful to read the whole problem first to see where it's heading. Finally, you have to pay careful attention to which quantity the problem is asking you for, and what this quantity corresponds to in your chosen setup.

Another sort of problem that we see less and less of in contemporary American math are those with multiple simultaneous constraints-- of the sort that require you to set up multiple simultaneous equations with multiple unknowns. Here, once again, you have to read through the entire problem before you set it up, and how best to set it up can take a fair amount of all those things to which today's education experts pay such enthusiastic lip service: grappling, grit, and higher-level thinking.

Most contemporary American math problems strike me as the equivalent of long sentences that consist of simple sentences strung together on a single strand. Their more traditional or Singaporean counterparts are instead like complex sentences with lots of subordination. It's the difference between:
Ten people got on the bus and then two people got on and then 1 person got on and 2 people got off and then 5 people got on and 1 person got off and then 3 people got on and 10 people got off and then 1 more person got on....
and:
Of all the people who got on the bus that morning, the one that made the greatest impression on those passengers who were paying attention--for, given how early in the morning it was, few people were doing so--was...
Even if each sentence boils down to "What was the name of the bus driver?" there's a huge difference in the amount of work you have to do to make sense of it all.

2 comments:

lgm said...

A 4th grade word problem presented in 8th grade is appropriate for the inclusive nonhonors classroom.

The issue is that honors may have been eliminated,so that the students who would have benefited from a more appropriate level, pace and depth have no opportunity at all. This scenario is sending middle class parents here to homeschool or online providers.

Anonymous said...

Claimed to be from a 7-th grade pre-algebra book:

A man is 3/8's of the way across a train bridge, when he hears the whistle of an approaching train behind him. It turns out that he can run in either direction and just barely make it off the bridge before getting hit. If he is running at 15 mph, how fast is the train traveling? Assume the train travels at a constant speed, despite seeing the man on the tracks.

Taken from http://www.ocf.berkeley.edu/~wwu/riddles/easy.shtml

This is what a real math should look like.