## When vocabulary isn’t the issue

With English as a Second Language students it is well known that known how to do mathematics is only half the battle; the other half is understanding what problems mean in the first place.

It’s generally assumed that vocabulary is the central problem, and when the gaps are filled in the students will be fine.

[Released item from the 2006 AIMS Math test.]

Neither the mathematics nor the vocabulary are issues here. Yet I had students do poorly, because they could not understand what the question is asking.

How would you handle teaching (the reading of) this question? This is at the fringes of my ability; while I can get the students to figure out vocabulary from context or wade through academic language, I couldn’t find a good way to lead students to the answer here.

### 20 Responses

1. Not only with ESL (or ELL as we call them here now) students, but many students in general would see all the words there and think, “Word problems? I HATE word problems!” and then give up. I would guess that some English speaking students would read through the problem and decide that the other words there don’t really matter and work the problem.

I wonder if the ESL students see words they don’t know and think they’re important, but can’t get to their meaning. Mathematically, almost none of the words matter, but if you didn’t know that, it can be overwhelming.

Is that what’s happening here?

2. Possibly relevant post:
http://samjshah.com/2009/10/26/nctm-day-2/

This blogger went to the NCTM conference and discusses a workshop on how to start with a set of data points and lead students to the associated equation (instead of the other way around, which is how it’s often taught). It might be useful in helping your kids.

Maybe there’s a way to teach your students how to recognize patterns among data. So when they see a table like this, they immediately start to think “how can I generalize a way to get from *t* to *C* ?” (i.e. make a rough guess as to the equation). And then when they look at the list of possible equations, they have a better intuition of which one is right.

The blog article gives an example of this using Pick’s Theorem and systems of equations.

• Keep in mind the math wasn’t a problem here: as soon as they understood what was going on they could solve this. So I’m not sure if adding another mathematical method would help.

3. What about relating it to something that most of them understand… like a cell phone bill? You could make up a situation like you pay 35 dollars a month for 1000 minutes and get charged 40 cents for every minute you go over. Go through solving that problem talking about constants and variables in terms of what you will always pay no matter what versus what amount will change because of you went over. Make a table and then transfer over the cellphone problem over to the plumbing problem.

This is a really tough question because you don’t want to get too technical. Maybe asking questions such as how much is “x” changing by in each row of the table; how much is “y” changing by in each row of the table; how can we use this information to tell how much more we’re paying the repairman with each passing hour? So if we’re paying the repairman “z” amount of money with each extra hour he’s at our place, then how would we find out how much money we’d pay him when he shows up to our house at 0 hours.

But then again, these questions are focusing more on the math side. Good question!!

• I think your formulation of the problem is a much better one than the one given on the AIMS test because the latter is amenable to “shortcuts” (which really isn’t good for learning mathematics) whereas the former, I believe, is what the test-takers really had in mind.

• To be honest, I’m not sure if the test-makers meant the question to be more than a plug-and-chug exercise. I only gave it to the class because it was an actual released question.

4. I am not sure what grade level the problem is meant for, but a student might perhaps reason as follows (on an actual test): “The choices (A through D) are equations to a straight line (with ‘t’ representing values on the “X-axis” and ‘C’ those on the “Y-axis”). Hence, we are talking about a graph of a straight line. Taking a pair of points, say, (1,75) and (3,145), we can then use one of the straight-line equations to find the answer.”

I would add that this problem isn’t a very good one, in the sense that, a student with a keen mind would be able to eliminate all the wrong choices by plugging in $t = 1, 3$ to find the corresponding values for \$C\$ without even knowing anything about straight lines. Of course, such “shortcuts” are inevitable given the nature of multiple-choice tests.

5. While the mathematical suggestions have all been lovely, remember the central issue is that the students were reading “Which of the following equations could be used to determine the repairman’s charges for a repair?” and (I guess) the prose came across to them as a formless blob.

6. […] When vocabulary isn’t the issue […]

7. “How would you handle teaching (the reading of) this question?”

I would rewrite the question! I’m a native English speaker, and it took me nearly thirty seconds to parse the first sentence. Here are some of the things that make this sentence hard for me to read:

1. The word “represents” is rather ambiguous in this context.
2. The symbol C is used before it’s defined.
3. C is defined by the appositive phrase “an appliance repairman’s charges,” but there’s no comma after “charges” to show that the appositive phrase is over.
4. The symbol t is said to represent “the hours it takes to make a repair,” when in fact it represents the number of hours it takes to make a particular repair.

Here’s my attempt at a rewrite:

A repairman uses the following table to determine how much money she will charge for a repair. In the table, t represents the amount of time it takes to do the repair, and C represents the amount of money the repairman will charge.

The rest of the question could be cleaned up a bit too (for example, the word “total” in the table title could be removed), but all of the really heinous problems are in the first sentence.

• Thank you, that was extremely helpful.

Given this was an actual problem on the standardized test the students have to take to graduate, how would you get them to be independent enough to make sense of the reading as you did?

• Aaack! Reading the sentence “How would you handle teaching (the reading of) this question?” again, I see that I have a reading comprehension problem of my own. 🙂

I don’t really know how I figured out how to parse the question, but I suspect that the most important part of my strategy wasn’t very sportsmanlike: I’ve seen enough of these questions to know that when you have a table with columns labeled C and t, and answers that are equations in C and t, you’re probably supposed to pick the equation that fits the table. Once I thought I knew what the text was supposed to mean, it was easy to go back and verify that I was right.

Of course, this is exactly what students aren’t supposed to be doing. Reading the numbers and ignoring the words can get you into all sorts of trouble, as may or may not be demonstrated by kids’ responses to “how old is the captain”-type questions. But if the words don’t make any sense, you don’t have any choice…

I think there’s an interesting analogy here with crossword puzzles. My grandpa is a crossword fiend, and I recall him saying that a big part of being good at crosswords is knowing the unspoken rules. A lot of the tricks used in the clues are conventionalized, as are a lot of the clues themselves. For example, if you see that the clue for 2 down is “wrath,” you immediately know, without even looking, that the answer is probably “ire.” Unfortunately, the only way to learn the conventions is to do a lot of crosswords!

8. Do the ELLs have the opportunity to have the test read aloud to them?

This issue is one that my students and I struggle with, too. My students aren’t technically ELLs, but in a way they are (I teach Deaf students who communicate through American Sign Language and receive all instruction through ASL, as well). Their IEPs have accommodations for the tests to be interpreted, but when questions having an assumed context pop up, many students flounder.

Do you think the students struggle with comprehending the wording as written, or is it an exposure issue: not having experienced payments dependent on time?

• Do the ELLs have the opportunity to have the test read aloud to them?

No.

They are only allowed a bilingual dictionary.

Do you think the students struggle with comprehending the wording as written, or is it an exposure issue: not having experienced payments dependent on time?

It seemed to be the former, although more experience with the latter would have helped.

9. I’m curious: how did native English speakers do with this question? Is this really an ESL issue, or is it an MSL (Math as a Second Language) question?

As this is a multiple choice question, part of the reading of the question must include the reading of the possible answers. If the students can look at the possible answers and realize that they are all linear and all have different slopes, then the way to approach this problem is to find the slope between any two points.

• The native speakers did fine. However, my sample size was small enough that I can only give that statement partial weight.

I would say in a sense it is a Math as a Second Language issue, but specifically Confusingly Written Math as a Second Language (see Aaron’s comments above), which is a lot harder to teach.

• It is a lot harder to teach and absolutely impossible to control or to plan for.

Have you asked the students who did well how they approached the problem? Or asked the students who did not do well what they thought the question was asking? Or where they became stuck/frustrated?

I would argue that the first sentence shouldn’t be read first. Find the question and read that first. This gives the table and the definition of variables some sort of context. Then, look at the table and see if you can make some meaning of that. Next, read the written definition of variables. Then it doesn’t matter if there is a missing comma or a dangling participle. Finally, since it is multiple choice, look at the answers to see if those help you make any deductions.

FWIW, I think if the test-writers stuck to non-multiple-choice questions, their wording would be much less ambiguous.

10. Are they allowed highlighters? Going after keywords may be a not so bad place to start.

Jonathan

11. […] learners read mathematics Posted on May 12, 2010 by Jason Dyer I have twice before now (here and here) pondered over the issue of how to help language learners past the […]

12. […] is a circumstance where vocabulary isn’t the issue, but phrasing […]