In Susan’s article titled “A Linguistic and Narrative View
of Word Problems in Mathematics Education”, she looks at the pragmatic
structure of word problems to try to find the “unspoken assumptions underlying
its use and nature as a medium of instruction”. She writes how word problems are typically structured using
three components: the set-up component to establish the characters and location
of the story, an information component that lists the needed information to
solve the problem, and lastly, the question component. This part was very interesting to me
because this is how I actually make up word problems, and yet if someone were
to ask me to describe the process I take in making word problems I probably would
not describe it this way. I would
say that I use real-life scenarios that ask students to use the mathematics we
just learned to problem solve.
This brings me to the next interesting part from Susan’s
article about looking at word problems using linguistic and metalinguistic verb
tense. While I don’t quite
understand what these terms mean (there are definitions of them within the
article), an example she gave really brought home to me the point about word
problems not really making sense at times. The example is “A truck leaves
town at 10:00am travelling at 90km/h.
A car leaves town at 11:00am
travelling at 110km/h in the same direction as the truck. At about what time will the car pass the
truck?” She goes on to explain how
“a truck leaves” and “a car leaves” are linguistic present tense and
“the car will pass the truck” is
linguistic future tense. She
explains how the tenses used in word problems are often self-contradictory and
how this takes away from the truth-value in word problems, which is the last
thing I will discuss.
Susan gives a word problem and then rewords it with the information
in parentheses added. The word
problem is this: Every year (but it has never happened), Stella (there is no
Stella) rents a craft table at a local fun fair (which does not exist). She has a deal for anyone who buys more
than one sweater (we know this to be false). She reduces the price of each additional sweater (and there
are no sweaters) … The problem continues on in this same way. She makes note how the truth value of
the word problem doesn’t actually change when this extra information is added
in.
I personally have never really struggled with solving word
problems, but this article brought to my attention how flawed word problems
are. It is no wonder people struggle
with solving word problems. Their
tenses are contradictory and there is no truth in their statements. This made me question the purpose of
word problems. I understand that
we use them to try to make the math seem more realistic and to show students
where they can use the learned material in everyday life, but if the word
problems are bogus, are we really benefiting our students?
I entirely agree and disagree at the same time! I wholeheartedly concur that the grammatical structure in word problems is often erroneous. When deciphering a word problem from the textbook with my students, I sometimes have trouble understanding what the question is truly asking as well. At times, the solution in the back of the textbook is contradictory to the setup of the situation the word problem suggests. In many cases, I end up spending more time decoding a word problem than to explore interesting ways to solve it. As a result, I ask my students to skip questions that are poorly worded. Unfortunately, many of the questions in our current mathematics textbooks are horribly worded. Grammar aside, I would argue that employing hypothetical situations in a word problem can be accomplished appropriately. Although I see much value in demonstrating real-life situations in all the word problems, these situations could seem dated and detached to our students in the classroom. Nonetheless, due to privacy concerns, situations that happen on a day-to-day basis are not presented to the students. Rather, hypothetical situations are created in the form of word problems so students can identify the mathematical problem with their personal experience. For example, I may want to use the Super Bowl as the setting of a word problem and a hypothetical hyperbolic trajectory of a football to create a quadratic equation. To me, creating a meaningful mathematical problem is independent of the "realness" of the problem.
ReplyDeleteThe tense problem is something I have not thought about, perhaps because my first language is mandarin. In mandarin, "leaves", "left", "is leaving" and "will leave" are all understood as "leave" with different time descriptions. This is something I should pay attention to.
ReplyDeleteThe hypothetical settings are widely used in math problems. It is even impossible to find a setting of ONE question that is "true" to most students. I still remember the time when I just arrived Canada and took a math test in Langara. Many words--most of them are nouns that are used to describe senerios--used in problems are strangers to me. I simply pretended I had not read these words, and used simbolic letters, i.e., A, B, C, etc., to substitute the nouns. Well, I was still able to solve most of the questions. Do the students really feel confused about the settings? Maybe. But do they really get confused about the math concepts? Sure.
It would have been interesting to take two identically matched groups of students (matched for age, gender, grade, school, etc…) and give them "equivalent" word problems, apart from "surface" features such as the tenses of the verbs, or the caveats mentioned by Susan above. I suspect this would have had a significant impact on the performance on word problems. I think students are often sensitive to "surface features" of word problems, such the presence or absence of a diagram, the ordering of information given, the language used, etc.. This kind of sensitivity to surface features is surprising to experienced mathematicians, who can boil a word problem down to its essence regardless of surface features.
ReplyDeleteWhat Shan mentioned above is an interesting case study. He was able to boil a problem down to its essence despite not knowing the meaning of words. I think this is a skill which comes with experience, and is an essential mathematical skill. Once a problem is boiled down to its essence, it is easier to see the mathematical similarities between it and others. In some sense it is isomorphic to another problem the student has solved. But how do we teach students to learn to distill a problem down to its essence? Is it just a matter of practice or experience?
Word problems are odd in that they combine language comprehension with mathematics. They are really testing two disciplines. When reading a word problem with a student I resist the temptation to reword the problem in simpler language, since this is one of the skills I want my students to develop for themselves.