Approaching word problems

My students tend to give up in frustration as soon as they see a word problem, and so I increasingly avoid assigning such problems for homework and make sure we spend class time on them instead. There's a strategy for working with word problems that I read about somewhere - can't remember where, unfortunately - that involves paraphrasing the word problem within the constraint of an upper word limit, then paraphrasing the shorter version with an even tighter word limit, and so on. After a sufficient number of iterations, use of mathematical symbols becomes necessary to condense the information further, and so the word problem becomes translated into algebraic formalism.

I have not tried this method as stated, but it would be interesting to do that some time. The graphic organizer* I used a few weeks ago for systems of linear equations is inspired by this idea, however. There are little boxes** for each of the following:

1. What exactly is the question? (What are you asked to find?)
2. What are your variables?
3. What information is given? List it or write a table.
4. What equations can you write relating the quantities?
5. Solve the equations.
6. What is your answer?
   

Circulating among the groups, I found that the instruction to be very terse needed to be repeated quite a few times. That might be due to the fact that I have otherwise encouraged full sentences and elaboration... it must have been confusing that I was now insisting that the students be brief to the point of ignoring rules for decent writing. Many students initially started copying the entire word problem into the first box, groaning as they did so, instead of writing down the shortest sentence fragment possible that would convey just what they were going to find out.

It took a while for most students to realize that the variables they were to define were directly related to the questions stated in the previous box, that the variables basically were symbols for these quantities. Many tried to assign variable names to known quantities instead. I might try and rearrange the worksheet to visually reinforce the idea that the box containing the question and the box where variables are defined belong together.

In response to the prompt to list the given information, students were again inclined to be somewhat long-winded, and we'll need to work more on extracting the essential information and writing a table. Maybe insisting on a table is moving a little too fast, actually - once that is done we're practically in the next box already. As an intermediate step, maybe just listing the numbers in the problem together with a key word for what they quantify might be better.

The next part, writing down equations relating the known and unknown quantities, remains somewhat hard - but at least it's easier now that the students don't jump directly from skimming the problem to this step! I've given the students 2-3 out of 5 points on test items just for completing steps 1-3 above. That may sound like watering things down, but it really has resulted in more students even attempting the word problems - and once they have completed the first 3 steps they are much more likely to be able to complete the rest anyway.

The "what is your answer" box is for a sentence answering the question in the first box, and this answer has to make sense in the real-world context of the problem: units are included, and answers of the kind "4 remainder 2 buses" wouldn't work there, of course.

*inconveniently on my school computer just now.
**there's nothing like little boxes for prompting students to write something and not skip a step!


When I make up my own "real-world" problems they often involve pink dragons with purple wings and silvery scales. Some students roll their eyes then, but the dragon problems make me happy, and at any rate it would take a lot to make problems more boring than the ones in the textbook. Why are they all about ticket sales, long-distance phone calls, and cars? Yawn.

Language acquisition and learning math

Just finished a class necessary toward clearing the credential, and in that context I've finally read through an actual textbook on teaching English learners, which was good. Thinking about strategies for teaching vocabulary, though, I think it would be helpful to distinguish between the kinds of words that can be taught by pointing to whatever the word refers to on the one hand, and the kinds of words where the referent of the word needs to be constructed from scratch. Learning math involves more of the latter kind, and the strategies involved for internalizing such words are so different from those required for the former kind that it seems misleading to even subsume the two under the same header of "vocabulary acquisition."

One lesson that I have not been very successful at teaching involves a lot of vocabulary of both kinds, so for one assignment I focused on this lesson on classifying numbers. I've been puzzled at how poorly my classes have done on this topic. Some of the problems, such as the inability to remember what "integer" means, would be addressed with the usual bag of vocab learning tricks of flash cards, personal dictionaries, repetition, and repetition - this would be a word of the first kind, where the learning task is simply to memorize a letter combination and to link the word to its referent. However, words such as "equivalence," "inclusion," and even "all," "some" and "none" are also necessary, and maybe such content area concepts are all so unique that no list of teaching approaches can be made - on the other hand, I'd be surprised if the field of linguistics does not provide some broad, general insights on acquiring such words. However, whatever these insights might be, they have not trickled down to the teacher ed classes I've taken, at least not while I've been paying attention.

All Algebra I and Algebra II textbooks I've seen introduce the topic of classifying numbers by presenting a Venn Diagram of the usual sets, and provide no further clarification about how Venn Diagrams work, assuming, it would seem, that this visual is self-explanatory and can replace a verbal discussion of basic ideas of sets. However, students do not generally know how Venn Diagrams work. In trying to clarify the idea I have asked multiple Algebra I and Algebra II classes to draw Venn Diagrams of statements such as

1. All high school students study math.
   
2. Some high school students study music.
3. No high school students are senior citizens.

Generally, students can deal with statement 2, but when presented with diagrams for statements 1 and 3 they are not only initially confused - they also frequently express disagreement, claiming that they have learned differently in English class. Basically, the only type of Venn Diagrams they are comfortable with are those for intersecting sets. So much for leaving interpretation of these diagrams to the learners.

After class discussions and multiple examples of Venn Diagrams for situations where one set is included in the other or where the sets are disjunct, a majority of students will get the point and draw accurate diagrams for natural language statements. However, many students do not get to this point, at least not in Algebra I, and will continue to diagram "All x is y" by drawing the circle for y inside the the circle for x, and I am wondering why. I have not yet tried to find out whether it is mainly a matter of not matching the appropriate visual representations with the words "all," "some" or "none," and whether they would be able to correctly answer questions about sets of numbers based on sentences alone. I have basically emphasized translation between Venn Diagrams and statements. Whatever the problem might be, I have a greater appreciation for how hard it must be to learn high school math without a pretty strong grasp of pretty abstract words for referents that can not simply be pointed to.

Love and Graph Paper

The last block on Friday is not the best time for Algebra, the girls are giggly and unfocused, they blurt out random things as weekend thoughts flit through their unwinding minds, and I am tired too and thoughtlessly answer the question when it comes from out of left field:

- Ms. C., are you married?

- Uhuh.

Murmur and wide grins all around. My brain starts scanning for ways of getting us back on task again as the girls start hurling follow-up questions:

- Ms. C., Ms. C., do you love him?

I wrongly see an opportunity for getting back to Algebra, and nod enthusiastically.

- Yeah. He's really good at math. I love him.

It's a wildfire of hilarity around the room - how could I not have anticipated that? I make a stern face and tell them to settle down. They actually do get serious for a moment, then ask again:

- Ms. C, how did he propose? Come on, please, tell how he proposed!

- No more questions now. If you're really focused for the rest of the block, you can ask off-topic questions during the last three minutes of class.

That's probably not a management strategy in accordance with the books. However, the girls do actually get into their graphing assignment then, and are pretty productive for a Friday afternoon. But they don't forget, and when the bell rings and they're free to go home they're asking again. I'm mystified by their romanticism. Seriously, who's into proposal styles when they're 15?

- What if I proposed to him?

Bright-eyed astonishment, and a wave of surprised laughter: - You did? That's cool! What did you say?

- Maybe I wrote a letter instead.
- On graph paper.

I shake my head at the uproar.

- You don't seriously believe everything I say?

The girls troop out and leave for the weekend, and I find myself grinning while recalling the dialog, as well as remembering those elements that make up the true half of the story. We have an anniversary today.

A good thing

This Thanksgiving I am grateful for the privilege of work. I have meaningful tasks and am capable of performing them. If one could ask for only one thing - would that not be it?

Next Vista - tiny math lessons online

In an article about teaching students about mean, median and mode during English class, Todd Seal casually links to Next Vista's collection of math videos. There are minute long clips on simplifying fractions, adding negative integers, comparing ratios, and other Pre-Algebra/ Algebra topics. They are very short and very clear. On Friday I assigned the watching of any three of these over the weekend for homework, and already an Algebra kid has e-mailed to say they were helpful.

What if similarly focused mini-lessons on all the California Math standards were available for free online? It seems such an obvious idea that it's almost surprising that it isn't there already - but I haven't seen anything more comprehensive around. Have you?

Mean, Median and Mode

A cartoon in my 8th grade math textbook in a different country made the notions of Mean, Median and Mode stick in my mind, and pointed toward the significance of these different measures of central tendency. Can't remember any author or title of that old textbook, unfortunately, but here's a reconstructed version assembled with tools from ToonDoo. I used this with my Algebra classes a week ago.


Referring to "Secretary 1" in the salary scale as "the favorite secretary" also generated some discussion about which if these measures of central tendency says anything interesting about the data and about our research question, which is whether this employer pays his subordinates decently.

Bar codes or something on homework papers

Grades were due this morning, and so I'm emerging from a few days of intensive data entry and wondering why, why, why we don't use technology for this mind-numbing part of the teaching job. The chore of entering those strings of numbers into the spreadsheet ranks high up there among factors that could drive me out of the profession - few things are as stressful or irritating. I invariably get some numbers in the wrong columns (and at my new school minor data entry errors generate time-consuming exchanges with students and parents about what could possibly have happened with that one assignment of two weeks ago), and the process of switching focus from a sheet of paper, to the screen, back to the sheet of paper, and back again to the screen makes me dizzy. And late assignments - the worst part about late work is going back through the spreadsheet, scrolling up and down and back and forth, to locate the column and enter the score. I'm not naturally good at this kind of task, probably rather poorer than average, and would never have signed up for an accounting job. On the other hand, why would we make strength at such a monotonous, tedious task a critical part of being a competent teacher? I wouldn't make a good robot - but there are reasons why robots were invented, aren't there?

I'm lucky in having small classes and a block schedule this year, so some three assignments per week per student - but many public school teachers have 150-180 students with daily assignments. How they get through the data entry part is a mystery to me. Here's hoping they all get competent TAs - or that someone introduces scanners and software for this task ASAP.

Of course going through the papers and checking for completeness must be done by a human, and this part has some interest. It teaches me a lot about what the students need more help on and what misunderstandings are typical. But when I've gone through the stack of papers and written my comments on them, what I'd like to be able to do is to feed the papers to a scanner and immediately have the assignments entered in the correct cell in the spreadsheet. Late assignments would be as straightforward to enter as today's assignment.

To make the papers scannable (is that a word?), maybe each student could just get a roll of stickers on the first day of school, and they could place this sticker on the top right corner of their paper. There could be bubbles on the sticker for filling in an assignment code and also for the grade. The computer could take care of the rest. In order to check that students completed the sticker correctly the program could maybe highlight the most recent entries before closing, giving the teacher a chance to quickly check that what was entered since last time really was in the correct column.

Now, wouldn't that be nice? Maybe some computer science students could start working on the program while fulfilling course requirements of some sort, and make the resulting code available for free. It would be worthwhile a contribution to mass education to free up time that teachers use for data entry. It would leave more time for designing interesting lessons and giving the students real, relevant feedback - tasks for which I have considerably more aptitude than I have for spreadsheet management.