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.

Learning styles

I've been puzzled by the fact that the IEP of one of my students describes the student as a "visual learner," while I would have said that visual information is what she has the most trouble with. This student is pretty good at algebraic manipulations, but understanding graphs has been a challenge. In particular, she quickly learned how to compute the slope of a line given the coordinates of two points, but learning to find the slope by inspecting the graph - by counting the units of vertical and horizontal change - required a lot of effort and repetition. And even then it seemed to be easier for her to first determine the coordinates of a couple points on the graph and then to plug these into the slope formula, rather than to just read the rise and the run off the graph. Also, she apparently really struggled with Geometry last year.

Her IEP recommends that she be placed near the front of the classroom so that she, being a visual learner, can see the board well, and all of that seems perfectly sensible and easy to accommodate. However, I am wondering about what exactly is meant by these learning styles categories: Are we talking about the mode of input - about which senses are used - or rather about the character of the information processed? I'll admit from the start that I have not done my homework in terms of reading the academic literature on learning styles - just those bullet-pointed short versions used in alternative credentialing programs - and maybe the most appropriate response to my questions would just be a reference to some standard text on these matters. Which would be fine.

In trying to distinguish 'mode of input' from 'character of information' (or whatever; if anyone knows any official terminology in this area, let me know) I'm reminded of Bach-y-Rita's little devices for transmitting visual information by tactile means. Starting in the 60s, he experimented with optical sensors attached to arrays of tiny electrodes attached to blind subjects' backs. Initially, candidates just experienced random tingling sensations from the electrodes, but with some time and practice this gave way to an immediate experience of rudimentary vision, enough to find their way while walking. They could, for example, "see" if an object was placed in front of another, blocking the latter from "view." When a candidate equipped with such an apparatus learns about his or her world by using information transmitted through tiny electrodes attached to the skin, is this candidate acting a as a "visual" learner or rather as a "kinesthetic" learner? I'd say the former, even though the mode of input is tactile.

Presumably the information needed to navigate a sidewalk could instead be given in verbal form, as a list of instructions about when to move and when to stop, and this language-based information could be accessed aurally, from directions read aloud, or the mode of input could be tactile, as when reading the list of steps in Braille. However, if the information were provided as a list of instructions about how to walk, reading these instructions would hardly make the information "visual" even though the mode of input were now visual, and reading them in Braille presumably would not make it more suitable for kinesthetic learners. Similarly, I would think that if a student learns most readily by a visual approach, then the student's needs are hardly met by the information being available as words on a screen or board, because even though the mode of input is certainly visual, the character of the information is verbal - and being able to see well is quite compatible with having a language-based learning disability.

In teaching my students about the number line I have come to wonder whether I am over-emphasizing a visual approach because such an approach happens to be mine. I guess I compare numbers by mentally placing them on the number line, and that is how I have always made sense of the rules for negative numbers. However, my (mathematically adept) husband insists that he does not think of numbers this way - and certainly the magnitude of a number and its position on a line are not the same thing. After all, the Ruler Postulate, which tells us that we can define a one-to-one correspondence between numbers on the one hand and positions on a line on the other, and in such a way that differences between numbers correspond to distances between points, is not a trivial statement. Besides, I have this unclear notion that Geometry developed for centuries without numbers attached in any important way, while Algebra did without Coordinate Geometry and number lines for the longest time (but I've never actually studied History of Mathematics, and don't really know). Anyway, students who have trouble distinguishing left and right might conceivably be more confused by the number line approach while still being perfectly capable of reasoning with signed numbers. It might be interesting to try and think of approaches to the material that are less visually oriented, instead of just assuming that this approach pretty much defines understanding, since that is the way I, um, see it!

Not that it matters that much in practical terms. I'm sure that in terms of learning outcomes for my students, they'd have been better served by my spending this time on getting their papers back to them quickly instead of writing all these words. But, whatever.

The electric eraser cleaner

My classroom has old fashioned black blackboards on three walls, and the chalk dust has been getting out of hand. A couple days ago I took two erasers to the bathroom and beat them against the sink for a while, sending up clouds of dust and choking on it, and the morning after I wiped the board with a wet cloth instead. Upon which a student gave out the inside information: C-dog*, there's a machine for cleaning those erasers!

I had never heard of such a thing, and was fully prepared for this to be a little joke on the part of the student (she is funny). But a colleague confirmed that there was indeed an eraser cleaner, and pointed out where it was located, in a tiny, tiny closet containing this machine on a shelf and nothing else. It's this powerful little vacuum cleaner with a rotating brush, and it sucks the chalk dust into a ballooning green bag as it works. And it makes some noise while it's on!

My erasers are very clean now. Here's a picture of the machine. Isn't it the neatest thing?

*No idea where that came from - again, she is a funny student.

Learning linearity from scratch

I'm teaching Algebra I for the first time this year, and having occasion to take a closer look at possible sources of great confusion and misunderstanding in later courses. Teaching Algebra II, I've often resorted to the number line to work with negative numbers and with fractions, always presupposing that the students would have a basic grasp of the very idea of a number line. Now I'm seeing what it looks like when students don't quite understand how a number line works.

In graphing scatter plots, it turned out a number of students - a significant fraction in a talented class - were 1) varying the distance between the numbers, and 2) placing the numbers according to their order in the table rather than sorting them in increasing order. I returned the homework assignments with a new deadline for redoing the plots, directing students to keep the spacing between the numbers on the number line constant. Several conscientious students redid the graphs, dutifully keeping a constant spacing of, say, 4 graph paper squares between the numbers - but the numbers thus placed would still be, say, 41, 37, 58 - the order would not be strictly increasing, and the differences between the numbers were not constant.

So the "simple" task of creating a scatter plot in the beginning of Algebra I actually presupposes some grasp of the Ruler Postulate, which they won't see for a year - the postulate that it is possible to assign numbers to the points on a line in such a way that number differences measure distance. After reiterating again and again that a certain distance on the axes must correspond to a fixed number difference, and copying student graphs onto transparencies and critiquing them in class, and conferring with students one-on-one - the graphs are finally looking pretty good.

I was initially surprised by what I saw as a puzzling deficiency in PreAlgebra skills. But my husband, a theoretical Physics grad student, pointed out that I was expecting students to understand linearity as a prerequisite for this unit on linear relations. The idea of a correspondence between distance and number difference contains what I am supposed to teach them.

After this lesson learned I am wondering about the Algebra II students I taught last year, whether many of them also never really knew how number lines worked either, and I just didn't get it - or whether a bit of Geometry taken in the meanwhile at least takes care of that issue.

'Graph' means 'picture'...

... as in 'geography', 'graphic novel', and so forth. Many students have a hard time grasping the idea of a graph as a visual representation of an equation; they confuse the coordinate grid with the graph, and think of graphing in terms of a list of incomprehensible 'steps'.

Dan Meyer is doing great things about that, and if I am not too late already my students may get to see the light, or rather to get the picture, some time over the next two weeks.

What's so hard about piecewise functions?

I'm teaching about absolute value functions these days, and about rewriting absolute value functions as piecewise functions. The students have a lot of trouble with this. In the hope that I'll remember to check out my notes on what they're struggling with before next time I teach this stuff, I'm storing them here.

The big problem seems to be to understand that the restricted domain is just that, a specification of what interval of the independent variable we are concerning ourselves with, and that the domain is not somehow "a solution" or part of the actual function or something. I am clearly unclear about just what is the issue, and how to go about filling in the missing parts. Apparently Dan Greene has figured out ways of approaching students' tendency to "flip back and forth between x and f(x) in their minds". I guess I should backtrack and spend a whole lesson or two on picking out the segment of a graph corresponding to a particular x-interval. Not that there's really time for that, but... it is needed.

A meeting of Bay Area math teachers

Dan Greene organized a meeting of Bay Area math teachers today. We were not many, but it was good. We met in a cafe in San Francisco and chatted about teaching basic skills and building classroom culture, about school administrations and useful software for math teachers. Hopefully this will develop into a bigger thing, a forum where math teachers can share ideas and materials, dissect concepts and debate how to teach them, learn more about technology from each other, and maybe find encouragement from knowing that others are struggling with similar challenges. It would make sense that there would be some kind of society of Bay Area math teachers. (Might there be one out there already? If so, it certainly is not very visible! Hello..?! Has anyone heard of such a thing?)

We also talked about elementary math instruction, and about finding ideas and resources by seeking out materials designed for K-4 teachers. Both math and English teachers in high-poverty high schools can benefit a lot from materials and staff development designed for elementary school educators. We have to teach place value and multiplication, reading and capitalization - and the pedagogical approaches designed for teaching these things are hardly dealt with in Single Subject credentialing programs. Not that these programs are generally all that helpful with grade-appropriate pedagogy either, but the point here is about where to look for what we need, and that may well be in the Multiple Subjects teachers' bags of tricks.

Given the level of skill that our students have when they enter 9th grade, one can't help wondering about what goes on in middle and elementary school classrooms. When the students enter without mastering 4th grade standards, is that due to major classroom chaos in earlier years? Long-term teacher vacancies? Lack of subject matter competency among teachers? Anything else? Maybe some time in the future a local math teacher society could have monthly meetings where elementary school teachers could get help to deepen subject matter knowledge from high school teachers, while high school teachers could get input from elementary school teachers on making foundational concepts accessible. Just a thought.

While we may be inclined to think of teaching basic skills as a bit of a hassle, Dan enthusiastically insisted that teaching his Numeracy course is a lot of fun for the instructor as well as for the students. Making remedial instruction as enjoyable as possible is certainly the way to go: Since we'll need to do a lot of it, we might as well do it cheerfully.

There's nothing here.

Sorry.