Sharing worksheets

Discovered Box quite serendipitously a few weeks ago, while figuring out something for my Midsession class - and this solved part of that problem of having multiple versions of worksheets on multiple computers, only some of which are connected to a printer. It turns out it can also solve the problem of sharing materials. My goal is still to contribute to I Love Math, but my materials are mostly written for the first time this year, and I'm constantly fixing typos, and constantly updating files once they're posted to I Love Math is just not going to happen this year. Meanwhile, the worksheet versions I'm using, warts and all, can be made available to anyone by a simple click if I store them at Box, as I had started doing anyway. Here are two assignments for my Algebra classes for this week - and when I get time to tidy up my files a little and tag them somehow, I'll put more materials in the public folder. It won't happen this week - there's an ed class deadline looming now - but hopefully sooner rather than later. Constructive criticism will be welcome.

Update (August 2): Moved everything except tests over to a public Box at after some cleanup this week. Not that rummaging around in other people's files is generally very edifying, but making decisions worksheet by worksheet about whether to share it or not has ended up meaning that it just hasn't gotten done, and besides the very idea that my files are public just might make me keep them a little more organized. Maybe. Without guarantees (of anything), though, they're at http://public.box.net/hmath

Pink Dragons and other Real World Applications

In order to determine what would make science classes appear relevant to learners, researchers of the ROSE project actually asked the students. What they found was that teenagers care little for learning about plants in their local area, how car engines function, and how chemicals interact. They are significantly more interested in learning about how atomic bombs function, why stars twinkle in the night sky, and phenomena that science still can not explain. And the item of most interest to the young learners was the possibility of life outside earth.

So much for the mantra that science must be made "relevant to students' daily lives."

I should not be surprised. I can dutifully and with some determination work up a bit of interest for the functioning of car engines, but only a little bit. I majored in Physics.

Why do we think that math problems will be more engaging to students if they are about bake sales, CD shopping, and other real world applications? And those little vignettes in the textbook that purport to explain how useful and applicable all this math will be - why do they always seem so contrived? Who thinks that a note in the margin stating that "If you become an ornithologist, you may use polynomial functions to study the flight patterns of birds!" will be more convincing to the kids than it is to us? And if the value of high school math for students' daily living were so clear cut, why isn't the case made more forcefully after so many years of textbooks?

Svein Sjøberg of the ROSE project argues that the reason why all students should learn science is not primarily that this knowledge will be so useful to them in their daily lives, nor should it be society's need for a sufficient supply of engineers and technicians. He instead emphasizes 1) the cultural argument and 2) the democratic argument. All citizens need to learn science because science, like arts and history and poetry, is a part of our common human heritage. Also, political decisions about issues involving science ought to be made by an informed electorate.

By the same line of reasoning, primary rationales for learning math could also be the cultural and political weight that this subject carries. Humans have calculated, devised and solved puzzles, and developed multiplicities of algorithms in all kinds of cultures throughout thousands of years. Accessing some of this heritage is part of the enculturation of a person in today's world - it is a privilege, not something we need to excuse or justify with awkwardly implausible future employment scenarios. As for the democratic significance of math, must not an informed electorate be able to interpret data displays and ask critical questions about statistical statements?*

There are times when I feel that my subjects are gatekeeper courses rather then essential components of a well-rounded education, as when I see a student aspiring to be a nurse struggling with logarithmic functions, and I wonder who ordered this, who has an interest in setting up this barrier between a dedicated and in many ways talented student and her choice of profession? On the other hand thinking of math in other terms than job training makes teaching it so much more interesting. I can happily create ridiculous word problems about pink dragons and syrup fountains, and remember that "relevance" for a teenager need not have much to do with usefulness in some narrow technical sense. The "relevance" of a math problem may have to do with the investment in completing it faster than the neighboring team, the joy of working together with a classmate on it, or the beauty of the graph when it is done in colored pencil.

*If we take the democratic argument seriously, maybe we should consider replacing most of Geometry with Applied Statistics as a graduation requirement and make formal, proof-based Geometry a college prep class rather than a course mandated for all citizens.

Emergency Math

Sarah at Mathalogical has suddenly gotten her course load increased to four preps (General Math being the latest addition) with little curriculum attached, and she's asking for suggestions. I'm responding here because the comment got too long.

First, four preps without textbooks or curriculum is rough. I did that last year, am veryvery glad it's over, and wasn't proud of the results. On the positive side, it gives you exposure to a large range of typical conceptual hurdles in a short amount of time, and your toolkit will grow very quickly. You'll know a lot more about just what your students in later courses aren't getting due to your experience with this course. In order not to get too discouraged it may sometimes be necessary to remind yourself of how much you're learning when you don't get enough time to prepare what it takes to have the students learning enough, selfish and futile as that may sound. And starting this marathon now rather than in August means you can try things out knowing that you can start over again in just one semester.

The three resources I found of most use last year were

1. I Love Math
2. The Math Worksheet Site (this costs $20 per year), and
   
3. The National Library of Virtual Manipulatives

I don't know that these are better than anything else out there, but these are the ones I returned to again and again, and where whatever did work usually came from. With only that basis for recommending the following, here's what I did:

There was no time for dreaming up a coherent curriculum with much by way of unifying themes or red threads, so in the General Math type courses I prioritized according to what skills I thought were hindering students the most in accessing more math. Some areas I focused on were

1. Integers on the number line. The Math Worksheet Site has neat pages of number lines with addition and subtraction problems that the students solve by diagramming the problem on the number line. A large number of 10th graders could not deal with negative integers, and in most cases these number line problems helped. The very idea of associating the numerical operations of addition and subtraction with the geometrical idea of motion along a line is the Big Idea that students just have to get in place, it's much less obvious than we like to think, and missing skills in this area really holds the students back.
   
   
2. Place value, and decimal numbers on the number line. First, placing these on the number line was a priority - though in many cases I did not succeed in teaching this. Dan Greene has great stuff on it (as you would already know) - but teaching place value just is not easy. It's awfully important, though, as the kids trip badly over this missing skill when they attempt to do more advanced stuff, so if you can do anything for them in this area, you're helping, even if it sucks up quite a bit of time. The Math Worksheet Site has lots of practice sheets for translating between Decimals, Percents and Fractions, and they're tidy and neat for what they do. As for resources for placing the numbers on the number line, the worksheets at this site aren't that satisfying. There must be animations out there that let you zoom in on a piece of the number line to study place value - but I haven't found anything great, and spent quite some time searching for it last year.
   
   
3. Solving simple linear equations. The common student error that bothered me the most was students' insistence on subtracting the coefficient of the variable instead of dividing by it - my explanations just did not work, and they were inelegantly wordy. What did work for many students was practicing with the Algebra Scale Balance at the National Library of Virtual Manipulatives. After working on this site the incidence of that error went down very noticeably, and it's the concrete representation that does the trick - doing a verbal version of this lesson, well, good luck. For practice problems, the "Partner Problems" worksheet for equations at I Love Math is great. It has two columns of problems of increasing difficulty, and horizontally aligned problems have identical solutions, so that the students can get near immediate feedback on their solutions. The students liked that sheet, and would gladly redo it if I photocopied it onto paper of a different color (and yes, they did need the repetition).
   
   
4. The basic operations. Many kids were more likely to settle down and do something when their assignment was a boring worksheet on practicing multi-digit multiplication, a fact that always puzzled me - my "interesting" discovery activities were much less likely to elicit absorbed concentration (they would involve reading a line or two of directions for each task - bad, bad idea :) The Math Worksheet Site has lots of practice worksheets, at various levels of difficulty, and the card game Top Deck at I Love Math (in the Middle School Folder) is a lot of fun. (Digression: The card games for practicing skills with fractions worked less well, because students tended to devise their own rules that defeated the purpose of the activity: for example, they'd agree to match denominators of different fractions rather than matching fractions for equivalence, as I wanted them to do!)
   
   
5. Area and Perimeter. If students can just get the difference between the two, nevermind formulas for calculating anything, that helps - it was a defining moment for me that October day when I realized that the students truly were unable to distinguish the two - that was when my ideologically rigid commitment to grade level standards started to give. A hands-on activity (measure the area of your desk in terms of number of colored paper squares you need to cover it; measure the perimeter of your desk in terms of number of standardized pieces of string you need to reach around it) did some good, but only some. A worksheet from a colleague, which involved drawing rectangles on a grid that all had the same area but different perimeters, or the same perimeter but different areas, did more good. There were still plenty of students who had plenty of trouble with just counting up line segments to find a perimeter of an irregular shape, though, and - well, I don't know what to do about that.

That was a sort of braindump of what for me emerged as priorities in a general math type course for underperforming students last year, without any authoritative pretenses. Never took a math ed class (and wonder whether they'd ever deal with 10th graders enrolled in Geometry who add 5 and 3 on their fingers and don't know multiplication from addition). If you are able to post about upcoming topics a week or two ahead of time ("We're doing the Pythagorean Theorem next week - what are the students going to struggle with?") you just might avoid some of my unpleasant surprises ("Squares? Square roots? What's that?") and do something relevant to what the students actually need.

Student-friendly blogs?

My school has a two-week "Midsession" between the fall and spring semesters, during which time we get to teach pretty much anything that we can persuade enough students to sign up for for two hours per day. It's one of those really-too-much-fun-to-get-paid-for things, for sure. I've got a gathering of some 10 students for "Technology for Communication," wherein we'll be reading and writing blogs, playing around with PowerPoint, and maybe - just maybe - creating a simple Podcast, though since I've never ever done that before myself and have no idea how to do it that might be wildly unrealistic. I was thinking it would be fun to teach a course that I'd learn a lot from myself, and for these two weeks anything that the students are enjoying as well as learning something from seems to be okay.

The students' familiarity with technology is going to be all over the place, with some students barely able to use e-mail and others - actually, I have no idea about the other end of the spectrum. My plans are still somewhat vague - in part because I'm half expecting to have to rewrite them in an intensive night after learning about the students during the first class.

One of the first things we'll do is subscribe to a few blogs, and now I'm looking for good reads for high school girls - preferably clustered around a theme or three. I was thinking Study Hacks, Cake Decoration (I used to be somewhat into novelty cakes before starting to teach), and - I don't know about a last theme. My question to the all-wise blogosphere is: what themes or blogs would you recommend for this reader group? I mostly read edublogs of various kinds, with a very small number of political blogs sprinkled in. Not terribly exciting for my students, I'm afraid. Of course, I could delay this part of the course and find out about the students' interests, first - maybe that would be better...?

So - any suggestions (on any aspects of the course, actually)?

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.