Monthly Archives: November 2006

Journal entry: “Collaboration”

Teacher Journal

11/15/06 Percents. I discovered that the district’s Chapter 3 test includes four problems on percent, when our pacing guide specifically skipped that system. So I developed what I thought was a nice, simple lesson on percent problems, including interesting stories from my own life. The stories were interesting enough, but the problems were too difficult. How dare I assume they would still be able to solve one-step equations, a full two weeks after we last studied the concept?? Rather than use the stupid system of the book – learn three different archetypes for percent problems, with a different formula for each – I taught the translation method. I showed them a nice two-column, four-row table, where “is” becomes “=” and so on. Fine, except then the second step is solve the equation you just created. They were left with a fraction, times an integer, times the variable x, and it all became too much math for their arithmophobic brains. “I don’t get it!” I could hear their eyes glazing over from the board, as I demonstrated canceling factors out of denominator and numerator.

Apparently, every time you teach The Right Way to solve percent problems, somebody pipes up and recites the way she learned it from her elementary school teacher. (This is the three-archetype method.) This is fine when you don’t know algebra, but it’s clearly inferior to the translate-and-solve method that you can use once you know how to work with equations. Just the same, some people would rather not give something up that works. Only both times I’ve had a girl stand up and announce that there’s an easier way, they either get the answer wrong or can only remember one or two archetypes, rather than all three.

11/16/06 I’m happy with a rewards system I’ve come up with. A minor branch of it is stickers: “WOW” stickers (which we like to call “MOM” stickers) which I put on papers for sitting down and beginning the work at the bell. To get one, you really have to have all your materials out and ready to go before the bell. This is inconceivable to a lot of the students, who feel you should be able to socialize, gaze off into space, and/or chase each other around the room until a little bit after the bell stops ringing, or until the teacher says, “OK, let’s get to work! You’re assignment’s on the board!” Whichever comes last.

The more important component of this system is a bag full of Room 14 Random Rewards. They’re like little Chance cards (of Monopoly fame) which the students get to pick for various good deeds: turning in homework (on an unannounced day), helping someone else, or articulating the perfect question or answer (vocabulary counts). Once I gave one to a student who put his hand on his paper so I could initial it without it spinning under my pen. The rewards include: a day working in my chair, a day as class factotum, three origami papers, one new pencil, one day operating the LCD remote, one half freebie on the day’s homework (full credit for half done), one free minute tardy. The chair, projector and tardy are popular, and they are distributed about the same as all the other cards (about four cards each per 32-card deck); I’m going to come up with a couple very special cards that will be more rare.

11/17/06 One of those classroom events that might have been handled better by an experienced teacher. Direct instruction went on way too long – mostly because of chatter and other distractions – and then the boys in Quadrant III started screwing around. This continued for about five minutes, in spite of my proximity and other subtle hints. I muttered, “Fine – we’ll start the phone calls home,” and then I walked to the computer, looked up one boy’s phone number. I dialed it and reached his grandmother; I put the boy on the phone and then attempted to resume the lesson. Soon, he was crying and saying he didn’t do anything. The conversation went on for another three or four minutes, and he finally hung up and gave me his father’s number. (Later on, when I reached his father, he acknowledged that his son is a talker but didn’t feel I handled it fairly.) I called another boy’s number and got an answering machine. There were two others in my sights, but I didn’t make it that far. Meanwhile, the neighbors of Boy #1 started telling me that he hadn’t been screwing around. I believe he was guilty of joining in, either this time or some other time; however, it was not my intention to humiliate anyone, just to get the attention of the distracted kids. I’ll probably abandon that technique.

11/27/06 In the interest of not “turning off” students who exhibit reluctant-learner tendencies, I had several private heart-to-hearts today: “Hey, uh, Jimmy. We need to talk. I’m trying to do my job here, which is to help you learn. Your job is to learn. When you choose to chat while I’m talking, it makes it hard for both of us to do our jobs….” Not sure if the somber attitudes of the students, as I spoke, was actual comprehension or just well-practiced “be quiet and let the old guy talk so he’ll let me leave” behavior.

I made an effort today to share tasks with students. One class clown in particular – who announced as he entered the room after lunch that he was “freezing his nuts off” – really warmed up to the responsibility of reading the night’s homework to the other students. He has his belligerent moods, and he definitely spent too much time acting like the teacher (“I’m going to have to write you up!”), but he surprised me by actually making an effort to read loud enough to be heard and to wait for class silence and attention.

11/28/06 Yesterday, I assigned two-paragraph essays about the state of things in the classroom. This was in response to (secondarily) the chronic sick state of behavior and (primarily) a four-day classroom-management workshop, which began last week and continues later this week. The first paragraph was meant to list three things that were broken in the classroom; the second paragraph listed three things that were working. I got the first set of these essays today and learned several things instantly. I believe the students gave me honest answers, and I was happy to see many complete sentences, along with a few topic sentences. I found that some people are very happy with how things are working. Of course there were generic complaints (too much homework, not enough opportunity to socialize), but many students had some very constructive things to say. I’ll try to incorporate some of these changes.

11/29/06 Got matched up with a buddy teacher at school this week, and he watched my worst group (period 6, after lunch) today. His impression was that it wasn’t as bad as I had led him to believe, but he had a few concrete suggestions, including giving the kids (returning after lunch to their second math period of the day) a required activity to get them settled down, practicing the basic classroom procedures (handing out and collecting materials, for example), and giving more wait time after asking questions.

The activity of this period was a step-by-step exploration of an equation’s intercepts, starting with “draw a coordinate plane” and “plot the origin.” Each student had a whiteboard: a transparent sheet protector containing a white card-stock sheet, with a dollar-store dry erase marker and a paper towel for erasing.

My buddy teacher specifically recommended that I give the students opportunities to discuss their answers with each other. He said that I should be requiring students to write their answers down before answering, both as a way to lighten a bit of their anxiety about answering questions and as a way to get them involved in the answers, rather than just zoning out or trying to get away with off-task behavior.

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Addressing Algebra Students’ Linguistic and Symbolic Preconceptions

When I began teaching math, my own pristine view of algebra was suddenly under assault by the incoherent and chaotic view that many of my students had of mathematical ideas. I thought the world (that is, the teachers and adult students I knew from several college careers) shared my understanding of equations, variables, constants, operations, properties and various key symbols. More accurately, I gave no thought to my understanding of the symbols and concepts of algebra (and the higher-math topics which depend on algebra), so consistent, reliable and solid was my understanding: I had nothing to question or explain about how algebra worked since I was as ignorant of inconsistency and incoherence as someone in a two-dimensional world is of a third dimension.

Imagine my astonishment when I first realized that my students had it all wrong. I’m not just talking about forgetting how to distribute, or getting the wrong product from 5×6. They used equal signs in ways I never would have considered: as short-hand for “and then the next step is” and as an indicator for “here comes the answer.” They added an amount twice to the left side of an equation and not once to the right side (apparently not understanding what a “side” was). They substituted a number into a variable with a coefficient and happily turned the whole variable term into a brand-new two-digit number (instead of multiplying the substituted variable by the coefficient).

The “ah-HA” moment for me came one spring afternoon when I was asking my students to read out loud some equations I had written on the board. I was attempting to focus in on and correct some problems with students who didn’t understand what the “sides” of an equation were. One girl read every element correctly on the left side of the equation and eventually read every element correctly on the right side, but instead of saying “equals” (when she saw the equal sign), she said, “…And then….” I heard this several other times during the week. A student would see the equation x + 5 = 2x – 7 and read it as, “X plus five … and then … two X minus seven.” There were usually pauses, which I’ve indicated with ellipses, which I took to mean both that the student was not comfortable with the equal sign and that the student recognized the equal sign as some sort of punctuation, to be carefully set off – like a comma – with a little pause in speech.

Way too late in the year, I’m sorry to say, I had discovered like so many before me that language was important to learning mathematics. This is commonly understood in the math-teaching field, although from outside the field, people commonly assume that a math classroom can be a sort of language-free zone, a place where the poor students of English and the English Learners should be able to flourish, “since math is all just symbols anyway.”

This view is refuted in the literature. MacGregor and Price (1999) conducted tests with over 1,200 students and found a positive correlation between language and algebra scores in each of three different grade levels. Vygotsky (1978) described how social interaction and communication are crucial elements in the process whereby children conceptualize words.

You often hear students express frustration over some other teacher who presents an idea and moves on too quickly, brushing off their complaints with some impatient statement such as, “I explained this already. You need to start paying attention.” The observant teacher recognizes that in many cases, the students are paying attention, but they see a completely different picture in their heads than the one in the instructor’s head, and of course a completely different picture than the one the instructor expected them to see. It’s as if the teacher is lecturing on palm-reading, thinking everyone in the class envisions a human hand, when what the students think he’s talking about is a giant tree.

One of the places where problems begin is that the teacher may think the students enter an algebra class innocent of all the material that the teacher will teach. Kaye Stacey and Mollie MacGregor (2000) discuss this situation and its ramifications in their paper Ideas about Symbolism that Students Bring to Algebra. They claim, “Teachers may think that students come fresh to algebra, not considering that they already have ideas about the uses of letters and other signs in familiar contexts.” In their research on more than 2,000 students, they discovered many misconceptions, which they collect into these four categories: “students’ interpretations of algebraic symbolism are based on other experiences that are not helpful; the use of letters in algebra is not the same as their use in other contexts; the grammatical rules of algebra are not the same as their use in other contexts; and algebra cannot say a lot of the things that students want it to say.”

In their paper, Stacey and MacGregor list the most common misconceptions, along with their explanations for the students’ misconceptions. Some of my favorites include:

  • Invented abbreviations which violate algebra customs, such as “Dh” to represent “David’s height”
  • Assignment of values to variables based on a variable’s position in the alphabet (e.g., h + 10 = 18, since h is the 8th letter of the alphabet and so must have a value of 8 )
  • Interpretation of a letter placed next to a number to mean addition, seemingly a best explanation for some students who are comfortable with the workings of Roman numerals.
  • Automatic assignment of 1 to any unspecified variable, apparently related to the idea that a variable without a coefficient means one times that variable.
  • Use of the equal sign to mean “makes” or “gives” (a habit from arithmetic), leading to such patterns as “3 + 5 = 8 x 7 = 56 ÷ 2 = 28.”
  • Application of natural-language grammar rules inappropriately in algebra, for example, interpreting a = 28 + b to mean “a equals 28, then add b
  • Invention of symbols to mean things where students have not been given a symbol, for example in situations requiring iteration which could not be represented by a simple equation (some students demonstrated creativity with constructions such as “x ↑ 1, y ↑ 1” meaning “as x goes up by 1, y goes up by 1”)

Interestingly, Stacey and MacGregor identified many of these problem misconceptions not only in students who were new to algebra, but also in students who were in their third year of algebra.

These examples are relatively specific. A teacher who successfully identifies one of these misconceptions being used by a student might be relieved to discover a problem as limited in scope as the natural-language grammar item above, since the solution begins with the simple explanation that algebra doesn’t obey the same rules as English. But what about more fundamental misunderstandings? Glenda Lappan (2006) writes about problems with the conception of variable itself. She points out that as in English, mathematical terms can have multiple meanings, and the listener or reader needs to choose the appropriate meaning from the context of the term. “Take, for instance, the concept of variable–something students must understand as they mature mathematically,” says Lappan. “Faced with a mathematics problem, students have to find ways to use mathematics to represent the situation, manipulate the representations to find solutions, interpret the solution in the original context, and look for ways to generalize the solution to a whole class of problems. Variables play a key role in the process of mathematizing a situation. But what meaning of variable for a given situation is appropriate? Is it a placeholder for an unknown? Or is it a domain of possible values for one of the phenomena? Or is it used in yet another way?”

Early in a first-year algebra course, we use letters mathematically a number of ways. We present an equation such as “y = a” and describe it as the equation of a horizontal line, casually pointing to the letters and calling them “variables,” without considering the sloppy definitions in play. In this example, although the letter y is properly called a variable, the letter a is actually what we as teachers know from our broader experience to be a constant, standing in for the domain of all possible values for y, but only for a single value from within this domain at any one time. It’s safer to call all letters “variables”; it would be extremely confusing to attempt to describe these highly abstract notions to a roomful of fidgety 13-year-olds.

In addition to what we present for our students to attempt to understand, we also ask our students to practice using the brand-new language we give them. As with the previously discussed dilemmas our students face using symbols, a student who must form his personal concept of what a variable is will often invent something that doesn’t match our concept, and it’s our job to recognize the flawed invention.

As Lappan accurately points out, as we teach our students the language of mathematics, “The challenge for our students is to learn both to use that language to show their ideas, … and to read that language to understand the meaning of someone else’s mathematical representations….” This requires us as teachers not only to listen to our students’ use of language but also to watch and judge how our students hear us and each other as we talk about mathematics.

References

1. MacGregor, Mollie, and Elizabeth Price. (1999). An Exploration of Aspects of Language Proficiency and Algebra Learning. Journal for Research in Mathematics Education. 30(4), 449-467.

2. Stacey, Kaye, and Mollie MacGregor. (2000). Ideas About Symbolism that Students Bring to Algebra. Algebraic Thinking, Grades K-12: Readings from NCTM’s School-Based Journals and Other Publications, 308-312.

3. Lappan, Glenda. (2006). The Language of Mathematics: The Meaning and Use of Variable. Retrieved November 27, from http://www.nctm.org.

4. Vygotsky, L.S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA. Harvard University Press.

Review: “Building Equations Using M&M’s”

The article describes an activity to support using algebraic techniques to solve linear equations in one variable, intended to directly follow instruction in equation-solving techniques. The author claims that the activity “actively involves students in identifying the variable, formulating an equation, and then solving the equation.” (Borlaug, 1997)

To begin, the teacher distributes small bags of M&M’s to each student and keeps a bag to use for him or herself. The teacher counts each color in his or her own bag and draws and completes a table on the board listing each color and the number of that color found in the bag. The students count their own M&M’s and make their own tables, but they keep their results secret.

The students then receive a list of questions, designed to be set up as equations and solved. The list begins with easy questions that the student will be inclined to “just answer” without algebra; the questions get progressively more difficult, encouraging the students to use algebra to solve them. The teacher selects a student and asks that student to complete one of the questions and then pose the question to the class for their consideration. For example, “I would have to add (or eat) ____ red candies to have the same number of red candies as the teacher. How many red candies do I have?”

The rest of the class works on a solution to this problem. As the questions become more complicated – “If I tripled the number of yellow candies I have, I would have ____ more candies than the teacher. How many yellow candies do I have?” – many students will decide that the algebra techniques they’ve been studying are – surprise, surprise! – useful for solving these problems. For those who don’t gravitate to algebra but who are stuck, the teacher provides friendly nudging. So, the next phase is to build and solve the appropriate equations. A side effect of this effort is that students sometimes discover that there is more than one appropriate equation, depending primarily on what color they choose for the variable x.

The author finds that after the main activity, some classes will be comfortable moving on and inventing their own problems. Here’s one actual example from the article:

I have a total of 61 candies in my bag. I have 9 more brown candies than orange candies. If I eat all my brown and orange candies, I will have 32 candies left. How many brown candies did I eat? How many orange candies did I eat?

This example represents an impressive level of involvement with the material; however, a number of issues come to mind as I consider the realities of using this activity in my own classroom. I would be concerned about my students just eating the M&M’s. I’m thinking of one student in particular, who has no self-control and who is frequently out on suspension for various discipline incidents. He has learned almost none of the algebra techniques that the rest of the class has picked up and so would not naturally be able to participate in this activity without hand-holding. An obvious approach would be to have students work in small groups; however, in this case I can’t imagine him contributing to a group larger than two. Even in pairs, it’s hard to imagine this student not just eating the materials. (I’ve seen him eat paper. Could he resist M&M’s?)

Another drawback I envision in a high-school classroom would be the non-traditional math student who prides himself on creative ways not to do the assigned work. If this student has access to the Internet, he could easily look up the distribution of colors in an average bag of M&M’s and use that to “cheat.” Of course, this would be more complicated than just doing the work and would only lead to approximate answers (due to the actual randomness of filling an M&M’s bag at the factory), but the student would have found another distracting way to avoid learning the material. This scenario is unlikely in most classrooms, especially a middle-school classroom such as mine.

Also, I’m not comfortable with giving candy to kids. Is that what the parents want? What about the teachers who have these kids later in the day? And do I want to fight the trash battle? But are there alternative manipulatives? They need to be similar in size and shape to each other and randomly distributed with a limited number of colors. I’ve seen little plastic colored circular pieces, like transparent tiddly-winks, that are made for overheads. Maybe the simplest solution is colored construction paper cut into one-inch squares and mixed together.

References

1. Borlaug, Victoria. (1997, February). Building Equations Using M&M’s. Mathematics Teaching in the Middle School, 2(4), 290-292.

Journal entry: “Origami, puzzles, and ‘slates'”

Teacher Journal

10/17/06 Students worked in pairs on individual “white boards”: transparent page protectors containing white card stock. Students could write on these and erase them. I supplied exactly one tissue and one pen for each team. We wrote down equations as they were modeled on the overhead (in +/- diagrams), and it was semi-competitive as students spontaneously tried to get the answers before the other teams. This method could be used for all kinds of learning – which I guess is what they were thinking of 150 years ago when individuals all had slates to work on. I’ll be doing this again in the future. There was no pen theft!

10/19/06 Students made playing cards and played in pairs to win tardy passes. Each team folded one blank piece of paper twice and tore on the seams to make four two-sided cards. The cards were labeled with eight potential “moves” in solving linear equations: + (meaning you will add a number to both sides of an equation), –, ×, ÷, CLT (combine like terms), DP (distributive property), MBR (multiply both sides by the reciprocal of a fraction that would otherwise be distributed), and CHECK (indicating solution is complete and all that’s left is to check our work. I displayed equations on the board in various stages of completion, and the team which displayed the correct next step first won two points. (I announced taking away a single point for displaying the wrong next step, but that turned out to be redundant.) During one period, I had the students just hold up their hands when they had the answer – first one up would get to announce their guess – and this turned out to be unrealistic, since most students quickly figured out that they could hold up their hands immediately and then use the next couple seconds to figure out the answer before they had to say it out loud. I like the game as originally planned except you need enough eyes to see everywhere in the room. Having a scorekeeper helped, but then that student is excluded from the game.

10/25/06 Reflections on problem solving. I present puzzles occasionally for transitions or sometimes for warm-ups. The students enjoy them. I also try to lead students to discover the skills and concepts we’re studying – with limited success. They get impatient with the indirect approach. Their attitude is, Can you just tell us how to do it? They’ve been conditioned over the years to believe that math is procedural: to get the answer, use these steps. They’re pretty good at learning this way: we just finished the unit on solving linear equations, and I’m happy with our progress. However, there’s no understanding behind most of their knowledge, and so the students on the fringes (who’ve been absent physically or mentally) have picked up a bit of information but have huge gaps in their ability to adapt or, some cases, to even do the basics. They will attempt to collect variable terms on one side of an equation, but will write down “–2” under each side of the equation, rather than “–2x”; or they’ll subtract a number from each side of an equation, producing a zero on one side but not writing down the zero because they don’t perceive it as even a number.

10/27/06 I presented a puzzle today – the fishing puzzle from our class Wednesday. A few students worked on the problem quietly, a few rowdy students blew off the assignment completely, and a few of the “troublemakers” got into it. Of course, it was hard for them to explain their thinking process, and nobody actually got the answer, but I think if you do this regularly, they might begin to recognize this kind of work as an actual math activity. In general, I’m not happy with any of the groups’ willingness to take turns talking and listen to each other. (One problem: as the level of discussion rises, the students who aren’t interested find the background noise serves as an excellent mask for their off-topic conversations.)

11/1/06 I keep looking for excuses to do origami in my algebra classroom. Rewards? Problem-solving? Instruction-following practice? (They’re so bad at following instructions. It’s all part of a general problem with listening to others – to me and each other.) I briefly taught a daily origami class to middle-school kids. It was delightful: almost 100% engagement each time. We worked simple models for a week and were just about ready to move into modular origami when the class ended. I realized as I was doing the class that we should have supplemented the geometry lessons the previous year with origami. My students never understood classical constructions: they would draw arcs free-hand, use the ruled markings on the straightedges, and pick endpoints arbitrarily. It was very frustrating. Would they have followed the origami instructions better? I know from experience that a lot of novices will go ahead and eyeball the origami folds rather than line up edges and creases per the directions. But I think the average user is more comfortable folding paper than using a compass, so we’d probably get more commitment.

11/3/06 What about using origami for teaching “non-graphic” math? Of course, you can use origami for various geometric ideas, but what about things like algebra and number theory? I researched this a bit and found a lot of discussion by mathematicians, math teachers and expert origamists regarding geometry (constructions and proofs), 2nd-year algebra (solving 4th-degree polynomials, folding parabolas), and various advanced topics, but really nothing about beginning algebra. Is all lost? Maybe not, if I make origami an ancillary activity and consider it valuable for teaching problem solving and prepping the algebra students for geometry.

11/7/06 Really bad lesson today on solving for a given variable in a formula. For starters, the students didn’t know how to use a formula, and then the first example in the book was the Fahrenheit/Celsius conversion – which I thought was a good example and which I illustrated with a nice table of temperatures but which completely overwhelmed the students. They didn’t know any of the basics about temperature, including where water boils and freezes. One girl even said that Celsius was “colder” than Fahrenheit. So, we were distracted by what I thought was fundamental. And then, although the kids were comfortable with the two basic equation-solving moves – adding/subtracting on both sides, and multiplying/dividing on both sides – they had completely forgotten (or never mastered in the first place) the alternate move of multiplying both sides by the reciprocal of a fraction, when the fraction “wants” to be distributed. (Many of them saw the “– 32” inside some parentheses and wanted to add 32 to both sides.) So, they got confused by the by-now unfamiliar multiply-by-a-reciprocal move. And then at some point they saw a variable in a denominator and really freaked out. I’m going to back up and re-teach tomorrow, focusing on the simpler examples, without any applications. Just: solve “y – 3 = x” for y.

Lesson observation: Algebra

Teacher: [name withheld]
School: [name withheld]
Class: Foundations of Algebra (8th grade), 5th period (immediately before lunch)
Goal: Review combining like terms, distributive property.

Duration Teacher Actions Student Actions
5 min. Post warm-up on white board: three-column table with pairs of consecutive numbers in first column, blank column labeled “sum” and blank column labeled “product.” Also, distribute individual “white boards”: 8.5”x11” card stock in transparent page protectors along with one Expo marker per student and one tissue per student. Students have worked with these materials before and copy table and begin completing it. Some discussion of meaning of “sum” and “product.”
5 min. Randomly check results of table. Review rules of the day: eyes front, take turns speaking, listen. Students display results and complete tables. Some ornamentation of tables.
5 min. Begin main lesson. Draw table on board with column heads: Expressions; Terms with 1st variable, Terms with 2nd variable, Terms with no variable; Rewrite expressions (grouping like terms); Rewrite expression (simplify). Write first expression in first column: 4x + 26 + 3 + 7x + 1 Students erase boards and copy table and first expression.
5 min. Guide students through breaking apart first expression into like terms, circling like terms in original expression. Prompt students to explain why terms are alike. Students list terms in proper columns, grouping and then simplifying into final form.
5 min. Repeat process with second expression: 3x – 2y – 3 + y – x Students complete next row in table.
5 min. Review distribution, continuing on white boards: 5(3+x) –> 15 + 5x. Remind students not to try to add 15 and 5x. Prompt students for justification. Use second example — 5(x+2) + 2(y-3) – to assess students’ ability to recognize that x and y terms cannot be combined. Students are proficient in distributive property, and some even explain why not to try to add 15 and 5x.
5 min. Review collecting variables terms from both sides of an equation, using the example 3x – 7 = 2x + 1 Students work on white boards, explaining to each other how to choose the side that will be the destination for the variable terms.

This lesson was not designed to cover a single new topic but to review several related topics once immediately before a unit test. I’m impressed with the extremely low-tech medium. Seemingly in direct conflict with the Technology principle of the NCTM, which focuses on digital technology to the exclusion of old-fashioned technology, the whiteboards work during power outages (which we had at our school just this week) and are inexpensive. Modeled on the individual chalkboards of Tom Sawyer’s time, they provide immediate assessment and equity within the classroom, easily addressing two of the six NCTM principles.

The lesson addressed several California Standards for the Teaching Profession, especially engaging and supporting all students in learning and assessing student learning. [Teacher] helped students manage their own time by giving them no opportunity to rest or become bored. They needed to constantly keep themselves and their classmates on task or risk being left behind.