Writing Inverse Functions

I like Sam Shah’s approach to teaching function inversion. It’s pretty much what I had intended to do this year, but somehow forgot about at some point. Hopefully writing these notes here will increase the odds that I get it right next time around. The original inspiration was Mr. K’s approach to solving two-step equations, and the following will make no sense without having read that entry, so do that first.

I showed my Algebra 2 students Mr. K’s series of boxes and arrows for solving equations early in the year, and they thought it was fun. We did more complicated cases such as 5-3(x+5)=2 and -2-(2x+3)=-5. It was great for reviewing Order of Operations in a novel way. Many students had started out subtracting 3 from 5 in the fist equation, and in the second case very few students were able to identify “multiply by -1” as one of the operations in the sequence. However, after working through a few cases they got quite fluent at writing sequences of operations such as for the equation -2-(2x+3)=-5, where the operations on x are:

1. Multiply by 2
2. Add 3
3. Multiply by -1
4. Add -2

The inverse operations, in the reverse order, are

1. Subtract -2
2. Divide by -1
3. Subtract 3
4. Divide by 2
   

Applying these operations to -5 solves the equation above. Applying the same operations to a variable yields the inverse of the function y=-2-(2x+3)

My plan was to bring this visual in again, as a recurring Opener problem perhaps, with every new operation we covered. The processes of squaring and taking a square root, of exponentiating and taking the logarithm, could presumably be organized nicely with the same kind of diagram that Mr. K used to organize the process of undoing linear operations on x. In Algebra 2, the reassuring string of little boxes could make for a beautifully transparent structure for inverting a function, with built-in checks at every step. Practicing with numerical input and output values first would likely be helpful.

Maybe I'll try this when reviewing just to see if it works.

Practice Problems, Patterned Practice

A characteristic of problem collections in standard textbooks is that each question tends to stand alone, that attempts at varying the problems in a systematic way in order to elicit patterns are typically few and relegated to the introductory section in each chapter, to be dropped after the rules have been listed and boxed. The incentive to add my own problems is usually a wish for more practice of the kind where each item belongs in a set that has some internal logic, and in the section on evaluating powers with negative and non-integer exponents this is particularly useful, I think.

The very idea of exponents that are not counting numbers causes the students a fair amount of puzzlement and headache. The common misconceptions that negative exponents change the sign of the expression, and that a base raised to the power of zero must be zero, seem very resistant to instruction. So far, my main strategy is to repeatedly place such powers in a sequence and to make a big deal of how they MUST have the values they have in order to follow the pattern we see with positive integer exponents. We complete these tables again and again, graphing the results to see how the rules for negative and zero exponents are nicely continuous with the ones the students know intuitively.

This year I’ve spent more time on treating rational powers the same way. We have, for example, evaluated 27^x for sequences of x-values that are multiples of 1/3, and (81)^x for sequences of x-values that are multiples of 1/4. In completing these tables, the powers with whole number exponents become familiar reference points that students can check their rational powers against, and it becomes clearer that the rules for non-integer powers are not random or contrived, but that they are what they are because they have to be that way:



This also, incidentally, reinforces a basic lesson on fractions that continues to be necessary in Algebra 2: that fractions just are numbers, that they belong on particular places on the number line, and that they behave just as you would expect based on what integers they fall between.

My worksheet on graphing exponential functions for Algebra 2 is here. I’ve just started using Sketchpad for the grids, and am finding it tricky to get them to look neat. Oversized dots, tiny numerals and inconvenient scales continue to be issues, and kids complain a bit about these things. I’d appreciate technical advice! My introductory worksheet on exponential functions for Algebra 1 focuses more on the sheer prettiness of the graphs of 2^x and its reflections when they are drawn on the same grid in colored pencil (this particular sample from a freshman of last year who elected to do her Arts project on Escher's work):

Notes from the Saturday at Asilomar

“Oh, I get it,” the students say, and I tell them that that's great, and that it is not enough. I tell them that learning math is a lot like learning to play a musical instrument or like learning to dance or to play a sport. Having once successfully played through a piece of music, or having once correctly executed a move, is all very well as a starting point - and only practice and repetition with feedback will ensure that they perform reliably and smoothly. “Getting it” goes only so far, and learning involves something more and other than that.

This is what I tell my students, so I do “get” that insight is not enough, that learning requires changing behaviors and habits, and yet it seems I have not learned this lesson at a behavioral level myself. And so it was that one of the most useful experiences of the Math conference at Asilomar yesterday was cutting the last class to stare at the ocean and think about what I had wanted to do and not do this semester, and about what is actually happening in my classroom. It’s been an unexpectedly difficult semester for private reasons (conditions at school would have predicted the best teaching year ever), with limited occasion for reflection on my work, and the Math conference provided a chance to take stock and regroup.

I “get” that assigning a large number of practice problems has little or no advantage over assigning fewer problems and is likely to even be counterproductive, I see that assigning homework problems that a large number of students won’t be able to do correctly on their own is silly, I know that having the students do more work than I have time to check is a waste of their time as well as mine, and I believe that there is much value in spending some minutes of class time every now and again on activities whose purpose is building relations rather than practicing math skills. Yet, judging from what I’m doing it would seem I disagreed with all of the above. Somehow, all kinds of resolutions made late last year got lost in the flurry of starting up again this year, and I’ve found myself teaching in ways I had made definite decisions not to.

So, what to do? Since it’s not a matter of “getting it,” not a matter of knowledge or conviction about what to do, I’m thinking about what kinds of feedback mechanisms to set up toward changing my behaviors. I could write down what I want to do and keep rewriting it daily to remind myself. I could talk to colleagues about what I want to do and ask them to do random five minute observations and let me know what things are looking like – and I am fortunate enough to have several fantastic colleagues who would be willing to do this. I could tell the students that there’s a cap of a certain number of homework problems and ask them to remind me if I forget my own policy. I’m sure they’d be delighted to oblige.

On another note, it’s been nice to rediscover the online math ed community recently after having neglected these readings for lengthy periods at a time this fall. New ideas and debate are like food and sleep in that they’re easily given lower priority during hectic times, and yet these things are all necessary for teaching and learning well in the long run. Glad for all of you who are still around and writing...

Algebra 2 is amorphous and has multiple heads

and defies reduction into compact and self-contained little parts. I am still trying to do just that, however. This continues the discussion about applying Dan Meyer's assessment system for Algebra 2, and anyone not deeply interested in this narrow topic may as well go on to the next item in their Reader.

Some time in June, Glenn Waddel wrote:

I have a rough draft of my skills checklist done right now. I am not sure I am going to post it yet. I am not happy with it. I think I am stuck in the “do I have to assess everything?” mode.

More than a month later, that's where I am still... and since time is running out I'm going to post the incomplete work in case that helps accelerate the process.

Glenn has meanwhile posted a carefully worked out list, and written about the tension between assessing specifically enough without introducing an intimidating number of concepts. Reading his list reminded me that the differences between the various versions of Algebra 2 around are significant, and that our final lists will have to be different to accommodate our respective course specifications and student groups.

In particular, my Intermediate Algebra course leaves conics and discrete math for the Trig/Precal course, which uses the same textbook and picks up where my course leaves off. I do not need to include assessments on these topics, then. On the other hand I must make sure that graphical features of quadratics, polynomials and exponentials are covered carefully, as this will not be repeated in Precal, and this increases the number of skills for these topics beyond what may be needed in Glenn's course. Also, Intermediate Algebra is for the students who do not make it to Honors Algebra 2, and so I need to include a lot of Algebra review. Dan Greene's version of Algebra 2 is similar to mine in the topics it covers, but students arrive directly from Algebra 1 without Geometry between, and so may need somewhat less review. I'm guessing that Sam Shah's course is for relatively advanced students. However, while our lists will need to be different in order to take these things into account, comparing notes could still be very useful.

Dan Greene and I met a few weeks ago, and made some progress on breaking down the chapter on Exponential and Logarithmic Functions, a unit where I am replacing all of my concept test items from last year. So far, the unit on Numbers and Functions has been the most demanding unit, I think. There are so many big, abstract and quite unfamiliar ideas there. On the one hand, the process of breaking down this chapter into parts that can be practiced separately may therefore be all the more necessary in order to make it accessible to students. On the other hand, much of the point is for students to recognize an abstract idea, such as the transformation of a function, across pretty different contexts, and this is just hard to assess in a piecemeal way. Or so it seems to me.

Finding a convenient format or platform for technical discussion of how to slice a topic into discrete skills and concepts is a challenge of its own, though. A series of blog posts, one for each chapter, seems both clunky and overly time-consuming (and school starts in just over a week over here). Instead, I've stored my work-in-progress on this Google Site, and if you have time and inclination to think some about what are the essential things to test for each topic or anything else, suggestions would be much appreciated.

Coffee and math ed readings

This summer I've met with a doctoral student of mathematics education a couple of times. Her area of interest is mathematical learning disabilities (MLD). Last time we met in a coffee shop to discuss an article by Geary et. al.¹ on students' placing of numbers on a number line, a topic I've been fascinated by for some time.

As it turns out, number lines constitute an active area of study in cognitive psychology and neuroscience, of theoretical interest

because magnitude representations, including those that support the number line, may be based on a potentially inherent number-magnitude system that is supported by specific areas in the parietal cortices ... (p. 279)

Geary's article cites earlier work by Siegler and Opfer² which suggests that young children use a more or less logarithmic scale when placing numbers. Children tend to perceive the difference between 1 and 2 as being greater than the distance between 89 and 90 in a semi-systematic way (p. 279), so that most numbers get clustered to the left hand side of the number line. This tendency is thought to reflect the postulated "inherent number-magnitude system."

Geary et. al. compared first and second graders' placements of numbers on a blank number line. They found some evidence that mathematically learning disabled students' placements not only failed to conform to the linear pattern at a rate comparable to that of their peers. In addition, their pre-instructional number placements also looked less like the logarithmic placement of non-disabled children:

Even when they made placements consistent with the use of the natural number-magnitude system, the placements of children with [mathematical learning disabilities] and their [low achieving] peers were less precise than those of the [typically achieving] children in first grade, that is, before much if any formal instruction on the number line. The implication is that children with MLD and LA children may begin school with a less precise underlying system of natural-number magnitude reprsentation. (p. 293)

Geary et. al. report correlations between performance on the number line tests with a battery of other cognitive tests. Unfortunately I know neither enough statistics nor enough cognitive psychology to extract terribly much information from these parts. Of rather more immediate interest to me as a teacher is, in any case, the question of how to go about eliciting the kind of cognitive change that's needed here. It certainly is not the case that all kids have the linear scale all figured out by the end of second grade - many of my 9th and 10th graders last year had not. The good news is that for this important topic, instruction tends to work. Cognitive Daily reports on more recent work by Siegler and Opfer showing that second graders responded quickly to some targeted feedback on their number placements, and that

once the linear form is learned, the transformation is quick, and permanent.

In other news, this thing of chatting about research in MLD over a morning coffee has been immensely enjoyable. The readings are demanding enough that I'd be much less likely to work through them if I were studying alone, but I'm awfully glad to be learning some of this.

I'm wondering how much interest there would be for some kind of regular math teacher/ math ed researcher meetups, such as a discussion of a predetermined article over coffee on Saturday mornings. Many new math teachers already have ed classes scheduled at that time, though, and older math teachers typically have family to take care of during weekend mornings, so how many would remain? And would there really be many researchers interested in talking with teachers? Still, given the curious absence of contact between researchers working on math education and math instructors working in schools, even in cases where their buildings are in the same geographical area, it would seem that some thinking should be done on how to afford more "vertical alignment" in Dina Strasser's sense of the term.

1. Geary, D. C., Hoard, M. K., Nugent, L. and Byrd-Craven, J. (2008). Development of Number Line Representations in Children with Mathematical Learning Disability. Developmental Neuropsychology 33:3, 277 - 299

2. Siegler, R. S. and Opfer, J. (2003). THe development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science, 14, 237 - 243.

Have you used Algebra tiles?

I introduced integer tiles to my Algebra classes early last fall, and then quickly gave it up. Several students balked at using such middle school measures for studying math, and my arguments that being able to represent math statements in many different ways, including with concrete objects, failed to persuade. In a class of insecure freshmen still figuring out their relative positions in the class, and in some cases still stinging from having been placed in Algebra rather than in Geometry after the placement test, using materials perceived as childish just wasn't socially acceptable.

I quietly dropped the project, only including a problem on modeling integer subtraction as an extra credit problem on a unit test some time later. Not one student got it right. Later in the year, when a number of students continued to demonstrate confusion about combining signed integers and combining like terms, I sometimes wished I'd stuck with the manipulatives a little longer.

Now I'm trying to make up my mind about whether - and, if so, how - to use tiles in my Algebra classes in the fall. Apart from the probable social issues to deal with, I'm wondering about the efficacy of Algebra tiles. A point I've picked up in passing while reading this summer (I'm sorry I can't recall where!) is that the same students who are likely to have much trouble with elementary Algebra are also likely to have difficulty picking up how to manipulate Algebra tiles.

There is, after all, no magic involved. The rules for representing addition and subtraction of integers with bi-colored tiles are not self-evident or trivial. Even for me, the representation of subtraction problems by adding the necessary number of "zero pairs" came as a bit of a surprise. And while I then found the very idea to be very cool and exciting, that is more than I can take for granted that my students will, even if they aren't unable or unwilling to master the rules of the game.

So, what are your experiences with Algebra tiles? How to you go about changing the image of tiles as being all too elementary? And assuming that you have gotten your young charges to take the tiles seriously, how much do you feel that the students learn this way that they do not learn just as well by simply reiterating the formalism of signed numbers and like terms?

Applying Dan's assessment system, Part II - scoring

Note: A discussion of more general lessons learned while applying this assessment system is posted here. This entry is a dry, technical discussion of scoring and grade calculations, of interest only to teachers thinking of applying this system themselves.

Dan Meyer assesses students at least twice on every test item, scoring out of 4 each time. At the second round of assessment, he alters the possible points from 4 to 5. If a student scores a 4 on both rounds of assessment, she nets a 5. Otherwise, her highest score applies. Dan makes the second round of assessments a little harder than the first, so that a second 4 indicates greater skill than the first 4.

Altering the possible points from 4 to 5 entails that students who do not actually improve their performance from one assessment to the next automatically see their grade drop. For example, if a student scores a 3 on the first assessment, and then another 3 on the second, more demanding assessment of the same skill, the grade on that particular concept drops from a 3/4=75% to a 3/5=60% - from a solid C to a D-. This caused problems that almost made me abandon the system early last fall. For one, students got upset when their grade dropped without their knowledge having changed. Secondly, having grades drop after a progress report has been issued is not actually legal - or so I was told by my Department Head. An evening shortly after the first progress reports had been printed found us manually going through all the scores in the gradebook, altering the scores so that the grades would be the same as before, for example by changing 3/5 to 3.75/5, since that is equal to 3/4=75%.

Another problem with this system was that a scale from 0 to 4 seemed fairly coarse grained. Students who made a mistake significant enough not to merit a top score on the first assessment would be marked down by 25 percentage points, and if they did not improve markedly by the second assessment they would net a D-. Improvement from this D- would be possible only if they subsequently scored a perfect score. I first thought that the large number of skills and the repetition of assessments would lead to an adequate continuity of the total grading scale, that students might average a C by scoring perfectly on some skills and poorly on others. However, some students seemed, even when working hard, to be unable to ever score a 4. They'd always make some or other significant mistake, but not enough to make a D- seem appropriate. Now I am sure that in the mutual adjustment of quiz difficulty and scoring practice there is some wiggle room for making this work in a fair way, and I assume Dan Meyer has figured out a balance here. However, I ended up changing my grading scale.

Solving these problems proved pretty difficult without losing important features of the original system, however, and I found no perfect solution. I wanted my score assignment to do what Dan's did, in particular, to make it necessary for students to take every assessment twice, in order to ensure stability and retention of knowledge. Dan's practice of increasing the possible points does just that - students can not just be satisfied with their 3/4=75% and decide not to attempt the second assessment of the same skill. In the end I decided not to report students' scores online until they had had both assessments. I made the two assessments of equivalent difficulty (which simplified things for me) and then grades were assigned based on students' best two scores according to the following table:

In summary: For scores of 3 or lower, the higher score applies. If both scores are above 3, the grade is the average of the two. If one score is above a 3 and the other below, the grade is the average of 3 and the higher grade. With this score assignment, students still had an incentive to demonstrate perfect mastery twice, in order to net a grade of a 100.

A disadvantage of this system is it's clunkiness compared to Dan's simpler system. Much of the appeal of this whole approach to grading was its transparency to students, the clarity it could afford them about what to focus on. Some of this is lost with this conversion table. Also, since the best two scores count, the system appears to have somewhat more inertia; poor scores don't go away as fast as they seem to in the original system, where the better score always counts. This slower improvement is more appearance than reality, since two 4's are necessary to achieve a 100 in Dan's system too, but appearance matters in this context. The main disadvantage, however, was switching to this different scale after the first progress report, which caused some confusion and, I think, some loss of buy-in from students. They seemed a little less enthusiastic about completing their tracking sheets after that.

As an alternative, I experimented a little with just entering both of the best two scores into PowerGrade this spring, labeling the entries "Skill 14A" and "Skill 14B," for example, and assigning half weight to each. I am undecided about whether I will do this in the fall or just enter the composite grade. It is of paramount importance that the students understand the relation between the scores on the papers they get back and the scores on their grade printout, and this system would help in that regard, but it would make for a large number of gradebook entries, which means more messiness.

Finally, a note on the scoring of any quiz item: In some cases it made sense to assign a point value to different components of the test item, and sometimes I wrote the test items to make this possible. Other times, I evaluated the complete response to the test item as a whole, and assigned scores as follows:


Frankly, for some skills that did not lend themselves well to decomposition into parts with point values for each, I'd score based on my mental image of what a D-, a B- and an A would look like. If grades are supposed to be derived from scores rather than the other way around, that introduces some circularity that one might argue about, but I don't care. I think grades as descriptors of performance levels rather than as translations of some numerical score make more sense anyway. But that is another story that would make for a separate discussion.

And since this scoring business turned out so much trickier than I'd anticipated, well-thought out suggestions for making it clearer and fairer would be appreciated.