Uncanny Valley

I have a love/hate relationship with Wolfram. For example, Wolfram Alpha is a wonderful resource for online calculations, but it's coupled with a very restrictive user agreement. Likewise, Mathematica has tremendous functionality, but I always hated its notation. Well, I now have a reason to both be afraid of Wolfram, and be scared for the future of high school homework. I present to you, dear reader,

Step-by-Step Math

The short of it? Wolfram Alpha can now solve most equations that have analytic solutions and can provide you with the proper steps to arrive at the solution. Oh woe!

Persistence Persistence Persistence Persistence

Maybe it is a bit inappropriate to reference Sisyphus when describing my work with the students at YHS, but it is somewhat unavoidable. Some days progress seems slow and halting and is inevitably followed by an equally large backslide. In light of my difficulties with the most recent problem of the week, I've decided to write a separate, but connected, piece.

Opposite to my tragic Greek brother's (after)life, a motif of 'lack of persistence' appears in many of my discussions with Mrs. Porter. In contrast to many that I know from college, the students in our classes find even the most modest challenges to be discouraging and disheartening. An equation that is slightly different than the one before is enough to render the student incapable of absorbing any information for the rest of class. All but a few are unwilling to acknowledge any difficulty, and in the worst cases, those confused turn to distracting the people around them. Most will disregard any words of encouragement and genuinely believe themselves incapable of solving certain problems. In these cases, I try to lead the student through a few example problems and have them tackle the remainder using those cases, but often they refuse to do any work independently. I've had several students throw their work to the floor as soon as it became apparent that I wasn't going to provide them with the answers followed by accusations of unfair treatment.

To their credit, many of the students are being asked to operate at a level much higher than they've ever experienced before. A large number have assessment scores which place them at elementary school levels, and yet we demand that they learn algebra. To make things worse, Mrs. Porter tells me that many have never had to do homework or take notes before, and must be taught the importance of both. Finally, each class is interrupted by futile writing/reading assignments that provide the students no feedback whatsoever. It is rather difficult to ask the students to do work that you yourself don't believe is useful or helpful. These factors contribute to a class whose content is pretty simple from a conceptual standpoint, but moves at such a fast rate through different assignments and subjects that few students can keep up.

The other day, Mrs. Porter asked me what it would take to properly teach the students. For many of the students in my class I believe that year-round school, with longer days and approximately half of the students per class are all requirements (there are many reasons for these, but they are out of the scope of this entry and for another time). Of course, there isn't a person in this world that could convince all the necessary parties to accept these changes, but for the current crop of students each step seems like a battle. I find myself wondering if any of what's being taught is retained. Most importantly, I ask how to convince the students that persistence pays off, because lacking that they'll be average in their achievements at best. Certainly, the concept isn't new to them, too many play sports or participate in other competitive events, but none of it is applied to their school work, or at least to math.

Fractional Learning

Over the past months, I've noticed that there is a severe deficit in the students' understanding of fractions and it has not been improving. Realizing that they're shorthand for division, how to reduce, reciprocals; each class I get a question about one of these subjects. In an effort to cement their understanding of fractions, I chose to do the PotW #5260, "Fraction Debate", over a period of two days. Usually, Mrs. Porter and I have attempted to finish problems within a single class period over the course of twenty minutes. However, this doesn't provide enough time for the students to produce a whole lot of work on their own. Spreading the problem out over multiple days is an obvious solution, but my days at YHS are not consecutive. By the time I return to the classroom, the students will have either lost the problem sheet, lost their notes, or forgotten about the problem entirely. In this case, we started on Friday, and I returned on Monday to complete the problem, as the students were off after Tuesday for Thanksgiving break.

At the start of the problem I broke the first class up into groups of three and the second class into groups of two. I attempted to choose groups whose students' strengths would be complementary. In addition, I tried to avoid placing the most talkative students together. This step alone proved to be exceptionally problematic in the first class. Every student wanted to work with their best friend and would outright refuse to work with anyone else. Regardless of any encouragement on my part, they just would not work together and kept on trying to establish their own groups. In the second class, I had the students work with the person to their left as Mrs. Porter's seating assignments had done most of the work in keeping problematic students apart. This session went much better, and the students seemed much more obliged to work together (though there were still a number that didn't try particularly hard).

The problem at hand is particularly interesting because it uses fractions, inequalities, and (I believe) a great introduction to proofs. For this reason, I started by explaining the difference between definitions and theorems. Definitions being the most basic rules of math that we accept as true without proof and theorems being the subsequent rules built by applying definitions. I then had several students read the scenario out loud. While some students dislike being put on the spot and having to read, there a fair number that are excited to participate and I think that more students pay attention when one of their classmates is doing the talking. We then proceeded to Noticing and Wondering as has been done in the past. This section appeared to work much better than before, the students seemed more apt to work together in generating ideas than by themselves. After giving them about five minutes to compile some ideas, I started asking them for their observations and proceeded to write them on the board. Their willingness to provide feedback has increased compared to the first problem I did with them, and I got several good suggestions. However, I found that both classes concentrated on the numbers in the example pictured above rather than key phrases in the scenarion; "any proper positive fraction," "add 1 to the numerator," "add 1 to the denominator," etc. In addition, I had notable difficulty in convincing the class that 4/5 > 3/4. Even after putting the decimal equivalent on the board, some were reticent to accept the statement.

In light of the progress made on the first day, I was very excited to have them work on the proof. I thought it would be optimistic if one or two students figured out how to complete the problem, but I hoped that they would be able to start work on it and write the comparison in algebraic form. I ran into timing issues in the first class that prevent me from giving the students time to work on their own. Instead, I tried to turn the problem into a class exercise with me guiding them through the problem. I lost the class almost immediately when I wrote the comparison of the two fractions with two variables and few numbers. Despite having already gone through an entire chapter on solving equations, the students do not seem to have developed a lot of flexibility in their ability to solve problems. The inequalities alone seemed to confuse most of the students and several didn't even know why variables were being used. The concept of a "general solution" was much more difficult to grasp than I initially expected, and only the students I spoke with individually seemed to see the point. There were several different ways of solving the problem and I had wanted to show the ones that I had found to the students so that they might see there was more than one way of approaching things. Unfortunately, it was difficult enough to get them through the first solution, let alone the subsequent ones. As has become common, there were maybe 5 or 6 students altogether that had genuine interest in understanding the problem, but their classmates proved to be too effective at distractions.

Engage!

Here's a short article on the gender gap between girls and boys in science education. It highlights some interesting results, unfortunately, the paper does not provide much insight as to why girls are not as engaged in these settings.

Studying the science gender gap at the high school level

Miscellany

A few brief notes:

After receiving some complaints of the comment system not working, I've changed the settings to allow anyone to comment. If problems persist, please contact me and I'll see what I can do.

The hip new thing on the internet these days is Google Wave. I've been playing around with it and I think it may be a useful place to discuss class specific topics. For example, I've created a "wave" with the word problem I plan to do this Friday. If you'd like access (even if it isn't necessarily for the TF program), I have several invites remaining, send me an email and I'll set you up.

A gRAPh

I saw this site about a year ago, but forgot about it until now. Maybe half of the graphs are appropriate for a high school setting, but they might be a neat way of explaining how to interpret visual data. It doesn't hurt that many of them are pretty hilarious.

rap represented in mathematical graphs and charts

Sixty Symbols

While looking for videos of Chladni plates for my presentation on sound, I came across a site with a lot of well produced videos on common physics topics. Called Sixty Symbols, it is a series from the University of Nottingham physics department. Most run from 7-9 minutes long and usually feature a demonstration of some sort. Like the PheT site, some of the videos are better than others and a few of the professors they talk to seem ill-prepared. Other videos are much more intriguing, my favorite so far is the one on Schrodinger's Cat. The level of the talks seem a tad high for my 9th graders, but may fit well in a science class.

Sound Principles

Today I gave my first large presentation to my two classes. On an earlier occasion I did a short talk on the LCROSS mission and had the class do basic calculations. That didn’t go over so well because I think many of them don’t share quite the same enthusiasm for space as I do. A few weeks ago I handed out a small piece of paper asking the students to write down a topic that they’d like me to talk about. I left the field open to any subject as I didn’t anticipate having any trouble finding math in even the most obscure suggestions. One of the most frequent suggestions was music; though many simply wrote ‘rap’, there were others that mentioned music, and how to build instruments. I decided that a presentation that covered the basic principles of sound and how it can be described mathematically would be the best place to start.

I used a few principles to guide how I put together the material. I decided on two “acts” in the talk. The first was rather basic and tried to imbue a qualitative understanding of the necessary principles. The second act used those principles as a foundation for real world topics that are difficult to analyze without an advanced degree, but would hopefully be understood intuitively. I also tried a form of multimedia blitzkrieg; almost every slide had a visual and audio component with several short videos near the end. It was my hope that this would keep the students’ attention much better than a straightforward, dry lecture.

My talk began with a discussion of what exactly sound waves are and some of their properties. I made a conscious effort to avoid words like transverse and longitudinal, but instead demonstrated the concept using a Slinky. By hooking my laptop up to the overhead speakers, I then played back various sine waves and combinations thereof. I did my best to show them how sines could be manipulated in much the same way as regular numbers.  I mentioned how the simple ideas behind superposition could be used for tuning instruments or for noise-cancelling headphones. These examples helped a fair amount in grabbing the attention of the students. I ended this first act with mentioning how combining a sufficient number of sine waves could added in such a way as to make any sound imaginable. This was a natural precursor to drum machines and electronic music which in turn led to music in general.

In the application portion of my talk, I went over many subjects. The first one was on turntablism. Here I used a microscopic view of record grooves to explain why scratching sounds like it does. I also used a video of an oscilloscope output to show how music is really just a collection of sine waves. My next target was how to use speaker response curves to select a good speaker. This slide did not go as smoothly as the turntablism one, probably because I did a poor job of discussing why a flat response curve is a good one and many found the topic as an excuse to discuss what speakers systems they wanted for cars. I also used a slow motion video of guitar strings vibrating to discuss how stringed instruments make noise and why many of them have large resonant bodies. I wrapped up the discussion with a slightly tangential video, but one that the students really enjoyed. I chose the collapse of the Tacoma Narrows bridge as an example of waves gone wrong. Like the resonation in a guitar body, I showed them how the bridge resonantly reacted to the winds.

Overall, the talk went very well. While there were a number of kids that got distracted, talked out of turn, or passed notes, many seemed genuinely interested. The best part of my day was when several told me how interesting the presentation was at the end of class. One student who is usually hostile and taciturn was very engaged by the talk, listened to me and volunteered answers. I was nonplussed. My only regret is not having a more interactive presentation; I’ve been meaning to make an ultra toriton for a long time and it would’ve been a perfect accompaniment to the talk. I’m planning to do my next talk on football which will likely be a bit more challenging to put together, but I’m hopeful. In the mean time, anyone who would like to look at my slides, can be download them  here (careful it’s nearly 20 MB).

John Travoltage

I'm in the process of putting together a presentation on sound and music. While searching for some good interactive examples of waves, I found this website hosted by the University of Colorado - Boulder. They host a large number of java simulations for common phenomena (including math, physics, chemistry, biology) and they're all free. The quality varies greatly between applets, but some of them are really clever and make for great visualization. Just try out the My Solar System and you'll see what I mean. I may try to work the sound wave one into my presentation, or take the students to the computer lab to use a few of the applets together.

Of Milk and Men

Molly and I have established a routine in problem solving and how we deal with the week. Her classes usually have to complete a quiz on Fridays, and after that quiz l use the remaining time to either give a presentation or do a word problem. This week, following a quiz that emphasized fractions, I decided to regale the students with problem #1790. The problem emphasizes the use of fractions in determining volume and introduces the concept of a factorial.

Once again, I find that the first class is much less receptive than the second class. I'm not sure if this is a result of applying my learned lessons from my first lecture to the second, or if the students in the second class are simply more involved. In both instances, I was struck by the lack of suggestions when we asked about what the students wondered. In this case, I assign some of the blame on the fact that I had copied the question onto the problem statements. However, I find that most of the students don't care enough to wonder anything about the question in question. After prompting, several of the students were willing to accede the presence of a pattern in the problem statement. Though the problem already explicitly indicates the pattern, I counted this as a win.

Solution strategies were once again lacking, the students either had no idea how to approach the problem, or just didn't care. I really would like to give the students the opportunity to work for several minutes on their own, but past experience has suggested that they just become distracted within a minute or two. Instead, I did my best to illustrate the pattern evident in the problem. Following that, I helped them construct an equation that would show how much milk was ultimately left.

At this point, several students were dismayed that we had spent nearly ten minutes in introducing the concept of factorials. While I don't regret a minute of the process, they were more upset that they were not allowed to work on their homework during that time. In fact, I'm surprised that more than one student opted to write out the final answer to the question in question.

While I can appreciate the philosophy that props up the instruction techniques from the Math Forum, I'm concerned that the level of engagement from the students is insufficient. While most of the problems are interesting from a calculation perspective, they lack any lasting impact on the students themselves. Mrs. Porter and I are in the process of discussing what the best technique is to engage the students in the problems. In particular we're trying to find some way of adding an incentive to the equation.

Animal Farm

References to the Russian revolution aside, a little over a week ago I did the Math Forum's Problem #5156 titled, Ostrich Llama Count. I've been remiss in my virtual live web blogging, so here is my belated account of how things went down.

As I described in Problems with Words, we began with Noticing and Wondering. I decided to continue my policy of writing everything down, with the sole exception of the student who wondered how large the ostriches' reproductive organs were. Sadly, that was one of the few suggestions for Wondering I got and we had to troop on (I'm all for suggestions on how to up the number of wonders). The students were more attentive through the beginning of the problem, and I felt like the stronger approach in teaching was producing results.

As we transitioned to solution strategies I had a last second realization that the algebra approach might be a bit of a reach as the concept of solving for a given variable hadn't been addressed in lecture yet. Several students successfully worked through the problem by using a table or by guessing and checking. For the remainder of the time I tried to explain how to substract the same number from both sides. As with last time, it was hard to tell how much the students actually understand. It also didn't help that I was using O as the variable for the number of ostriches.

Destructive Tendencies

I have a few ideas that I've been throwing around about presentations for class. Any input on these ideas would be wonderful.

1. Super Bouncy Balls - United Nuclear provides the necessary chemicals to make your own Super Bouncy Balls. While it's more of a chemistry demonstration, it would present the opportunity to go from a chemical equation and molecular masses to how much of each chemical is needed. There's also the downside of giving 50 high schoolers each their own superball. I suppose only one needs to be made, but I'd actually like to involve the students somehow.
2. Something Small - When I was in high school I began the trek to my current profession after working in a biology lab. Being able to manipulate items too small to see and retrieve observable results was amazing. I'm not sure where this would go; while plasmas offer plenty of opportunities in this respect, most of the math is a bit too involved for class.
3. Math Rap - I recently conducted a survey of the class on what topics they'd most like to learn about. Top of the list was football and rap, while football has plenty of opportunities to involve math, I've been tearing my hair out day and night about rap. Off the top of the head, a few topics come to mind. Rhythmic structures could be used to discuss how a consistent number of syllables in each line gives the impression of better flow. On the other hand, I have an oscilloscope that would allow me to talk about waveforms and superposition (in a pretty intuitive way). Similarly, using a computer and audio software to do FFTs would let us analyze an audio sample for its beat pattern.
   

The Uncertainty Principle

Over the last several weeks of classes, I've noticed that it's impossible to predict whether a given student be in class on any particular day. In fact, there are several students on the roster whom I haven't seen yet. Weirdest of all, in this menagerie of truancy, is the steady appearance of new students.

After speaking with Molly, it seems that there's a large variety of reasons for this. In some cases, the students were being shuffled to smaller classes where they'd get more attention, in others they needed to switch course times for scheduling reasons, so on and so forth. I can't help but find it strange and off-putting that students have not yet settled their schedules over a month into the school year. The needless distraction of having to learn a new teacher's protocol, getting used to the new schedule, familiarizing themselves with the new class, all work together to take away valuable time from actual education.

I find that our classes are usually struggling to make net progress on a day to day basis. This issue is particularly noticeable in the fourth hour class which has over thirty students. A typical class period is split into the following sections:

1. The Starter - A short set of ~5 problems meant to emphasize important aspects of the previous lecture. During this period, Molly will generally walk around and check to see if the students have completed their homework.
2. Checking - Time during which the students correct their starter, and correct their own homework.
   
3. Lecture/Quiz - Meat of the class, often the time during which new topics are introduced. Quizzes are frequently held on Friday in order to cement the previous week of learning.
4. Homework - If time permits, Molly will help the students get a head start on the homework.

The schedule doesn't appear too hectic, but when the class period is 58 minutes long it gets a bit tight. When you add in the time it takes for students to get out notes, hand in assignments, and other small but necessary tidbits, the class seems like it's a non-stop whirlwind of activity. Whenever I have some free time, I've been trying to come up with suggestions on improving the class efficiency, but no eurekas just yet.

Problems with Words

Almost all Teaching Fellows have the same set of responsibilities; we all do presentations, a field trip, demonstrations, and several other things. As a math TF, I have the added fun of integrating new ways of teaching problem solving. Last week was my first attempt at using some of the methods provided by the Math Forum. Herein, you will find out how well that attempt went.

For those counting, I chose Problem #3520 to try. Simply put, it involves solving a system of five simple algebraic equations. I wanted to try this one because the development of the equations was simple, but the problem was long enough that it'd be difficult for the student to see the immediate conclusion. Each class began with a session of Noticing and Wondering. Here we all wrote down and discussed what we noticed and wondered about the problem (without yet knowing the question). This was followed by some attempts to prompt suggestions from the class on solution techniques. Finally, we ended with a discussion of the actual solution.

The Noticing and Wondering did not go entirely smoothly. Many of the students didn't see the importance to thinking about the problem statement, and instead tried wisecracking their way through the section with silly suggestions. In response, I decided the best thing to do was to write down their silly suggestions on my list. After this, I was glad to hear the students start chastising each other over meaningless statements. Unfortunately, this led to only two or three very forward students saying anything, and the rest quickly became detached from the lecture.

While I tried to get some discussion going on how to solve the problem, no one was willing to say anything, even after some hints on how to approach the problem. When I was teaching a college lab, this is the point at which I'd stand around saying nothing until someone volunteered a solution no matter how long it would take. With less than an hour to do practice problems, the starter, take attendance, hand back papers, and lecture, each minute not spent teaching feels like one lost. By the second class there was little more than a brief pause and comment between the noticing/wondering and the solution. Thankfully, I got some help from the students on writing out the equations.

At the end of the class, Mrs. Porter and I picked up the problems from each student. I've just finished grading them; you might ask,"Ben, how do you grade a problem that you solved for the students?" I would reply, "I treated them like notes, and the students who wrote down all major steps of the solution got full credit." This might seem like an easy grade for most students, but I counted a total of 29 papers when there are 48 students enrolled in these classes. How can I convince the students that there's a reason behind all that's going on? I wish I could have two hours with them to answer all the questions they have, but don't feel like asking.

As an epilogue, I'll be trying another word problem tomorrow. This one should be a bit shorter and I actually worked through this one with the instructors from Drexel. Counting ostriches and llamas, maybe I can come up with a slightly more interesting alternative...

First Class

I forgot how much of a rush high school was. As soon as each bell rings you see people spilling out into the halls as if the levies just broke.

Mrs. Porter had me give my lecture at the beginning of both classes and it went about as well as I expected. You can find a link to a pdf version of the powerpoint at the end of this post. There were a few scattered questions and quiet comments throughout the talk that made it a bit difficult to focus on what I was saying. In particular, the word 'nuclear' sparked a bit of conversation that mostly centered around bombs and toxic waste. While I acknowledged those topics, I did my best to shift their attention to other applications, for example: radiography, material interactions, computer chip production, etc.

After the talk, I split my time between trying to help Mrs. Porter answer questions, and observing the class. The hardest part about answering questions is trying to guide the student to a solution. When teaching undergraduates, this was much less of an issue, as many of them were putting forth a lot of effort to understand the material. At the high school though, many of the students were quick to get frustrated and give up on problems altogether. Talking to them, I see that they understand much more than they let on, but even the smallest difficulty is enough to throw them entirely off track.

In some cases, we were lucky to have the students at least reading the problem. Several were content just to sit in front of a blank page of paper unless prompted. Many were only convinced to write when it was for a grade; note-taking is the exception not the standard. The levels of their patience is lacking, and each time I talk I feel like I'm competing for their attention with a dozen other things in the room.

It did not take long to notice which students were already well ahead of the class. Almost as a rule, they kept to themselves, but were the first ones to answer any question. I've got to admire their enthusiasm, but at the same time, I think it is discouraging to the rest of the class. It also pleasing to have several students actively seek me out for help on some of their problems, but of course, those aren't the ones that I worry about in the long run.

These are just some brief impressions for the moment, I had class again today, which I'll elaborate on in a short while.

Introductory Presentation

Postponed

Until today or tomorrow depending on your timezone. Just making sure that my formulation of the three dimensional Schrodinger equation is right on my slides, and the assumptions with the Gamow approximation are clearly stated.

Once more unto the breach...

 

This picture is the last remaining evidence that I ever took algebra, or even existed before I do now. I'm not even sure if I was in the class mentioned on the whiteboard, but that's beside the point. To get to the point, a little background is necessary. 

After years of struggle and heartache, I scratched my way out of high school and found my way to college. Following a series of fortunate accidents, I'm now in the PhD program of the Nuclear Engineering & Radiological Sciences department at the University of Michigan. Somewhere along the way, I developed a strange sense of civic responsibility. Further along the way, I noticed that virtually every problem I saw around me could be solved by improving public education. While not ready to give up my research yet, I decided to get involved in some way.

As a result, I became part of an outreach program run by the College of Engineering meant to drive interest and understanding of STEM fields (science, technology, engineering and math). I will be working with Molly Porter in two Algebra I classes for this coming year. My duties will primarily entail composing/giving presentations on the applications of STEM topics, putting together a field trip, and integrating problem solving techniques from Drexel's Math Forum. My first day in class will be this Friday.