Madder 'n Hell

Actually, I'm not mad at all, but it fits in with the word problem for this week. As mentioned in a previous post, I had decided to do the Math Forum problem #3340 with the students. It was the first word problem I've done since returning from break, and had quite varied success with. I also assumed that the students would be less interested in charity donations, so the problem was altered to be about the Madden video game. The rewritten version can be found here.

In both classes, I decided not to go through the explicit process of Noticing & Wondering. The students did not really seem to buy into it. I replaced it with a nearly equivalent process where I projected the problem statement on the board. I then had the students prompt me to circle what they thought was important. The process more closely emulates what they might do on a homework assignment or test, and reduces the amount they have to write which lead to a noticeable improvement in participation. On the flip side, if they're not writing are they still learning?

After the last attempt at encouraging the students to work out a solution themselves, I did not feel that allowing them to work together would be fruitful. Instead, I tried to lead the entire class to a solution. I accomplished this in different ways for each class, and the results were similarly different.

During third hour, with a total of 10 students, I threw them into the deep end without floaties. After having them read the problem statement, I immediately tried to convince them that the number of donors and total budget represented a coordinate point. I found the step to be logical, but if anything, I've learned that you can't force students to reach a result. Instead, you have to leave a trail of breadcrumbs and let them arrive to the result themselves. As soon as I talked about replacing the x-y plane with a donors-budget plane, I had lost nearly everyone. Trying to connect the problem to y = mx + b, was also futile. Usually, the quicker students can help to pull the class along, but most of them were missing on this day. It got to the point where I asked one student a simple question and he ignored me. For several uncomfortable minutes. Last year I taught a lab class at the undergrad level, and the NCRTL told us that you just have to wait it out. I don't think they considered a case where you only have 15 minutes to make your point. I pushed ahead, but had lost any momentum that I had started with.

Determined to correct my mistakes in third hour, I spent the lunch break rethinking my approach to the problem. This time, I deliberately ignored the linear relation. Instead, I had them rewrite the statement,

The company started by setting aside  a  certain  amount  of  money  to  produce  the  game.  To  encourage  their 60  richest  fans  to make  individual contributions, the company pledged to also provide an additional fixed amount for each fan who made a personal donation to the budget.

in the form of an equation. We ultimately ended up at something like B = FA+C, where B represented the company's part of the budget and F was the number of fans who donated. All it took to convince them that this was the same as a line was writing the slope-intercept form directly beneath it. Despite the success in this initial portion, I still think many of the students had difficult thinking of the data in as coordinate points. However, there were definitely several students that grasped the concept and helped move the class forward. While we were able to find A, time ran out right before we could finish the calculation for C. Nevertheless, change in the students' attitudes was palpable.

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.

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.

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...