Student Learning

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

Student Profiles

As giving pseudonyms confuses me hopelessly, I've decided on the dubious expedient of dubbing them Student #1 and Student #2.

Student #1

Student #1 is a Caucasian who, during the first semester, was failing or near-failing in every class but Guitar. He is also a junior and, despite professing to want to become an engineer, rarely if ever turns in homework. A policy implemented second semester whereby students could opt to take a quiz on the material instead of handing in the homework has benefited him greatly. He is very articulate in writing and speech, and when he feels comfortable enough to speak, has a wry and offbeat sense of humor.

In this unit I hoped to appeal to his creative streak and verbal abilities; though he was almost completely disengaged all through the first semester despite showing up to every class, I was curious as how much he would like or dislike the topics. I thought that the more nontraditional activities, such as PEG SOLITAIRE, would appeal to him and keep him engaged.

Student #2

Student #2 is a Latina who is also one of four seniors in the class. She speaks Spanish by preference and her spoken English is slightly accented; her written English shows some non-native features but overall her skills are on par with the "average" writers in the class. During class she also gravitates naturally toward the other Latinas in the class. She has generally gotten C's in her prior classes, but was barely passing this class during the first semester when instruction was far more traditional, and every so often she has missed class. She is quite organized in the work she turns in, is comfortable with computational procedures (many in the class are not) and also devises clear, easy-to-follow examples of concepts. It became apparent after I took over the class that opportunities to work collabaratively benefited her greatly, as she can then explain problems or solicit explanations from others.

I wanted to examine this student's work more closely as I was wondering how her non-native proficiency in math was affecting her ability to pick up the material. Sometimes the fact that a more traditional math class isn't as strongly language-based can benefit such a student, so I wanted to examine the impact of the shift in emphasis (from first to second semester) to a more communication and collaboration-based.

Commentary

Expectations for Student Success

Despite the occasional reversion to complaints about "copying," by the time I taught this unit, the majority of the class was used to my roundabout answers and referrals to other students. Groupwork and metacognitive questions such as "Why do you think I asked you to do that?" sent the message that students' thought processes was not only valuable to me, but something that they should value among each other.

I provided feedback both through grades, which I give because I must, and comments, which I give because even if only half the class reads them, they allow me to address particulars of students' visible thinking far more effectively than a number. I also provided indirect feedback on the quality of their work by having students comment on each others' work, then commenting on their comments.

Student Understanding

Analysis of the Student Work Samples reveals that both understood the tasks and topics to some degree. Student #1 became confused between Cartesian-to-polar and polar-to-Cartesian conversions, which is not unsurprising, as most of the class had trouble with it from time to time. Student #2 had little trouble with the computations involved despite some confusion on how to write them.

In the LETTER TO ZAP assignment, Student #1 demonstrated the ability to explain a coordinate system and how to use it. I think it's really partly my fault that I only assigned them one point (students drew randomly from a selection), the origin, which obviates a lot of the computational aspects that I imagine students will see in the future. In any case, this resulted in an clear explanation of the (novel) polar system. Student #2 once again demonstrated her computational ability and confusion over notation. Because of the way I constructed the assignment for pairs, I did not have the opportunity to see deeper conceptual understanding on their part. The shortcoming was mine more than theirs.

Assignment Modifications

I did not make any large-scale modifications to the activities based on these two students. I made some adjustments that I felt would benefit these students because they would benefit the entire class, such as streamlining steps on activities (mine tend to be unfortunately wordy, as the Curriculum Unit attests) or adding or subtracting structure from activities in accordance with the class' overall comfort level and performance. Other than those, I wasn't sure what I could do that would be more effective than continuing to encourage students to rely on each other. This is something I have attempted to develop by designating "experts" on particular narrow topics, referring students to other students, and asking them, "Who did you ask before you asked me" when they have a question. If they can't get an answer from another student that satisfies them, then I'm happy to step in.

Comparison with the Class in General

Student #1 is one of my baselines for the class' engagement factor: if I can make an activity interesting enough to hold his attention, there's a strong chance that the rest of the class will be more interested. After all, even the students who politely go along are capable of being bored out of their minds. Ability-wise, he seems to need a little more individualized "push" to talk to other students, but once that happens he grasps the ideas fairly quickly. Gradewise, he is in the lower half of the class, but it currently seems likely that he will emerge with a C of some stripe.

While Student #2 is not a native speaker of English, many of the aspects of teaching that benefited her in particular tended to benefit the class in general. Indeed, she is stronger than much of the class in quickly grasping and simplifying computational procedures, which suggests that she has good "equation sense." (Given the number of students who need prompting on how to solve equations of the form x/3 = sin(30 degrees), I don't consider this trivial.) While not as active in class or group discussions, perhaps because of the language barrier, her written work shows her thought processes fairly clearly. She, too, will emerge with some form of C.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 6 June 2002