## Posts Tagged ‘Math 121’

### The Many Ways of IBL Conference

June 26, 2013

I attended the University of Chicago’s “Many Ways of IBL” conference last week. Here is a brief list of my thoughts for the week, in no particular order.

1. It was utterly great to see a couple old friends. I have been blessed to have had good colleagues everywhere I have been, and I wish that I could have taken many of them with me to my current position.
2. It was great to meet a bunch of new friends. I hope to stay in touch with many of them.
3. Part of the conference was to watch John Boller teach an IBL class on real analysis to a bunch of super-motivated high school students. Both John and the students did a fantastic job. I told John that it was so enjoyable that he could charge admission.
4. One big thing I was failing at with IBL last year: I did not discuss the statements and meanings of the theorems before students presented. Boller did this, and it must help students understand everything about the course better.
5. Paul Sally continues to be amazing. He is also hilarious.
6. In many classes, I have students read the textbook rather than lecture. I have no idea how to mesh this with IBL, but it is something I value. I realized from the conference that the reason why I value this is that it helps students learn how to learn on their own.
7. Even though I have been calling my recent hybrid classes “a mix of Peer Instruction (PI) and IBL,” I no longer think that I have been doing IBL. At best, it is IBL-Lite, although it is probably just “students presenting problems.”
8. This will lead me to alter a paper that I recently wrote on a PI/”IBL” calculus class; I will now qualify that my IBL is pretty weak.
9. I am now fairly certain that my courses for pre-service elementary education majors are IBL.
10. I might do IBL in my abstract algebra course this spring. If so, I might interweave IBL and PI differently: I might mainly do IBL, but then have some PI days to make sure students understand the ideas that have already been presented.
11. In abstract algebra, I might also create a class journal, where students can submit homework problems to an editorial board (of students) for peer review.
12. In IBL classes, have students take pictures of the board work. They can then upload the pictures to the course website as a record of what happened.
13. Matthew Leingang gave me a nice way of communicating course rules. He has “The Vegas Rule” for his class: “What happens in Vegas, stays in Vegas” where “Vegas” is defined as “the world outside of this classroom.” This is a nice concise way of reminding students to not use previous knowledge and outside sources.
14. Leingang also got me excited about paperless grading. Now I just need to find \$1200 for an iPad and scanner.
15. Ken Gross uses an “adjective-noun” metaphor for fractions, where the adjective is the number and the noun is the whole. That is, you can explain common denominators by doing something like: $2/3$ “units” $+ 1/2$ “units” equals $4/6$ “units” $+ 3/6$ “units,” which is equivalent to $4$ “sixths of a unit” $+ 3$ “sixths of a units” $= 7$ “sixths of a unit” $= 7/6$ “units.” Most of the work then is just changing the “noun” and finding the appropriate “adjective” for each of the new nouns.

### What oral exams taught me

June 8, 2012

In my course for elementary education students, I once again gave oral exams—this time for the final exam. Here are two take-aways from the oral exams.

First, I need to do some peer instruction next time. In particular, students had a difficult time understanding the difference between the “whole” of a fraction and the “denominator” of a fraction (Consider “$\frac{1}{2}$ of a mouse” and “$\frac{1}{2}$ of an elephant.” Both have a denominator of “2,” but the whole of the first is “mouse” and the whole of the second is “elephant.” This leads to different meanings. I think that three clicker questions would eliminate this.

Second, I was shocked at how ineffective my lectures were. The oral exam questions (which they also had to create screencasts for) were ones that were previous done in class (for example: why does inverting and multiplying give the correct answer to a division problem?). The process was this: students would figure out why the algorithm works, and then present at the end of a class period. I begin the next class period by giving the same argument. Other class periods begin with students presenting on similar questions, the class evaluating the presentations, and—if needed—me presenting the correct explanation.

Furthermore, I gave the answers to each of the oral exam questions on the last day of class. Test test So students saw the answer to each oral exam question at least three times, and probably more (especially since I had students view other students’ video solutions).

I was concerned that students would simply memorize these explanations. This simply did not happen. Either students understood the algorithm (I can tell from the oral exams—these students could answer any question that I had on the algorithm) or students did not understand any portion of the algorithm.

Most puzzling is that, in my student evaluations, some of my students complained that they were never shown how to do the algorithms correctly. This is in spite of seeing a completely correct solution to every problem between 3 and 10 times. I can only explain this in two ways:

1. Somehow students did not understand that the solutions they saw were solutions to the problems from the oral exams and screencasts. This would mean that I did not clearly communicate the intent of presenting the solutions.
2. Lecture was monumentally ineffective in helping them learn—so much so that students did not even remember that they occurred.

Do you have any other ideas?

### Jigsawing

April 19, 2012

My elementary education students are creating vlogs that explain why different algorithms work for different operations. They have been creating roughly one video per week, posting them, and then getting feedback from the course grader. The only graded part of this is at the end of the semester after many drafts.

This week, we did a jigsawing-type activity to improve the videos (like most everything else, this idea was inspired by Andy Rundquist. On Tuesday, I split the students into four groups: one for addition, one for subtraction, one for multiplication, and one for division. The students came to class having watched all of the videos on their particular operation, and the class period was spent deciding what makes for a good explanation for that operation. At the end of the class, we split into new groups where one member of the group had just studied addition, one subtraction, one multiplication, and one division.

Today, we spent the entire class period reviewing videos in these teams. One team member was an “expert” on each operation from Tuesday, and they made suggestions on how to improve the explanations.

I asked everyone if this was useful enough to repeat on our fractions algorithms, and every student said that it was (most were emphatic). This appears to be a success.

My one reservation: although I am not sure, it appears that some students are trying to memorize a good explanation rather than understand. I know that I will be able to tell which students really understand from the oral exams, but I am wondering if it will be clear from the videos. Does anyone have any experience with memorizers?

### Assessing with Student-Generated Videos

January 17, 2012

I regularly teach a course for future elementary education majors. The point of the class is for the students to be able to do things like explain why you “invert and multiply” when you want to divide fractions. This involves defining division (which, itself, requires two definitions—measurement division and partitive division are conceptually different), determining the answer using the definition, and justifying why the “invert and multiply” algorithm is guaranteed to give the same answer. At this stage, I simply tweak the course from semester to semester. This semester, though, I am making a major change in how I will assess the students.

Since this class is for future teachers, it makes sense to assess them teaching ideas. So there are three main ways of assessing the students this semester:

1. The students will have two examinations. Part of each examination will be standard (a take-home portion and an in-class portion), but there will also be an oral part of the examination. The oral portion will require students to explain why portions of the standard arithmetic algorithms work the way they do.

I only have 31 students in this class (I have two sections), so hopefully this will be doable. Moreover, I am going to distribute the in-class portion of the exams over a period of weeks: many classes will have a 5 minute quiz that will actually be a portion of the midterm.

2. The students will regularly be presenting on the standard algorithms in class. This is only for feedback, and not for a grade. I am hoping that the audience will listen more skeptically to another student than they listen to me.
3. The students will be creating short screencasts explaining each of the standard algorithms (Thanks to Andy Rundquist for this idea). Students will be given feedback throughout the semester on how to improve their screencasts, but they will create a final portfolio blog that contains all of their (hopefully improved) screencasts for the semester. This portfolio blog will be graded.

I will keep you posted. I welcome any ideas on how to improve this.

### Scholarship and Creativity Day 2011

May 5, 2011

As I did last semester, I had my students (all elementary education majors) do mini-research projects and present at a small poster session.

As before, these posters were optional, although a student cannot get an A for the semester without doing one. I have 37 students, and 24 choose to do a poster. Unlike last semester, there was no paper that accompanied the poster.

Also unlike last semester, I did not hold the poster session during class time. Instead, I integrated it into the campus-wide “Scholarship and Creativity Day.” There were no classes this day—it is a day completely devoted to showing off students’ creative projects.

Here were my suggested projects:

1. Note that $\frac{1}{2}=0.5$ and $\frac{3}{4}=0.75$ do not have repeating decimals; we say that they “terminate.” How can you tell which fractions in Martian arithmetic will terminate?
2. Consider extensions of our Last Cookie game (basically, a Nim game). What is you could remove either 2 or 3 cookies per round, but not 1? What if you could do 1,2, or 4 What about other combinations?
3. There is a division algorithm called “Egyptian division.” Explain (as we have been doing) why this gives the correct answer to a division problem.
4. Learn about “casting out nines,” a method that helps you determine if you did an arithmetic question correctly. Explain why this method works.
5. There is a fast and easy way to determine if a number is divisible by 3 in base ten. Explain why this method works.
6. There is are not-so-fast and not-so-easy ways to determine if a number is divisible by 7 in base ten. Explain why one of these methods work.
7. Explain divisibilty results for other bases (can you easily tell when a number is even/divisible by 3/5/7/etc in base six? Base eight?)
8. Research one algorithm from the Trachtenberg System, and explain why it is guaranteed to give the correct answer.
9. Teach Mayan students how to use our number system.
10. Come up with your own topic (talk to me about it first).

By far, most students choose the “divisibility by 3” or “casting out nines” problems, a reasonable amount choose “teach Mayan students about base ten” “the Last Cookie” problem. Three others did a Trachtenberg problem, one student chose to explain “Egyptian Division,” and two explained why a finger trick works for multiplication by nine.

Many of the presentations were excellent, and many still had trouble understanding what the question is. This was expected. What was not expected was the number of students who participated: I expected about half the number I had.

Finally, many professors from other departments approached me to compliment the poster session. In fact, the dean of the college referenced one of my students’ posters in an address later that evening.

I must remember to try to do this again in most of my classes.

### Peer Assessment

May 2, 2011

In my ongoing attempt at helping my students to understand the difference between arithmetic and mathematics, I had my students do a peer assessment exercise on the papers they are writing to explain why certain arithmetic algorithms give correct answers (e.g. why long division gives the correct answer to a division question).

I had all of the students bring drafts of their papers in, and the students had self-assessed their papers by “traffic-lighting:” a mark of green at the top of the paper means that the student thinks that the paper is close to being the final draft, a “red” means that they think they have a long way to go, and a “yellow” is somewhere in between. I then grouped the students by “traffic-light,” planning on having the “green” group and “yellow” group read each other’s papers and offer feedback, while the “red” group would work with me directly to get them on track. The reality is that pretty much everyone gave themselves a “yellow,” so this was not much of a differentiation. (I stole this whole idea from Assessment for Learning: Putting it into Practice).

Here is what I learned:

1. This seemed to be extremely helpful to some students. I asked some students what they learned, and they told me exactly what I had hoped they had gotten out of it.
2. I am not very good at organizing peer assessment sessions yet. I got the sense that many students did not know what they were supposed to be doing, and consequently they were off-task and/or left a couple minutes early. I also think that this might not warrant an entire class period.

I am hoping that I look back on this in five years and laugh at how hard this was for me in 2011. In the meantime, I would love any advice that people have on peer assessment—I really do need to improve on this.

### Arithmetic vs. Mathematics

April 28, 2011

I am teaching a content course for elementary education majors this semester, and I had them do presentations yesterday. Their task: “explain why an unusual multiplication or division algorithm (e.g. “Lattice Multiplication”) gives the correct answer to a multiplication/division problem.” The idea, of course, is to show how the algorithm is really just doing a definition of multiplication (usually “repeated addition”) or division (either partitive or measurement).

Several presentations were good, and one was dazzling. However, most of the presentations missed the question entirely, and simply demonstrated how to do the algorithm (and assuming that it is an algorithm worth doing). This has been an on-going challenge this semester (and every other semester I have taught this course). I have had some successes, but they are usually short-lived for most of the class.

I am also giving the students feedback on drafts of papers that explain why some algorithm gives the correct answer (according to the definition). We will see how much this helps.

Does anyone have any suggestions on how I could communicate the question (or that a question even exists)? The students are very focused on the arithmetic, but ignore the mathematics.