Math by Making

An entrepreneurial endeavor in designing for mathematical experience by the students of the PhD program in Mathematics Education at Montclair State University

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This is how my project turned out. Couple of issues, but it is a working prototype. I was able to test it out, as well as my ability to conduct a teaching experiment.

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The finished product.. Very please with my work... Used velcro to stick the pieces together.. #abiodunbanner #discoveringgeometry #derivingarea

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The final print for 2d general neighborhoods. 

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Here is the final numberline product.  It turned out better than expected. 

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These are the final products for the 2D models of neighborhoods that I studied.  The bottom is a numberline with neighborhoods and open intervals, while the top is a general open neighborhood for 2D.

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...of Invention

Well, it seems that the limitations of my design ability led to a more simple, elegant and versatile form of my Normal curve tools. The final draft form of the two curves, and the original prototype in transparency, are depicted below.

On question I would like to study is whether the more solid manipulative leads to a different learning experience than the transparency form.  That would be an interesting follow up to the study I have planned!

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Here's a print of my number/factor strips. They look good, so now I just need to re-print them with colors. My interview questions are all set, and I intend to conduct both of them within the next few days. I want to focus on the student's interpretation and use of the tool, so my questions do not have much to do with particular mathematics content. I am interested in the meanings they make of the physical knowledge of the tool and any connections they see for themselves to mathematics. I also would like to ask questions about the ways in which they see items like this in their daily lives. In this way, I'm curious as to whether other cultural knowledge, outside of Yup'ik, can be seen in the tool.

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Finally

So... I finally got an almost perfect slash decent enough print to proceed with my teaching experiment. As it turns out, printing my flat object sideways/vertical was the way to go. Unfortunately, one of the arms didn’t quite hold up... Ceire is trying to fix it for me with her 3D pen. I created and practiced my questions with Zareen... she seemed impressed and was very encouraging. Unfortunately I was so angry and frustrated prior that I really was not able to relax and change my mood despite her kind words. I still won’t get to conduct my experiment as originally intended, as a series of experiments with increasingly complex iterations of my coordinate slider... but I guess you have to save something for the dissertation... once you’re more familiar with and practiced in the teaching experiment methodology (you being me, of course).

What else? I do have to experiment with my design and see if I can improve the flimsiness of the product (the arms move around a bit too much) and improve the smoothness of the moving arms (they get stuck and you have to use two hands where I was hoping for more fluid motion). I think I also have to stick rubber feet on the bottom to create more friction between the base and the surface it’s placed on. Given the effort it takes to move the pieces, the base was sliding on the table. I could always hold it while the participant in my teaching experiment moves the arms... but that would be weird.

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Neighborhoods and problems.

The tools are ready, the questions for discussion are ready, and the consent and assent forms are signed, however, the school needs board approval.  I intended to speak with students and perform my study last Friday, but was turned away pending board approval.  This is extremely frustrating for me.  However, there may be a positive.  If I cannot receive board approval this week, I can interview three friends.  I may complete this interview regardless of board approval since it presents an interesting situation.  My friends all have different levels of knowledge about mathematics.  One is my fiance, who has taught mathematics at the elementary and middle school level, another is an English major, and the other is a middle school elementary teacher with a B.S in Mathematics Education.  I believe it will be very interesting to interview the differing levels and compare the results, however I still wish to see the knowledge gained from the Calculus student’s perspective.

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It’s done, its finally done… There’s something so satisfactory about seeing your ideas come to life. I remember struggling to formulate my project idea… then struggling to design my idea…. struggling to print the idea and now to see it in reality.

As I walked into a graduate class, my project, fresh off the printer and in hand I was abruptly stopped by a classmate. As she excitedly shifted my project pieces from hand to hand she exclaimed, “This is so cool! Do you know what you can do with this?” As I sat silently, an amused grin on my face, I listened to her explain how this model can be used to derive the area formulas for polygons. She pulled me over to a nearby table and showed me how my printed shapes could be transformed and manipulated so as to derive these formulas. Afraid to trample her excitement and feelings of accomplishment I feigned surprise. In my most grateful voice I thanked her for her innovative ideas and said I would give it some thought. As the professor began class, I couldn’t help the feelings of achievement that washed over me. This will be a great project I thought, everyone can see its benefit…. As I packed up my pieces to focus on the instructions being given a small voice whispered inside… but she is a math major with a mathematical way of thinking.

With that in mind I’ve decided to do 2 different experiments with my project… One will be with freshmen students who have not been introduced to geometry and the other will be with other math teachers who do not teach geometry. Will I be able to replicate the “Aha” moment I witnessed earlier within theses students or will it come as easily to my colleagues. Will they immediately see the usefulness of my project… I am excited to find out…

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Slicing a cone

Finally our manipulative: a cone and it’s slices is ready. We will conduct interviews with two people in the next two days. It has been a wonderful experience to design the tool on tinkercad. It started from a simple three dimensional cone and later took varied forms. We encountered  problems in designing , printing, joining the slices9 tried magnets, double side scotch tape and velcro tape) and each of them has been a learning experience for sure. here are the different tools that we printed.

model 1 

model 2 

model 3 ( was too delicate and broke while we tried to put magnets)

model4 ( printing was stopped mid way because of 100% fill; which meant it would take days to print)

final model ready for interviews

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The final product is finally done!  This week I plan to question students about what learning takes place.  I am very excited that the product turned out as well as it did.  Thanks to Iain and his idea of splitting the sphere, it turned out fantastically.  I hope to encourage a deep understanding of neighborhoods using a physical manipulative instead of the abstract “picture a sphere” definition that has always existed.

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The interviews on the 5-year old and high school math teacher were conducted.  The 5 year old’s interview completely shocked me.  I don’t know how much to share without giving our paper away, but it was not what we were expecting.  The teacher interview was also not what we were expecting with the teacher focusing more on the questioning instead of creating an activity/experience.  

I was considering interviewing another teacher that might not be as jaded.  

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So... I heard that my vertical print went well... just as I was about to call it quits. I went to pick it up but no one was in the lab. Actually almost emailed you, Dr. Greenstein asking for advice on the whole ‘printing with feet’ thing... but then Adam told me he saw my print. Guess we’ll see. Just means I have a little less time to test it out, and maybe only one of the iterations... I don’t know...

Motivation is at a low... could do a lot with my idea... keep tinkering, working with the design, and I think work out a decent dissertation... I just don’t know how this all fits in with life, with family goals, personal goals, professional goals... I’m sort of noncommittal in that sense right now... it will all pan out.

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Interviewing...

I had my first interview with two 8th grade students today using my tool.  Below are some questions and things that came out of that 45 minute interview/task exploration. This post is just based on my notes from the observation, I have not started transcribing yet.

- They quickly found the triangle and circle but had difficulty finding the parabola and ellipse.  After some time working on the first task of finding conic sections from the clay model they created, we had a short discussion about all conic sections and I set up the next task (design an interesting space that someone could occupy using the pre-printed conic section pieces and the 3D pen).

- I wish I had videotaped the students completing the tasks instead of audio-taped.  The students were talking about the shapes they were using without identifying them.  Although I took as many notes about their decisions as I could, I feel like having the video tape to refer back to would be really helpful.  

- Because I only had one 3D pen, I asked them to collaborate on the design task.  I’m not sure if I lost anything by doing this.

- They thought deeply about the design but decided they would create something better next time. 

- I plan to go back to interview/present this task again.  I’m deciding whether i want to explore what the students meant by “better” and see what other design they may come up with after thinking about design (and learning how to use the 3D pen) or if I want to see what a new set of students will complete the task from beginning to end.

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Methods

Assuming I get a decent print... or even if I don’t... I intend to test my slider on my high school students. The upcoming testing schedule is actually pretty convenient. I should be able to film teaching episodes as part of a teaching experiment. Like I said, I haven’t been able to get a good print... I think at this point I’ll just shoot for the first iteration or two of my project. I may not get as far as I initially intended, but I’ll at least be able to conduct a teaching experiment, get a feel for that, and the whole transcription bit (maybe), and give my slider a trial run at the same time.

It’s crazy how sitting down to write really does help my thinking... I’m only just now realizing that I was always trying to print the later iterations of my project (the more complex ones)... at least that’s what I’ve been focusing on lately. I’m going to go request some prints now then :o)

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Tori and I started interviewing our participants. We are very excited by our initial data and we feel that we will have a lot of interesting things to report. For our methodology we are going with Phenomenography because we are interested in our participants perceptions of mathematical experiences.