Hello friends and colleagues,

To begin this online conversation, we’d like to start with some concrete video examples of mathematics in making and tinkering settings, taken from the in-person workshop at the New York Hall of Science. Here are the links to the short clips on Vimeo (four to five minutes each):

**After watching the videos, please share your reflections with the group (specifying specific time codes in the video whenever appropriate).**

*What do you see as the mathematics currently in these experiences?**What is the potential for mathematical reasoning beyond what is currently present? How would you bring out this potential (e.g., changing the design, asking follow-up questions)?**In what ways are these videos examples of broader o**pportunities and challenges related to integrating mathematics with making and tinkering?*

For this discussion, we're lucky to have Jan Mokros from the Maine Mathematics and Science Alliance to help us frame and facilitate the conversation.

Looking forward to your thoughts and reflections!

Best,

Scott Pattison, Andee Rubin, and the *Math* *in the Making* team

Hi Folks,

I'm Josh Gutwill, Director of Visitor Research and Evaluation at the Exploratorium. I attended the Math in the Making workshop in May. After watching the videos, I see that I was feeling

veryenthusiastic during the 3D Sculptures activity. (I'm the bearded guy who seems overly excited in the video.) This is why I don't drink coffee!In terms of mathematical thinking, I believe that Bronwyn was showing such thinking during the Scribbling Machines activity, when she connected the goal of making a "tighter" (smaller) circle with the strategy of shortening the length of the lever arm holding her marker. But this exercise of coding the videos raises questions for me, which I hope we can wrestle with during this forum: People use mathematics in their everyday reasoning all the time. Is there something special about using math in making? Is making particularly conducive to mathematical thinking? I guess these questions reflect my own ambivalence about seeking math in the making -- on the one hand, I think it's very important for us informal educators/practitioners/researchers to identify, value and foster rich STEM thinking in our exhibits, activities and programs. On the other hand, I want us to do that in an authentic way. What I mean is that I want the mathematical reasoning to genuinely be a part of the making activity, not an add-on or overlay. Coming back to the video example, I think Bronwyn was authentically using mathematical thinking, in that she probably would not have made her next move to shorten the arm length without that mathematical reasoning. It's hard to know what she was thinking, but her comments implied both an understanding of the relationship between radius and circumference and the use of proportional reasoning for shrinking the circumference. Could a facilitator have helped Bronwyn articulate those ideas? Might that conversation have led to something new or deeper in Bronwyn's making? I'm not sure, but I'm curious about how deepening the math can authentically enrich the making.

What I'm seeing in the two videos is several examples of people making hypotheses and/or raising mathematical questions about the objects. Lots of predictions about "if I do this, I think this is going to happen." In the sculpture example, people are making predictions about how the arm and the main cavity are related. They are asking questions like "what would happen if we had a hole over here?" and "what could we do to make it spin?" In the scribbling machine examples, people are specifying goals, like "I want to make a tighter circle" or "I want to see if it will continue to spiral or will it come back to the inner space." It was amazing to see this level of articulateness about goals (though of course, we are used to doing this in our work!) Regarding Josh's question about whether this is "something special" or another example of everyday reasoning, I think it is something special because there is a mathematical goal that is defined by the participants. Once this is set, you naturally have to use mathematical reasoning to achieve the goal that you set. think the facilitator might be helpful in articulating these mathematical goals with the people who are working on them. In the examples we saw, the goals sprang up pretty spontaneously, but I don't think this would always be the case. I think many people would explore in a more free form way, which is fine, but the facilitator could jump start the process of mathematical goal-setting. But facilitating mathematical goal-setting may be easier said than done!

I’m interested in this point about “mathematical goals” in these activities, Jan. It seems to me that the goals stated by the folks in the video are instrumental, maker goals that just happen to require mathematical thinking to be achieved. To me, this suggests that one of the tricks of integrating mathematics and making is finding compelling, authentic maker goals that naturally require mathematics and giving learners tools that enhance and support mathematical thinking as they pursue their goals. Of course, I’m also wondering if it would have felt too constraining if the facilitator at the beginning of the 3-D sculpture activity had said something like, “your challenge is to make a sculpture that includes a ‘bent elbow’ shape.”

Scott

Hi all,

This is Tracey Wright, Senior Researcher/Developer at TERC, and part of the Math in the Making Team. While watching the Inflatable Sculptures video I was struck by the role of experimentation, both thought experiments and actual empirical testing of ideas. To me, this is an area where math/science overlap. (If you look up “thought experiment” in Wikipedia, <<https://en.wikipedia.org/wiki/Thought_experiment>> you can see more about its history in both fields.) I see a lot of important mathematical reasoning going on already in the sculpture examples, and experimentation is one aspect of this. “Let’s try it out….” (empirical testing) is heard as a wish, and enacted with the bird. Molly does more out loud reasoning, “Don’t you think “x” will happen if_____ ? “ and gestures while predicting what could occur. I find that there is potential in this environment for people to move between thought experiments and actual testing (although we don’t see this back and forth movement in these short clips). Experimentation can be a valuable way for people to address their own goals (some of which are stated explicitly, “What we’re trying to figure out is how to make the wings flap.” “I want to see if it will spin.”). I wonder what it is in THIS making environment (and the Scribbling Machines example) that encourages experimentation in its multiple forms and how this can be supported in other making environments….

Tracey

Hi. My name is Steven Greenstein and I'm an assistant professor at Montclair State University in NJ. My field is math education. I did not attend the workshop at the NYHoS, but I wish I had. Earlier this summer I attended a Making and Learning conference which took place at the Children's Museum in Pittsburgh.

I'm taking a different approach to making and learning that has the teachers doing the making, and what they're making are tools to support the math learning of their students. This approach contrasts the one I hear about on this thread and with which I myself have struggled: Where's the math in the making? I get some sense that mathematical activity is going on (e.g., discovering and representing patterns, making and testing conjectures, constructing examples and counterexamples, devising and defending arguments), but I get a bit stuck when I try to attach those things to a respectable range of mathematical concepts.

Consistent with a math educator perspective, in this work I've started with the math and then looked at 3D design and printing as a possible venue for thinking in new ways about what teaching and learning math might look like. Honestly, I started the other way around. I thought 3D printers were cool and I tried to find a way to incorporate them into my research. "Manipulatives" was an easy place to start. Manipulatives have a long history in mathematics education, especially in elementary school where they've been used to teach such concepts as number, fraction, and place value. The way learning works with manipulatives is the way learning works with any physical tool, so I figured we could develop new tools to teach any mathematical concept to any mathematical learner, from the elementary concept of number to the advanced concepts of topology and beyond. So I taught a doctoral level course called Designing for Mathematical Experience, which I describe more fully here. You can see their final projects and trace their development here.

I offer this story as an alternate view of what math and making might look like.

An idea that comes to me, on the face of the three initial questions, is that we should revise/study/discuss math ed literatures that deal with them in other fields or practices. Perhaps the most significant one is the literature on ethnomathematics. The original program of ethnomathematics has been to pluralize mathematics through cultural ethnography: to find the other mathematic(s) that western-centric traditions have ignored and/or dismissed in the name of colonialism. The ethnomathematics program is alive and well, but in the course of a few decades it has generated a healthy amount of criticism and exchange of ideas. The most extreme critique has been published by Rowlands and Carlson (2002) arguing for the universality of mathematics. A more nuanced critique based on cultural alienation is the one of Pais (2011). At the last ICME, Veronese et al presented a paper entitled ‘The evolution of ethnomathematics” (in press) which is an effort to re-define the aims of ethnomathematics in the course of addressing these criticisms. Another relevant body of literature is the one on mathematics in the workplace, such as the multiple interpretations and revisions of ideas that Analucia and David have produced about street sellers, or the ones that Celia and Richard wrote about the ways nurses deal with drug dosages.

In any case, the first notion that I want to antagonize is a tacit expectation, which may not be shared by us but it is common enough, that one can find mathematics “there” in the same way that one can find pieces of apple in a fruit salad. I’d say that it is not a matter of finding the mathematics happening “currently in these experiences” but of envisioning mathematical investigations that could grow from them. As any other vision, these en-visions are supposed to be open-ended, ambiguous, hesitant, and weak enough to be able to adapt to unforeseeable circumstances. The scribbling machines video stimulates me to think about randomness, continuity, periodicity, and so on. The 3-D sculptures video suggests to me, among other things, the interplay between volume and surface, or how maximizing volume determines shape and vice-versa. However, there is a lot that has to happen to transform these perceptual suggestions (I say perceptual because they arise from seeing), into productive visions for mathematical investigations. I believe there are, at least, two conditions important to be part of such transformations. One is that whatever we feel stimulated to pursue does not leave behind making, turning into a textbook-type of mathematical study. It is likely that some kind of constraints will occur to us (e.g. what kind of sculptures can we create by joining cylindrical shapes?) but these must be constraints that push the envelop for our work on 3D sculptures as well. The second one is that the ensuing mathematical investigations should be such for all the participants, including educators, professional mathematicians, parents, etc. To me, an en-vision from a mathematically educated participant for an investigation, the conclusion of which he deems fully explained in a certain webpage or chapter of a book, is neither inspiring nor thought provoking. It has to also be an investigation for him or her.

I could go on and on. But these are supposed to be little nuggets.

A hug

Ricardo

Pais, A. (2011). "Criticisms and contradictions of ethnomathematics."

Educational Studies of Mathematics76: 209-230.Rowlands, S. and R. Carson (2002). "Where Would Formal, Academic Mathematics Stand in a Curriculum Informed by Ethnomathematics? A Critical Review of Ethnomathematics."

Educational Studies in Mathematics50(1): 79-102.Just wanted to clarify that Analucia and David are Analucia Schliemann and David Carraher. One of their signature works is Street Mathematics and School Mathematics.

Book_Author:Terezinha Nunes, Analucia Dias Schliemann, David William CarraherBook_Title:Street Mathematics and School MathematicsReference:1993, Cambridge University PressCelia and Richard are Celia Hoyles and Richard Noss who worked with Stephano Pozzi on the following 2 articles:

Proportional Reasoning in Nursing PracticeCelia Hoyles, Richard Noss and Stefano Pozzi

Journal for Research in Mathematics EducationVol. 32, No. 1 (Jan., 2001), pp. 4-27

Abstraction in Expertise: A Study of Nurses' Conceptions of ConcentrationRichard Noss, Celia Hoyles and Stefano Pozzi

Journal for Research in Mathematics EducationVol. 33, No. 3 (May, 2002), pp. 204-229

Just wanted to clarify that Analucia and David are Analucia Schliemann and David Carraher. One of their signature works is Street Mathematics and School Mathematics.

Book_Author:Terezinha Nunes, Analucia Dias Schliemann, David William CarraherBook_Title:Street Mathematics and School MathematicsReference:1993, Cambridge University PressCelia and Richard are Celia Hoyles and Richard Noss who worked with Stephano Pozzi on the following 2 articles:

Proportional Reasoning in Nursing PracticeCelia Hoyles, Richard Noss and Stefano Pozzi

Journal for Research in Mathematics EducationVol. 32, No. 1 (Jan., 2001), pp. 4-27

Abstraction in Expertise: A Study of Nurses' Conceptions of ConcentrationRichard Noss, Celia Hoyles and Stefano Pozzi

Journal for Research in Mathematics EducationVol. 33, No. 3 (May, 2002), pp. 204-229