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Reasoning with Yarn

Authors:

Abstract

This work is part of a larger project examining mathematical practices within traditional textile crafts, and how they might be leveraged in middle school classrooms. Through interrogating the practices of expert knitters, we determined that much of the mathematics was most salient—at least in explicit form—in designing projects. While designing is often reserved for more expert knitters, this study sought to determine whether novice knitters could begin designing from the outset, and if so how the knitting might become a resource for the proportional reasoning necessary for designing patterns.
This material is based on work supported by the National Science Foundation under Grant No. DRL-1420488; the views expressed herein do not necessarily reflect those of the National Science Foundation.
The author also wishes to thank Melissa Gresalfi and the rest of the Vanderbilt Interactivity Lab.
BACKGROUND
RESEARCH QUESTIONS
FOCAL CASES
SUSIE– SEPARATE SPHERES AMY– GENRE BLINDERS TORI AND STEPHANIE– TRANSPARENCY
This work is part of a larger project examining mathematical
practices within traditional textile crafts, and how they might be
leveraged in middle school classrooms.
Through interrogating the practices of expert knitters, we
determined that much of the mathematics was most salient—at
least in explicit form—in designing projects. While designing is
often reserved for more expert knitters, this study sought to
determine whether novice knitters could begin designing from
the outset, and if so how the knitting might become a resource
for the proportional reasoning necessary for designing patterns.
One of the first mathematical ideas that comes up when
designing is calculating gaugehow many stitches per inch,
or rows per inch, in a particular piece of fabric. Gauge is
affected by needle size, yarn weight, and an individual knitter’s
tension, and thus each knitter must create a small sample—
called a swatch—and calculate her own gauge for each new
project.
RQ1: How is mathematics evident in these instances of
novice knitting and beginning design?
RQ2: How do we undermine reasoning with knitting?
RQ3: How might we support mathematical reasoning
with knitting?
Susie
RQ1: Enthusiastic about the ways proportional reasoning is
useful in knitting; still no evidence that it pushes back on her
pencil-and-paper reasoning—math and knitting remain
separate spheres, though she can leverage both.
RQ2: When calculating gauge, simply looked at one inch and
counted the number of stitches, rather than measuring the
whole swatch and determining a unit rate.
RQ3: Knitting fluency afforded many more opportunities to
calculate gauge.
Amy
RQ1: Talking about math pulls her out of the knitting sphere
—when prompted for calculation, she literally sits back from
the table and asks for explicit instructions.
RQ2: Disfluency with the knitting itself limited the number
and kind of encounters with mathematical ideas.
RQ3: Using knitting as a site for mathematical thinking allows
her to ground reasoning and step out of “magical
manipulations” mode (simply guessing about operations).
Tori and Stephanie
RQ1: Engaged in spontaneous proportional reasoning both
within and beyond what was designed for. Mathematical
reasoning is a transparent system for organizing their
emerging work, both in their design, but also in time and
space. The two spheres are mutually reinforcing.
RQ3: Some of the emergent questions from this group could
be incorporated into future designs, including speed and on-
the-fly adjustments to designs.
REASONING WITH YARN
KATHERINE CHAPMAN
METHODS FINDINGS AND IMPLICATIONS
All five days were videotaped, with four days including
individual recordings of each of four participants.
All participants completed a mid-week written assessment of
proportional reasoning in the context of knitting (also
videotaped).
All participants but one were briefly interviewed on the last
day about their impressions of any overlap between math
and knitting.
Videos were initially coded for explicit evidence of
mathematical reasoning in dialog, with chief focus on didactic
episodes with the researchers, including the mid-week
assessment.
Next, individual videos (usually from individual cameras)
were coded for evidence that the reasoning from the initial
didactic interaction was either repeated or adjusted, explicitly
or otherwise (e.g. through material constraints).
Participants were youth, ages 10-16, who self-selected to
participate in a week-long “learn to knit” program offered over
the summer at a public library lead by three researchers
Interactivity Lab
RQ1: For Susie, math was leveraged in service of knitting;
whereas for Amy “math talk” seemed to hide the knitting,
though knitting eventually served to anchor mathematical
reasoning. For Tori and Stephanie, the two genres were more
mutually reinforcing—they demonstrated spontaneous
proportional reasoning related to their knitting but not
specifically included in the original design.
RQ2: Elements of the learning ecology that undermine
reasoning include simplified ways of calculating gauge, and
disfluency with the physical materials, which lead to more
limited opportunities for mathematical exploration
RQ3: Mathematical reasoning is supported through reference
to knitting, particularly for younger students who were
motivated to think through problems beyond grade-level, as
well as being pulled out of rote algorithmic guessing through
access to the material. Additionally, future designs will explore
incorporating discussions of precision in measurement to
address findings from both RQ1 and RQ2.
Tori: I’m trying to get this to be I
don’t think this is going to be big
enough for a purse, because I’m on
24 now, and I have 28 [in my pattern].
It’s gonna be way too small.
[conversation with researcher]
Tori: So, maybe I’ll make it to 40?
Stephanie: Um, it depends on how
long you want. Right now you have
24, and if you wanted it like 48
would be double that size, wouldn’t
it? It’d be about here.
Tori: Yeah. That’s pretty good.
Amy: Wait it’s a math problem, right?
What, how do I write it, then? I don’t
know how to write that.
Researcher: Well, let’s think about it.
There isn’t really one way to write it,
actually.*
Researcher: let’s just think about this,
instead of let’s just think about, on
this ruler, if we are only looking for one
inch [ ]
Amy: Let’s see
Researcher: So Yep, look down here.
Here you can use this knitting needle.
One, two, three, four, five stitches in
one inch. And now I want to show you a
pattern. So if it’s twenty five stitches in
five inches, and you also know it’s five
stitches in one inch, how many, um,
how many stitches do you think there
you would count
Amy: Ten!
Susie—Part of the older group; significant knitting experience
from final-day interview:
Susie: [I used math this week] when I was trying to figure out, like, if
my gauge whenever I like, take my gauge, like for rows and
stitches, and like figure out how many stitches it will take and like, like
you said, it will take me, somebody said it will take like seventeen
hours to finish my pillow because I like did one row in six minutes. []
It was definitely different [than math class] especially the way you
approached it, like when you first see knitting it’s like clack clack clack
there’s no math. But there’s, there’s more math than I thought there
was in knitting, and I mean I’ve been knitting for a long time. []
Interviewer: When we asked you to do the knitting problem on paper,
did that seem like it was the same thing we’d been doing?
Susie: No, it was different. Um, I don’t know. It’s just—it seemed a lot
more complicated on paper to me. ‘Cause I hadn’t thought about the
way that they’d written it exactly, I guess. [] I just tried to think, um, I
tried to think logically and hope for the best.
Part of the older group; neither had experience with knitting.
Amy—Part of the younger group, some experience with crochet
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