## Abstract

Design-based research studies are conducted as iterative implementation-analysis-modification cycles, in which emerging theoretical models and pedagogically plausible activities are reciprocally tuned toward each other as a means of investigating conjectures pertaining to mechanisms underlying content teaching and learning. Yet this approach, even when resulting in empirically effective educational products, remains under-conceptualized as long as researchers cannot be explicit about their craft and specifically how data analyses inform design decisions. Consequentially, design decisions may appear arbitrary, design methodology is insufficiently documented for broad dissemination, and design practice is inadequately conversant with learning-sciences perspectives. One reason for this apparent under-theorizing, I propose, is that designers do not have appropriate constructs to formulate and reflect on their own intuitive responses to students’ observed interactions with the media under development. Recent socio-cultural explication of epistemic artifacts as semiotic means for mathematical learners to objectify presymbolic notions (e.g., Radford, Mathematical Thinking and Learning 5(1): 37–70, 2003) may offer design-based researchers intellectual perspectives and analytic tools for theorizing design improvements as responses to participants’ compromised attempts to build and communicate meaning with available media. By explaining these media as potential semiotic means for students to objectify their emerging understandings of mathematical ideas, designers, reciprocally, create semiotic means to objectify their own intuitive design decisions, as they build and improve these media. Examining three case studies of undergraduate students reasoning about a simple probability situation (binomial), I demonstrate how the semiotic approach illuminates the process and content of student reasoning and, so doing, explicates and possibly enhances design-based research methodology.

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## Notes

It seems only a matter of difference in discursive norms of two communities of practice – mathematicians, designers – that mathematical problem solving is currently held to higher standards of accountability than design-based problem-solving. Yet in a climate of public accountability, I am concerned, under-conceptualization might lead a designer or reviewer to devalue or even reject a potentially sound decision because it apparently cannot be rationalized.

See other publications for literature reviews and details of the design’s rationale, evolution, and implementations that herein are necessarily condensed.

The ratio of sample size, four marbles, to population, hundreds of marbles, renders the ‘without-returns’ issue negligible.

In expressing their judgments, students alternatively used figures of speech such as ‘more/less likely,’ ‘greater/smaller chance,’ and ‘you’ll get more/less of that.’ Whether students’ judgments could be called ‘intuitive

*probability*’ or ‘intuitive*frequency*’ remains a moot question (Gigerenzer 1998, calls this ‘natural frequency’; Xu and Vashti 2008, call it ‘intuitive statistics’).All ProbLab computer-based simulations are built in

*NetLogo*(Wilensky 1999).This response changes the situation for Rose from one of statistical investigation – attempting to determine the green-to-blue distribution in the marbles “population” – to a probability experiment, where one can apply the Law of Large Numbers and/or compute expected values (see Abrahamson 2006a, on nuanced relations between statistics and probability).

A future study is necessary to determine whether students would use similar gestures for other-than-half-half proportions. Also, if an actual number line, running from 0 to 100, were attached to the rim of the marbles box, students’ spontaneous part–part gesture could index a part-to-whole numerical value, thus bridging from the preverbal to the numerical.

Mark uses the mathematical term ‘expected value,’ yet his explanation does not abide with the formal procedure for determining an expected value but rather it is a qualitative argument for why the mean sample should have 2 green balls and 2 blue balls. (To calculate the expected value, Mark should add up the four independent .5 probabilities of getting green, one for each concavity, to receive the sum of 2 green balls as the expected value of a single scoop.)

Note how the palindrome scale is co-centered with the marbles box. This spatial co-positioning of artifact and gestured construction may appear epiphenomenal to normative bio-mechanics of tool use and, thus, barely pertinent to that which we may wish to call mathematical reasoning. However, a cognitive-ergonomics approach to artifact-based mathematical learning and design should tend to this spatial relationship and monitor its emergent consequences.

At the limit, the histogram signifies both, and explicating this theoretical-empirical homomorph in terms of properties of the random generator – its combinatorics and probabilities – could be regarded as the apex of coordinating theoretical and empirical aspects of the binomial. The Law of Large Numbers predicts that the empirical outcome distribution will converge on the theoretical expectation. After several thousand trials in a simulated marbles-scooping experiment (see Applet 1), the frequency distribution stabilizes at the expected shape.

Searching for a new material anchor that would enable us to run the blend for all

*p*values, I attempted various design variants on the combinations tower, such as a stretchable combinations tower, and I have examined the affordances of different media for implementing this and related design solutions (see ‘Sample Stalagmite,’ Abrahamson 2006c and Applet 2, for another solution, which is only glossed over in this article).I am grateful for the support of a NAE/Spencer Postdoctoral Fellowship and a UC Berkeley Junior Faculty Research Grant. Thanks to members of EDRL (http://edrl.berkeley.edu/), Eve Sweester’s Gesture Group, and CCL (Uri Wilensky,

*Director*; http://ccl.northwestern.edu/; Paulo Blikstein,*4-Block engineering & production*; Josh Unterman,*NetLogo modeling support*). The paper expands on previous publications (Abrahamson 2007a, b, 2008a). Special thanks to Betina Zolkower, Norma Presmeg, and three anonymous reviewers for very constructive comments.

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Abrahamson, D. Embodied design: constructing means for constructing meaning.
*Educ Stud Math* **70**, 27–47 (2009). https://doi.org/10.1007/s10649-008-9137-1

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### Keywords

- Cognition
- Reasoning
- Problem solving
- Semiotics
- Gesture
- Multimodality
- Probability
- Binomial
- Combinatorial analysis
- Outcome distribution
- ProbLab
- Media
- Computer
- Model
- Simulation
- College