# New Paper: Decomposition LT

Hello, everyone. I hope you all had a great summer!

I’m using my first blog post of the year for a little self-promotion: I have a new paper out!

In this year’s International Computing Education Research (ICER) conference proceedings, Carla Strickland, Diana Franklin, Andrew Binkowski, and I share our recently developed learning trajectory intended to guide instruction on the computational thinking (CT) concept of decomposition for students in K-8 (Rich, Binkowski, Strickland, & Franklin, 2018).

That was a lot of ideas in one sentence. Here’s a brief introduction to several of the ideas I just mentioned.

First, a learning trajectory (LT) is a possible pathway from a student’s existing knowledge to a desired learning goal. Martin Simon (1995) first used the term while describing the ways teachers must negotiate between knowledge of their students’ thinking and knowledge of the mathematics they are intending to teach. One purpose of an LT is to manage the tension between the needs for advanced instructional planning and for spontaneous, responsive instructional decision-making in the classroom (Simon, 1995). LTs have since become a popular theoretical construct among curriculum developers and professional development providers seeking to base their materials on research in student thinking (Clements & Sarama, 2004; Sztajn, Confrey, Wilson, & Edgington, 2012). We — meaning a group of colleagues at UChicago STEM Education — were interested in developing instructional materials for K-8 students to learn computational thinking concepts, and so we first set out to develop some LTs for CT through the LTEC project.

Second, faithful readers of my blog right now will certainly know that computational thinking is loosely defined as the thinking processes used by computer scientists (Wing, 2006). CT is quickly becoming a new kind of literacy all students will need to be productive and engaged citizen in our technology-oriented world.

Lastly, decomposition, or more specifically, problem decomposition, is a process of breaking down problems, objects, or phenomena into smaller, more manageable parts. We think of it as a computational thinking practice. Just as modeling, pattern-finding, and generalization, for example, are mathematical practices, decomposition is a computational thinking practice.

So, to return to the paper: It shares our work in developing a decomposition LT intended to guide CT instruction in K-8. Check out the full paper to read about the processes of synthesis and theoretical frameworks we used to guide the development of the LT. Our aim was to make the best possible use of existing research evidence about students’ learning of decomposition to form a starting point for curriculum development.

Spoiler alert: There is a lot more research out there on students’ use and creation of procedures and functions then there is research about students’ overall processes of problem solving through decomposition. Our LT-building efforts made a start at connecting use of procedures to broader decomposition ideas. For example, the LT suggests that a productive intermediate learning goal might be to fluently and flexibly connect existing functions to decomposed parts of complex problems. It may be that such ideas are taught in CS courses, but according to our review, they have seldom been mentioned or studied in the K-12 CS education literature. Future research may well reveal other core but tacit ideas that would be productive to explicitly address in decomposition instruction.

Our LTEC research team has a lot of experience in the K-5 space, and so many of us are particularly interested in how decomposition ideas could be addressed with young students before they begin programming. Two particular ideas seem worthy of mention here.

First, the LTEC team is curious about the relationship between early work with decomposition and early work with another CT idea for which we already developed an LT: sequence. We previously used research evidence about students’ abilities to parse stories into steps to support our sequence LT. Through the development of the decomposition LT, we also came to see this as a kind of decomposition. We are not bothered by the duality in principle, but the double-use of this idea made us wonder whether the difference between sequencing and decomposition will feel meaningful to young students — and what the implications of the answer to that question might be for K-5 CT curriculum development.

Second, I have been thinking a lot about how decomposition in CS/CT relates to decomposition in mathematics. In both the LTEC project and my work at MSU with the CT4EDU project, one of the goals is to develop integrated mathematics and CT instruction for students in K-5. A big part of this work is to identify key ways that ideas are used similarly in the two disciplines and figure out how to leverage the similarities in instruction. Decomposition seemed at first quite similar in CT and math, but close scrutiny has led me to examine an interesting divergence.

In mathematics, the thing being decomposed is usually some kind of mathematical object — a number or shape, for example — and not the problem itself. Students decompose 25 into 20 + 5, or decompose a rectilinear figure into rectangles. The purpose of this decomposition is often for the purpose of solving a complex problem, like multiplying 25 by another number or finding the area of a rectilinear figure. However, the connection between the decomposition of the mathematical object and the decomposition of the problem is not always made clear. In the Common Core State Standard for Mathematics (CCSS-M) 3.MD.7d, the connection to the problem is clear: “Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts.” In the CCSS-M 3.NF.3b, however, students decompose fractions with no purpose stated: “Decompose a fraction into a sum of fractions with the same denominator in more than one way.”

So, decomposition is not always discussed in CT-friendly terms in mathematics. Oddly enough, I think this divergence makes decomposition one of the strongest candidates for integration of CT ideas into elementary mathematics. In this case, adopting the CT practice of focusing the decomposition on the problem has the potential to be a subtle, achievable instructional change for teachers that:

• Makes certain mathematical tasks more meaningful to kids. (You practice decomposing fractions because you can use those decompositions to help you add later!)
• Gives kids an introduction to a basic CT idea in a context that fits easily into core instruction in elementary school.

Cool, right?

This and other math-CT connections will be the focus of my contribution to the ICER doctoral consortium. You can check out my abstract for that here.

Thanks for reading, and hope to see some of you in Helsinki!

References

Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81–89.

Rich, K. M., Binkowski, T. A., & Franklin, D. (2018). Decomposition : A K-8 computational thinking learning trajectory. In Proceedings of the 2018 ACM Conference on International Computing Education Research (pp. 124–132). ACM.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114–145.

Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147-156.

Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35.