In my last post, I talked about the value of using precise vocabulary in curricular resources. Today, I’m going to talk about (potentially) problematic vocabulary used in research and development.
There are twin problems of vocabulary in academia. Referring specifically to research on student engagement, Reschley and Christenson (2012) called these the “jingle, jangle” problems. The first problem is that we sometimes use the same word to refer to multiple ideas (jingle). The second problem is that we sometimes use different words to refer to the same thing (jangle).
I can think of plenty of examples of jangle — in particular, I think we use the terms real-world, relevant, contextualized, and authentic to refer to problem contexts when we really just interested in engaging problems. But I’ve been wrestling all year with an example of jingle. In short, I think that the word abstraction, as a learning goal, means rather different things to mathematicians versus computer scientists.
In general terms, abstraction can be used as a noun or a verb. As a noun, an abstraction is a representation that reduces complexity in order to focus attention on the core, essential elements of a situation or phenomenon. Usually an abstraction exists independently of any specific example or instance. The fraction ¾, for example, is an abstraction of three out of four equal-size pieces of pizza. The fraction can be used to represent a specific portion of any whole — it exists independently of the examples.
As a verb, abstraction is used to refer to the process of creating such representations.
For both the noun and verb meanings of abstraction, I do not think that mathematics and computer science differ too much. However, I do think there are subtle differences in the way the disciplines talk about abstraction as a learning goal. In short, I think mathematics is focused on the noun, and computer science is focused on the verb.
Admittedly, the Standards for Mathematical Practice in the Common Core State Standards for Mathematics (CCSS-M; Common Core State Standards Initiative [CCSSI], 2010) do highlight the process of abstraction. Standard for Mathematical Practice 2, Reason abstractly and quantitatively, says, “Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically …” (p. 6). (The other ability is contextualizing.)
However, when looking across grade levels within the content standards, I can’t help but notice a trend toward working with abstractions, rather than creating them.
Consider the following three standards about place value (CCSSI, 2010):
1.NBT.2 Understand that the two digits of a two-digit number represent amounts of tens and ones.
2.NBT.2 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones.
4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
The first- and second-grade standards are focused on understanding numbers written in base-10 notation as abstractions of quantities. The fourth grade standard highlights an even higher level of abstraction. It generalizes the base-10 place value system, independent of the examples of two- and three-digit numbers that are the focus in first and second grade. So there are two themes here: Understanding abstractions, and moving from less abstract to more abstract understandings and representations of mathematics. The movement is in one direction, and because that movement is separated across grades, it’s not clear that students will even be aware of the process of abstraction happening.
Getting to the highest possible abstraction is an implicit goal in a lot of mathematics education. In addition to sequences of standards like the one above, some instructional frameworks highlight this specifically. Two examples are the concrete-representational-abstract framework (Agrawal & Morin, 2016) and concreteness fading (Fyfe, McNeil, Son, & Goldstone, 2014).
By contrast, my read of computer science education literature is that there is a greater emphasis on using multiple levels of abstraction, and that necessitates greater focus on the process of moving among those levels. For example, the Computer Science Principles (CSP) framework references the process of abstraction first (and the result last) in its description of Abstraction as a Big Idea: “In computer science, abstraction is a central problem-solving technique. It is a process, a strategy, and the result of reducing detail to focus on concepts relevant to understanding and solving problems” (College Board, 2017, p. 14). The process also is highlighted in the specific learning objectives (which I’d consider parallel in specificity to the mathematics content standards): “Develop an abstraction when writing a program or creating other computational artifacts” and “Use multiple levels of abstraction to write programs” (College Board, 2017, p. 15).
Specific instructional approaches for abstraction in computer science also emphasize moving between levels of abstraction. For example, Armoni (2013) outlined a framework for teaching abstraction to computer science novices. She emphasized being explicit about moving between levels of abstraction in order to help students learn to move freely and easily between levels as they problem solve.
Thus, mathematics focuses on abstractions. Computer science focuses on abstracting. Both refer to abstraction, but often mean different things in terms of the goals of learning.
I’m not claiming that one focus or the other is inherently better. But I do think the difference is important to keep in mind, especially when thinking about integrated instruction.
Agrawal, J., & Morin, L. L. (2016). Evidence-Based Practices: Applications of Concrete Representational Abstract Framework across Math Concepts for Students with Mathematics Disabilities. Learning Disabilities Research and Practice, 31(1), 34–44.
Armoni, M. (2013). On teaching abstraction in computer science to novices. Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284.
College Board. (2017). AP Computer Science Principles Course and Exam Description. Retrieved from https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap-computer-science-principles-course-and-exam-description.pdf
Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/Math/
Fyfe, E. R., McNeil, N. M., Son, Y., & Goldstone, R. L. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9–25.
Reschly, A. L., & Christenson, S. L. (2012). Jingle, jangle, and conceptual haziness: Evolution and future directions of the engagement construct. In S. L. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of Research on Student Engagement (pp. 3–19). New York: Springer.