Bringing Levels of Abstraction to Mathematics Problem Solving

• In Event: Computational Thinking in Elementary and Secondary Classrooms: Models for Teacher Professional Development

Abstract

Objectives
To explore if and how incorporating instruction in shifting among levels of abstraction into elementary mathematics problem-solving activities could improve mathematics performance.

Background
Abstraction is highlighted in standards documents for both computer science (College Board, 2017) and mathematics (Common Core State Standards Initiative, 2010), but the skill of moving among levels of abstraction—that is, choosing the best balance between high detail versus wide scope—is discussed frequently in CS education literature (e.g,. Armoni, 2013) and rarely in mathematics education literature. We examined fourth- and fifth-graders’ work on a commonplace mathematics task to identify if and how attention to moving among levels of abstraction could improve students’ performance.

Methods & Data Sources
We identified three levels of abstraction involved in a task that asked students to use a bar graph to determine how many cars were parked on various floors of a parking garage: (1) Considering the problem context (wide scope, but low detail), (2) making sense of the bar graph (mid-level scope and mid-level detail), and (3) computing with numbers derived from the graph (narrow scope, but high detail). We analyzed how students moved from the mid level (making sense of the graph) to the low level (computing with numbers from the graph) in three steps: (1) identifying students’ strategies for shifting the level of abstraction, (2) identifying errors they made in making the shift, and (3) comparing the number of errors made while shifting the level of abstraction to the number of errors made will doing mathematics after the shift.

Results
About 70% of students used two strategies that involved an abstraction level shift followed by a strictly mathematics step: Adding Bar Heights, wherein they first read the bar heights from the graph (shifting from the graph to a set of numbers) and then added the numbers, and Counting, wherein students used the graph scale to determine the value of each square of shading (shifting from the graph to a counting problem) and then counted the squares. An additional 12% used other strategies that did not reflect an attempt to shift from the mid-level to the low-level of abstraction.
Around 40% of students made an error in the abstraction step of the Adding Bar Heights or Counting strategies, by misreading bar heights or misinterpreting the graph scale. By contrast, only 13% of students made errors in the mathematics step of their strategies, either incorrectly adding the numbers they read as bar heights or incorrectly counting the squares.

Scholarly Significance
These results suggest that the difficulties students had in correctly solving the mathematics task were more likely to be associated with abstraction errors than strictly mathematical errors. Attention to moving among levels of abstraction therefore has potential to support mathematics achievement while also exposing students to a skill used by computer scientists. Further research is warranted to explore the feasibility and efficacy to incorporating levels of abstraction into elementary mathematics instruction and if and how exposure to this skill in elementary mathematics instruction supports later computer science learning.

Authors

• Kathryn Rich, Michigan State University

• Aman Yadav, Michigan State University

• Marissa Zhu, Michigan State University