Nekeisha Johnson


We find evidence for five groups of responses that indicate students make consistent mistakes when adding and subtracting one dimensional vectors.

Examining consistency of student errors in vector operations using module analysis

Nekeisha Johnson

John B. Buncher

North Dakota State University • PERC2020


Successful manipulation of vector quantities is critical to success in introductory physics courses. Prior work has demonstrated key aspects of these struggles, highlighting that more than 1 in 4 students were unable to add two-dimensional vectors following a year of instruction [1] and students tend to stick with one method of solving, even when another method may be more appropriate [2]. We use module analysis to examine if students make consistent mistakes across different types of vector problems.


  • Data were collected from four classes of introductory physics students (algebra-based, physics I and II) who responded to a series of multiple-choice, one-dimensional vector addition and subtraction questions
  • Using Module Analysis for Multiple Choice Responses [3] the results for each individual course were plotted as a network and local communities were found.
  • These communities were then analyzed across all four courses, to examine stability of the groupings. This stability is represented visually as a heat map (at right)

Naming Scheme

The naming scheme defines responses as the operation prompt (±), the set of original vectors A and B (1-6), and the directions of A and B that could be properly added to yield the given resultant vector (:±A±B).


The brightest patches in the heat map were examined to identify any possible trends. Of these bright patches, we find five groups:

  • 7 responses, with 4 that follow the same pattern (-Q:-A+B)
  • 6 responses, with 5 that follow the same pattern (-Q:+A+B)
  • 6 responses, with 4 that are related as both (+Q:-A-B) and (-Q:-A-B) for questions – that is, two identical sets of vectors
  • 6 responses, with 4 that match a single pattern (+Q:+A-B)
  • 5 responses, with 4 that follow the same pattern (-Q:-A-B)

We offer plausible explanations for the trends in each of these groups in the full paper

More Details

Future Work

  • Two of the classes that took this assessment also had a variety of two-dimensional questions, which are being analyzed using the same process.
  • A later class was given handwritten versions of some of the two-dimensional questions, which we hope will aid in classifying the groups we find from the two-dimensional multiple-choice questions

Additional Images

Example question

Question (“+1”) with each of the four responses. In this case, option a) is (+1:+A+B), option b) is (+1:+A-B), option c) is (+1:-A-B), and option d) is (+1:-A+B).

Percent Correct

Comparison of percentage of students getting each problem correct across all four courses. Students performed much better on addition than subtraction.


[1] Nguyen and Meltzer, Initial understanding of vector concepts among students in introductory physics courses

[2] Hawkins et al., Students’ Consistency of Graphical Vector Addition Method on 2-D Vector Addition Tasks

[3] Brewe et al., Using module analysis for multiple choice responses: A new method applied to Force Concept Inventory data


Material based on work supported by NSF DUE 1560142 Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.