# A study comparing effects of calculator usage on later math skills?

Each year my university tries to decide whether or not it will have calculator and CAS based introductory math courses (the calculus sequence, linear algebra, and ODE) or not. Other than some hearsay or personal preference (which looks like "You need to do the arithmetic to get a real feel for how things work" vs. "Wolfram|Alpha and MatLab can do all their real work for them too - might as well get them used to using it now and not duplicate the effort"), there is no basis for the decision.

I was wondering if there were any studies indicating a trend (or lack of a trend) between calculator usage in middle and high school and success in college math courses (or perhaps another measure of post-secondary school mathematics).

• My gut feeling is that good students are not hurt (at least not much) by having used calculators a lot, but mediocre students are hurt because e.g. their "sense" of how fractions work is left underdeveloped, and consequently they have severe problems manipulating rational functions and such. It did make me sad, when I was working with a relatively good student on a multivariable integration problem. In the end we reached the correct answer $\frac14+\frac16$, at which point she started rummaging her bag searching for her calculator. – Jyrki Lahtonen Mar 27 '14 at 6:37
• Searching Google Scholar for "calculator education" gives many results. The top one points to this paper: sciencedirect.com/science/article/pii/S0732312396900279 ("The graphics calculator in mathematics education", Drijvers and Doorman). Unfortunately I don't have time to read it or to search for an ungated copy right now. – Neil Strickland Mar 27 '14 at 8:23
• I find that if you are going to use calculators then you have to teach students how to use them: "remember, radians and not degrees!", "Brackets...", "$-1^2=-1\neq1$", and assorted rounding errors. I am still unsure if teaching the students how to get around these issues counts as teaching them maths or not... – user1729 Mar 27 '14 at 10:10
• +1 ... asking for a study, and not merely opinions and anecdotes. – Gerald Edgar Mar 28 '14 at 13:55
• A quick summary of the article Neil Strickland has in his comment. They tested 16 year old kids with an experimental lesson with cardioids. They found the kids were able to see what they were doing better and explore the concept more, but they had trouble integrating together all the information they had.The calculator offered them dynamic feedback which could be used to create new and more interesting lessons, and finally it did allow the students to see the affects of changing parts of the equation, giving them more flexibility with the concept. – ruler501 Mar 29 '14 at 3:52

Brief Remarks: It is difficult to find longitudinal studies on calculator use as specified by the OP. One of the reasons for this is that tracking students from, e.g., high school till college is quite complicated. Another reason is that studies on technology use are often fodder for theses, which are completed in too short a timeframe to provide such an account (i.e., one over several years). A third reason is that technology changes pretty quickly, so that, e.g., computer use five years ago and computer use today look very different.

The National Center for Education Statistics has some data on their page Calculators and Computers.

Some information is available there from the National Assessment of Educational Progress (NAEP) administered in 1990; for example, the site says:

Teachers and administrators reported that computer access was difficult for about half of the students at grades 4 and 8. Teachers and students agreed that school use of computers was greater at grade 4 than at grade 8, but usage in general was quite limited. A positive overall relationship was seen to exist between the availability of computers and average mathematics proficiency at grade 4; this relationship tended to hold across ability groupings at that grade level. At grade 8, however, there appeared to be no relationship between the availability of computers and mathematics performance.

Calculator use in Calculus: With regard to Calculus and calculators, one paper of interest is:

Penglase, M., & Arnold, S. (1996). The Graphics Calculator in Mathematics Education: A Critical Review of Recent Research. Mathematics education research journal, 8(1), 58-90. Link.

The text provided at the link above says:

Reviews recent research into the effectiveness of the graphing calculator as a tool for instruction and learning within precalculus and calculus, specifically in the study of functions, graphing, and modeling. Contends that much research fails to provide clear guidance or informed debate regarding the role of graphing calculators in mathematics teaching and learning. Contains 103 references.

Further Remarks: Though longitudinal data are not easy to find, there are certainly a lot of studies on calculator use (consider that the one source above has over a hundred references). Below I include six theses that I found on the first page of ProQuest while searching for calculator calculus mathematics education. Even with just six studies and only excerpts from their abstracts, you can see that the length grows very quickly. Overall, I think the lesson is that there is not substantial evidence for the use of calculators in teaching Calculus; perhaps this may change moving ahead, but these dissertations (from a range of years between 1993 and 2011) do not provide conclusive data for calculator adoption.

Searching Google Scholar (as suggested in the comments) is one way to find a lot of sources. Just to give a slight feel for some of the literature out there, here are some abstract excerpts from recent dissertations about calculator use as it relates to Calculus instruction. Bold text was added by me.

Everett, K. M. (2011). Does the use of a graphing calculator tutorial affect the attitudes, achievement, and calculator ability of non-mathematics majors in a calculus course? (Order No. 3477163, The University of Southern Mississippi). ProQuest Dissertations and Theses.

Abstract: This study shows the effects of a graphing calculator tutorial for non-mathematics major students taking calculus as a required course. Qualitative methods were used in order to provide information about the types of students taking a calculus course for nonmathematic students. The tutorial was used to help the student better understand the use of a graphing calculator while learning calculus. The study tested changes in attitudes, achievement, and ability with respect to the use of a graphing calculator through the use of a post-survey, a pre/post-quiz, and overall grade comparison. The tutorial was not heavily used by the students. This finding suggests that students are not dependent on the use of a graphing calculator to better understand calculus and its applications.

Hunter, J. S. (2011). The effects of graphing calculator use on high-school students' reasoning in integral calculus. (Order No. 3463473, University of New Orleans). ProQuest Dissertations and Theses.

Abstract: This mixed-method study investigated the impact of graphing calculator use on high school calculus students. reasoning skills through calculus problems when applying to concepts of the definite integral and its applications. The study provides an investigation of the effects on reasoning when graphing calculators are used, since it is proposed that, through reasoning, conceptual understanding can be achieved. Three research questions were used to guide the study: (1) Does the use of the graphing calculator improve high school calculus students' reasoning ability in calculus problems applying the definite integral? (2) In what specific areas of reasoning does use of the graphing calculator seem to be most and least effective? and (3) To what extent can students who have used the graphing calculator demonstrate ability to solve problems using pencil and paper methods? The study included a quantitative, quasi-experimental component and a qualitative component. Results of the quantitative and qualitative analysis indicate that (1) graphing calculators had a positive impact upon students' reasoning skills (2) graphing calculators were most effective in the areas of initiating a strategy and monitoring progress (3) students' reasoning skills were most improved when graphing calculators were used together with the analytic approach during both instruction and testing and (4) students who used the graphing calculator performed equally as well in all elements of reasoning as those who used pencil and paper to solve problems.

Nasari, G. Y. (2008). The effect of graphing calculator embedded materials on college students' conceptual understanding and achievement in a calculus I course. (Order No. 3296875, Wayne State University). ProQuest Dissertations and Theses.

Abstract: The two types of treatment used were: (a) Using TI-83 graphing calculator embedded materials to teach four units of Calculus I course to the experimental group. (b) Using traditional methods that did not use TI-83 graphing calculator embedded materials to teach four units of Calculus I to the control group.

Data analyses revealed the following: (a) there was a statistically significant difference in favor of the experimental group found on the skills oriented problems in the limit and their properties unit; (b) there was statistically significant difference in favor of the experimental group found on the conceptually oriented problems in the limit and their properties unit; (c) there was no statistically significant difference between the experimental and control groups found on the skills oriented problems in the differentiation unit; (d) there was a statistically significant difference in favor of the experimental group on the conceptually oriented problems in the differentiation unit; (e) there was a statistically significant difference in favor of the experimental group on the skills oriented problems in the application of differentiation unit; (f) there was a statistically significant difference in favor of the experimental group on the conceptually oriented problems in the application of differentiation unit; (g) there was no statistically significantly difference between the experimental and control groups on the skills oriented problems in the integration unit; (h) there was a statistically significant difference in favor of the experimental group on the conceptually oriented problems in the integration unit. The final analysis showed that the use of graphing calculator embedded materials improved students' grades in the Calculus I course.

Ocak, M. A. (2006). College calculus students' experiences with, attitudes towards, and uses of the graphing calculator: Interactions from cognitive flexibility perspective. (Order No. 3221094, State University of New York at Albany). ProQuest Dissertations and Theses.

Abstract: This study found that mere availability of the graphing calculator in the problem solving process does not affect or change students' solving problem strategies. Rather, the results suggest that using the graphing calculator and mathematical understanding must work together for the solution of the problem.

Castillo, T. F. (1997). Visualization, attitude, and performance in multivariable calculus: Relationship between use and nonuse of graphing calculator. (Order No. 9824883, The University of Texas at Austin). ProQuest Dissertations and Theses.

Abstract: Results indicated that the treatment and nontreatment groups had no statistically significant difference $(\alpha = .05)$ in their mathematical processing preference. The four sub-scales used from the Fennema-Sherman Mathematics Attitudes Scales also did not register a statistically significant difference between groups. Assigned grade for overall performance between the treatment and nontreatment groups was statistically significant at $\alpha = .05$ in favor of the treatment group [TI-85 group].

Emese, G. L. (1993). The effects of guided discovery style teaching and graphing calculator use in differential calculus. (Order No. 9316149, The Ohio State University). ProQuest Dissertations and Theses.

Abstract: The research design was a three group experimental study in a university freshmen differential calculus course with the following group specifications. Group 1: Use of graphing calculators and discovery approach, Group 2: Use of graphing calculators without discovery, Group 3: Traditional instruction. In the discovery section, part of the new material was covered using worksheets, where a sequence of questions/problems led to the new concept, relationship or technique. Students worked in groups, pairs or individually. They could get help from the hint-sheet, solution-sheets, their classmates and/or from the instructor.

No statistically significant differences were found on the computational, conceptual, and transfer skills parts of the pretest, nor on the following background variables: placement level, the year in which students took the placement test, their precalculus grade and the year in which they took precalculus.

Analyses of covariance were used for student achievement comparisons. The scores on the corresponding subtest of the pretest served as covariates. Students' time spent on the course and the extent to which students worked with their classmates outside of class were compared. Statistically significant differences were not found between the groups on any of these variables. No instructional method proved superior to the others on comparison.

• It would be nice if you put a summary of results in your answer. It's rather hard to read a solid wall of text. – ruler501 Mar 30 '14 at 5:45
• @ruler501 I did: See the paragraph entitled Further Remarks. (You will note I also wrote there: Even with just six studies and only excerpts from their abstracts, you can see that the length grows very quickly.) Moreover, I think it would suffice just to read the bold text in the six excerpts; however, it seems to me that someone genuinely interested in the answer might prefer to read more. – Benjamin Dickman Mar 30 '14 at 6:17
• Sorry I didn't read enough. – ruler501 Mar 31 '14 at 3:28

Studies such as Suydam (1979) indicate that there is not an adverse effect. However, most of the cited studies on the effects go back to the late 70's or early 80's. Anecdotally, as a high school teacher, I think that the early results are no longer valid. Additional studies need to be carried out using today's technology. The early calculators required a considerable mathematical knowledge to operate correctly. Today's calculators (especially with CAS) do not. As a result, many high school students require a calculator for items as simple as $-3 \times 22$, or for decomposing numbers. Many of my Algebra II students (11th Grade) no longer remember how to conduct long division, because they have relied upon calculators. They have lost fluency in many calculations that should be done with automatically. Often, I can solve problems much faster than the students because I do not need to use a calculator for fairly simple calculations.

In short, there are many studies, but I do not think that their results are valid for today's electronics.

• The inapplicability of studies from past years is increased given the invention of apps which allow students to just take a picture of the problem. They don't face any obstacle of data entry to get their easy answer. – James S. Cook Nov 11 '20 at 17:11