A Metric for Improved Teaching of Hard Parts of Courses
By Russell Jay Hendel
Expanding Horizons, August 2024
Note: This paper was presented at the 2024 Actuarial Research Conference.
This paper presents a simple, easily applied metric that allows instructors to quickly identify the course components that will be challenging to weaker students and then shows how the course can be designed to facilitate improved student learning, performance and satisfaction as well as decreased failure rates. First, however, we must introduce the concept of Stroop interference.
Stroop Interference
In 1935 Stroop performed an experiment in which the subjects saw words that denoted colors (red, green, blue and yellow) printed in a corresponding color (e.g., the word red was written in red ink) or in a noncorresponding color (e.g., the word blue was written in red ink).[1] The subjects were required by instruction to identify the ink color either orally or by pressing a button.
The response time for each identification was measured. Averages for recognizing congruent items (where the meaning of the word and its ink color were the same) and incongruent items (where the meaning of the word was different than the ink color in which it was presented) were calculated. Stroop discovered a faster average response time for congruent items than for incongruent items. Stroop named this psychological phenomenon “interference,” and it frequently is called Stroop interference in the literature in recognition of his discovery.
Later studies, besides measuring average response time, also measured omission rates (not responding to a particular trial) and error rates (e.g., saying that the word red in a blue font has a red font).[2] Stroop interference affects all three of these measures: incongruent items elicited slower response rates, higher omission rates and higher error rates.
An Interference Metric for Course Problems
We can define the interference metric of a course problem as the number of potential interferences occurring in its solution. Notice that the metric depends on the solution approach. The illustrative examples that follow clarify calculation.
Example 1: Level Monthly, Increasing Annually
Problem: Using an annual effective rate of 3%, price a 30-year monthly annuity with level begin-monthly payments of $2,000 in the first year, and with the level monthly payments increasing 0.5% each year over the previous year.
Analysis: The interreference metric is 3 since there is interference in (a) the number of periods, 12 months per year and 30 years per annuity; (b) the period rates, monthly and annual; and (c) annuity types, level and geometric.
Example 2: Pricing Bonds
Problem: Price a 10-year, 4%, $1,000 bond with semi-annual coupons yielding 5%.
Analysis: This can be solved by plugging into the bond formula. Its interference metric is 2, since there is interference in (a) duration, 10 years and 20 periods, and (b) rates, 4% and 5%.
Example 3: Binomial Probability
Problem: For a student taking a a 10-question multiple-choice test with 5 choices per question, calculate the probability of the student getting at least a C (at least 7 questions correct) if the student randomly guesses on each question.
Analysis: The question may be solved by plugging into the formula for binomial probability. There are two sources of interference: (a) probabilities, of success per question and overall student score per test, and (b) counts, of choices per question and needed successful responses per test to achieve a C.
Example 4: Normal Probability Problems
Problem: Calculate the probability of a standard normal variable being at most 1.645.
Analysis: We can solve this problem using a normal probability table. There are no sources of interference here, so the interference metric is 0.
Note the subtlety that we might call Example 4 a plug-in and dismiss it as too easy; but the other problems are also plug-ins. The factor differentiating the problems is their interference.
The interference metric may be calculated prior to teaching a course. Additionally, as the instructor tabulates frequent student errors during the teaching of a course, their understanding of sources of interference may lead to a modified calculation.
We note that each of the interferences presented in Examples 1 through 4 are in fact typical errors of weaker students. Stronger students frequently can master course contents independent of the interference content.
Instructional Pedagogy Addressing Interference
The following strategies have led to improved teaching, improved student satisfaction and decreased failure.
- Time allocation: When designing the course syllabus, allocate time for particular modules based on the interference score of the problems of that module. This is consistent with the original Stroop interference experiments. The examinees could successfully identify colors; however, they needed more time when interference was present. This implies a teaching strategy: allocate more time to high-interference course modules and allocate less time to low-interference course modules.
- Disclosure: Explicitly, both in the syllabus and during teaching, explain to the students about interference. For example, “Some of you may find solving problems like this difficult; this has nothing to do with your mathematical ability. Rather, it reflects that it is easy to confuse parts of problems. While practicing homework exercises, be certain to pay attention to these parts with interference; practice until you achieve mastery. Mastery for these problems may, however, take longer and require more practice.”
Success Stories
My earliest success with this method happened in 1995.[3] I had noticed that weaker students in my probability and statistics classes frequently did not get binomial probability problems correct because of interference issues. I approached a fourth-grade mathematics enrichment teacher and asked her to teach a module to her class on binomial probabilities. Because the students were fourth-graders, we allowed computation of binomial probabilities using trees. The teacher created a module lasting 10 days. The experiment was successful; the fourth-graders were able to independently create and solve problems such as the one in Example 3. Their diaries reflected that they enjoyed the project. This story, with the contrast of fourth-graders learning a college-level topic, highlights the importance of correct time allocation for successful pedagogy.
I had another major success in the spring of 2024 in a service-course statistics course for health majors. Teaching such courses can be very challenging because of the verbal problems, which require both mathematical and verbal skills, as well as the fact that the students are poor in mathematical skill competencies because their chosen major is not mathematics.
I traditionally have many failures. However, I used the interference metric to highlight the hard part of the course. Hypothesis testing typically involves multiple dimensions where interference can take place: (a) tests for means versus proportions; (b) minimal sample sizes for ensuring the statistical validity of using a normal distribution approximation for mean versus proportion problems; (c) multiple definitions of the standard error for single and multiple means and proportions; (d) three types of hypothesis statements; (e) two approximating distributions, normal and t; (f) two formulas for p values, depending on whether a test is one- or two-sided; and (g) two formulas for the critical line, depending on whether the test is one- or two-sided. Thus, the interference metric for this part of the course is 7. No wonder weaker students do poorly. By allocating three weeks to hypothesis testing and exclusively focusing on four tests, single and multiple means and proportions, I was able to successfully teach the class whose scores on the corresponding exam were comparable to their other tests in this course.
Conclusion
This paper has presented a simple-to-use, easily calculated metric that can be skillfully applied to reallocate time for course modules. This enables, especially for weaker students, increased mastery and satisfaction and lower failure rates. While students may take longer to learn, the extra time allocated reduces error rates in exam questions as well as blank answers consistent with Rezaei’s findings.[4] We urge other instructors to implement this method in their own courses.
Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries, the editors, or the respective authors’ employers.
Russell Jay Hendel, Ph.D., ASA, is an adjunct faculty III member of the Department of Mathematics at Towson University, where he assists in teaching Actuarial Mathematics, and a former chair of the Education and Research Council of the Society of Actuaries. Russell can be reached at RHendel@Towson.Edu.