When I was in elementary school, each year my mathematics class always started with the same topic—namely, sets. (No, this article is not a piece about mindless repetition in education, though perhaps a topic for another day). The definition of sets, the examples, and the basic operations were always the way the math year began. One year before the initial lecture, I distinctly remembering thinking, "Will we see something new this time?" The answer always seemed to be no, yet again, I was disappointed to have to study the topic of sets. It's not that it was bad or difficult; it was just old (and I was probably eight years old at the time). What's the big deal? You can construct some interesting Venn diagrams and answer a few questions, but really, do we need to see this concept every year?

Then one year I recall connecting the set operations with binary logic. Yes, it was still elementary school and we did not use either of the terms "binary" or "logic." We simply tried to map set operations to English sentences where we would use the words "and," "or," and "not." For example, the size of the set of "people who are smart or happy" should be at least as large as the set of "people who are smart and happy" (and likely larger). That observation, that connection, was new to me.

I realized that "or" was not exactly defined in mathematics (i.e., inclusive) as it is in English (i.e., exclusive). I also remember my first time using this new knowledge with my parents. They would ask if I wanted to have ice cream or cake for dessert. Of course I took both (to their surprise). In secondary school, the notion of sets was not explicitly covered as much as the new ideas of algebra, trigonometry, geometry, and calculus. I realized that sets were a needed concept in each of these topics, and started to appreciate engaging with sets throughout primary school.


I asked my children where they first saw this type of figure, and no one recalled seeing it in any mathematics course. ... So where did they recall seeing a Venn Diagram? They replied with language arts, history, and English courses.


Onto college, which covered or assumed sets in almost every mathematics course. This was particularly true in the discrete mathematics course, which explicitly connected with computer science. After some recent investigation [2], I have learned that some undergraduate mathematics programs considered using discrete mathematics as an alternative to calculus as entrance into the major. I would assume the budding field of computer science helped to motivate this consideration.

In fact, I was able to find the textbook [5] I used in college for discrete mathematics. After an initial chapter on models (that was numbered zero!) and a second chapter on reasoning, the topic was once again sets. I became hooked on computer science at this time as we explored the classic set paradoxes as well as the connections between recursion and induction. For most of the last twenty years, I have used these experiences with sets in my introductory courses to introduce recursion, as well as proof techniques. The concepts of domain sets and range sets serve modular design when considering function preconditions and postconditions as well.

I have also expected that my students had a similar experience with sets as I taught computer organization, architecture, and logic. An understanding of basic set theory is certainly assumed for all core courses in computing. It appears that all those years of learning about sets in elementary school were important, at least to my career in computer science.

I recently attended my first ever conference in mathematics—namely, Math-Fest 2017 in Chicago—and I plan to write about my experiences there in subsequent articles. While I was at the conference, I started to question one of the assumptions I had made about the mathematical readiness we can expect in the classroom: is our students' comprehension of sets on a par with mine from my undergraduate years? If not, were they ever comparable?

The Common Core for State Standards in Mathematics [1] outlines the mandated topics in mathematics for students grades K-12 as they prepare for university study in the United States. I did find sets covered in a few places, such as starting kindergarteners using sets of objects to teach counting. However, sets are mostly studied in the context of the "data set." Sets are also mentioned in secondary school topics covering domains and range for functions. I also reviewed the Pennsylvania Standards for Mathematics [4], a concise overview of the topics covered in the Commonwealth's public schools in grades K-12. A quick search of the seventeen-page document found a single use of the term "set"—but as a verb.

Did they really only cover sets briefly in kindergarten and a few times in secondary school? I was a bit taken aback, so I decided to ask a few questions of my own children, all now in secondary school. I know, not a rigorous study, but a quick start. I sat down with each of the four and asked them to tell me about the math topics they remembered learning in elementary school. Each listed addition, subtraction, multiplication and division; two noted fractions and their associated operations; one talked about area and circumference in geometry; and one even mentioned prime numbers. No one stated sets explicitly, and yes, I was surprised.

I then proceeded to take a blank sheet of lined paper and drew the classic Venn Diagram with two circles (with a non-empty overlap) inside a rectangle, and asked, "Does this look familiar to you?" Three of the four immediately identified the drawing as a Venn Diagram; however, and delightfully, all of them could use it appropriately to classify objects and their relationships, including where they overlapped. I asked each where they first saw this type of figure, and no one recalled seeing it in any mathematics course.

So where did they recall seeing a Venn Diagram? They replied with language arts, history, and English courses. I asked each to provide any example of how to use a Venn Diagram, and I got "ingredients for pizza and cheeseburger" (overlap of cheese), "presidents Lincoln and Washington" (overlap of male and deceased), "fruits and vegetables" (see Figure 1), and hitters and pitchers on the Philadelphia Phillies 2008 World Series baseball team (with Cliff Lee in the overlap, a decent hitting pitcher). These examples suggested that although my kids did not recall seeing Venn Diagrams in a mathematics class before secondary school, they had experienced sets and set relations in other contexts.

Finally, I started to shade in the Venn Diagram, initially only the overlap and asked, "How would you describe the shaded area?" Again, no one used the conjunction "and" for this intersection, though they all knew that's what it implied. I extended the shading to cover both circles entirely, and asked again for each to describe this shaded area. This shading confused each of my children, with two replying that they did not know how to describe it at all.

I then drew another classic Venn Diagram with non-empty overlap and shaded the equivalent of "circle A XOR circle B." Even with the examples, my kids had difficulty expressing what this shading implied as much as the previous shading. Either they had not much experience with "exclusive or" in Venn Diagrams, or they were tired of the exercise (or both!).

I am lucky my kids trust me (and I only took five minutes each of their summer). I believe my assumptions about student preparation for sets may be incorrect, or at least out of date. I plan to extend my pre-course survey for the introductory courses this fall to explore this question further, and to work on other ways to ensure my course does not assume too much regarding student preparation involving sets. And I do believe, at some point in their academic careers, students need to have the "sets talk."

References

1. Common Core State Standards Initiative, 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers, 93 pages; http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf. Accessed 2017 Aug 8.

2. Dougherty, J.P. 2016. "Computational Maturity." ACM Inroads 7, 2 (May 2016), 19–20.

3. Life Science Staff, 2012. "What's the Difference Between a Fruit and a Vegetable?" Life Science Online; https://www.livescience.com/33991-difference-fruits-vegetables.html. Accessed 2017 Aug 7.

4. Pennsylvania Department of Education, 2014. Academic Standards for Mathematics: Grades Pre K-High School, 17 pages; http://static.pdesas.org/content/documents/PA%20Core%20Standards%20Mathematics%20PreK-12%20March%202014.pdf. Accessed 2017 Aug 7.

5. Stanat, D.F. and McAllister, D.F. Discrete Mathematics in Computer Science. (Englewood Cliffs, New Jersey Prentice-Hall, 1977).

Author

John P. Dougherty
Computer Science
Haverford College
370 Lancaster Avenue
Haverford, Pennsylvania 19041-1392 USA
jd@cs.haverford.edu

Figures

F1Figure 1. A Example Set Relationship [3]

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