Let’s Talk About Sets
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 stu-
dent preparation involving sets. And I do be-
lieve, at some point in their academic careers,
students need to have the “sets talk.”
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.
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
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/
Accessed 2017 Aug 7.
5. Stanat, D.F. and McAllister, D.F. Discrete Mathematics
in Computer Science. (Englewood Cliffs, New Jersey:
John P. Dougherty
370 Lancaster Avenue
Haverford, Pennsylvania 19041-1392
Figure 1: A Example Set Relationship [ 3]