Emphasis on Student Learning Objectives (SLOs) and grades should not divert us, the math faculty at community colleges, from our main goal: a meaningful and quality education for our students. SLOs and evaluations are necessary but we need to recognize that other factors are also important. One such is reading comprehension, the ability of students to understand what they are reading, particularly word problems. While students can answer straightforward questions like “Let A and B be events with P(A) = 0.8, P(B) = 0.1 and P(B|A) = 0.2, Find P(A and B)” or solve quadratic equations like x2 – 7x + 11 = 0, they are sometimes unable to parse sentences in word problems to figure out what needs to be done, far less solve them.
Yet it is word problems that help students connect with the real world, encourage them to think about relationships between numbers, and reveal interdisciplinary connections between mathematics and subjects such as English, physics, astronomy, chemistry, biology and environmental science.
Here is an example from statistics that
illustrates how a lack of reading comprehension becomes a barrier for students to
solve word problems.
Statistics (section 7.2, Elementary Statistics by Navidi and Monk): According to the National Health Statistics Reports, the heights of adult women in the United States are normally distributed with a mean of 64 inches and a standard deviation of 4 inches. If three women are selected at random, what is the probability that at least one of them is more than 68 inches tall?
The first difficulty students face is the phrase “At least 1”. The second is with the meaning and implication of the word “random.”
Students had learned one of the probability
formulas in a previous section: “Probability (At Least 1) = 1 – Probability
(None)”. They have no difficulty running “normalcdf” in their calculators to determine
the probability when the parameters are explicitly given. However, connecting the
formula and the idea of randomness and “normalcdf” to this problem seems beyond
the capacity of most students. It comes down to a reading comprehension issue.
After carefully parsing the sentence “If three women are selected at random,
what is the probability that at least one of them is more than 68 inches tall,”
they slowly begin to make the connections. To ensure comprehension, I ask
students to write complete sentences describing the steps they use to solve
word problems such as this “if you want full credit.”
That last clause gets their attention.
This is a typical writing sample from approximately 60% of the students (the other
40% struggle to express themselves) who write complete sentences to describe
the steps:
a) Find the probability that any one of the three randomly selected women is shorter
than 68 inches by running (TI-84) normalcdf (0, 68, 64, 4) = 0.841. That is, the
probability that a woman picked at random has a height between 0 and 68 inches
is 0.841.
b) Since the three women are selected at random (no connection between them,
that is, they are “independent” of each other), the probability that ALL three
women are shorter than 68 inches is, by the multiplication law of probability,
P (A and B and C) = P (A) x P(B) x P(C) = (0.841)3 = 0.595.
c) Apply the “At Least 1” formula: Since the sum of all probabilities = 1, and
since “At Least 1” includes all possibilities other than 0 or None, “At Least
1” and “None” include ALL possibilities between them. They are complements of
each other. Therefore,
P(At Least 1) + P(None) = 1; P(At Least 1) = 1 – P(None)
Probability (At least One Woman taller than 68 inches) = 1 – 0.595 = 0.405
(Occasionally, a few students will go further and fill in more details. This is
typical of what they write: To calculate P (At Least 1) directly requires the calculation
of 7 different probabilities for this particular problem.
1. A is taller than 68 inches but not B and C
OR
2. B is taller than 68 inches but not
A and C
OR
3. C is taller than 68 inches but not A and B
OR
4. A and B are taller than 68 inches
but not C
OR
5. A and C are taller than 68 inches
but not B
OR
6. B and C are taller than 68 inches but not A
OR
7. A, B and C are all taller than 68
inches
The only other option is
8. All of them (A, B, and C) are equal to or shorter than 68 inches, that is,
NONE are taller than 68 inches.
The sum of all 8 probabilities = 1. So a) either I calculate the probabilities for options 1 through 7 individually and sum them, which is tedious and can lead to mistakes, or b) I do option 8 and subtract it from 1, which gives me the sum of probabilities for 1 through 7. It's easier to use option b, a neat trick!)
One or two students who take meticulous notes of what I emphasize in class will
also add something like this:
“Even though entering actual heights between two boundaries gives the area
under the bell curve, which is equivalent to the relevant probability, the
calculator converts the heights into their corresponding z-scores ‘behind the
scene.’ The area under the curve can be interpreted as probability only when
the actual values, the heights in this case, are converted to their
corresponding z-scores.”
I insist on complete sentences to explain the solutions to word problems because it becomes a test for students to see how well they understand the problems, that is, how good their reading comprehension is. Reading carefully clarifies their thinking, which, in turn, leads to clear writing. Reading and writing reinforce each other in a creative loop. Since language is the basis of thought, reading and writing well allow students to think well too. Students discover that this is true not just for English but also for math.
I find it helpful to emphasize to students that they can often figure out solutions to hard problems as they go along. Many students, at least initially, have the mindset that they can only start when they have figured out the entire solution, so they never start!
(Other examples from statistics: Write
complete sentences explaining the meaning of a confidence interval or the
implications of rejecting or not rejecting the null hypothesis in a given
context. Explain why switching events in conditional probability (“confusion of
the inverse”) leads to different probability results. Describe a “black swan”
event and whether or not you have experienced one that had a significant impact
on your life. Should you buy that warranty or that lottery ticket? Why or why
not?)
There is a lot of resentment in the beginning (typical reaction: this is not an
English class!) but gradually students come around to appreciate the symbiotic relationship
between reading comprehension and clear thinking and writing.
Precalculus word problems are good examples of showing interdisciplinary connections. Example: Throwing an object upward to calculate the highest point reached and the time it takes to get there and fall back to earth under the influence of gravity shows the connection between math and physics. Exponential functions describing radioactive decay and carbon dating show the connection between math, physics, chemistry, archeology and paleontology. Extinction of species shows the connections between math, biology, environmental science and climate change. A mathematical model for how we forget what we learn over time shows the connection between time and memory. And so on. One writing exercise I assign students is to describe how the irrational number “e” harnesses the power of infinity in a limiting sense, in situations where things happen continuously, like birth and death in a population. (Unintended humor: A student wrote that “e” captures eternity rather than infinity!)
Some students ask for extra-credit projects because they are falling behind and
want to bring their grades up. One project I often assign is to define the meaning
of 10 words in both day-to-day context and mathematical contexts and to
construct a sentence for each. Example: “irrational” usually means unreasonable
or illogical but in mathematics, an irrational number, such as pi or e, is a number that cannot
be expressed as a ratio of two integers. As a decimal, an irrational number neither terminates nor
repeats.
Example:
Define the following words in their mathematical and
non-mathematical contexts and write a sentence for each: Function, Eccentricity, Rational, Random, Sample, Population, Outlier,
Probabilistic, Deterministic, and Complex.
We faculty are constrained by time. We
have to teach courses, grade tests and quizzes, assess SLOs, maintain and
monitor Learning Management Systems such as Canvas, track attendance, tutor students, maintain office hours and perform
a host of other activities. Where is the time to raise the level of reading
comprehension and encourage writing with clarity and precision? How can we instill
the habit of paying deep attention and cultivating such skills as patience,
curiosity, discipline and grit, necessary for academic and
professional success, when we are constantly juggling time to complete so many basic
faculty duties and responsibilities?
There is no easy or single answer to this. Perhaps the first step is to recognize that we need to look beyond SLOs, grades, performance and achievement by integrating some habits and practices in our teaching that can help students think clearly and independently and live courageously and confidently. One such practice, in my opinion, is to improve their reading comprehension by paying attention to what they read (difficult, given the continuous digital distractions) and writing the steps clearly and precisely as they slowly work their way toward solving word problems.
Good mathematics, like good reading and writing, requires an appreciation of structure, beauty, rhythm, and pattern. If we can make this idea an integral part of our teaching, as best fits our respective temperaments, we may consistently experience the joy that comes from shaping minds, semester after semester.
