Raising the Level of Reading Comprehension of Students at Community Colleges

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.

Math-English Synergy: Student Response

In a previous article, I suggested that making a connection between English and Math words can help students overcome their fear of math word problems. Is it true? Does this really work? Here are some student responses to the hypothesis, as they experienced it themselves.


Amanda believes that when one understands the meaning of certain words both in their everyday context and in their mathematical context, it can make both subjects flow more seamlessly. “Creating a table of words that show their everyday meanings and their mathematical meanings right next to each other made me think more about how the words correlate in both subjects. It gave me a new tool to study when presented with word problems. Rather than avoiding math word problems at all costs and only studying exactly what I need to before an exam, maybe I need to spend more time studying the linguistics of the words so I have a better comprehension of what the words are presenting. By doing this, I hope to achieve the skill of being able to decode the problems and understand how to get the algebraic equations.”

For Brian, the synergy between English and Math has obvious benefits: “I enjoy learning new words every day.  Bahare, however, is not completely sold: “It’s a good idea as long as we are not tested for our vocabulary!” For both Linda and Axel, “learning to increase our vocabulary in Math class is important because it helps us understand word problems. It sure makes math more understandable.” Athena is equally emphatic: “Vocabulary is very important to express ourselves. One of my favorite books is the thesaurus.” Desarae finds the connection “good, almost necessary. It helps you better understand what we are learning. Knowing the vocab makes the concepts less scary!” Lizeth goes so far as to say that “a weekly vocal quiz in our math class would be a great idea!” It is an opinion shared by Kathy, who finds that “sometimes a word has a different meaning in the context of algebra than in regular usage, and you need to know the difference to solve word problems.” Leslie says flat out that “if you don’t understand the words, you don’t understand algebra!” As Dana sees it, “it is important to know what the words mean in math. If I didn’t know the meaning of function or factoring, I wouldn’t know how to solve the problems!” Alexander has a different angle: “Knowing these words can honestly impress girls!” Alma finds that “math is useful everywhere, in school, job, shopping etc. Knowing what the words mean and increasing vocabulary can only help.” Nico feels that “learning new words and using them properly can actually make you smarter.” But Eduardo will have none of it. “This is a math class, not an English class!”

For Hess, the issue is more nuanced. “I am one of those people who have a hard time with word problems. It’s not that I don’t understand what the terms mean, because I get when it says a number increased by 6 is x+6 but somehow when it comes to writing the equation I suddenly have short-term memory loss or something. Maybe it is PTSD, Post Traumatic Solving Disorder! Whatever, I need to find a way to overcome it. One way is to understand math words like “rational” and “radical” in both their regular usage and in their math context. I think a lot of students understand the terms. It is not the language barrier but more of a sentence structure issue. Also, like when you learn Spanish, you may know all the terms but put the adjective before the noun like you would in English, and the sentence becomes grammatically incorrect in Spanish. It is the same thing in math: If you put the equation in the wrong order, you may end up with the wrong answer.”

Alyssa knows from personal experience that when there is an equation in front of her, she knows where to begin “but as soon as it is surrounded by words I’m completely lost. The ironic part is that English happens to be my strongest subject and Math my weakest. I do like the idea of having a better understanding of terms and phrases and how they relate to in Math and English. Looking at math and English phrases in depth will certainly help me overcome my fear of math word problem. I think the two questions in a word problem are always: ‘Where do I start’ and ‘How do I know what the equation is asking?’ Word problems are tricky, because they ask a question at the end but there are a lot more steps before you can solve it. Breaking them down to certain key phrases is already helping me solve them with more confidence."

Ashley learns best “when I can learn something from two or three different approaches. Making the link, when I hear, for example “reducing” or “reduce” at work place to fractions is really helpful. If I can see a fraction in my head when I hear that word, I can learn to practice the mathematical term “reducing” more and become better at it. I also think some of the words in math are rather beautiful and if I could use them in everyday language, I would sound more educated. ‘Exceeds’ is one of those words. I use ‘difference’ a lot but not really thinking of math directly, unless I am working on a word problem.  Using words in daily language that can be applied to math is turning out to be a very useful concept for me.”

Brandon finds that a clear understanding of words in their math and regular usage context “helps me understand word problems and math itself a lot more. It definitely helps with learning everyday language. Not only are you learning more about the word itself but you are also learning how you can use it towards math and real life. For example, the word ‘rational’ is used in life to explain reason. When it comes to math, it means a ratio, either of numbers or polynomials. It is really amazing to me how math can be so connected to the English language in such a weird but helpful way. It helps me understand algebra a lot more and at the same time helps me learn the English language even better. I never realized how important it was to connect English and math together!”

Yannick agrees that “understanding meanings of words contributing to everyday and math context will increase math skills, but I disagree it will help with English skills. In every math textbook, there should a good handful of word problems. But if you open up an English textbook, there might be a few math problems, but not as detailed or as skilled as the problems in the math textbooks. However, it may not apply to somebody whose first language is not English. I feel this way because in math, there are sections and chapters in the book, with each one building its way up to more skilled formulas and methods, challenging individuals as they go through the book. When one has to solve a problem from an English textbook, that person is already used to the language and reading a simple sentence or a paragraph will not be an issue, unless it is asking the person to apply the problem with numbers and formulas. We are too used to speaking English every day. That’s why English texts are meant for English and Math textbooks for Math. In my opinion, the only reason people hate to solve word problems is not because the way it is worded, but rather the way it is printed on the paper, which is much longer than the equation problems.”

For Tyler, “learning the meanings of the words that have context in both English and Math would greatly improve the skills in both subjects. In Math, to know how to form a problem would help in a few situations in the real world, say, trying to figure out how much yarn you need to make a sweater. If you don’t know how to structure the equation, then the problem becomes impossible to solve. However, in English, one should constantly attempt to learn new, more complex words and various meanings so that in the future you could describe a complex sentence or in this instance, a mathematical problem. In my own accounts I use mathematical vocabulary to figure out pay in my job to make sure I get the right pay. As a referee I am paid different amounts depending on the games refereed. For example, I am paid $20 for a regular 45 min game, but I am paid one-and-a-half time more for a high school or junior high game. So if I work 20 regular games and 8 High school games then the expression should be: 20x + (20 x 1.5) y which becomes 20(20)+(20x1.5)(8). Since 1.5 times 20 is 30, so the equation ends up being 400+240=640. This is not my actual pay, but I wish it was! To form this equation requires the knowledge of a mathematical equation structure, as well as knowledge of how to form a proper English sentence with mathematical terms. Thus I can honestly say that the knowledge of Mathematical vocabulary will help students learn how to decrypt and solve word problems far easier than they would before they knew what the words structuring the sentence mean!”

As an ESL student, Pariya is convinced that knowing and understanding English is critical to understanding Math word problems. “My first language is Farsi. During the first days of class, understanding word problems becomes very hard for me. I try to learn the words that are the most useful in math as I came to appreciate the connection between English and math. It helped me a lot and now I have less difficulty with math word problems. Understanding such words also helped me to speak English more fluently at work that I could before. An Architectural interior designer and I deal with numbers and math problems most of the time at work, for example, for calculating the occupancy of a building. Also, knowing and understanding these words have helped me to think more logically.”

Jessica believes that “having command of the English language is absolutely crucial to understanding any type of math. Even though I have a strong understanding of the English language, as I have been speaking it my entire life, I find that I still struggle with understanding math word problems. Not being able to solve word problems is a huge issue in the real world. Outside of the classroom, real life issues are not presented in nicely laid-out equations for you to solve. In real-life situations you’re presented with a number of variables that you are tasked to put together to solve a problem. Though equations and formulas are helpful to know, students should be able to solve real situations with real numbers. For instance, when an anesthesiologist is tasked with administering anesthesia a patient, they must take into account a number of different variables in order to make sure that the patient is receiving the correct amount of medication. In the classroom it is important to practice the concepts of math and understand how to correctly solve problems but it is also critical that students understand how to solve problems that are not completely spelled out for them.” 

English-Math Synergy Helps Students Overcome Math Fear

Community college students generally dislike math word problems. The difficulty is particularly pronounced in introductory algebra, intermediate algebra, and introductory statistics courses. Students often cannot make sense of what the question is asking them to solve. It frequently comes down to a difficulty with the English language. The challenge is not restricted to ESL students but to native speakers as well. Students can solve algebraic problems if they are expressed in straightforward mathematical notations but if the same problem is expressed in words, they are lost.

Consider the following:
Solve for x in the equation
x – 0.2x = 320
Students find it easy to solve:
0.8x = 320
Dividing both sides by 0.8, x = 400

Now consider the problem: After a 20% reduction, you purchase a camera for $320. What was the camera’s price before the reduction?

Suddenly this problem looks strange and difficult. There is that 20 percent reduction. There is that word ‘before.’ How exactly do they translate into mathematical notation?

Consider another problem: Solve for x and y.

x + y = 146
x = y + 12

Again, this appears to be an easy problem to solve.
Substitute the value of x from the second equation into the first:
y + 12 + y = 146
2y + 12 = 146
2y = 134
Dividing both sides by 2 give y = 67
Thus, x= y + 12 = 67 + 12 = 79

However, suppose the problem is presented like this:
In two consecutive games, the college basketball team scored a total of 146 points. The team scored 12 more points in the first game than in the second. How many points did the team score in each of the two games?

It is the same problem but writing down the two equations, in which x = points scored in the first game and y = points scored in the second game, pose a problem for many students.

Finally, consider this problem:
Solve for x and y:

x + y = 16000
0.06x + 0.08y = 1180

Students can substitute the value of x from the first equation into the second and solve for y and then solve for x.
0.06(16000-y) + 0.08y = 1180
Solving for y gives y = 11000. Hence, x = 5000.

However, suppose the problem is stated this way:

You invest part of $16,000 at 6% interest and the remainder at 8% interest. If the annual yearly interest from these investments is $1180, find the amount invested at each rate.

Again, creating a set of linear equations to answer the question seems as remote as the moon, visible but beyond reach.

It may be helpful for math and English teachers to compile a list of words and their meanings in everyday context and in the context of mathematics, as well as a list of mathematical phrases and their translation into mathematical notations. By working together, math and English faculty can help students overcome their fear of math word problems, enrich their vocabulary and enhance their critical reading and writing skills.

The larger goal is to help them see the connection between Math and English. It is through such interdisciplinary connections that students can discover new insights and make new connections of their own.

A partial list of words may include:
constant, variable, ratio, proportion, fraction, slope, factor, rational, irrational, commutative, percent, percentile, integer, decimal, compound, absolute, perimeter, area, volume, coefficient, term, monomial, binomial, trinomial, polynomial, simplify, evaluate, solve, equation, inequality, linear, non-linear, base, power, exponent, exponential, hypotenuse, numerical, numeracy, innumeracy, round-off, round-up, sequence, series, intersect, intercept, radical, elliptical, radius, circumference, circular, parabola, parabolic, ellipse, elliptical, quadratic, imaginary, complex, conjugate, matrix, unknown, vertex, model, prime, square, cubic, parallel, horizontal, vertical, grouping, precision, accuracy, dependent, independent, function, one-to-one, one-to-many, many-to-one, domain, range, average, mean, median, probability, hypothesis, regression, correlation …

Fragments may include:
at least one, at most 4, ratio of a to b, x split into k equal parts, golden rectangle, golden ratio, margin of error, confidence level, significance level, confidence interval …

Serendipity occurs at the intersection of disciplines. It is something that has been missing in America’s schools for far too long. The time has come to address this urgent issue. A first step will be to create synergy between English and math teachers. There is plenty of data that show how student performance in solving math word problems increase when they are clear about the precise meaning of words as they are used in their everyday context and in mathematical context.