“I had no clue what that problem meant.” “I got confused — what does the mean mean?” “They can do the math, but they can’t understand the word problems.” “That problem was way too wordy, so I skipped it.”
When you work with students in an SAT or ACT math prep course, you realize something quickly: you’re suddenly spending a lot of your time as a reading teacher. Solving math problems presents a host of reading pitfalls—from decoding technical jargon to making sense of convoluted prose.
A Student’s Perspective
You suddenly must accept that “mean,” for example, no longer applies only to how your older brother treats you, but also to the arithmetic average of a set of numbers. You must agree that a statement like “a number squared is equal to 7 less than 35 more than that number” is both a sentence that can be understood and one that you actually care to understand!
In short, you are learning a new language. But here’s the rub:
Learning math as a language is not necessarily invested with all the fun and purpose of becoming fluent in French, so you can travel to Paris, explore, and enjoy touring the Louvre. Instead, all too often learning this language looks a bit more like training a puppy to sit, shake, and roll over by cueing up discrete behavioral actions with verbal commands.
Doing Math Stuff
Consider a student learning word-problem translation.
It often begins with providing a lexicon or translation key. Students are taught that “of” means “multiply” and “is” means “equals,” etc. However, this form of instruction is largely procedural: follow this recipe, and you’ll produce an equation that will make sense. In the end, students can be trained to respond to these cues and “do math stuff”… but can they make real math meaning? Doing math stuff—executing procedures, using recipes, writing out steps—does not necessarily lead to a meaningful outcome.
In fact, we often see students “do math stuff” in an SAT or ACT math prep course but produce some outrageous, illogical conclusions:
In a problem that involves a series of discounts applied to the value of a $100 dress, a student concludes that the dress costs more than $100! Yes, the student did math stuff, but that stuff lacked contextual meaning and any truly incisive check back from the student.
Plants growing according to regular increments suddenly start shrinking? Athletes running foot races suddenly reach break-the-sound-barrier rates of motion? And a student with 10 equally weighted test scores – consisting of nine 80s and one 100 – enjoys the happy fate of earning a 90 average for the semester? What luck!
All these scenarios are so magical as to be kind of funny, expressing some witty adolescent desire to be subversive. But, sadly, they are not. Instead, they reflect a common gap between translating math in a perfunctory manner and interpreting math for meaning.
So, while it’s important to teach terminology and translation tactics to math students, teaching strategic reading for true comprehension is essential in an SAT and ACT math prep course.
Strategic Reading for Comprehension
Some clear and very practical guidelines around strategic reading in math were put forth in 1945 by mathematician George Polya in his book How to Solve It, which went on to sell over a million copies and has been translated into 17 languages.
Polya identifies 4 basic principles of problem solving and provides guided reading prompts for math teachers to use to help students read and reason strategically through math problems:
- Understand the problem
- Do you understand all the words used in the problem?
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or diagram to help you understand?
- Devise a plan
- Have you seen this before?
- What approaches would you predict are best and why? Some approaches to draw on:
- Guess and check
- Make an orderly list
- Use direct reasoning
- Solve an equation
- Look for a pattern
- Draw a picture
- Work backwards
- Solve a simpler but related problem
- Carry out your plan
- What steps do you follow?
- Can you prove that each step is correct as you go?
- Look back
- Can you check your result and confirm it?
- Can you derive the same solution another way
Polya’s framework is timeless, providing a thoughtful scaffold for prompting strategic reading comprehension in math through guided reading instruction.
In “Reading in the Mathematics Classroom,” Diana Metsisto explains how guided reading strategies can be developed and deployed effectively by teachers. She provides an example in the following Figure, which provides a simple example of a possible guided reading approach for a lesson from an algebra text.
The text would be unveiled one paragraph (or equation) at a time rather than given to the students as one continuous passage.
|Solving Systems Using Substitution||1. What does the title tell you?|
|From a car wash, a service club made $109 that was divided between the Girl Scouts and the Boy Scouts. There were twice as many girls as boys, so the decision was made to give the girls twice as much money. How much did each group receive?||2. Before you read further, how would you translate this story problem into equations?|
|Translate each condition into an equation.
Suppose the Boy Scouts receive B dollars and the Girl Scouts receive G dollars. We number the equations in the system for reference.
|3. What do they mean here by “condition”?|
|The sum of the amounts is $109.
(1) B + G = 109
Girls get twice as much as boys.
(2) G = 2B
|4. Did you come up with two equations in answer to question 2 above? Are the equations here the same as yours? If not, how are they different? Can you see a way to substitute?|
|Since G = 2B in equation (2), you can substitute 2B for G in equation (1).|
|B + 2B = 109
3B = 109
B = 36 1/3
|5. How did they arrive at this equation?
6. Do you see how it follows?
7. Does it make sense? How did they get this?
|To find G, substitute 36 1/3 for B in either equation. We use equation (2).||8. Do this, then we’ll read the next part.|
|G = 2B
= 2 × 36 1/3
= 72 2/3
|So the solution is (B, G) = (36 1/3, 72 2/3).
The Boy Scouts will receive $36.33, and the Girl Scouts will get $72.67.
|9. Did you get the same result?|
|Are both conditions satisfied?||10. What conditions do they mean here?|
|Will the groups receive a total of $109?
Yes, $36.33 + $72.67 = $109. Will the boys get twice as much as the girls? Yes, it is as close as possible.
|11. How would you show this?
Where did they get this equation?
|Note: Text in the left column above is adapted from University of Chicago School Mathematics Project: Algebra (p. 536), by J. McConnell et al., 1990, Glenview, IL: Scott Foresman.|
Developing Independent Math Readers
Guided reading is a great practice for one-on-one instruction or small group work in an SAT or ACT math prep course. Ultimately, the goal is to build independence by scaffolding the reading process thoroughly in dialogue.
A good SAT or ACT math reading coach first serves as a dialogue partner, rehearsing a process of questioning—a systematic, repetitive questioning that builds comprehension. Then, once the student understands that dialogue, the questions are internalized, and the student develops his or her “internal coach.” The internal coach helps the student talk it through.
I often say to students, “My job is to write myself out of my job.” That is, as a teacher, I recognize that my job is to help my students become autonomous and self-directed. Along the way, I try to make the mean less mean by helping students make meaning.