Why Math Class is Harder than English Class

Most students complain that math is their hardest subject.

Many adults agree.
Including teachers.
Students’ scores on quizzes and tests and their class grades seem to bear this out.
Why?

Poetry, that’s why.
Well, poetry and polynomials.

Factoring & Versifying

A couple years ago I had a gig tutoring Fiona, a 12^th grader, in algebra, specifically in the area of factoring polynomials, and I gained insight into why students come to believe that math can kill you while English class is a breeze.

Fiona was failing Algebra 2, but together we found ways to transfer her strengths in understanding English literature and interpreting poetry to math, thereby successfully navigating the last semester and avoiding disaster and insanity. It turned out that poetry and polynomials have a lot in common!

The Poetry-Polynomial Connection

At that level, polynomials come in three general flavors—two-term (binomials), three-term (trinomials), and four-term (four-term polynomials)—and Fiona was having trouble mastering the six methods used to factor them, as well as identifying the patterns that cue her as to which method to use when. Teachers typically spend a month on the topic, taking up about ten 90-minute blocks of classroom time of extended torture for many teenagers.

Despite Fiona declaring that she “hates math and math hates her,” we began working on factoring by Greatest Common Factor (GCF), Grouping, deFOILing, Difference of Squares, Difference of Cubes, and Sum of Cubes methods.


Fortunately, the steps for factoring polynomials are standard and consistent, and each polynomial presents obvious “pattern clues” that narrow the options for solving available to students. For example, factoring out the GCF first applies to all factoring problems. Then, if it’s a trinomial, deFOILing is the way to go, and four-term polynomials usually are pulled apart initially by Grouping, followed by one or more of the other methods.

If the binomials often seem challenging, then it might be that they have some interesting shortcuts that allow students to determine the factors quickly with minimal math calculation (just knowing square and cube roots), but require them to memorize the shortcut patterns. These are known as difference of squares, difference of cubes, and sum of cubes. For example, if the student notices three things – that there are just two terms (it’s a binomial), that it’s subtraction, and both terms are perfect squares – then the two factors can be written out in less than five seconds.

As we worked on these special binomials, it struck me that there was a way to connect to a demonstrated strength of Fiona’s, one that would help her with math – her ability to interpret and perform poetry at state level, competitive Forensics (Speech). It seemed obvious that picking the right factoring shortcut for these special binomials was just a matter of noticing the binomial’s properties (Are there just two terms? Is it subtraction or addition? And are perfect squares involved?) and was not dissimilar to the techniques Fiona used to interpret poems. Drawing upon her success in the literature world, Fiona and I discussed the steps she takes in analyzing a poem and how they could be applied to factoring. She quickly identified rhyme scheme and meter as two tools she employs in interpreting a poetic piece; both help the reader and listener recognize the key lines and words to focus on for discovering theme and meaning.

The short version of this story is that Fiona realized that her search for flags in her poetry pieces – flags such as whether the meter is iambic pentameter or trochaic tetrameter and which lines are connected by the poet’s rhyming – is little different from recognizing whether a binomial employs subtraction and has two perfect squares or two perfect cubes as its terms. This important connection enabled Fiona to transfer her English class skills to Algebra class for the last few months of her senior year and survive math and avoid insanity.

It’s hard not to notice that many students see math classes as being way more challenging than other courses, and even teachers notice that many students who are getting Ds and Fs in math are simultaneously getting As, Bs, and Cs in English. And their perceptions are not without foundation: I have access to hundreds of high school students’ report cards all year long and it’s obvious that the majority of teens who are struggling and failing in Algebra and Geometry classes are passing their English classes, most of them comfortably; a closer analysis shows that even the failing math students are doing at least C work on their English class assessments.

What makes English so much easier for those students? What prevents a bright student like Fiona, who can look at line 70 in Victorian poet Alfred Tennyson’s “Ulysses,” recognize that the writer is employing iambic pentameter meter, and therefore focus on the words “strive,” “seek,” “find,” and “not yield,” but not see that two perfect squares separated by a subtraction sign factor into an easily memorized and applied pattern?

Math is More Complex than Literature

An obvious answer that many of us arrive at is that the nature of the elements of math problems and concepts are more cognitively complex and demanding than those the students are encountering in English class. The combination of a couple of variables (usually limited to x or y or a), the four arithmetic operations (+,–, x, and ÷), the concept of squares or powers (x², really just another form of multiplication), and memorization of a few static patterns represents a significant cognitive leap from explicating a poem such as “Ulysses.”

To explore this idea a bit, let’s look in greater detail at the first few lines of “Ulysses.” The first challenge to Fiona’s classmates is understanding all the words – in context – of these five lines (a common concern of students when reading poetry, especially when in non-contemporary English). Aside from common articles, conjunctions, and pronouns, out of the 39 words in the initial five lines, there are about two dozen words whose meanings the reader must know as a first step in understanding what Tennyson is doing here. Referring to the (abridged) The American Heritage Dictionary (4^th edition), these 24 or so words combine to a total of over 130 possible individual meanings. We can safely assume that most students would or should know at least eight of these 24 (king, by, wife, unequal, laws, sleep, feed, and know, although, curiously, the word “by” by itself yields an amazing 17 definitions!). This brings the number of unfamiliar words that may require looking up and discussion to 16, combining to a more manageable 80 definitions, which, of course, could produce thousands and tens of thousands of permutations of larger meanings for the student to consider.

In reality – and fortunately for the students – the teacher will direct the students down narrower explorations of meaning for the sake of time, and typical literature anthologies provide glossaries as footnotes and sidebars on the same page as the poem. After all, these are just the first five lines of a 70-line poem.

Tennyson was an expert practitioner of meter, so knowing the poem’s rhythms is key to fully understanding “Ulysses.” Most high school students learn about four common metrical patterns: iambic, trochaic, anapestic, and dactylic. We could equate recognizing these patterns to knowing whether a polynomial in algebra is a binomial, trinomial, or other type of –nomial; however, whereas “nomials”(or terms) are easily counted since they are always separated by either a “+” or “–“ sign, the poetry reader must find a way to identify where the writer has placed stressed or accented syllables in each line – and the poet doesn’t roadmap these with symbols. To know if these stress patterns (“feet”) are made up of iambs, trochees, anapests, or dactyls helps the student to know whether each line is in (or varies from) monometer, dimeter, trimeter, tetrameter, pentameter, hexameter, and heptameter (the number of feet in a line).

(Interestingly, here is another obvious and useful overlap between poetry and math, since many of the prefixes denoting quantity are used in algebra and geometry as well.)

At a simple level, knowing the meter helps the student focus on key words in each line. As a gift for high school students, in “Ulysses,” Tennyson uses the most common form of meter in the English language, iambic (it sounds or feels like a heartbeat), thus letting us know, for example, that he considers “strive,” “seek,” “find,” and “not yield,” to be the most important words/concepts in the closing line and the student discovers the intent of the poet using a process similar to looking for flags in polynomials that tell her which factoring method to use.

In addition to iambic feet (unstressed/stressed or “duh-DUH”), the student must also be able to recognize trochees (stressed/unstressed), anapests (unstressed/unstressed/stressed), and dactyls (stressed/unstressed/unstressed).

Fortunately, for the student, poets provide other clues for interpretation, rhyme and other sound combinations chief among them. High school students learn about end rhymes, internal rhymes, slant or near rhymes, rich rhymes, and visual rhymes; of these, end rhymes are the most recognized thanks to pop music and eye or visual rhymes are some of the most challenging (“bough” and “rough” sound nothing alike, but the author may still use them to support theme and structure).

Tennyson doesn’t use common rhyming in “Ulysses,” so let’s look at another popular poet in high school English courses, a guy named Will Shakespeare. Beside his plays, The Bard is celebrated for his sonnets, a verse form highly dependent on rhyme structure for delivering meaning. Shakespearean – or English – sonnets (like Petrarchan or Italian ones) are chiefly recognized by their end rhyme structure, usually abab cdcd efef gg (the most common – but not only – rhyme scheme for Petrarchan sonnets is abbaabba cdecde). For the high school reader, this structure helps create connections between words and ideas that identify possible themes, at the micro-level (such as tying a pair of lines together) and at mid- and macro-levels; Petrarchan sonnets typically introduce a problem in the initial quatrain (again, easily identified by the abba rhyme scheme memorized by the student), develop it further in the second quatrain, and provide a solution in the closing sestet.

If there is time, English teachers may explore other areas of sound patterns related to comprehension, such as internal rhyme and alliteration. For example, a close look at Shakespeare’s use of internal rhyme in his “Sonnet III” might reveal ideas and themes of renewal and regeneration in his first two quatrains, given the repeated and regular use of “re” at the ends and beginning of words, which, in combination with understanding the meanings of “uneared,” “tillage,” and “posterity,” leads to an interpretation that Shakespeare is urging a young man to ensure his immortality by procreating before he becomes too old.

Sonnets, by the way, offer a shortcut to a “solution” for students similar to the opportunities offered by differences of squares (or difference of cubes or sum of cubes) in binomial factoring. If the student notices that a poem has fourteen lines (analogous to seeing two terms separated by a subtraction sign), then sees that the final two lines are a rhyming couplet (like confirming both terms are perfect squares), then the student knows to look to the couplet for the theme.

As we can see, the English student possesses a large array of tools for analyzing a poem, ranging from the meanings of the words (individually, in groups, in stanzas, and as a whole work – and often many of them new) to meter and rhyme to identify genre, structure, mood, and key words and phrases, all combining in shifting and alternate ways to develop and present a theme (or themes). Admittedly, what we have here is a necessarily incomplete inventory of investigation and analytical strategies and techniques, since this paper would become a book if we attempted to discuss all of the methods a skilled poet utilizes to create a poem – even if we focused only on a single work, such as “Ulysses.” We could go on and look at how other factors and elements contribute to the full enjoyment, appreciation, and comprehension of “Ulysses” – such as world history (knowing who Ulysses is in Classical Greek history and who Arthur Henry Hallam was to Alfred Tennyson), a more extensive knowledge of literature ranging from Homer to Dante, and the importance and value of figurative language to both the author and the readers/listeners, including a good understanding of metaphors, personification, hyperbole, and imagery.

If we were to return to discussing Shakespeare, a fuller understanding of wordplay and the political and moral atmosphere of Elizabethan England would be necessary. Since we’re not writing a book, let’s return to our mission, that of showing how polynomials are more complex than poems, or, on a larger scale, that math and math class is harder than literature and English class in general.

How Do Factoring and Poetry Analysis Compare?

To recap, we see that a typical Algebra 2 class spends about ten 90-minute class periods learning how to use six well-defined methods to factor binomials and trinomials comprised of a limited number of elements (variables, numbers, exponents, and four operations). There are about six different flavors of these polynomials, since teachers also focus on special binomials that contain perfect squares and cubes and some attention is paid to four-term expressions, and ultimately students are expected to solve problems requiring multiple methods in multiple steps.

As for poetry, based on syllabi published on high school websites, students may spend as little as one class period and as many as three or four on a single poem (or set of related poems). The methods they employ to comprehend the poem’s significance and possible meanings require an extensive existing and sometimes newly developed vocabulary and understanding of words in small and large units of context (lines, verses, and complete works), the ability to recognize at least four different metrical patterns in combination with seven types of foot repetition and apply it to identifying such properties as genre, form, mood, meaning, and theme (while recognizing the additional effects that variations on these have on comprehension), a practical knowledge of figurative language and its ability to convey meaning and connections outside words encompassed in the work (e.g., allusions), and the ability to identify at least five types of rhyme by sight and sound and use it to develop theories about the poem’s possible meanings. Discussions of the historical and literary contexts of the work are also usually key to authentic apprehension of the poem’s meanings.

Both factoring and poetry analysis benefit from prior years of classroom experience: factoring relies on practice in factoring simpler, non-polynomial expressions (something the students have been doing since elementary school); adding, subtracting, multiplying, and dividing (also something they have done every day since elementary school); and regular recognition of ten numerals that combine to form a large collection of numbers; similarly, poetry comprehension requires a foundation of a solid English vocabulary, and practice in interpreting grammar patterns and sentence structure.

We should note, however, that the basic elements of polynomials (numerals, variable letters, and operations) are limited; especially in the context of Algebra 2 factoring, no new polynomial ever presents an element the student is unfamiliar with. Poems of significance, on the other hand, differ in that they often present substantial amounts of new vocabulary, they also tend to play games with word order and meaning, and they constantly experiment with structure and form.

For comparison, here’s a single polynomial factoring problem and its solution, which requires using three of the six factoring methods in multiple steps.

Fully factoring 4x⁴ – 34x² + 72 in these five steps pretty much encompasses all of the methods an algebra student needs to successfully factor any binomial or trinomial. The beauty of solving math problems like this is that the student can always check the result’s accuracy by reversing the factoring process (called multiplying); in this example, multiplying (2) (x – 2) (x + 2) (2x² – 9) should equal 4x⁴ – 34x² + 72; and it does!

Note how this multi-step process compares to the amount of knowledge, practice, and number of rhyming and form patterns and structures needed to analyze a single poem. And also note that despite using over 1,200 words to address the tools used to analyze “Ulysses,” that we haven’t even mentioned the possible themes and meanings of the work. And, regrettably for the student, poetry analysis does not possess the easy “answer-check” methods math does; in fact, multiple and even conflicting interpretations of a single work are not only possible, but likely and desirable, as evidenced by the libraries of literary criticism books and articles generated by popular works of literature.

The Inevitable Conclusion

It is clear that both activities – polynomial factoring and poetry analysis – share common techniques and cognitive activities and a student’s success in one can help her improve her performance in the other, something Fiona experienced.

It’s also clear that analyzing and interpreting poetry involves far more memorization of terms (or “words”) and pattern recognition relative to what’s required for factoring polynomials, and that the patterns present in poems are not as obvious and clear as the patterns presented by polynomials. Additionally, the sizes of the “problems” in English class are significantly greater and more complex than those typically encountered by students in an Algebra class (and Geometry, as well). Can you imagine a high school math teacher presenting a 70 line math problem to a classroom – with the expectation of solving it in one class period?

Even for far simpler math problems and literature pieces, a student’s experience in the English classroom is significantly different from what she experiences down the hall in math class. Reflect on just one common aspect of these circumstances: In her math class, the student knows there is only one “right” answer, one that is shared by all the successful students in the room (and usually verifiable by a familiar “check-your-answer” process); in her English class, however, it is possible for the student to hear five (or 10 or 20) other answers about a poem’s (or short story’s or novel’s) meaning from her classmates – and they all are right!

This huge disparity in complexity of tasks, skills, knowledge sets, and expectations/outcomes between what is needed for success in English class and what is needed in math class explains why high school students such as Fiona conclude that math courses like Algebra are so much harder than English courses, a conclusion seemingly supported by their teachers, based on the generally higher grades assigned in English over math.