Archives For Mathematical Practice Standard #7

Today is the third Monday in January, a national holiday commemorating Martin Luther King, Jr., and if you have not already seen Nancy Duarte’s visualization of Martin Luther King’s “I Have a Dream” speech, then here it is below on YouTube (or the Vimeo link here):

If you have not heard of Nancy Duarte or how she happened on this form of presentation, here is the her TED Talk link. In this presentation, she explains how she compared the simple “structure” of a story as first suggested by Aristotle (having a beginning, middle and end) to the structure suggested by Gustav Freytag’s in his story “pyramid”.

You may remember Freytag’s structure as something called a “plot mountain” from 4th or 5th grade:


Freytag diagrammed the strict dramatic structure that the Roman critic Horace defined in his Ars Poetica:

“Neue minor neu sit quinto productior actu fabula” (lines 189-190)

“A play should not be shorter or longer than five acts”

Good drama, Horace maintained, is based on a five act structure with an exposition, a rising action, a climax, a falling action and a denouement (unraveling or resolution) of the story. Freytag’s model provided the visual to Horace’s critical analysis.

Duarte praised Freytag’s visual in her TED Talk saying:

“I love this shape. So we talk about shapes. Story has an arc, well an arc is a shape. We talk about classical music, having a shapeliness to it. So I thought, hey, if presentations had a shape, what would that shape be? And how did the greatest communicators use that shape or do they use a shape?”

She wondered about this connection between story arc and how a presenter is the same as someone telling a story when she came up with the idea to overlay two great speeches to see if they followed the same story arc that Frytag suggested:

“So I took the obvious, I took Martin Luther King’s “I Have a Dream” speech, and I took Steve Jobs’ 2007 iPhone launch speech, I overlaid it over it, and it worked. I sat in my office, just astounded. I actually cried a little, because I was like, “I’ve been given this gift,” and here it is, this is the shape of a great presentation.

In her TED Talk, she explains how the shape of the both presentations follows the pattern established in Freytag’s pyramid.

Now, I could go one step further and make another connection from Duarte and Freytag to the Mathematical Practice Standards as outlined in the Common Core State Standards. These eight Mathematical Practice Standards “describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.”

It is Mathematical Practice Standard #7 (MP7) that connects to Duarte’s visualization of text. It states that students should:

Look for and make use of structure.

In explaining how “mathematically proficient students look closely to discern a pattern or structure,” educators are developing the interdisciplinary and cross-curricular skills required to discover the patterns in other subjects as well, the patterns in literature and the patterns of history.

In her analysis of Martin Luther King’s speech, Duarte brought attention to the patterns created through his figurative language: the call and response, allusions, metaphors, etc., and she lays them out in multi-colored vertical bars for audiences to see. There is a geometric shape, there are patterns, and so, there is math.

From speeches as stories, to stories as visualized patterns, and to visualized patterns as part of mathematical practice, helping students understand the structure of  Martin Luther King, Jr’s speech can help them better appreciate the brilliance of his craft in both creating and then in delivering his unforgettable message, “I Have a Dream.”




I am seeing patterns.

My recent fascination with looking at crossovers from the Mathematical Practice Standards to the English/ Language Arts classroom has me seeing patterns everywhere. In poetry and in prose, I am seeing applications to the Mathematical Standard #7 where “proficient students look closely to discern a pattern or structure.”

Take, for example, the poem I assigned this morning to the Advanced Placement students. The students are studying how the structure or form of the poem helps to convey the meaning of the poem. The poem under discussion was Elizabeth Bishop’s “Sestina.”

Before reading the poem, however, an understanding of the sestina is in order. This form of poetry is highly structured; 33 lines constructed with five stanzas of six lines each with the final stanza as a tercet. The pattern is in the repetition of the intital six end-words of the first stanza; the last end work in the stanza before becomes the first end word in the following stanza.  The final tercet is called the envoi which contains all of the end-words.

 The form is as follows, where letters represent end-words:

  • Stanza 1: A, B, C, D, E, F
  • Stanza 2: F, A, E, B, D, C
  • Stanza 3: C, F, D, A, B, E
  • Stanza 4: E, C, B, F, A, D
  • Stanza 5: D, E, A, C, F, B
  • Stanza 6: B, D, F, E, C, A
  • Tercet: AB CD EF
    • First line of Envoi: B, E
    • Second line of Envoi: D, C
    • Third line of Envoi: F, A

The pattern above looks to me like sets of algebraic equations, and I mentioned that when I passed out copies of  the sestina by Elizabeth Bishop titled, appropriately enough, “Sestina.”

Listen to the poem:


September rain falls on the house.
In the failing light, the old grandmother
sits in the kitchen with the child
beside the Little Marvel Stove,
reading the jokes from the almanac,
laughing and talking to hide her tears.

(READ THE POEM continued…)

animated houseThis poem always reminds me of some small child’s drawing of a house, in which everything suddenly comes to life, to dance or to hover or to fall. These images seem too animated to place under multiple choice microscope I use to prepare students for the Advanced Placement test. But the new semester has begun, and test preparation is necessary, so we took our time studying the questions that had been prepared in dull, unanimated standardized testing rooms.

The grandmother and the child in the poem are portrayed primarily through descriptions of their:
(a) actions
(b) thoughts
(c) conversation
(d) facial expressions
(e) physical characteristics

Reading the question aloud, I was not entirely sure that Bishop would care about the primacy of her descriptions. Given all the action verbs in each stanza, we settled on “(a) actions,” and we were right.

One of the questions dealt directly with the poem’s pattern:

Which of the following literary devices most significantly contributes to the unity
of the poem?
(a) Use of internal rhyme
(b) Use of epigrammatic expressions
(c) Use of alliteration
(d) Repetition of key words
(e) Repetition of syntactic patterns

A discussion of what syntactical patterns appeared ensued.
“There is dialogue,” one said.
“But, not as a pattern,” another replied.
We settled on (c) repetition of keywords, and we were right. Recognizing the pattern was helping with the questions.

 “There’s a lot of crying in this poem,” remarked a student, “What are ‘equinoctial tears’?”

I explained that knowing what “equinoctial” means, a violent storm of wind and rain occurring at or near the time of the equinox, is important to the understanding of the poem. Bishop’s association of tears with “equinoctial” suggests that there is some emotional storm that is the result of an annual event or anniversary. Once the students knew that, they were able to answer the last question:

 The mood of the poem is best described as
(a) satiric
(b) suspenseful
(c) reproachful
(d) mournful
(e) quizzical

“The repetition of words means there is more emphasis on them,” said a student.
“All that crying,” agreed another student, “Someone must have died.”
We decided on (d) mournful, and again we were right.

In the end, students got about 60% of the questions right on the poem, a high score this early in their practice. They had employed both the Mathematical Practice standard #7 and “looked closely to discern a pattern or structure” as well as the English Language Arts Literacy Anchor Standard #1, “Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when writing or speaking to support conclusions drawn from the text.” 

“I liked it,” said a student, as she left.
I know she did not mean the multiple choice questions.
I am sure she meant the poem,“Sestina,” a crossover between English and mathematical practice standards.

A poem with a pattern:

Screenshot 2014-01-24 17.02.08 

Poetry Friday hosted this week by Tara at:

Come join her!

The year 2013 provided one of the best examples of real life detective work as well as real-life application of the Common Core Mathematical Practice Standard #7:

CCSS.Math.Practice.MP7 Look for and make use of structure.

The investigation was initiated because of structures and patterns, specifically the writing patterns of the author J.K. Rowling of Harry Potter fame. This mathematical practice standard MP#7 calls for students to “look closely to discern a pattern or structure,” and noticing a pattern was exactly what a computer program did in unmasking Rowling as the author of The Cuckoo’s Calling. The mystery novel, had been published under the name Robert Gailbraith, and the novel had begun to generate some critical acclaim. Only there was no Robert Gailbraith; Gailbraith was the pseudonym Rowling had chosen for her new foray into the mystery genre.

The ruse did not last long. In true detective fashion, two university professors, acting on a anonymous tip, wrote a computer code that used algorithms to compare patterns in the writing from The Cuckoo’s Calling with titles from Rowling’s Harry Potter series and the works of other mystery writers. The algorithms targeted several possible mystery writers, but Rowling’s name came up most consistently with language patterns that matched word length, 100 most common words, pairs of words, and the patterns of letters, spaces and grammatical marks known as “four character strings.”

The steps to identifying were outlined in an article in Popular Science, “How Computer Algorithms Uncovered J.K. Rowling’s Pseudonymous Novel.” Writer Francie Diep explained that, “Some of the individual tests found authors other than Rowling were the best match. Nevertheless, Rowling came up the most consistently.” 

The methods of the professors investigating Rowling belong to a practice known as the digital humanities, a field of study that “aims at developing and using the digital resources and tools for solving the research questions in the Humanities.”— Takafumi Suzuki (For other definitions check out

As texts become available digitally, they can be deconstructed into parts in order to answer research questions such as word origins (etymology), locating primary sources, and determining authorship. In the journal A Companion to Digital Humanities an article titled “Stylistic Analysis and Authorship Studies” author Hugh Craig points out that,

“There are enough successes to suggest that computational stylistics and non-traditional attribution have become essential tools, the first places one looks to for answers on very large questions of text patterning, and on difficult authorship problems.”

Yet, these patterns can do more than identify authorship. Patterns can be used to support an author’s purpose. For example, in the play Julius Caesar by Shakespeare, the character Marc Anthony, a wily politician, deepens the character Brutus’s involvement with the murder of Julius Caesar through the use of the phrase “honorable.” Here, the actor Marlon Brando plays Marc Anthony and recites the speech (1951 film):

The famous speech begins “Friends, Romans, Countrymen; Lend me your ears..” (3.2.) and Shakespeare employs the rhetorical device, an antistrophe or repetition of the same word phrase at the end of successive clauses, repeating “that Brutus is an honorable man.” In the opening 30 lines of the speech, Marc Anthony also connects “ambition” with the death of Julius Caesar.  Four times, Marc Anthony refers to Brutus “an honorable man,” but links each mention of honor with an “ambitious man”. By the end of his oration, Marc Anthony’s rhetorical accusations have inferred Brutus’s less than honorable behavior was an ambitious grab for power, and an incensed mob storms the streets of Rome seeking revenge. A final analysis reveals that Shakespeare’s use of rhetoric, a textual pattern, provided the tool that Marc Anthony used to attack Brutus very publicly for political gain.

Employing a pattern of repetition can serve an author’s purpose, and understanding this purpose requires the stylistic analysis that is embedded in the English Language Arts Literacy Reading Standard 4 where students “interpret words and phrases as they are used in a text, and analyze how specific word choices shape meaning or tone.” Patterns reveal the author’s craft; patterns also reveal author’s purpose.

images (1)You can find a textual pattern in any one of seven basic sentence types. You can find a textual pattern on any one of the seven days of the week on any one of the seven continents. Using Mathematical Practice Standard #7, helps find the purpose of a text or find the author of a text…like J.K. Rowling, who wrote seven books in the Harry Potter series. Coincidence? No. Pattern.

The villanelle is a nineteen-line poem with two repeating rhymes and two refrains. The form originated in France in the late 1800s, and the structure is comprised of five tercets followed by a quatrain. The most famous villanelle is by the Welsh poet Dylan Thomas, “Do Not Go Gentle into That Good Night.”

Do not go gentle into that good night, 

Do not go gentle into that good night,
Old age should burn and rave at close of day;
Rage, rage against the dying of the light.

Though wise men at their end know dark is right,
Because their words had forked no lightning they
Do not go gentle into that good night.

Good men, the last wave by, crying how bright
Their frail deeds might have danced in a green bay,
Rage, rage against the dying of the light.

Wild men who caught and sang the sun in flight,
And learn, too late, they grieved it on its way,
Do not go gentle into that good night.

Grave men, near death, who see with blinding sight
Blind eyes could blaze like meteors and be gay,
Rage, rage against the dying of the light.

And you, my father, there on the sad height,
Curse, bless, me now with your fierce tears, I pray.
Do not go gentle into that good night.
Rage, rage against the dying of the light.

I recently taught this poem to a class of Advanced Placement English Literature seniors who after some discussion were able to determine audience (“And you, my father..”), the form of address (imperative “Do not go“) and the poem’s paradox (“Curse, bless, me now“). They were intrigued by the most striking element of the poem, the repetition of the lines “Do not go gentle into that good night” (1, 6, 12, 18) and “Rage, rage against the dying of the light” (lines 3, 5,15,19). The ferocity of Thomas’s repeated exhortations increase because of the structure of the villanelle.

Structure matters. Structure, that “fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities” is not limited to poetry. Structure is in all the arts, and in all the sciences.

Structure matters in mathematics. Common Core Mathematic Practice Standard #7 requires students to:

Look for and make use of structure.

This standard details that “proficient students look closely to discern a pattern or structure,” and suggests, “young students, for example, might notice that three and seven more is the same amount as seven and three more.” Promoting this practice standard beyond the math classroom will have students noticing structure in other classes in every grade level. Students will be better equipped to recognize and use the structure of the periodic table or the color spectrum; they will be be prepared to identify and to employ patterns in pronouncing vowels and consonants. Students will be empowered to discover coordinates in longitude and latitude or to categorize the ingredients in food groups.

Recognizing and using structure is the critical academic skill that can help a student unravel both a problem in geometry or the complexity of a poetic structure like the villanelle. Once they understand the structure, they can evaluate the poet’s purposeful choice such as the one made by Dylan Thomas who selected the villanelle for this powerful poem.

The Common Core’s Mathematic Practice Standard #7 is in math and poetry. Note the same letters MP? Pattern? PossiblyCoincidence? Maybe not!

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