Saturday, April 30, 2016

Social emotional learning for everyone, or special interventions for disruptors?

Grit, growth mindsets, social emotional learning (SEL): these latest edu-fads are flourishing as never before--the more so as No Child Left Behind is succeed by the more optimistic Every Student Succeeds Act. In assessing our school children, states must now include at least one “non-academic” measure. The claim, of course, is that non-academic factors ultimately influence academic performance. And who would argue with the idea that how much you persevere and how engaged you are affect how much you learn?

But when schools divert students away from learning activities in order to engage in “social emotional learning,” (SEL), it’s reasonable to be skeptical--even when we encounter “research” that “shows” that some of these SEL programs are increasing academic test scores.

In particular, we must rule out:

1. The Hawthorne effect

2. The possibility that the extra staffing and investment involved may, independently of any SEL-specific activities, have positive ripple effects on classroom academics

3. The possibility that SELs programs improve academic achievement only inasmuch as they improve classroom behavior.

This last factor strikes me as the most likely reason for the efficacy of those SELs programs that are in fact effective. Disruptive, distracting behavior imposes a tremendous drain on teaching/learning—for perpetrators and victims alike.

But then the question becomes: is having the entire school population participate in weekly/daily SEL programs really the most efficient way to improve the behavior of the specific students who disrupt learning? How about instead doing the following:

1. Split the classroom teaching/classroom management positions into two separate jobs.
2. Put highly qualified teachers up front, and highly qualified classroom managers in back.
3. Give the latter the authority to remove disruptive students (temporarily or for the long term).
4. Offset the expense of extra adults in classrooms with substantially larger class sizes.
5. Spend the money that would have been spent on SEL instruction for the entire student body on special psychiatric and academic services for disruptive students.

Thursday, April 28, 2016

Classroom grammar instruction, fallacy II: teaching grammar rules to native speakers

So what predominates in public school writing instruction is neither:

- useful, explicit grammar instruction that facilitates the understanding of style rules (dangling modifiers, parallel structures), foreign language grammar, and complex sentences in English


-opportunities for implicit learning that come from expert feedback on multiple drafts.

In terms of writing, the results of this are evident in student papers, in written instructions, in promotional materials, and even in published articles.

What keeps most of us complacent about this are two phenomena

1. to some extent, it’s mostly the good writers who recognize bad writing for what it is (keeping the general malaise about the State of Writing in equilibrium)

2. there will always be a decent number of decent, self-taught writers: people who read enough high-quality prose to pick up the conventions; people write intuitively by ear.

It occurs to me that, besides the false choice between part-of-speech drills and peer-editing, there’s a second fallacy afoot. People forgot that, when it comes to native speakers, only certain aspects of grammar need to be taught. No native English speaker needs to be taught how to conjugate English verbs or form the comparatives and superlatives of English adjectives—and yet, I’ve seen this happen.

Self-taught writers aside, what English speakers need to be taught isn’t the syntax of their native language, but how to make active use of that syntax: call it “applied syntax.” One example of applied syntax is identifying and fixing dangling modifiers and un-parallel structures. Another is deploying options like active vs. passive voice (“I was astounded by his tone”), clefting (“what particularly astounded me was his tone”), and inversion (“particularly astounding to me was his tone”) to control emphasis and flow.

Tuesday, April 26, 2016

How good does your English have to be to score well on Pennsylvania math?

To earn full points on the Open-Ended questions on Pennsylvania's assessment tests (the PSSAs), you have to provide satisfactory explanations for your answers.

From the Grades 3-6 PSSA Released Practice Items:

Marco cut his cake into 8 equal pieces. Nikki cut her cake into 16 equal pieces. Niki says that her cake is bigger than Marco’s because it has more pieces.

EXPLAIN why Nikki is not correct.

A "sufficient" explanation:

An "insufficient" explanation:

Friday, April 22, 2016

Math problems of the week: 6th grade Common Core-inspired math question

A sample question from Pennsylvania's state test (PSSA) released items and soring rubric:

Sergei knows the bicycle he wants to buy will cost more than $84.00. He has already saved $26.75 for the bicycle. His aunt has given him $20.00 to use to buy the bicycle. Which inequality describes all of the additional amounts of money (m), in dollars, that Sergei could save to be able to buy the bicycle?

m > 37.25

m >  46.75

m >  57.25

m >  84.00

[69 words]


Sergei wants to buy a bike that costs over $84.00. He has $26.75 and his aunt gives him $20.00. Complete the inequality for the amount he still needs in dollars.

m  > _____

[33 words]

Extra Credit:

Is anything of mathematical significance lost in translation?

Wednesday, April 20, 2016

Classroom grammar instruction, fallacy I: either parts of speech drills or child-directed writing

Today’s classrooms tend to assume that grammar amounts to parts of speech: nouns, verbs, prepositions, conjunctions, and so on. Enthusiasts of such "grammar" treat memorizing these labels as an end in and of itself; detractors, rejecting this, reject grammar instruction in general.

Memorizing parts of speech and stopping there is, indeed, a waste of time. But, taught sensibly, the basic labels give students the vocabulary they need to proceed on to the meanings of more useful grammatical terms like “sentence,” “clause,” “subject,” “predicate,” “subordinate conjunction,” and “coordinate conjunction.” It’s hard to explain what these terms mean without using the more primitive terms “noun,” “article,” “main verb,” “auxiliary verb” and “conjunction.” The reason for learning the meanings of “sentence,” “clause,” “subject,” “predicate,” “subordinate conjunction,” and “coordinate conjunction,” in turn, is to learn the conventions specific to written language that most students don’t pick up on their own: particularly punctuation and style rules. It’s hard to explain when to use (or not use) a period, comma, semi-colon; or which modifiers are dangling; or which structures are or aren’t parallel, without reference to “sentence,” “clause,” “subject,” etc.

Those who reject part of speech instruction as pointless seem to assume that proper punctuation and style will emerge organically, through text-rich environments and frequent writing assignments.

In principle, frequent writing assignments could improve punctuation and style, but only if students get frequent expert feedback. Through regular feedback from discerning teachers--circles around misplaced punctuation marks and dangling modifiers, etc.-- students might learn implicitly, without reference to entities like clauses and subjects, the various rules of style and punctuation. But this requires teachers to spend time thoroughly marking up written work, and to assign multiple follow-up drafts in which students fine-tune their corrections.

Unfortunately, schools have moved quite far from this once commonplace practice. Instead of expert feedback, students now mostly get feedback from fellow novices—in a trendy practice known as “peer editing.” Its appeal is obvious: not only is it a lot less work for teachers (marking up a stack of student papers is one of the most tedious teaching tasks out there); it’s also pretty to think that, true to the Constructivism dreamworld, students learn best not through explicit teaching from sages on stages, but implicitly, organically, on their own and from one another.

So, while today’s students may get a smattering of speech labels some time in elementary school because some authority somewhere said something about grammar being important, what they learn next is anyone’s guess.

Monday, April 18, 2016

The dark side to disability sensitivity

In theory, it sounds great to be as sensitive to disabilities as we are here in America. In theory, every child gets educated in the least restrictive environment; every child gets what s/he needs; no child is triaged out or given up on.

In practice, the more sensitive we are to disability, the more people end up with disability labels. Labels bring accommodations: extra time on tests, note takers, etc.. But the more people use accommodations, the less they help those in the greatest need of them. The more students get extra time on tests, the less of a relative boost it is to those who most need that extra time. The more students are entitled to scarce resources like note takers, as we’ve discovered, the less likely it is that those who are most in need of those resources will receive them.

Also, the more we broaden challenging disorders like autism to include kids at the milder end of the spectrum, the more our most sought-after special needs programs (e.g., autism summer camps) can favor the easy kids and screen out the hardest ones--and the more the world of disability services can claim growing success in overall outcomes. In lessening the average severity, of course, we also raise the average prognosis--along with false hopes.

Another side of our disability sensitivity is a disabling political correctness: a tendency to deny the severity of impairments. We like to ignore how deeply undermining certain disabilities are of cognitive functioning, and/or how deeply entangled they are with core personality. Thus, we get euphemisms like “have autism” in place of “is autistic" (which neurotypical thought leaders consider dehumanizing), and "have an intellectual disability" (at the same time that no one "has intellectual giftedness"). We also get memes like “normal child trapped inside”--along with sensational recovery stories--bringing more false hopes, along with ill-targeted interventions. And we get ideologies like “every child should be able to meet the same high academic/Common Core standards at the same calendar age-based grade level"--and more false hopes and ill-targeted interventions.

Such is the price we pay for our self-proclaimed, self-congratulatory sensitivity to disabilities.

Saturday, April 16, 2016

Let’s have required writing classes taught by actual writers

Three of my recent claims:

1. Students with autism need writing classes that don’t involve reading and writing about literary fiction;

2. English professors and literary theorists, some of them “Bad Writing Context” laureates, aren’t necessarily particularly good writers;

3. an under-used tool for appreciating literary fiction is to reverse-engineer the mechanics of plot and plotting

…all point towards a fourth:

4. both writing and literature classes should be taught by professional writers, and not necessarily by professional professors.

Creative writers would teach creative writing; essayists would teach essay writing; students would get to choose which genre best suits them. And creative writers would also teach the literature courses.

Where does this leave literature professors? Let them continue offering their courses, and let the market decide. Perhaps I’d choose “Coincidence and timing in comedies of manners” over “Engendering the body and destructive spectatorship,” but I don’t want to speak for everyone else.

Friday, April 15, 2016

Math problems of the week: 5th grade Common Core-inspired math problem, plus rewrite

 A 5th grade test question from Engage NY:

There are 12 players on a new softball team. Before the team starts playing games, the team must pay a total registration fee of $572. Along with the registration fee, the team will also need to spend a total of $1,240 on equipment.

To pay for the cost of the registration fee and the equipment, the players held a car wash and raised $786. They then decided to sell candles for $9.50 per candle to cover the remaining costs. If each player sells the same number of candles, how many candles must each player sell?

[95 words]


A new softball team must spend $572 to register and $1,240 on equipment. To cover these costs, the 12 players raise $786 and then sell candles at $9.50 per candle. If each player sells the same number of candles, how many must each player sell?

[45 words]

Extra Credit:

Assuming away any concerns about English Language Learners and children with language and/or reading impairments, are there reasons for giving kids math problems with gratuitous verbiage?

Do the demands of the 21st century require that we measure reading speed and working memory along with math skills?

Might we accomplish the same goals by blasting distracting noises and conversation at students--or by letting them keep their iPhones turned on--while they do practice problems and take tests?

Wednesday, April 13, 2016

Engineering plots, and reverse-engineering literature

Hester Prynne’s husband, left behind long ago in an Old World city and later held captive in the wilderness of the New, shows up in Boston just as Hester Prynne has appeared on the scaffold, along with her baby, to be publically shamed as an adulteress. How on earth does Hawthorne get away with such a contrived beginning?

Several chapters into Pride and Prejudice, you can guess that Mr. Bingley and Mr. Darcy and going to end up marrying Jane and Elizabeth, respectively. How does this not make the novel feel hopelessly predictable?

When we analyze literature, we focus on small-scale “devices”: an image here, a symbol there, a smattering of alliteration. When we expand to the big picture, it’s mostly to look at character development and themes. But what about plot and plotting?

In none of my high school or college literature classes was plot considered something worth analyzing. But, as any real story teller can tell you, coming up with an interesting, believable, character-driven plot is one of the hardest things about writing fiction.

Then there’s the intricate “plotting out” of the plot, including decisions about how exactly to set things up and what to reveal when.

Good plotting is difficult. It’s like an extended magic act: you create a diversion in one direction while stacking decks and contriving coincidences in another. The way that Hawthorne gets away with his first big coincidence is by distracting us with what appears to be the main focus of the book (the heroine on the scaffold) and making the new arrival among the onlookers seem like someone who serves simply as our temporary proxy in inquiring about what’s going on. Only some dozen pages later, when this coincidence is no longer so fresh in our minds, do we learn Roger Chillingworth’s true identity.

The centrality of Mr. Bingley and Mr. Darcy is less in doubt. We’re pretty sure we know what will happen, but Austen sets things up so that we don’t know how, and that yes, but how is what drives the novel’s suspense. There are the barriers of social class; of insufferable relatives; of pride, of prejudice; of geographic separation; of the fact that Mr. Darcy has already proposed once to Elizabeth and been resoundingly, humiliatingly rejected. And yet the plot is so engineered, with carefully timed revelations, and coincidences that manage (as in The Scarlett Letter) not to be too self-evidently contrived, that everything convincingly works out. The inevitable yet surprising resolutions seem entirely driven by believable characters, and not by the clever author behind the curtain.

Deconstructing how all this is engineered involves a left-brained, analytical approach to literature in which you take apart not just the stereotypically “literary” elements, but also the mechanical ones. Demystifyingly reductionist though this is, it would, I think, significantly enhance students' literary appreciation—at least when it comes to those classic novels whose intricacies (reminiscent of those of their musical contemporaries) extend far beyond multiple syntactic embeddings and sentential clauses suspended by appositives and parentheticals to the embeddings and suspensions and other complex machinations of plot.

After all, everyone wants to tell a good story. Isn't the best way to learn how to do so to spend time reverse-engineering those of the world's best story tellers?

Monday, April 11, 2016

"Eat food get fat:" left-brain vs. right-brain comprehension strategies

In a recent post, I drew a contrast between what makes modern prose challenging and what makes prose by classic writers challenging. What bedevils the latter (cf., e.g., Mary Shelley, Jane Austen, Franz Kafka, and George Eliot) are long sentences with lots of embedding that, while difficult to parse, ultimately make sense once you work them out syntactically. What bedevils the former are jargon, excess verbiage and repetitions; ambiguous references; and syntactic incoherence: in short, bad writing.

One reason I've been carrying on about this here is that it strikes me as part and parcel of our culture's "left brain" to "right brain" shift. That is, making sense of the classic prose that used to dominate our reading lists involved lots of analysis; making sense of contemporary prose involves less analysis (because the sentences are so much simpler), and, especially in the worst cases, much more socio-contextual reasoning (pragmatics).

For example, to make sense of Judith Butler's jargon you need to be part of her social milieu of critical theorists; and to make sense of Ronald Reagan's speech extract you have to get inside his head and figure out what he probably meant. If you're convinced he's an idiot, he meant that missiles can be recalled once launched; otherwise, the less alarming interpretation may win out.

True, you can also take a pragmatic approach to figuring out what the Jane Austen sentence means. Some commenters predicted that their kids wouldn't be able to answer my grammar questions while still correctly answering my comprehension questions. They could accomplish this by doing a "shallow parse": figuring out the individual clauses but not worrying about how exactly they fit together syntactically, and letting pragmatic reasoning about the characters' likely perspectives and motivations do the rest.

The tradeoff between complex syntax and complex pragmatics also occurs cross-linguistically. Different languages strike different balances between the two; in the process, they also strike different balances between the relative demands they place on communicator vs. receiver. In languages that use less complex (shallower) syntactic structures, the literal messages are often ambiguous. Less work for the communicator; more work for the reader or listener.

Consider the serial verb construction in Chinese. In this highly ambiguous construction, a series of verbs strung together without any intervening function words can either be analyzed as coordination ("eat [and] drink"), subordination ("eat [in order to] drink"), or causation ("eat [and as a result] drink"). Does the Chinese equivalent of "I eat food get fat" mean "eating food causes me to get fat" or "I eat food in order to get fat"? It's up to the listener/reader to figure that out based on what he or she knows about the communicator.

These days, "eat food get fat" no longer sounds so foreign. Perhaps, even in languages like English that have traditionally been more communicator-centered, our texting/tweeting shorthand is shifting the linguistic demands in a Chinese direction: even further away from the communicator (and his/her thumbs), and away from syntactic elaboration and analysis--and, therefore, more towards the social, contextual reasoning skills of the recipient.

Friday, April 8, 2016

Math problem of the week: Common Core Math Goal 5.OA1-inspired problem

A sample 5th grade problem from the Common Core resources of the New York-based Lake Shore Central School District:

Operations and Algebraic Thinking:

5.OA1: Write and Interpret numerical expression.

Maxwell bought a new pair of skis for $350. He put $110 down and received a student discount of $30. His mother gave him 1/2 of the balance for his birthday. Which of these expressions could be used to find the amount Maxwell still owes on the skis?

A. 350−110+30÷2
B. 350−(110−30)÷2
C. [350−(110−30)÷2]
D. [350−(110+30)]÷2

Answer: ________

Extra Credit:

Besides “write and interpret numerical expression,” what other skills does this problem involve? Consider the ESL student, the language-impaired student, or the student on the autistic spectrum, and:

1. Idiosyncratic meanings that override literal meaning: “put money down;” “balance.”

2. Knowledge of transactional “schema”: the special discount; the down payment.

A rewrite that more specifically measures 5.OA1:

A pair of skis costs $350. Students get a price reduction of $30. A student has $110. His mother gives him 1/2 of what he still needs to buy the skis. Which of these expressions shows the amount he now needs?

A. 350−110+30÷2
B. 350−(110−30)÷2
C. [350−(110−30)÷2]
D. [350−(110+30)]÷2

Answer: ________

Tuesday, April 5, 2016

Unpacking the hegemonic problematizations of contemporary prose

While I waited for my Wi-Fi to return, I was thinking about a comment that Anonymous recently posted on my latest blog post about the Jane Austen sentence I’ve been showcasing. S/he proposes that we engage with sentences that are similarly complex:

struggling through a user’s manuals written by risk managers; hoping to get to the end of a Help function for a software we’re using; trying to understand the fine print in a jury summons notice.
I agree with Anonymous that the 21st Century presents us with plenty of difficult prose. But I'm guessing that most of that prose is difficult not because of Austenian levels of syntactic complexity, but because it's poorly written. Poorly defined terms and jargon; excess verbiage and repetitions; ambiguous references: that’s what I typically see in the modern prose I struggle with the most.

The big culprits? Academic writing, legalese, and bureaucratese.

Consider the winner of the fourth annual Bad Writing Contest, sponsored by the scholarly journal Philosophy and Literature: Judith Butler, a Guggenheim Fellowship-winning professor of rhetoric and comparative literature at the University of California at Berkeley. Her first-prize sentence, from “Further Reflections on the Conversations of Our Time,” (Diacritics, 1997):
The move from a structuralist account in which capital is understood to structure social relations in relatively homologous ways to a view of hegemony in which power relations are subject to repetition, convergence, and rearticulation brought the question of temporality into the thinking of structure, and marked a shift from a form of Althusserian theory that takes structural totalities as theoretical objects to one in which the insights into the contingent possibility of structure inaugurate a renewed conception of hegemony as bound up with the contingent sites and strategies of the rearticulation of power.
Syntactically speaking, this sentence is not particularly complex. Consider a syntactically homologous counterpart with the jargon removed:
The move from a country house in which draftiness was felt to undermine bodily comfort in relatively subtle ways to a house in the city in which climate control is easy to achieve, maintain, and readjust brought the family in question into a situation of ease, and caused them to adjust from a type of lifestyle that involved occasional sacrifices of physical well-being to one in which the enjoyment of the regular possibility of comfort aroused a perception of life as independent of the occasional whims and impulses of the vicissitudes of nature.
Also not particularly complex syntactically, for all its long-winded repetitions and anal-retentive hair splittings, is this example of legalese:
In case any building or structure is erected, constructed, reconstructed, altered, converted or maintained, or any building, structure or land is used, or any land is divided into lots, blocks or sites in violation of this article or of any local law, ordinance or other regulation made under authority conferred thereby, the proper local authorities of the town, in addition to other remedies, may institute any appropriate action or proceedings to prevent such unlawful erection, construction, reconstruction, alteration, conversion, maintenance, use or division of land, to restrain, correct or abate such violation, to prevent the occupancy of said building, structure or land or to prevent any illegal act, conduct, business or use in or about such premises.
Here’s a bracketed version that marks off the embedded phrases-- of which there is much less than we see in the Jane Austen sentence:
In case [any building or structure is [erected, constructed, reconstructed, altered, converted or maintained]], or [any building, structure or land is used,] or [any land is divided into [lots, blocks or sites]] in violation [of [this article] or of [any local law, ordinance or other regulation [made under authority conferred thereby]]], [the proper local authorities of the town], in addition to [other remedies], may institute any appropriate action or proceedings [[to prevent [such unlawful [erection, construction, reconstruction, alteration, conversion, maintenance, use or division of land]], [to [restrain, correct or abate] such violation], [to prevent [the occupancy of [said building, structure or land]]] or [to prevent any [[illegal act, conduct, business or use] [in or about such premises]]].
Of course, one needn’t look to academic papers and legal documents to find modern prose that’s hard to make sense of, especially if one is a teacher or professor--and/or interacts with modern curricula and curricular goals. Reform Math word problems are a gold mine. Consider:
Draw several groups of an item that comes in groups of some number between 2 and 12. Describe the picture in words. Then, on the back of the student sheet, use that information to make a riddle. 
In your description, remember to identify the three key pieces of information: the number of groups, the number of items in a group, and the total number of items. In your riddle, remember to use only two pieces of information from your description. Your riddle is the question you ask about the missing third piece of information!  
(TERC Investigations)
Then there are politicians. Here's some verbiage from a bit further back in time (the Cold War):
A land-based missile is the missile sitting there in its silo in which there could be the possibility of miscalculation. That is the one that people know that once that button is pushed, there is no defense; there is no recall . . . Those that are carried in bombers, those that are carried in ships of one kind or another, or submersibles, you are dealing there with a conventional type of weapon or instrument, and those instruments can be intercepted. They can be recalled if there has been a miscalculation.
These pieces are difficult (and, in some cases, alarming) not because syntactic complexity makes them difficult to parse, but because their ambiguities and jargon confuse us, and/or because their jargon, excess verbiage, and repetitions make our eyes glaze over.

Friday, April 1, 2016

Math problems of the week: Common Core-inspired word problems plus commentary and rationale

In the age of Common Core, wordiness reigns supreme: whether in Common Core-inspired word problems, Common Core-inspired commentaries on word problems, or Common Core-inspired rationales for word problems.

Here are two sample 4th grade problems from New York's Common Core-inspired EngageNY curriculum,  Each is accompanied by one commentary and one rationale. I've followed each problem, commentary and rationale with an edited counterpart.

Problem 1

Candy wants to buy herself a new bicycle that costs $240. Candy has already saved $32, but she needs to make a plan so she can save the rest of the money she needs. She decides to save the same amount of money, x dollars, each month for the next four months.

Part A: Write an equation that helps Candy determine the amount of money she must save each month.

Part B: Solve the equation to find the amount of money she must save each month to meet her goal of buying a bicycle.

[94 words]

Problem 1, edited

Candy wants to buy a $240 bike. She has $32 and will save x dollars each month for the next four months.

Write an equation that describes this situation. Solve the equation.

[32 words]

Commentary: This question aligns to CCLS 4.OA.3 and assesses a student’s ability to solve a multi-step word problem posed with whole numbers. It also assesses the ability to represent a problem using an equation with a letter standing for the unknown quantity.

[41 words]

Commentary, edited: This question assesses the ability to solve a word problem involving whole numbers using an equation involving one variable.

[19 words]

Rationale: In Part A the equation includes the subtraction of $32 from $240 to identify how much is needed to be saved in four months and the division of the remaining amount, $208, by four to represent the amount to be saved each month. Likely errors may include dividing $240 by four without subtracting the already saved amount of $32 (…) or using $32 dollars as the amount of money saved during the first month and dividing the remaining amount by three (…). In Part B errors may occur during the computation of the equation in Part A or may be the result of accurate computations based on an inaccurate equation from Part A.

[113 words]

Rationale, edited:

[0 words] (The rationale is obvious.)

Problem 2

Students from three classes at Hudson Valley Elementary School are planning a boat trip. On the trip, there will be 20 students from each class, along with 11 teachers and 13 parents.

Part A: Write an equation that can be used to determine the number of boats, b, they will need on their trip if 10 people ride in each boat.

Equation: b =______________________________________

Part B: How many boats will be needed for the trip if 10 people ride in each boat?

Part C: It will cost $35 to rent each boat used for the trip. How much will it cost to rent all the boats needed for the trip?

[110 words]

Problem 2, edited:

Students from three classes will go on a boat trip. There will be 20 students from each class, along with 11 teachers and 13 parents.

a. Write an equation that describes the required number of boats, b, if 10 people ride in each boat.

Equation: b =______________________________________

b. How many boats will they need?

c. If each boat costs $35 to rent, how much is the total cost?

[69 words]

Commentary: This question aligns to CCLS 4.OA.3 and assesses a student’s ability to solve a multi-step word problem posed with whole numbers. It also tests the student’s ability to represent the problem using an equation, with a letter standing for the unknown quantity. It tests a student’s ability to interpret the remainder of the division problem and use this interpretation properly to determine the number of boats as well as the total cost.

[72 words]

Commentary, edited: Same as in Problem 1, above. This question also assesses whether the student applies the fact that boats can’t be fractional.

[21 words]

Rationale: The equation in Part A includes a calculation for the number of students who went on the trip (20 × 3 = 60) plus the 11 teachers and 13 parents, bringing the total to 84 individuals on the trip. The number of boats, b, needed is the sum of all individuals divided by the number of people able to sit in a single boat. In Part B, students perform the calculation—84 is divided by 10, to get 8 R 4. The remainder of 4 indicates that an additional boat is needed, so the number of boats needed is 8 + 1 = 9 boats. In Part C, the total cost is the number of boats required multiplied by the cost per boat, $35 × 9 = $315.


[0 words]