Friday, April 18, 2014

Math problems of the week: Rates problems in Chicago Algebra vs. 1900s algebra

I. From the "Rates" section of the "Division in Algebra" chapter (Chapter 6) of The University of Chicago Mathematics Project Algebra: [click to enlarge]:

II. The rates problems in the "Simple Equations" chapter (Chapter 2) of Wentworth's New School Algebra [click to enlarge]:

Wednesday, April 16, 2014

Casualties of "balanced" literacy, years later

pra sno vla pni smu gra

These are some of the nonsense syllables that students in the after school program I teach in were recently asked to read aloud. This list was part of a French language literacy test, and the point of this test was to assess the kids’ French skills (many of them have French-speaking parents). But what was most revealing about the results of this particular test component had nothing to do with French.

The kids did ok with two other lists of syllables: ones that consisted of consonant vowel combinations (ta, le…) and vowel consonant combinations (ame, ette). But when it came to the consonant clusters in the above list (pr, sn, vl, pn, sm, and gr), they stumbled. Invariably, if they pronounced the second consonant at all, they placed it at the end of the syllable, such that “pre” became “par”; “sno” became “son,” etc.

In other words, these students, now in 3rd and 4th grades, were totally stumped by consonant clusters. What does this mean for their ability to read words like “presque” and “pneue”?

As it turns out, none of these students can read any French. As for English, while they are able to read it, they do so less fluently than an unsuspecting person might predict, given how many years they’ve attended ELA classrooms. How would a new word like “protracted” or “gravely” sound in their mouths?

The difficulty these kids have sounding out single syllabus with consonant clusters has nothing to do with their unusual backgrounds, and everything to do with the continued de-emphasis on phonics instruction in America’s K12 classrooms. I didn’t get to test the older kids, but I’m guessing that they, too, had problems with “pra,” “sno,” etc. If you never learn how to sound out arbitrary consonant clusters, you’d think this would get only marginally easier over time.

In fact, it would be really interesting to take a random sample even of high students and see how they would do sounding out random nonsense syllables with consonant clusters. Let alone erstwhile SAT words like "phlegmatic" and "punctilious."

Monday, April 14, 2014

The lost arts of listening and learning

In this week’s New York Times Education Section, we see a continuation of the love affair between education journalists and “interactive” classrooms that minimize extended reading and listening. In Ten Courses with a Twist Laura Pappano characterizes such instruction as “inventive,” explaining that it treats students not just as “sponges soaking up content,” and citing an expert to elaborate further:

They apply lessons to life, says C. Edward Watson, director of the Center for Teaching and Learning at the University of Georgia. He adds that “faculty are trying to be more engaging in the classroom” because, for one, “competition is greater than it used to be.”
Why are these changes occurring now? Usually people cite “21st century skills,” claiming that today’s world requires “real-life” skills rather than the core knowledge and academic skills that, for centuries, have formed the basis for intelligent thought and effective communication. But Pappano instead cites the rise of online courses and the decline in listening skills:
The proliferation of online content means in-person courses must offer more than just another lecture “video.” Professors also face challenges in getting and keeping the attention of students raised on quick takes. Some weave in ways for students to use restless fingers and splintered focus; every few minutes during Prof. Perry Samson’s “Extreme Weather” lecture at the University of Michigan, students must respond to questions by phone or laptop. Others design courses with gaming features.
The less experience people have attending to real-live lectures, the worse their listening skills become, and the more they tend to assume, as Pappano does, that listening is passive, that live lectures are as canned as canned lectures are, and that lecturing precludes q & a and other interactions between lecturers and audiences. As I noted earlier:
Each year that a teacher opts out of exerting the energy needed to hold students’ attention for major chunks of class time, whoever teaches these students the next year will find this even harder.
These concerns aren’t shared by writers at the Education Times, who shift their focus elsewhere:
We looked around the nation for courses with buzz, according to campus newspapers, higher education experts and enrollment numbers. Students still file into lecture halls and classrooms, but once they’re seated, it’s clear that these courses are different. They mess with the old models. And they give students an experience that might change how they think, what they care about or even how they spend their lives.
Thus, in the Introduction to Cultural Anthropology at Kansas State University:
The … class is broken into groups of indigenous peoples and colonizers. They get bins of limited supplies and must trade for other items to make weapons, following rules they devise in advance. Colonizers typically get blowgun-like tools to launch marshmallow-tipped straws while indigenous peoples may only use rubber bands.
Jordan Thomas, who took the course in 2012 and is now a teaching assistant, felt the impact of being colonized and made to string marshmallows on rubber bands. When you get “taken over and are forced to sit around and assemble and manufacture a necklace for the entire hour, you engage in the emotions that come with that,” he says, adding that this was something he never would have gotten from a book.
But what drove the professor to teach this way wasn’t the desire for students to experience emotions that they “never would have gotten from a book,” but frustration with their attitudes:
Dr. Wesch started the simulations in 2004 after growing frustrated that most student questions were about grades and how much something was worth on a test. “Those are terrible questions,” he says. “I realized I needed to change everything.” Yes, there is a final exam, but it’s only one question: Why are you here? (He’s expecting you to tell the 12,000-year history of mankind and what you plan to do for the planet.)
One finds a similar de-emphasis of core content knowledge in another of the Times’ picks: Professor Samson’s Extreme Weather class at the University of Michigan. Here, anecdotes about extreme weather prevail (a car being bounced across a highway when to close to a F4 tornado), and all exams are “open book, open computer, call a friend.”

Another of the Times’ pics is Professor Monger’s Introduction to Oceanography at Cornell, where:
a third of the course is activism. Dr. Monger keeps a website for the course, (sample post: “Why you should avoid eating shrimp”), and a listserve of 1,700. “I want to stimulate these guys to raise their voices,” he says. “I tell them, ‘That ocean is as much yours as anybody else’s.’ ” The final assignment is to write Congress, though students are not required to mail the letters.
And no, I’m not worried about indoctrination; I’m worried about how much students are actually learning and retaining.

Relatedly, in his World Regions class at Virginia Tech:
Mr. Boyer wants students to “get excited about the world” and lets them choose how they engage. Students participate through Twitter, in-class smartphone surveys and old-fashioned microphones. They earn a course grade by doing assignments with point values; collect 1,100 points for an A, 1,000 will get you a B. They also decide what class will cover (this spring, it’s the Middle East, Russia and China), and Skype with international figures.
When he put up a map to talk about Egypt and the Arab Spring, someone said, “How come Jordan doesn’t have anything going on?” His reply: “Maybe we should ask someone from Jordan.” Less than six hours after a YouTube appeal to King Abdullah II of Jordan, the king’s office responded.
In another of the Times’ Top Ten, the Global Jam Forum at the Berklee College of Music:
Students jam with [prison] inmates, put inmate poems to music and respond musically to art, poetry and even health issues like malaria in Africa. Caili O’Doherty, a pianist, says the class “changed the way I think about music,” adding: “I think about playing for those different audiences. We are playing for them, not for ourselves. The music isn’t about me.”

Then there’s The Art of Walking at Centre College:
Wear comfortable shoes because this environmental studies class covers serious mileage. Walks take several hours and typically cover 15 to 25 miles. Readings include philosophers like Martin Heidegger and are discussed during nonwalk days. Dr. Keffer, who began teaching the course in 2002, has offered it on campus in Danville, Ky., and as part of Centre’s study abroad program.
Last January, in Strasburg, Germany, students walked 17 miles between two villages in the Black Forest, what he calls “Heidegger’s office.” There is nothing goal-oriented or prescribed in the walks; students don’t phone or text (it’s not banned, they just don’t). Covering distance by foot, Dr. Keffer says, opens “a temporal branch of environmental studies.”
Meanwhile, in courses in Philanthropy at Princeton and the University of Virginia:
Having real money, and a deadline for giving it away, lets students feel both the power and the challenge of charitable donations. Since 2011, the Once Upon a Time Foundation has provided some $2.5 million for hands-on learning at 13 campuses, including the University of Virginia and Princeton. Fueling the trend, Warren Buffett’s sister Doris began an online course last year through her Sunshine Lady Foundation in which participants give away $100,000.
At Princeton, Dr. Katz’s freshman seminar is as much about learning to reach a consensus with 14 others as it is about tackling big questions. “Some of the disagreements are quite profound,” says Dr. Katz, whose students research charities and must persuade classmates to align with them. “Some students feel it makes no sense to give a gift in the United States,” while others find value only in “giving gifts close to home.” Last fall’s class had $25,000 to give away.
Last but not least in Self-Theories at Stanford:
Prepare to take on your demons in this freshman psychology seminar. Dr. Dweck’s groundbreaking research has helped shape current wisdom about success and achievement — that failure and recovering from it are more valuable than sticking with what you already know how to do. Dr. Dweck tells students to tackle something “they have never had the guts to try.”
A student belted out “The Phantom of the Opera” on a public bus; another struck up conversations with strangers in San Francisco. Ricardo Flores, a self-described introvert, challenged himself to run for dorm co-president and, though filled with anxiety, give a campaign speech. He spoke, and won the election. For his next task, Mr. Flores is honing his salsa skills in hopes of performing with Los Salseros de Stanford.
When it comes to the lost art of listening in education, Diana Senechal has posted some wonderful comments on Joanne Jacobs' recent post  about an OILF post:
The people who aggressively disparage the “sage on the stage” don’t realize what a mess they are causing. Students, too, are getting the message that they shouldn’t have to listen to the teacher (or, for that matter, to anything or anyone). Sometimes the message is subtle, sometimes direct–but it’s there.
The “achievement gap” is in many ways a listening gap. The kids who will fly off the handle if they aren’t given something concrete to do every minute–these tend to be the ones who do poorly. (There are exceptions: students who focus and listen but don’t do well, and students who seem perennially distracted but somehow ace their courses and tests.)
Guess who’s more likely, overall, to get into a good college and do well there? The student who can listen. Not because this student is “docile” or “passive”–but because he or she has developed the discipline of focus and attention, which are essential for most intellectual fields.
It’s inadvisable for a K-12 teacher to teach by lecture exclusively. Even in college, lectures are complemented by discussion sections, labs, etc. But the campaign against “teacher talk” is misguided and destructive. Not only does the teacher have something to convey, but the student benefits from learning to take it in.
Senechal adds:
What worries me is the “turn and talk” impulse–the tendency of many students to start talking to their neighbors at random moments (about anything at all). Students who do that are rarely focused on the subject, in my experience; they’re more concerned about what’s going on socially in the room.
I don’t see this as their fault entirely; they are receiving many messages that the classroom is a place for socializing.
If it were established that students should listen in class, then much of the problem would disappear (not all, but a lot). Unfortunately, teachers are told over and over to avoid talking and to have students constantly “turn and talk.” That feeds the problem, unless the discipline of listening is already established.
Sadly, the dying art of listening applies to adults as well: if only more people would listen to Diana Senechal!

Saturday, April 12, 2014

No, it's not a therapist, it's a "teacher-researcher"

"Can you let me in to what's going on? Into your thinking?"
No, this isn't a therapist talking; it's a "teacher-researcher." As the opening paragraphs of a recent article in Edweek (Teachers May Need to Deepen Assessment Practices for Common Core) explain:
For Olivia Lozano and Gabriela Cardenas, team teachers at the UCLA Lab School in Los Angeles, understanding what each of their students know and can do at any point in time is so integral to their practice that they call themselves "teacher researchers."
Over the 10 years they've worked together, the two have put formative assessment at the center of their instructional routines. Each day during workshop time, they pull students aside one-on-one or in small groups to ask open-ended questions about the lesson at hand and to gain insight into each 1st and 2nd graders' thinking.
And as one of these teacher-researchers tells us:
"I have a conferencing binder where I'm taking copious notes on each individual student. I analyze their work and see where they're at."
Known as formative assessment, this process potentially improves student learning, so long as:

1. It doesn't consume too much instructional time.
2. It is used *only* to inform and tailor instruction, and not to determine report card grades.

(As I've argued earlier, report cards should be based on what students can do on their own at the end of a given unit, not on their works-in-progress or their thought processes. Report cards should measure a student's degree of ultimate mastery of instructed material, not how they got there or vague things like the "depth" of their thinking, or how "critical" or "exploratory" of "creative" that thinking was.)

Transforming teachers into so-called "teacher-researchers" risks unwarranted intrusions into student's thought processes. Some intrusions--forcing students to share personal reflections that they might rather keep private--violate privacy and provoke anxiety. Other intrusions--making students who can do math automatically and nonverbally in their heads (which should be the ultimate goal!) fake their way through verbal "translations"--make things tedious, decelerate learning, and disadvantage kids with language delays.

The opening quote falls mostly into the latter category. Here's its context:
The common standards are asking students to do that and more. They are aimed at "building childrens' [sic] capacity to think, and analyze, and communicate, and reason," said Margaret Heritage, the assistant director for professional development at the National Center for Research on Evaluation, Standards, and Student Testing at UCLA. "We need to know if [students are] grappling with complex ideas," said Heritage, who mentored Lozano and Cadenas. "Where are they? Is the idea beginning to consolidate? What do I need to do to go deeper and really help them get this?"
All of that may be tough to measure with quick-answer questions or exit slips. Instead, to get a full picture of student understanding, teachers need to ask open-ended questions and push students to explore ideas aloud, the UCLA educators say. "When [students are] solving problems mathematically, they say, 'I did it in my head,'" said Cardenas. "And you ask, 'Can you let me in to what's going on? Into your thinking?'" 
With the common standards, "classrooms will look different," said Heritage. "We'll need a lot more talking, more focus, more discourse, more depth."
Cardenas and Lozano spend conference time asking guiding questions and posing strategies to help lead students toward an answer—and to get them talking about their thinking. "You're developing their metacognition skills, helping them think about 'What kind of a learner am I? What's going to help me learn better?'," Lozano explained. "It helps to give them a voice."
[Nancy] Frey of San Diego State University tells teachers that, when listening closely to students, "The question you have to ask yourself is not whether the answer is correct or incorrect, but rather what is it likely that that student knows and doesn't know in this moment in time that would lead him to that response?"
Rather than asking multiple-choice questions or scanning quickly for right and wrong, teachers will need to be attuned to what students are saying during those discussion and debate sessions. "If you're walking around with a clipboard or notebook as kids are working through application, you're hearing, are they using mathematical thinking? Are they attending to precision? How well are they using the mathematical practices?" said Pecsi.
Notice how all of this is being justified by the supposedly pedagogically neutral Common Core Standards.

These Standards, apparently, justify thought-process intrusions not just by teachers, but also by peers:
Another technique for potentially deepening assessment practices—and complying with the new standards' focus on collaboration and communication—is to have students assess each other.
Amanda Pecsi, director of curriculum at the Washington, D.C.-based Center City Public Charter Schools, pointed out that one of the mathematical practices required by the common standards is to "construct viable arguments and critique the reasoning of others." She said this may lead to teachers using more peer review during their lessons. well as a shift of responsibility from teachers to students:
"Ideally we want to be moving into a place where students are doing that heavy lifting and their formative assessment is how they evaluate someone else and how they talk about it." well as a tremendous inefficiency in math instruction that risks leaving American students even further behind than they already are with respect to the developed world:
In light of the math common standards' emphasis on performance tasks and constructing arguments,... Pecsi said teachers will need to begin using more inquiry-based problem-solving. That might entail "20 minutes of students digging deep into one problem and debating," she said. "Ideally that could be an entire lesson eventually." well as an expansion of the Educational Testing Industrial Complex, in which ever more money flows from impoverished school districts to the testing companies whose consultants comprised the majority of the authors of the Common Core State Standards:
Meanwhile, the two main common-core assessment groups—the aforementioned Partnership for Assessment of Readiness for College and Careers and the Smarter Balanced Assessment Consortium—are planning to support teachers with formative assessment.
Smarter Balanced is putting out a "digital library," which Chrys Mursky, the group's director of professional learning, emphasized is "not a test bank of items" but a group of digital resources aimed at helping teachers build their own formative assessments. The library will be available by the time the Smarter Balanced assessments are ready to use, but only for teachers in states that purchase the full suite of tests.
PARCC plans to have adaptive, online "non-summative" tests for students available to all teachers in PARCC states. However, Bob Bickerton, co-chair of the PARCC non-summative working group, said the consortium is still currently looking for a vendor for some of the formative tools, so those will not be available until the 2015-16 school year.
 ...all of it, of course, for the sake of those "teacher-researchers" and, ultimately, the guinea pigs populating their laboratories--um, classrooms.

Thursday, April 10, 2014

Math problems of the week: word problems in 1900s vs. 2000s algebra

Word problems with one or more unknown, following systems of equations practice.

 I. From Wentworth's New School Algebra, published in 1898: the Problems chapter immediately following the the Linear Systems chapter:

II. From College Preparatory Mathematics, published in 2000: the Cumulative Review section at the end of the Graphing and Systems of Linear Equations chapter:

Tuesday, April 8, 2014

Deconstructing the sample Keystone Exam, continued

I've been mulling over some great comments I received on my most recent post on the sample algebra exam for Pennsylvania's new Keystone tests. As I commented in response, I'm intrigued by the idea that these problems are each intended to measure a particular skill rather than, say, the general skills of setting up word problems algebraically and manipulating the resulting expressions to solve find solutions. It makes sense that the test-designers are attempting to align particular problems to particular Standards, and also that they are trying to ensure that students who failed to master skills from pre-Algebra and below are still able demonstrate that which they can nontheless do.

However, it seems to me that there's a significant overlap in what these 10 problems measure, and that the skills they measure boil down to:

1. passively recognizing the correct set up of an equation
2. correctly plugging in numbers and doing the arithmetic
3. using estimation to rule out unlikely candidates
4. knowledge of ordinal pair conventions
5. understanding of the inequality sign
6. facility with arithmetic

This list, unfortunately, includes few, if any, of the core skills of first year algebra.

I've ended up with a more cynical take on what, at least in part, is going on here. To most people, unless they actually sit down and do the problems, this sample exam looks to be testing much more advanced skills than the 6 I've listed above. It looks like we have word problems that need to be set up; systems of equations to solve algebraically; expression upon expression to manipulate symbolically; possibly complex relationships involving ordered pairs, or vertical and horizontal distances. In other words, it looks like a student's performance on this test is a function of how well he or she mastered first year algebra. Or, using ordered pair notation: (test score, mastery of Algebra I).

This gambit will quickly appease many people will might otherwise worry about what students are learning--or not learning--in today's schools. Enough kids will do OK, and it will look like they're thinking deeply and critically about algebra.

I'm reminded of the few problem sets in Math Investigations that appear at first glance to actually involve complex calculations, and that only on closer inspection clearly involve none. Consider, for example, this one, excerpted from one of my Problems of the Week.

STOP. Don't start yet. Star problems that may have odd answers.
× 7


× 65
× 37





Sunday, April 6, 2014

Word problems in a sample in Keystone Algebra exam--deconstructed

In an earlier post, I posted some word problems from a sample exam for Pennsylvania's new Common Core-inspired tests (the Keystone Tests). I found these problems vaguely troubling, but hadn't taken the time to figure out what was bothering me.

As Auntie Ann pointed out:

1) They're wordy.
2) They give you far more information than necessary. If kids really are supposed to understand this stuff, why does almost every problem write out the equation for the student? Why not have them generate it themselves? Shows a lack of faith that the kids *have* actually learned what to do.
Indeed, all cases, the expressions are set up for you. Furthermore, as I noticed upon going through these problems and doing them myself, no algebraic manipulations are necessary in order to solve any of them.

Here, again, are the problems, with my commentary below:


All that’s involved here, obviously, is a passive identification of the correct setup. Actively translating words into algebra is significantly more challenging.


Here, not only is the equation already set up, but the variables are explicitly defined. All you have to do is map the equation to word problem to determine what the coefficients stand for, and then apply your knowledge of the conventional x, y ordering of ordered pairs.


This one is not really a word problem: you can answer the question without even reading through the scenario. Since the four choices for x are all easily plugged into each equation with the resulting values for y easily calculated (from the second equation) and checked (via the first equation), the correct value of x can be determined by guess and check alone—i.e., by simple arithmetic—along with simple realization that the second equation is easier to start with.


Here, you’re given the equations and told what the variables stand for. You can determine the answer by seeing which one of a handful of obvious whole number pairs works for the first equation. What works for the first equation, (4,3), also works for the second equation.

Here, there’s just one variable, and what it stands for is obvious from the given scenario. Choices b-d look suspicious (it would be a coincidence if 185 is also the value of b; more likely, the 185s here are red herrings). Choice a is easily confirmed by plugging in 204 and doing some simple arithmetic.

Here again, not only is the problem set up for you, but the variables are also identified for you. All you have to do is plug in the pairs of values in the different choices until you find one that works. Again, the problem boils down to simple arithmetic.

Problem 18 involves passively picking out the correct expression; choices c and d can be eliminated instantly because the intervals are obviously too large, and choice b can be eliminated because the constant term (75 times 4453) is obviously too large. In problem 19, the equation is set up, the variables are defined, and the correct answer is readily determined by a quick inspection of the only plausible choices: c and d.

Assuming an understanding of similar triangles and simple ratios, the answer to this problem is obvious (and the triangle diagram, as with the scenario in problem 12, is a pointless distraction).

Here again the equation is already set up and the variables defined; one simply recognizes that x equals 0 when the machine is full, and that one can therefore eliminate x and solve for y.

Facility with algebraic manipulation is crucial for calculus. The idea that one can get sail through the Keystone algebra test by passively interpreting someone else's algebraic expression, and by plugging numbers into them, is deeply disturbing. What incentive is there left for algebra teachers to prepare students for higher-level math?