Tuesday, June 30, 2015

Confusing math with math education

It strikes me that much of what is wrong with math education results from a confusion of math with math education. Is the goal to teach kids how to do math, or how to be math teachers?

Consider two tasks common to today’s math assignments but rare before Reform Math: explaining answers verbally, and explaining what’s wrong with other people’s solutions. Variants include having third graders write letters to second graders about why, say, 1/3 is bigger than 1/4, or to Jack “telling him what he did right, and what should do to fix his mistake.”

I and others have argued here and elsewhere that explaining your answers verbally is often a counterproductive waste of time that, in particular, shortchanges second language learners and students with language delays. Similar arguments apply to explaining why someone else’s answers are wrong. But, if you’re in a teacher education program training to be a math or a K6 general education teacher, then suddenly being able to provide these types of verbal explanations is absolutely essential.

Ironically, these explanation demands are especially common in elementary school, when students are least able to verbalize things clearly. Perhaps this has to do with the profile of the typical elementary school math teacher, who, in his or her teacher training program, has had to take courses in math education, but not in actual math. To some extent, however, such teachers are simply following the math curriculum that others have written and/or selected for them. So what about those most responsible for creating and selecting math curricula--the Deborah Balls and Andy Isaacs and Jo Boalers of the world? Is it possible that most of them get more training in math education than in math?

When it comes to educating K12 students, math education should primarily involve math, and not the infinite regress, as it were, that comes from educating students in math education.

Sunday, June 28, 2015

Problematizing grit, II

How hard you work on something isn’t the only effort (or grit)-related variable. Also key—and what Angela Duckworth’s questionnaire doesn’t probe—is how you direct that effort within the project. I realized that recently when, for the first time in over a decade, I decided to learn a new piano piece. Having allocated myself a mere 15-20 minute window on weekday mornings (a rare stretch of quiet solitude), I was determined to practice as efficiently as possible.

And this meant resisting all sorts of temptations that as a student I often succumbed to: the temptation not to bother working out fingerings and using them consistently; the temptation to interrupt my work on the sections I knew the least well, or found the most difficult, for the satisfaction of breezing through easier or more familiar sections; the temptation to play the piece too fast, too soon. It’s not just the distracting temptations outside a project, I realized, but also the distracting temptations within a project, that need resisting.

Directing your efforts appropriately involves brains as well as brawn. In learning a piano piece, for example, it helps to realize that muscle memory is essential, and that muscle memory will develop fastest if (1) you use consistent fingering and (2) you play slowly enough to minimize errors.

Teachers, too, can be smarter about grit. Neither should they try, vaguely, to "teach" it (e.g., by spending lots of class time on "growth mindsets"); nor should they simply give students tons of work or make them "grapple" indefinitely without guidance. Rather they should give students frequent advice and feedback about performance--and about how best to allocate their efforts.

Friday, June 26, 2015

Math problems of the week: Common Core-inspired "algebra" test problem

A problem from the "calculator section" of Algebra II  Performance Based Assessment Practice Test from PARCC (a consortium of 23 states that are devising Common Core-aligned tests).





Extra Credit:

Based on the given information, determine the ratio of algebraic to verbal challenges in this problem. Describe the steps used and explain any assumptions made. Create a model and describe the steps used to create it. Enter your answer, model, explanation, and assumptions in the space provided.

Wednesday, June 24, 2015

Problematizing grit

In her Ted Talk on “grit,” Angela Duckworth offers the following definition:

Grit is passion and perseverance for very long-term goals. Grit is having stamina. Grit is sticking with your future, day in, day out, not just for the week, not just for the month, but for years, and working really hard to make that future a reality. Grit is living life like it's a marathon, not a sprint.
All this, Duckworth finds, predicts long term success. So far so good—but (dare I say it?) hardly surprising.

What’s a lot less obvious is whether grit can be taught. Of course, this hasn’t stopped the education establishment, ever eager to focus on something other than academic instruction, from jumping to conclusions. Here, on the other hand, is Duckworth:
Every day, parents and teachers ask me, "How do I build grit in kids? What do I do to teach kids a solid work ethic? How do I keep them motivated for the long run?" The honest answer is, I don't know.
Duckworth says the best idea she’s heard is Carol Dweck’s “growth mindset”: “the belief that the ability to learn is not fixed, that it can change with your effort.” Duckworth cites Dweck’s finding that:
when kids read and learn about the brain and how it changes and grows in response to challenge, they're much more likely to persevere when they fail, because they don't believe that failure is a permanent condition.
Again, so far so good—but (dare I say it?) hardly surprising.

Plus, there’s only so far mere beliefs can get you. Indeed, the questionnaire that Duckworth uses to measure grit (and predict success) addresses how distractible you are, how fickle vs. sustained your interests are, and how hard and how diligently you work on things; not what you think about failure.

Given this, perhaps a better way to raise students’ perseverance is to provide extra incentives for hard, concentrated work. Ideally these incentives would be built into the work itself. You make sure that it’s interesting; that students get timely feedback about their progress through it; that completing it results in a satisfying final product, set of revelations, set of new skills, and/or sense of accomplishment. As far as these things go, much school work (whether because it’s busywork, easy work, group work, vaguely defined, and/or lacking in timely feedback) comes up short.

But even with some of the best types of assignments, and/or with certain types of students, there may be insufficient incentives for perseverance. In that case, as we’ve seen with J, why not resort to extrinsic incentives? For those who fail the marshmallow test, why not incentive them with marshmallows?

Monday, June 22, 2015

All about meteors or all about MEteors?

According to Michael Tscholl, a researcher at the University of Wisconsin (as reported in a recent article in Edweek):

Most students harbor fundamental misunderstandings about how forces such as gravity and acceleration operate in outer space. That's because their beliefs about physics tend to be based on their experiences in their own bodies.
Bodies on earth, Tscholl explains, need energy to keep moving; objects in space don't.

How to overcome these fundamental misunderstandings? Guess what Edweek/Tscholl propose? Is it:

1. Enhance students understanding of the concepts of friction and inertia.

2. Give students "embodied cognition" exercises in which they move their bodies around through earthly friction?

Hint: the solution proposed by Edweek/Tscholl is MEteor,
a room-size "simulation environment" that calls to mind a space-age version of the popular space-age version of the popular arcade video game Dance Dance Revolution.
Still stumped? Here's more:
In MEteor, planets and other space objects are projected on the floor and walls. The students must predict the trajectory of an object moving through space by physically moving along the path they think a meteor (projected on the floor) will travel. Laser scanning technology tracks their movements, offering real-time feedback on whether their predictions are correct. Based on that feedback, students adapt their beliefs about scientific principles, then adjust their movements to reflect what they are learning.
Final hint: it's probably reasonable to assume that these MEteor-facilitated embodied cognition exercises don't take place in outer space.

Another problem reported by Tscholl: "students are scared of symbolic representations." Given this, what do you think his solution is?

1. Give students more practice with symbolic representations and their relation to physical phenomena.

2. De-emphasize symbolic representations.

Stumped? Consider: (a) how facility with symbolic representations, and with manipulating these mathematically, is essential to doing physics, and (b) how little sense there is in anything in this article.

Saturday, June 20, 2015

You need to do some graphics to make it look like they’re flying, when they’re not really flying

I recently came across this un-facilitated, unedited, in-class assignment that J wrote for his graphic design class. Somehow, with its earnest attempt to cope with whatever the prompt was, and with his years in high school now weeks away from their conclusion, I found it quite endearing. I reproduce it here with permission from the author.

Given what I’ve written recently about Facilitated Communication, I should note that, in a sense, the author’s in-class communication is facilitated. J’s handwriting being so bad that often even he can’t read it, he regularly uses an AlphaSmart keyboard. But the keyboard remains stationary, sitting on his desk rather than on the palm of someone else’s hand; it offers no text-completion software with pop-up windows of likely next words and grammatical corrections; and no one would even consider hovering over J and supporting his wrist while he types. This is an author who feels strongly about being left alone while the creative juices flow:

There are some people who becomes a graphic designer. Like making a fictional movie, you’ll have to do some graphics on some objects. Like when Violet turned into a blueberry, people had to do some graphics since you obviously can’t inflate people into a ball. 
You have to be good at programming. Graphics require some programming. When you make a movie, you’ll want it to look real, and not make it look like it’s edited. Like when we see Violet turning to a blueberry, it looks real, and has not been edited.
You have to be good at painting to make some cartoon movies. In cars, Lightning McQueen and other cars look like they’re real, but they were actually painted. You would want to make it look real, and not look like they have been painted.  
You have to have a software to do some graphics. Photoshop is one of the software. It can edit some things out, and put some new things in. Like if you want to change some of the words, you’ll want to remove the words, and put new words in, and you’ll want to make it look like real, and has not been edited.  
So if you want a graphic designer, you need to be prepared. You want to make a movie look real, and not been edited. Like in Harry Potter movie, quidditch is obviously not real. You’ll have to make some graphics to make it look like they’re flying, when they’re not really flying.

Thursday, June 18, 2015

Math problems of the week: Common Core inspired 8th grade functions problems

From the Smarter Balanced Assessments, a Common Core-inspired, standardized test consortium now consisting of about 12 states.


The Common Core goal in question?

Grade 8 » Functions » Use functions to model relationships between quantities. » 5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.


Extra Credit:

Labels vs. Concepts

If you factor out the hurdle of knowing the meanings of the various labels ("linear", "non-linear," "positive slope," "negative slope"), how much mathematical challenge is left, and at approximately what grade level would you put it?

Tuesday, June 16, 2015

What we forget about history textbooks

It’s fashionable these days to decry traditional history as all about names and dates and powerful people, and history textbooks as inferior to primary sources. But our collective historical memories are short.

People forget that there are textbooks, and then there are textbooks. Some of them are written by committee, are dumbed down for a very general audience, and, written to offend no one, are dull as doorknobs. Others are written in the single voice of a learned historian and raconteur: someone who knows how to make even the driest facts as interesting to you as they are to him or her.

People forget that to really appreciate primary sources, you need historical context; that survey courses are the best way to acquire and retain this; and that there are some really good survey-based textbooks out there written by learned historians/raconteurs who know how to make even the driest facts interesting—particularly if you go back in time.

Because, finally, people forget—or probably never knew—that there are all sorts of really good history textbooks that were published ages ago, and that aren’t all about names and dates and powerful people.

Here’s how one of them opens:

Could Louis XIV now see the France he once ruled, how startling the revolution in politics and industry would seem to him! The railroads, the steel steamships, the great towns with well-lighted, smoothly paved, and carefully drained streets; the innumerable newspapers and the beautifully illustrated periodicals, the government schools, the popular elections, and his deserted palaces; the vast factories full of machinery, working with a precision and rapidity far surpassing those of an army of skilled workmen; and most astonishing of all, the mysterious and manifold applications of electricity which he knew only in the form of lightning playing among the storm clouds: all these marvels would combine to convince him that he died on the eve of the greatest revolution in industry, government, and science that the world has ever seen. It is the purpose of this volume, after describing the conditions in Europe before the French Revolution, to show as clearly as possible the changes which have made the world what we find it today.  
If a peasant who had lived on a manor in the time of the Crusades had been permitted to return to earth and travel about Europe at the opening of the eighteenth century, he would have found much to remind him of the conditions under which, seven centuries earlier, he had extracted a scanty living from the soil…  
The houses occupied by the country people differed greatly from Sicily to Pomerania, and from Ireland to Poland, but, in general, they were small, with little light or ventilation, and often they were nothing but wretched hovels with dirt floors and neglected thatch roofs. The pigs and the cows were frequently better housed than the people, with whom they associated upon very familiar terms, since the barn and the house were commonly in the same building…

Even in the towns there was much to remind one of the Middle Ages. The narrow, crooked streets, darkened by the overhanging buildings and scarcely lighted at all by night, the rough cobblestones, the disgusting odors even in the best quarters—all offered a marked contrast to the European cities of today, which have grown tremendously in the last hundred years in size, beauty, and comfort.
(From James Harvey Robinson’s Outlines of European History, which my daughter and I started reading a couple of months ago.)

Sunday, June 14, 2015

Modern-day Calvinism: predicting predestination

One longstanding frustration for “autism families” is how much more public money funds research on causes and early signs of autism than interventions and assistance. What’s the point of finding out when your kid is 3 weeks old that he or she is autistic if little is known about what to do next?

Something similar might be said of all that K12 assessment. Consider the amount of public money (and public discourse) spent on educational testing--from Common Core tests to assessment technology to the man-hours that teachers spend weekly on assessment forms and “formative assessments.”  How does this compare with the amount of money (and discourse) spent on follow-up measures? Education experts praise the new Common Core tests for predicting college and career readiness; they say little to nothing about what specifically to do on behalf of those who, on one or more of the hundreds of standards and sub-standards, don’t fully measure up. What’s the point of making predictions about future success if you have nothing to offer those who need help?

In the case of autism research and autism funding, I’ve often suspected that part of what’s going on is the allure of the easy. I’m guessing it’s a lot easier to fish around for genetic and neurological correlates and early infancy symptoms (and to tout early detection as the prerequisite for early intervention) than it is to create and efficacy-test the kinds of early interventions that would truly justify all that’s spent on early detection.

Now, as the edworld seems more and more focused on assessing everything from “mathematical thinking” to developmental skills (e.g., organization, attention) to personality traits (sociability, grit, “risk taking”) to readiness for college and careers, with little thought about going beyond measurement to measurement-informed interventions, I have to started to suspect something similar about education. It is, just maybe, a heck of a lot easier to score tests than to teach skills.

Friday, June 12, 2015

Math problem of the week: 6th grade Common Core-inspired test question

More from the Smarter Balanced Consortium and its Common Core-aligned tests: the final items on a 6th grade number sense assessment.

Stimulus: The student is presented with statements about two rational numbers and their position on a number line in relation to each other.

Example Stem: Select True or False for each statement.

The numbers 7 and –12 are both located to the right of 0 on the number line.

The number –12 is located to the right of 5 on the number line.

The number –12 is located to the left of –8 on the number line.


Extra Credit:

Discuss how confusing mathematical conventions with mathematical concepts is similar to confusing mathematical labels with mathematical concepts.

Wednesday, June 10, 2015

One thing I don't miss while homeschooling:

The mandatory science fair project.

But here's a great idea for a project:

Is holding a middle school science fair a worthwhile endeavor? A team of science educators and researchers funded by a $2 million National Science Foundation grant is hoping to find out.

The group is collecting data on science fairs' cost effectiveness, as well as their impact on learning and on students' interest in science.
According to Abigail Jurist Levy, the principal research scientist for the four-year project, "Science Fairs Under the 'Scope," science fairs have "never been really rigorously researched":
"As valued as they are by some, and as criticized as they are by others, we really don't know what they offer students in terms of learning experiences and engendering enthusiasm in science."
(Education Week)
Here's another science project idea, courtesy the Huffington Post:


From Susan Messina, designer of the above poster:
Any elementary school project that requires a lot of parental time, energy, resources, support, cajoling and financial investment is just BAD. Such projects privilege students from higher-income families for all the obvious reasons.
They also privileged the extraverted and artistic self-starters over other types of students (among them, some of our future scientists).

Monday, June 8, 2015

Who speaks for children with special needs?

Who speaks for autism? This was the question I raised a couple of years ago in an earlier post, noting the tensions between what is advocated for by certain high functioning individuals with autism vs. the parents of their lower functioning counterparts.

Current events inspire me to ask a much broader question: who speaks for children with special needs?

The special needs category, obviously, is several orders of magnitude greater both in terms of numbers, and in terms of diversity, than autism alone is. It includes a large number of individuals who are cognitively neurotypical, but have sensory or motor impairments (e.g., visual impairments or mobility impairments). Even among those with cognitive differences, it includes a large number of very high functioning individuals whose impairments don’t significantly affect, say, their comprehension of written language or of algebraic equations.

Despite all this diversity, a large consortium called the Consortium for Citizens with Disabilities, a consortium of approximately 100 organizations ranging from the American Association of People with Disabilities to the World Institute on Disability, has spoken out with a single voice on one particularly controversial issue. That issue has to do with America’s new Common Core-aligned tests.

For some time now, I’ve been arguing that the Common Core Standards are tough on kids with special needs. The big problem, I argue, is that they impose a one-size-fits-all, calendar-age based sequence on nearly everyone.

Currently only the most severely cognitively impaired 1 percent of the student population (about 10 percent of students with disabilities) is exempted from Common Core-aligned testing. But that 1 percent does not come close to including all the children who are reading, writing, or computing well below grade level.

Given this, you would think that disability advocates would crying out more exemptions, both from the tests, and from the calendar-aged-based, Common Core-aligned curricula that are proliferating around the country. After all, the more these curricula raise standards for neurotypical students, the more they deprive those with cognitive impairments and learning disabilities of access to appropriate instruction at their Zones of Proximal Development.

But nope. As it turns out, the Consortium for Citizens with Disabilities (again, approximately 100 organizations ranging from the American Association of People with Disabilities to the World Institute on Disability) has spoken out with a single voice to denounce a provision that would allow an additional 2 percent of students (or about 20 percent of students with disabilities) to be tested on “modified academic achievement standards” and measured for proficiency on these.

This provision, they argue, is a way to get around teaching students with disabilities on the same academic standards as their typically-developing peers .

And, yes, so it will. These anti-exemption advocates are exactly right about that.

But, given everything we know about optimized learning environments, not teaching students with disabilities on the same academic standards as their typically-developing peers is a good thing. No matter who you are, starting at just above your current level of mastering results in faster long term progress than starting beyond your current level does.

So I ask, who are these anti-exemption advocates who claim to be speaking for all people with disabilities? Who are the real spokespeople here, and what do they have in the way of standing, and/or expertise, and/or experience? Their website doesn’t say.

So I can only guess. Perhaps these anti-exemption advocates include self-advocates whose only challenges are sensory or motor impairments or other non-intellectual impairments: impairments that can and should be straightforwardly accommodated to provide access to calendar-aged based curricula and testing. Perhaps these anti-exemptions advocates include other self-advocates whose learning disabilities are at the mild end of the spectrum: people with ADHD or dyslexia or high functioning autism who have largely overcome the various impediments to academic success. And perhaps these anti-exemption advocates include hopeful parents and other caregivers that are in some state of denial (a denial perhaps facilitated by “facilitated communication”) about the current and future academic readiness of those closest to them.

But there’s one thing I’m pretty sure of, and that’s that these anti-exemption advocates don’t include those who actually struggle to teach math, reading, and writing to students with significant learning difficulties--many of whom are extremely frustrated by the requirement that they teach the students calendar-aged-based material instead of material they can actually handle. Nor, or so I’d venture to guess, do they include the students themselves—for example, the language-impaired 11th grader forced to “Analyze how an author’s choices concerning how to structure specific parts of a text (e.g., the choice of where to begin or end a story, the choice to provide a comedic or tragic resolution) contribute to its overall structure and meaning as well as its aesthetic impact” (CCSS literacy goal RL.11-12.5) or the dyscalculic 11th grader forced to “graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior"(CCSS math goal HSF.IF.C.7.C)

The special needs of these populations need to be heard by many more people. Wouldn’t it be nice if some of the scores of disability rights organizations would break ranks from the Consortium and help make this happen? Now more than ever, in our Procrustean Age of one of one-size-fits-all Standards and Universal Design for all, where those who don’t span what needs to be spanned are stretched till they snap, the help of those who would be disability advocates is sorely needed.

Saturday, June 6, 2015

Depression era whole language and project-based learning

"Now you tell your father not to teach you any more. It's best to begin reading with a fresh mind. You tell him I'll take over from here and try to undo the damage--"

"Ma'am?"

"Your father does not know how to teach. You can have a seat now."

...

"Don't worry, Scout," Jem comforted me. "Our teacher says Miss Caroline's introducing a new way to teaching. She learned about it in college. It'll be in all the grades soon. You don't have to learn much out of books that way--it's like if you wanta learn about cows, you go milk one, see?"

"Yeah Jem, but I don't wanta study cows, I--"

"Sure you do. You hafta know about cows, they're a big part of life in Macomb County."

I contented myself with asking Jem if he'd lost his mind.

"I'm just trying to tell you the new new way they're teachin' the first grade, stubborn. It's the Dewey Decimal System."

Having never questioned Jem's pronouncements, I saw no reason to begin now. The Dewey Decimal System consisted, in part, of Miss Caroline waving cards at us on which were printed "the," "cat," "rat," "man," and "you." No comment seemed to be expected of us, and the class received these impressionistic revelations in silence. I was bored, so I began a letter fo Dill. Miss Caroline caught me writing and told me to tell my father to stop teaching me. "Besides," she said. "We don't write in first grade, we print. You won't learn to write until you're in third grade."
...
 The remainders of my schooldays were no more auspicious than the first. Indeed, they were an endless Project that slowly evolved into a Unit, in which miles of construction papers and wax crayon were expended by the State of Alabama in well-meaning but fruitless efforts to teach me Group Dynamics. What Jem called the Dewey Decimal System was schoolwide by the end of my first year, so I had no chance to compare it with other teaching techniques. I could only look around me: Atticus and my uncle, who went to school at home, knew everything--at least, what one didn't know, the other did. Furthermore, I couldn't help noticing that my father had served for years in the state legislature, elected each time without opposition, innocent of the adjustments my teacher thought essential to the development of Good Citizenship... [A]s I inched sluggishly along the treadmill of the Macomb County school system, I could not help receiving the impression that I was being cheated out of something. Out of what I knew not, yet I did not believe that twelve years of unrelieved boredom was exactly what the state had in mind for me.

--Harper Lee (1960).

Thursday, June 4, 2015

Math problems of the week: 6th grade Smarter Balanced "number sense" problems, cont:

More from the Smarter Balanced Consortium and its Common Core-aligned tests: the final items on a 6th grade number sense assessment.

Stimulus: The student is presented with a real-world or mathematical context and a graph of ordered pairs.

Example Stem 1: This grid shows the location of three points.


Enter the distance, in units, between point A and point C.

Rubric: (1 point) Student enters the correct numeric value for the distance (e.g., 7). Units of measure should be assumed from the stem.

Example Stem 2: This grid represents the layout of Tom’s neighborhood. Each unit on the grid represents 1 square mile.
• Tom’s house is located at (4, 2)
• A store is located at (–3, 2)
• Tom’s neighbors are located at (4, 4).


Enter the distance, in miles, from Tom’s house to the store.

Rubric: (1 point) Student enters the correct numeric value for the distance (e.g., 7). Units of measure should be assumed from the stem.

OILF Extra Credit:

1. Follow up to extra credit on last week’s POW: discuss the correlation between conceptual “depth” and mathematical challenge.

2. Discuss the lost opportunity in the above problems vis-a-vis a real-world application of the Pythagorean Theorem.

Tuesday, June 2, 2015

It’s not just the Common Core: how vagueness and complexity entrench current practices

One of the problems with the Common Core Standards is their vagueness. Some people see this as a virtue: the Standards, they reassure us, don’t spell out how or what to teach. Local schools and teachers, they say, still have as much autonomy as ever. The downside is that the CC provides no guidelines on how to attain its goals—the more so because the goals are often vague, not just about how they are to be met, but about what precise skills they involve.

Worse, this vagueness can be used by the powers that be to further entrench current practices like Reform Math, which many experts outside the power structures find highly problematic. Because the CC standards are so vague, anyone can argue that their preferred curriculum and pedagogy are supported by them. While theoretically this empowers everyone, in practice it particularly empowers those who already have power and influence over today’s classrooms.

But the Common Core isn’t the only vague factor out there that particularly empowers the Powers that Be. There’s also the testing data. Consider the declines in U.S. test scores, or our poor rankings relative to other developed countries. Particularly when this occurs on measures that, like the PISA or this new test, emphasize conceptual understanding, reasoning, and applied problem solving, Reform Math advocates say it’s because we’re not doing enough Reform Math; student-centered discovery-learning advocates say it’s because we’re not doing enough student-centered discovery learning; and technology-in-the-classroom advocates say it’s because we’re not making enough use of classroom technology. So obvious are these solutions that their advocates find no reason to look at what’s happening, or not happening, in the countries that outperform us.

It’s the same with the economy—another highly opaque set of factors (opaque, especially, in their complexity). When the economy is perceived to be in bad shape (or, for that matter, in good shape), advocates of current practices say we need more of these practices. The difference, of course, is that advocates of whatever current economic practices are don’t generally have quite the power monopoly enjoyed by those in the educational industrial complex.