One of the most underappreciated of sins is wasting people’s time. Frittering away an hour of someone else’s time, if you think about it, is not that different from shortening that person’s life by one hour.
Time wasting is all the more toxic in environments that are difficult to escape, and/or where the opportunity cost (the value of what you could potentially be doing instead) is high. For example, K12 classrooms. Here, the most obvious symptom of time wasting is boredom.
Enthusiasts of today’s technology-enhanced, “child-centered,” “real-life relevant” classrooms often describe traditional classrooms as boring. Surely desks in rows, rote drills, and pen, paper, textbooks and chalkboards are inherently duller than manipulatives, interactive screens, multi-media projects, student-led discussions, and students facing one another in desk pods.
But some of the core features of today’s classrooms make them more boring than ever. Culprits include having students work in groups rather than independently (a.k.a. cooperative learning), and assigning students of different ability levels to each group (a.k.a. heterogeneous grouping).
1. Combining group assignments with heterogeneous grouping ensures that few students are working within their Zones of Proximal Development. This makes the assigned tasks too easy for some students and too difficult for others—disengaging both parties and slowing down everyone’s progress.
2. Students are generally less engaging as teachers than teachers themselves are, and student-led, teacher-decentered discussions are often rambling, confusing, and repetitive: slow to move through the material and/or to get to the point.
3. Many of today’s solve-in-multiple-ways-and-explain-your-answer math problems and be-colorful-and-creative multi-media projects involve a very high ratio of busy work to actual learning.
4. The pod-based seating that facilitates group work means that half the students now have their backs to the front of the classroom. Factor in that today’s teachers spend less time in front and more time moving around, and it’s no longer possible for bored students to shield behind their textbooks and notebooks the more engaging material (the cartoons, the puzzle books, the adult novels) they once snuck in from home.
5. Boredom, in turn, is a much-underappreciated source of misbehavior. Aggravating this, student-centered classrooms foster in fewer and fewer students the habit of listening to and learning from their teachers. Students, in short, are harder and harder to teach, and are more and more distracted and distracting (no, you can’t just blame this on extracurricular technology and social media!).
For those who nonetheless still do want to learn academics, six plus hours daily in classrooms of restless and distracting classmates is a terrible way to waste a brain.
Sunday, May 31, 2015
One of the most underappreciated of sins is wasting people’s time. Frittering away an hour of someone else’s time, if you think about it, is not that different from shortening that person’s life by one hour.
Friday, May 29, 2015
It turns out the Montgomery Public Schools isn’t the only institution going deeper with K12 math. So is the Smarter Balanced Consortium and its Common Core-aligned tests. Here are two sample items from a 6th grade number sense assessment (one from the beginning, one from the end):
Stimulus: The student is presented with a context involving a negative number or zero.
Example Stem: A Fahrenheit thermometer shows that the temperature is 15 degrees below zero.
Enter the integer that represents the temperature in degrees Fahrenheit.
Stimulus: The student is presented with statements involving absolute value in a real-world context.
Example Stem: Sea level is defined as being at an elevation of 0 feet. Objects can be above or below sea level.
• Submarine J is 35.6 feet below sea level.
• Submarine Q is 21.5 feet below sea level.
• Submarine Z is 43.8 feet below sea level.
Determine whether each statement comparing the submarines is true.
Submarine J is deeper than Submarine Q because |–35.6| > |–21.5|.
Submarine Q is deeper than Submarine Z because |–21.5| > |–43.8|.
Submarine J is deeper than Submarine Z because |–35.6| > |–43.8|.
OILF's Extra Credit:
Is it possible that what’s challenging about negative numbers and absolute value aren’t the “deep” concepts that underlie them, but, rather, the more complicated operations on them that emerge in a curriculum less focused on “number sense” and more on (shudder!) mathematical procedures?
Wednesday, May 27, 2015
In theory, today’s Constructivist classrooms, inspired as they are by educational progressivism, are supposed to favor child-centered discovery learning. And yet, in many ways, they are less child-centered than ever.
We see this, for example, in Reform Math and Common Core-inspired math classes. Here, children are told not just to solve problems, but how to solve them. And they are often required to solve them several times over in multiple ways. Typically, the standard algorithm is only one of several options, and preferred options are things like counting forwards or backwards from “landmark numbers,” “splitting” numbers via “number bonds,” repeatedly adding, repeatedly subtracting, or (if you’re lucky enough to be using Everyday Math) multiplying via “lattices”. Not to mention explaining, verbosely, how you did what you did and why.
A more child-centered, discovery-oriented approach to math problems—indeed, a more “authentic” and “organic” approach—would be to let the problems themselves, rather than the verbal directions, suggest one particular type of strategy or another.
999 + 77 vs. 956 + 77
The first of these invites a “landmark” numbers approach: solve it by converting it to a similar problem involving 1000 instead of 999. The second problem is much more rapidly solved using the standard addition algorithm, left to right, with regrouping.
1000 – 7 vs. 956 – 77
The first invites a counting back approach (count back 7 from 1000); the second is much more rapidly solved using the standard subtraction algorithm with borrowing.
9 × 1004 vs. 9 × 1234
The first invites a “splitting”/distributive approach (split 1004 into 1000 and 4 and multiple each by 9 separately, then add). The second is much more rapidly solved via that standard algorithm for multiplication.
8032 ÷ 8 vs. 8032 ÷ 7
The first of these invites a “splitting”/distributive approach (split 8032 into 8000 and 32 and divide each by 8 separately, then add). The first is much more rapidly solved vis that standard algorithm for division.
Offering a large number of problems that invite different strategies is the approach taken by so-called tyrannical, teacher and textbook -centered traditional math. Here, it’s much less common for kids to be told how to solve particular problems. Yes, the standard algorithms are “privileged” as the most efficient ways to solve most problems. But students weren’t generally forced to use these—so long as they solved the problems correctly.
But the problems themselves were different. There were many more of them than what kids get today; they involved more digits and fewer “friendly” numbers. The result: many more problems for which the most efficient strategies were the standard algorithms. Also, calculators weren’t—so to speak—part of the equation. But speed often was. Timed math tests and timed problem sets were frequent. As I discussed in my earlier post:
Many people assume that speed tests (especially multiple choice speed tests) measure only rote knowledge. But they’re also a great way to measure conceptual understanding. Performance speed reflects, not just rote recall, but also efficiency, and efficiency, in turn, is a function of reasoning, strategizing, and number sense.In particular, time pressure will inspire you to use the standard algorithms on problems like 956 + 77, and nonstandard shortcuts (landmarks, splitting, etc.) on problems like 999 + 77.
This brings up another difference between traditional math and Reform Math. Traditional math doesn’t belabor the shortcuts—or, indeed, even teach them. After all, if you have enough timed tests involving problems like 999 + 77, you will figure it out on your own—in the spirit of true, child-centered discovery.
It’s only when you drastically lower the number of problems that you assign, allow calculators, and dispense with speed tests, that you find yourself having to start spoon-feeding students the shortcuts and other ad hoc strategies. (At the expense, of course, of the standard algorithms).
Monday, May 25, 2015
The Educational Technology Industrial Complex offers yet another reason to make students show their work
It helps sell technology!
Specifically, “screen casting technology.” As a recent article in Edweek notes, such technology can get “students to create a multilayered record of their thinking while attempting to solve math problems.” According to one paper presented a few weeks ago at the annual meeting of the American Educational Research Association
Such an approach could help teachers “go beyond determining whether students correctly solved the problem, to understand why students solved the problem the way they did.”Here's the screencast example showcased by the article:
In an ongoing study in which “students generate screen casts of their problem solving processes” and “recorded themselves as they verbally explained their work,” one particularly remarkable result was recorded:
One student… incorrectly solved a word problem that required division. By reviewing the screencast of the student’s work in conjunction with her audio-recorded narration, the researchers were able to ascertain that the student had used a sound problem-solving strategy, but made an arithmetic error caused in part by her haste to finish quickly (and thus demonstrate that she was “good at math”).The authors go on to highlight the crucial role played here by the screen casting technology:
Without the screencast… “it would have been difficult to pinpoint where exactly the mismatch took place, and it could have been incorrectly concluded that [the student] did not understand the problem from the start.”It really makes you wonder how people functioned back in the dark ages, when all you could do was talk to your students face to face and look directly at the sheets of paper they did their work on.
Of course, back in the really dark ages, when students weren’t required to do arithmetic in multiple steps and explain their answers verbally (which, incidentally, allowed them to do at least 10 times as many math problems per problem session as kids do today), there must have been no way to tell who didn’t understand the problems and who was simply prone to stupid mistakes.
Saturday, May 23, 2015
Behold Albert Einstein: not the tidy young patent clerk, working through his most groundbreaking theories, but the scraggly eccentric of his later years. This image speaks volumes about our conception of scientific geniuses. We view those we most admire more as crazy, intuition-driven, mold-breaking, wild-haired artists than as meticulous researchers and rigorous analyzers. We imagine their greatest mathematical and scientific breakthroughs occurring not at desks or in laboratories; instead, we see Archimedes in his bathtub, Newton under and apple tree, and Franklin in a storm with his kite.From an early draft of Raising a Left Brain Child in a Right Brain World, then called "Out in Left Field in a Right Brain World."
I regretted eliminating this section; it didn't fit in with the publisher's reconception of my project as a parent-oriented advice book rather than as a broader cultural critique. But the more I think about it, the more I think that this right-brained conception of science and scientists has contributed to the demise of science education in ways that specifically shortchange left-brained, scientific minds.
--The notion that the way you get kids interested in science is to showcase the epiphanies rather than the puzzle solving--downplaying the importance, and the fun, of solving hard puzzles.
--The notion that the way to prepare kids for science careers is to promote "creativity" and "out of the box thinking" rather than the analytical and mathematical skills that scientific competence depends on.
So it was nice to see physicist Leonard Mlodinow's Op Ed in Sunday's New York Times. As soon as I read the first two paragraphs, I knew just what he was getting at:
The other week I was working in my garage office when my 14-year-old daughter, Olivia, came in to tell me about Charles Darwin. Did I know that he discovered the theory of evolution after studying finches on the Galápagos Islands? I was steeped in what felt like the 37th draft of my new book, which is on the development of scientific ideas, and she was proud to contribute this tidbit of history that she had just learned in class.
Sadly, like many stories of scientific discovery, that commonly recounted tale, repeated in her biology textbook, is not true.Noting that "The popular history of science is full of such falsehoods," Mlodinow writes:
The myth of the finches obscures the qualities that were really responsible for Darwin’s success: the grit to formulate his theory and gather evidence for it; the creativity to seek signs of evolution in existing animals, rather than, as others did, in the fossil record; and the open-mindedness to drop his belief in creationism when the evidence against it piled up.Perhaps most compelling is Mlodinow's critique of the recent Steven Hawking movie
The mythical stories we tell about our heroes are always more romantic and often more palatable than the truth. But in science, at least, they are destructive, in that they promote false conceptions of the evolution of scientific thought.
Of the tale of Newton and the apple, the historian Richard S. Westfall wrote, “The story vulgarizes universal gravitation by treating it as a bright idea ... A bright idea cannot shape a scientific tradition.” Science is just not that simple and it is not that easy.
In the film “The Theory of Everything,” Stephen Hawking is seen staring at glowing embers in a fireplace when he has a vision of black holes emitting heat. In the next scene he is announcing to an astonished audience that, contrary to prior theory, black holes will leak particles, shrink and then explode. But that is not how his discovery happened.
In reality, Mr. Hawking had been inspired not by glowing embers, but by the work of two Russian physicists.
According to their theory, rotating black holes would give off energy, slowing their rotation until they eventually stopped. To investigate this, Mr. Hawking had to perform difficult mathematical calculations that carefully combined the relevant elements of quantum theory and Einstein’s theory of gravity — two mainstays of physics that, in certain respects, are known to contradict each other. Mr. Hawking’s calculations showed, to his “surprise and annoyance,” that stationary black holes also leak.Not glowing embers; difficult mathematical calculations.
Mlodinow notes that "the oversimplification of discovery makes science appear far less rich and complex than it really is." He also touches on broader consequences:
Even if we are not scientists, every day we are challenged to make judgments and decisions about technical matters like vaccinations, financial investments, diet supplements and, of course, global warming. If our discourse on such topics is to be intelligent and productive, we need to dip below the surface and grapple with the complex underlying issues. The myths can seduce one into believing there is an easier path, one that doesn’t require such hard work.To see this in action, one need look no further than the education world--including, of course, the subworld of science education.
Thursday, May 21, 2015
From a Algebra II Performance Based Assessment Practice Test from PARCC (a consortium of 23 states that are devising Common Core-aligned tests).
Discuss the relative challenges of the mathematical labels (i.e., for types of methods) vs. the mathematical concepts vs. plain old common sense.
Tuesday, May 19, 2015
Auntie Ann makes a great point on my last post:
Giving many problems and demanding wordy answers on a test are mutually exclusive. In the time it takes to explain in words one problem, a student could demonstrate their proficiency on several problems with different mathematical concepts. Writing wordy explanations is much slower than giving a student a variety of different questions.Assuming that the point of making students explain their answers is to distinguish those who really don't understand the math from those who've simply made stupid mistakes, then there are two possible approaches.
1. Assign a smaller number of problems so that students spend time explaining their answers.
2. Assign a larger number of problems.
Back in the day, we got perhaps ten times as many problems per session as students do today.
A student who is prone to stupid mistakes won't get nearly every answer wrong; a student who doesn't understand the math will. The type of answer generated by stupid mistakes often looks different from the type of answer generated by conceptual misunderstandings. Assign enough math problems, and a competent teacher can easily distinguish between the two types of student. Include harder problems that involve more mathematical steps than today's problems do, such that more students will naturally write down their mathematical steps, and it's even easier to distinguish those who understand from those who don't.
Doing lots of math problems (and getting timely feedback on them) is probably also a better way for students to overcome conceptual misunderstandings than explaining a much smaller number of problems is.
And its a great way for everyone to get better (especially more fluent) at math.
Sunday, May 17, 2015
Barry Garelick and I have a piece up on Education News.
At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.”
The girl threw her arms up in frustration and said “Why can’t I just do the problem, enter the answer and be done with it?”
[For some problems] the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious.
Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus— doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?
Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?
Friday, May 15, 2015
I. The final problem in the Common Core-inspired Engage NY 5th grade Fractions module:
1. Lila collected the honey from 3 of her beehives. From the first hive she collected 2/3 gallon of honey. The last two hives yielded 1/4 gallon each.
a. How many gallons of honey did Lila collect in all? Draw a diagram to support your answer.
b. After using some of the honey she collected for baking, Lila found that she only had 3/4 gallon of honey left. How much honey did she use for baking? Support your answer using a diagram, numbers, and words.
c. With the remaining 3/4 gallon of honey, Lila decided to bake some loaves of bread and several batches of cookies for her school bake sale. The bread needed 1/6 gallon of honey and the cookies needed 1/4 gallon. How much honey was left over? Support your answer using a diagram, numbers, and words.
d. Lila decided to make more baked goods for the bake sale. She used 1/8 lb less flour to make bread than to make cookies. She used 1/4 lb more flour to make cookies than to make brownies. If she used 1/2 lb of flour to make the bread, how much flour did she use to make the brownies? Explain your answer using a diagram, numbers, and words.
II. The last two fractions problems in the 5th grade Singapore Math Primary Mathematics 5A Workbook (in Unit 4, Multiply and Divide Fractions, pp. 98-99):
3. After giving 1/3 of his money to his wife and 1/4 of it to his mother, Mr. Li still had $600 left. How much money did he give to his mother?
4. Lucy spent 3/5 of her money on a purse. She spent the remainder on 3 T-shirts which cost $4 each. How much did the purse cost?
III. Extra Credit
One of the biggest challenges found in Singapore Math problems (and not just in the one that recently went viral) is in figuring out what the first step is.
Compare the obviousness of the first steps in the EngageNY problems to those of the Singapore Math problems above.
Wednesday, May 13, 2015
1. Tell teachers to arrange students into heterogeneous-ability groups, assign most work to the group as a whole, and give everyone in the group the same grade on this work.
Justify this by saying that this is how things work in the collaborative, 21st century work place.
Justify this to teachers in particular by pointing out how much it reduces the amount of grading they must do.
2. Tell teachers (and educational testing companies) to minimize the cognitive and traditional academic challenge in the various assignments/assessments so that most kids ceiling out on these measures and earn more or less the same number of points on them.
Justify this by saying that in today's world, where you can look everything up on the Internet, and where more and more of the computational and analytical work is done by calculators and computers and technicians in Asia, knowledge and computational/analytical skills matter less and less.
3. Tell teachers to maximize (in the various assignments/assessments) factors based on inherent personality traits like extraversion and sociability ("makes appropriate eye contact;" "engages the audience") and other subjective factors like creativity and outside-the-box thinking ("takes risks;" "shows innovation;" "includes colorful, pleasing illustrations")--where the skills/traits involved are fairly evenly distributed across the academic spectrum. Give these factors an aura of objectivity by making them the headers of columns in quantitative-looking assessment grids called "rubrics."
Justify this first by saying that, since computational/analytical skills matter less and less in the 21st century workplace, creativity and interpersonal skills matter more and more.
Justify this also by touting the inherent virtues of allowing the "type of student" who wouldn't have thrived under traditional measures to shine as never before.
4. Subtly incentivize teachers to use appropriate discretion in assessing the more subjective factors so as to boost the scores specifically of those whom traditional measures might deem the "weaker" of the students.
5. When these erstwhile "weaker" students later fail to thrive in the real, 21st century world, blame it on poverty, prejudice, and the chronic under-funding of public schools, and say that such outcomes are therefore beyond the schools' control.
Looking beyond the "stakeholders" of the 21st century educational-industrial complex, one finds more promising, outside-the-box, 20th century ideas about eliminating the achievement gap. Take linguist John McWhorter, an increasingly prominent spokesperson for disadvantaged children. In an article he wrote for the New Republic over 6 years ago, he reminds all of us about Project Follow-Through:
A solution for the reading gap was discovered four decades ago. Starting in the late 1960s, Siegfried Engelmann led a government-sponsored investigation, Project Follow Through, that compared nine teaching methods and tracked their results in more than 75,000 children from kindergarten through third grade. It found that the Direct Instruction (DI) method of teaching reading was vastly more effective than any of the others for (drum roll, please) poor kids, including black ones. DI isn't exactly complicated: Students are taught to sound out words rather than told to get the hang of recognizing words whole, and they are taught according to scripted drills that emphasize repetition and frequent student participation.
In a half-day preschool in Champaign-Urbana they founded, Engelmann and associates found that DI teaches four-year-olds to understand sounds, syllables, and rhyming. Its students went on to kindergarten reading at a second-grade level, with their mean IQ having jumped 25 points. In the 70s and 80s, similar results came from nine other sites nationwide, and since then, the evidence of DI's effectiveness has been overwhelming, raising students' reading scores in schools in Baltimore, Houston, Milwaukee, and other districts. A search for an occasion where DI was instituted and failed to improve students' reading performance would be distinctly frustrating.
...schools of education have long been caught up in an idea that teaching poor kids to read requires something more than, well, teaching them how to sound out words. The poor child, the good-thinking wisdom tells us, needs tutti-frutti approaches bringing in music, rhythm, narrative, Ebonics, and so on. Distracted by the hardships in their home lives, surely they cannot be reached by just laying out the facts. That can only work for coddled children of doctors and lawyers.
But the simple fact of how well DI has worked shows that "creativity" is not what poor kids need. At the Champaign-Urbana preschool, the kids--poor kids, recall, and not many who were white--had a jolly old time with DI, especially when they found that it was (hey!) teaching them to read.McWhorter was talking, specifically, about the reading gap. But Direct Instruction's efficacy is seen in all subjects, and performance in all subjects, of course, is partly a function of reading skills.
Monday, May 11, 2015
Four feet deep, to be precise.
In the 21st century, a deep understanding of mathematics, and the ability to apply that understanding, is more important than it has ever been. In Montgomery County Public Schools (MCPS), and across the country, mathematics instruction is changing to make sure we provide our students with the skills and knowledge they need for success in college and the workplace. From the MCPS’s math website.Sound familiar? Yes, it’s all about our friends the CCCSS and the PARCC, along with the collaborative, 21st century workplace all our kids are going to end up in:
The improvements to the math curriculum are in response to several factors and will results in MCPS students having a stronger, more comprehensive understanding of mathematical concepts.Reasons include:
• The adoption of the internationally-driven Common Core State Standards (CCSS) and new, more difficult assessments being developed by the Partnership for Assessment of Readiness for College and Careers (PARCC), of which Maryland is a member…
• The changing demands of the work force, including 21st century skills, such as, collaboration, persistence, critical thinking, and creative thinking…The CCSS, the website notes, “demand a higher level of thinking in math for all students”:
Computation and procedures were sufficient to reach success in previous curriculum [sic] and assessments. The CCSS requires students to show greater depth by demonstrating their Understanding, Computing, Applying, Reasoning and Engagement (UCARE) in mathematics. As a result, the math content at each grade level is more difficult than previous curriculum [sic; boldface, here and elsewhere, mine].So difficult that much more is required in order to advance through it:
Following the CCSS, the elementary program is designed to go deeper in the topics of number (counting, addition, subtraction, multiplication, division, fractions, and decimals) to ensure that students have a strong foundation before moving on to more advanced content.
…Students at all levels are expected to express a deep understanding of the math content they are studying before moving to more advanced content. This means students will need to demonstrate their understanding in multiple ways, beyond just memorizing a formula or single procedure for solving a problem.Despite these hurdles, and despite all the ways in which the new curriculum is “more difficult” for all students, gifted students will get even more challenge:
C2.0 [Curriculum 2.0--the new curriculum] includes enrichment and acceleration options added by MCPS to ensure that students who demonstrate understanding of a topic will be able to deepen and extend their learning.Indeed, MCPS’s C2.0’s enrichment opportunities “exceed the requirements” of the CCSS. In particular, students who “demonstrate readiness” in grade 3 will have the option to enroll in the 4/5 Compacted Math class. However:
due to the increase in the rigor of the grade level curriculum, far fewer students than in previous years will need to skip a grade level in elementary mathematics to be challenged.Readers who’ve read this far must be burning with curiosity about just how deep a deep new, CCSS-exceeding curriculum goes for those select few who’ve already demonstrated exceptionally deep understanding. So here’s an example from the compacted 4/5 curriculum—a problem so deep that the class spent two weeks on it:
What is the opposite of 4 feet ABOVE sea level? What is the opposite of the opposite of 4 feet ABOVE sea level?"The parent who shared this problem (on a listserv for MCPS parents) adds that:
Kids who learned negative numbers years before do not get any acceleration through this, and group work is a huge part of every math concept. If your kid learned long division years ago and understands the concept completely, they will still have to memorize all the different, laborious "strategies" and spit them out verbatim…She concludes:
MCPS is not a place for rapid learners anymore… Our daughter has been bored to tears in math and science, as she has been every year. Most of the kids at the top 1% or so come home from school and then do a private math program in order to keep them engaged in math…For more examples of what the MCPS website calls “Changing Expectation in Curriculum 2.0 Mathematics,” see this past week's Problems of the week.
According to a 2014 article in the Boston Globe:
many, or even most [gifted kids]... aren’t identified early, and they don’t necessarily get special attention from their schools. [Researchers at Vanderbilt] have .. found that those who weren’t challenged in school were less likely to live up to the potential indicated by their test scores. Other research has shown that under-stimulated gifted students quickly become bored and frustrated—especially if they come from low-income families that are not equipped to provide them with enrichment outside of school.One of the researchers, Vanderbilt psychologist David Lubinski, worries about the broader impact of shortchanging our most academically capable students.
“We are in a talent war, and we’re living in a global economy now,” Lubinski says. “These are the people who are going to figure out all the riddles. Schizophrenia, cancer—they’re going to fight terrorism, they’re going to create patents and the scientific innovations that drive our economy. But they are not given a lot of opportunities in schools that are designed for typically developing kids.”
Saturday, May 9, 2015
In the next few days, J will be taking AP exams in BC Calculus and Computer Science. He earned a 4 on last year’s AB Calc exam, and whether he earns 5s, in general, depends on how well he slogs through the verbiage on the open-ended problems, and whether he shows sufficient work on them.
I used to think that the challenge, for J, was exclusively a verbal one. For understanding what to do with verbose problems, it certainly is. But showing your work is different. As far as the AP is concerned (at least for now), showing your work means what it did a generation ago: showing the key mathematical steps that lead to your solution. This requirement isn’t problematic for language-impaired kids in the way that the Reform Math and CCSS-inspired “explain your answer” is.
But for certain ASD kids like J--kids who do math in their heads and have difficulty taking other people’s perspectives--this requirement, I’m realizing, is problematic nonetheless. A neurotypical kid who can do a multi-step problem in their head has some sense of what steps she should write out for others—in particular, for whoever is evaluating her. But this kind of perspective taking (stepping out of the shoes of someone who has just solved the problem and into the shoes of someone who is evaluating your answer) does not come naturally to people with autism.
The open-ended problems on the BC Calc exam can often be broken down into dozens of steps; you’re not expected to write them all down (and doing so takes time). The trick is to figure out which combination of steps to spell out in order to satisfy a particular problem’s requirements, and this is where even the most autism-friendly math programs can shortchange those on the spectrum.
Thursday, May 7, 2015
In this week’s Problems of the Week, we revisit some claims found on the website of the Montgomery Counter, MD public schools (MCPS):
Computation and procedures were sufficient to reach success in previous curriculum [sic] and assessments. The CCSS requires students to show greater depth by demonstrating their Understanding, Computing, Applying, Reasoning and Engagement (UCARE) in mathematics. As a result, the math content at each grade level is more difficult than previous curriculum [sic].To illustrate this, the MCPS makes the following comparison:
The MCPS website asks us to note that the PARCC question “assesses similar content a year earlier”; that it’s “a multi-step question that requires application of content learned in earlier grades”; and that “it cannot be successful ly completed by memorizing a procedure, it requires reasoning.”
Note, however, that while the most obvious strategy for the MSA problem involves dividing a three-digit number, the most obvious strategy for the PARCC problem involves repeatedly multiplying and adding one and two digit numbers. It’s less the case that the PARCC problem is similar in its content to the MSA problem, or that it requires more reasoning, than that that it requires a heck of a lot more busy work.
The vice of simply memorizing a procedure is inadequate for either problem: in both cases, you have to figure out which procedure(s) to apply. What about the virtues: reasoning and “number sense”? Assuming that speed affects test performance, these are at least as important in the MSA problem as in the PARCC problem. In the latter, where only one choice is possible, reasoning tells you to use number sense to eliminate choices, and number sense immediately rules out A and B as too small, and D as ending in the wrong digit.
Side note: many people assume that speed tests (especially multiple choice speed tests) measure only rote knowledge. But they’re also a great way to measure conceptual understanding. Performance speed reflects, not just rote recall, but also efficiency, and efficiency, in turn, is a function of reasoning, strategizing, and number sense.
Not that the MSA 5th grade problem is something to be proud of. For a math curriculum that is actually more challenging than the old MCPS curriculum, one need only look backwards a decade at a something that several MCPS schools tried out for four years and then, despite positive results, abandoned (for more on that, see this very illuminating article by Barry Garelick).
That spurned something was Singapore Math.
Tuesday, May 5, 2015
As I discussed in my earlier post, autism as imagined by pop culture is at odds with autism as explained by scientists. This is not so much the case with high functioning autism: the person most famous for being autistic, after all, is Temple Grandin, and she presents a highly accurate exemplar of high functioning autism. But low functioning autism, particularly the world of nonverbal individuals, leaves much room for imagination and wishful thinking.
One result are miracle stories: stories in which the nonverbal individual emerges, typically via some sort of facilitated communication system, as higher functioning than Temple Grandin is. That is, communicative utterances are attributed to the nonverbal individual that show a greater level of eloquence, empathy, and introspection that we hear from Temple Grandin. In (among other things) “explaining” to us what it’s like to “have autism,” these communications belie everything that scientists have discovered about autism. It’s just that, unlike Temple Grandin’s communications, those of nonverbal individuals with autism are (by definition) neither spoken nor typed without assistance.
Few people seem to wonder how someone can have low functioning autism (characterized, not just by verbal difficulties, but by difficulties with eye contact and social attentiveness and reciprocity) and yet end up communicating more neurotypically than one of the highest functioning autistic people in the world. Or perhaps Temple Grandin, too, could sound fully neurotypical if only her communications were facilitated.
Of course, this idea should ring false even to the most stalwart proponents of facilitated communication. Indeed, take any autistic person who communicates independently and fluently (and here I include J): how likely is it that such a person would suddenly start communicating at a neurotypical level of eloquence and empathy if we simply provided “facilitation”? Is a child who carries on 24/7 about ceiling fans really going to share intimate, empathetic feelings with me if I start holding up a keyboard to his fingertips?
The problem (“problem”) with kids like J is that they leave little room to the imagination.
But there’s room in plenty of other places. Indeed, the tendency—and temptation—to see communicative intent where none actually exists is nearly everywhere. When our babies cry, our cats meow, or our plants wilt, they are “telling” us that they need something; when we figure out how to deal with a difficult child, he has “taught us” something new; and when a nonverbal child pushes a button on a keyboard (with or without assistance) in order to make something happen, he or she must be “expressing” an actual desire for that something rather than merely executing a particular stimulus that gets a particular response.
Sunday, May 3, 2015
Proponents of cooperative groups in K12 schools cite the rise of cooperative groups in 21st century workplaces. As I’ve frequently pointed out, however, professional collaborations differ from classroom-based groups in a number of key ways, not least of which is the amount of time people spend working in groups. While students are generally expected to do most of the work together, professional collaborators generally divvy things up and work separately.
Working not only separately, but also in solitude is what many professionals prefer. It's also what makes them most productive. But that doesn't necessarily mean it's happening. Here's what Lindsey Kaufman writes in a recent article in the Washington Post (thanks to Auntie Ann for telling me about it):
A year ago, my boss announced that our large New York ad agency would be moving to an open office. After nine years as a senior writer, I was forced to trade in my private office for a seat at a long, shared table.
…All day, there was constant shuffling, yelling, and laughing, along with loud music piped through a PA system.
…A 2013 study found that many workers in open offices are frustrated by distractions that lead to poorer work performance. Nearly half of the surveyed workers in open offices said the lack of sound privacy was a significant problem for them and more than 30 percent complained about the lack of visual privacy… In a previous study, researchers concluded that “the loss of productivity due to noise distraction … was doubled in open-plan offices compared to private offices.”
The New Yorker, in a review of research on this nouveau workplace design, determined that the benefits in building camaraderie simply mask the negative effects on work performance. While employees feel like they’re part of a laid-back, innovative enterprise, the environment ultimately damages workers’ attention spans, productivity, creative thinking, and satisfaction…However, given that “about 70 percent of U.S. offices have no or low partitions,” with newer, trendier companies like Yahoo, eBay, and Facebook leading the charge, a classroom group-work enthusiast could claim that that’s all the more reason for classroom-based group work. How better to prepare kids at school for the challenges of the 21st century workplace: the challenges to their attentions spans, productivity, creative thinking, and satisfaction?
Friday, May 1, 2015
1, The final problem in the Ratio and Unit Rates unit of the 6th grade Common Core-inspired EngageNY curriculum:
2. The final problem in the 6th grade Singapore Math (U.S. Edition) Speed unit:
a. Compare the difficulty of the two problems.
b. Compare the difficulty of part b of the 6th grade EngageNY problem to the difficulty of a second grade math word problem (with or without the extra verbiage).
c. For part a of the EngageNY problem ("How far did you and your mother travel altogether?"), should points be taken off for answers that add the distance traveled by "you" to the distance traveled by "your mother"?