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

Right-brained science, again: the myth of the finches

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 Gal├ípagos 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.

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.
Perhaps most compelling is Mlodinow's critique of the recent Steven Hawking movie
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

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

From a Algebra II  Performance Based Assessment Practice Test from PARCC (a consortium of 23 states that are devising Common Core-aligned tests).


Extra Credit:

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

Two approaches to math assessment: quantity vs. "quality"

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

Knowing, Doing, and Explaining Your Answer

Barry Garelick and I have a piece up on Education News.

Some excerpts:

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

Math problems of the week: Common Core-inspired math vs. Singapore Math

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

Five 21st century ways to eliminate the achievement gap (vs. one 20th century way)

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.