Wednesday, April 30, 2008

Math problem of the week: Everyday Math vs. Singapore Math

1. From unit 2 of the Everyday Math grade 2 Student Math Journal, volume 1, p. 44  (volume 1, of 2, 164 pages total)

There are 10 houses on Jerry's block.  There are 15 houses on Nancy's block.  How many more houses are on Nancy's block?

Write a number model

A pack of gum costs 25¢.  Sean bought 3 packs.  How much money did he spend?


2. From unit 2 of the Singapore Math grade 2 Primary Mathematics Workbook, volume 1 (2a), p. 54 (volume 1, of 2, 192 pages total)

Mrs. Bates bought an oven for $393.
She also bought a refrigerator for $438.
How much did she spend altogether?

She spend $____ altogether

There are 468 desks in a hall.
There are 156 more chairs than desks.
How many chairs are there in the hall?

There are ___ chairs in the hall.


Place value and borrowing/carrying (regrouping): the bigger picture:

Grade 2 Singapore Math teaches place value in Unit 1 and regrouping in Unit 2.  

Grade 2 Everyday Math teaches place value in Unit 3.  Not until Unit 12, the final unit of the the grade 2 curriculum, do students do problems comparable to the Singapore problems above.  I.e., problems like 356 + 275 (cf. p. 302), which are specially classified as "harder" problems, and for which you are invited to use "your favorite addition strategy."

Tuesday, April 29, 2008

Autism, extreme maleness, and male culture

This past Saturday, as I listened to my 11-year-old autistic son recount the day's TV highlights ("At 5:00, Phillies game.  At 8:00, Nascar Racing [where did he get that idea???]"), and again later as I watched him, crouched forward towards the TV on his large shoulders and bulging biceps, yelling "Go Phillies!  You! need! to! score! a! point!," I thought of Simon Baron-Cohen.  

Autism, says Baron-Cohen, is a form of extreme maleness.  Baron Cohen's maleness includes a preference for objects over people, reduced empathy, and linguistic delays.  He notes that:

--newborn males, unlike most newborn females, gaze longer at mechanical mobiles than at human faces.
--baby males make less eye contact and respond less to other people's distress.
--toddler males talk later and expand their linguistic repertoires more slowly.
--adult males score higher on Baron-Cohen's Systematizing Test and lower on his Empathy Test

Underlying all of this, Baron-Cohen believes, is testosterone. The higher the concentration in the fetus's amniotic bath, research suggests, the lower his or her eye contact, and the slower his or her language development.  Too much testosterone, Baron-Cohen speculates, exaggerates these traits into autism.

So far as I'm aware, Baron-Cohen doesn't mention biceps, sports fanaticism, and Nascar racing. While I'm pretty sure that testosterone promotes muscle growth, I'm not so sure about the other two. Where does biology end and culture begin?

To that age-old question, my autistic son, as wild about baseball, basketball, football, hockey, and Nascar as he is oblivious to culture in general and gender stereotypes in particular, poses an interesting challenge.

Monday, April 28, 2008

How left-brained (male brained?) are you?

Responding to my first left-brain epiphany installment, Tim Worstall sent me a link to an updated, Americanized version of the popular left-brain/"male"-brain/systematizer vs. right-brain/"female"-brain/empathy test (MaleFemale.asp, to the left, on the sidebar).  

I took it this morning.

Here are your EQ SQ results:

EQ : 33
SQ : 72
Brain Type : Systemizing

What does your score mean?

Click Here to learn more.

Though I think of myself as strongly systematizing, I was surprised at how high I scored on this particular measure. As with the British version, the questions seemed biased towards certain types of systems. Many address visual ones like machine mechanics and how objects are put together. Others focus on detail-intensive systems like accounting and history. Neither of these subtypes grab me--not because I don't love systems, but because I'm weak at spatial reasoning and terrible at keeping track of details.

In a systematizing test biased more towards more conceptual systems like language, math, cognition, and the mechanics of ecology, evolution, and social interaction, I imagine I'd be off the charts.

In his book The Essential Difference, Simon Baron-Cohen, the author of these tests, argues that males tend to be systematizers and women, empathizers.  Perhaps if he asked fewer questions about automechanics, camera lenses and sound systems, and more about grammar, cognition, and social systems, the gender difference would look less systematic.

Sunday, April 27, 2008

Left-Brained epiphanies, part II

In an earlier post about two new movies featuring introverted eggheads who emerge from transformative experiences more passionate and socially engaged, I asked whether all epiphanies are right-brained. How often, for example, does an outgoing musician learn that what makes him truly happy is retreating to his study to analyze business cycles and market equilibria?

So far, three people have shared their left-brained epiphanies.  Last week, Dawn of Day by Day Discoveries shared hers.

This past week, Liz Ditz of I Speak of Dreams offered two:

1984 was a watershed year for me: I acquired an Osborne portable computer (which I named Ozzie). Suddenly, I discovered the satisfactions of quantitative analysis! I did not have to undertake the huge cognitive burden of mathematics my own self -- my faithful friend Ozzie did the calculations, and I could now play with assumptions. I was in an MBA program at the time. I'd read a case study, and have a way of quantifying my sense that "hmmmn, something's wrong here". I had enough math to set up the parameters, and Ozzie's abilities let me play until my intuitions were quantified.

This story makes me wonder how many other people would discover the joys of left-brained thinking if only the lower-level analysis were less tedious.  

How many more grade school students, for example, would enjoy higher level math if only they had more automated command of the lower-level stuff--the standard algorithms and multiplication tables?

Liz's second epiphany:

I'm also thinking about the cognitive demands of interpreting behavior. I have a dog (a middle-aged, female Golden Retriever) and I'm currently taking care of a friend's dog (a young, female Border Collie). Today I had a guest who is naive about dogs; he was very much concerned about the "aggression" the dogs were displaying. Well, they were playing--play biting, play growling, play chasing. To him, it was fearful aggression; to me, I could easily see by the dogs' respective demeanors that it was just play.

This made me think of Malcolm Gladwell's Blink, often cited as endorsing intuition over analysis.  In fact, it argues that naive intuitions are often wrong, and can be sharpened by long term experience.

Meanwhile, PaulaV wrote:

I certainly never thought I would be interested in reading graphs or stats on cognitive behavior. Yet, one day this changed when my son's principal told me he was unfocused and had a disconnect in math. I've became quite obsessed with finding out all I can on working memory, executive function, processing speed and how it relates to various subjects.

Paula's story shows yet another way in which, as our schools continue abandoning analytical rigor--in both their teaching and their evaluations--we parents must continue picking up their slack.

If you have a left-brained epiphany you'd like to share, please post it as a comment below.

Friday, April 25, 2008

Real world data on real world problems

For years I've seen the "real-world" word problems and "applied math" activities that dominate Reform classrooms shortchanging kids with autism. Their elaborate scenarios and multi-step directions stymie those with language delays and gaps in worldly knowledge.  Their multi-media, multi-sensory, interdisciplinary sprawl is too full of distractions for those who require streamlined, structured learning environments.

Now a new study by Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State, discussed in today's NYTimes, suggests that, for all students, there's such a thing as too much real-world math:

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

Clever word problems have their virtues, but it would seem that students must also learn what's no longer fashionable to teach:  math concepts plain and simple.

Thursday, April 24, 2008

Anything but the schools

We Americans readily hold schools responsible things that are largely beyond their control: children's emotional and moral development; their sexual behavior; the achievement disparities between those from educationally nurturing vs. educationally impoverished home environments.

When it comes to factors schools can control, on the other hand, we readily blame everything else. Consider the most recent books lamenting the spotty civic knowledge, and abysmal math and science skills, of Generation Y.

For William Damon, author of The Path to Purpose: Helping Our Children Find Their Calling in Life, the culprit is existential aimlessness: as Charles McGrath puts it in his review in last Sunday's NY Times Education Life, "most students are drifting aimlessly, with no clue as to what they want to do or become in the future."

For Mark Bauerlein, author of The Dumbest Generation: How the Digital Age Stupefies Young Americans and Jeopardizes Our Future, it's the digital revolution, which, in McGrath's words, "has eroded attention spans and analytical abilities."

For Susan Jacoby, author of The Age of American Unreason, it's not just the digital age, but the rise of religious fundamentalism, the triumph of the self-esteem and self-help movements over the earlier self-education movements, and inadequate education funding.

And for David Anderegg, author of Nerds: Who They Are and Why We Need Them, it's anti-nerd stereotypes: the growing antipathy that Americans in particular direct at so-called nerds or geeks, one of the few remaining groups which it’s still socially acceptable to mock.

Existential aimlessness; the digital revolution; religious fundamentalism; self-esteem; education spending; prejudice against nerds. Haven't we covered it all?

But what about Reform Math, Reform Science, and influential education experts like Deanna Kuhn, who shrug off knowledge and above-basic skills, asserting that the main concepts today's students need are inquiry and "argumentation" (which, if Kuhn is preaching what she practices, means argumentation by assertion, not by logical analysis of empirical data).

When it comes to that for which schools are most obviously responsible, why are so many of us--not just those with vested interests--so ready to point fingers everywhere else?  

Wednesday, April 23, 2008

Math problems of the week: linear equations in reform vs. traditional algebra

1. From "Say it with Symbols," Investigation 4, about 2/3 of the way through the 8th grade Connected Mathematics curriculum:

The school choir is selling boxes of greeting cards to raise money for a trip.  The equation for the profit in dollars, P, in terms of the number of boxes sold, x, is
P = 3x - (100 + 2x).

a. How many boxes must the choir sell to make a $200 profit?  Explain how you found your answer.
b. How many boxes must the choir sell to break even?  Explain how you found your answer.


2. From "Systems of Linear Equations and Inequalities," about 2/3 of the way through McDougal and Littel's Algebra 1, a traditional text:

A gold and copper bracelet weighs 238 grams.  The volume of the bracelet is 15 cubic centimeters.  Gold weighs 19.3 grams per cubic centimeter, and copper weighs 9 grams per cubic centimeter.  How many grams of copper are mixed with gold?


OILF's assessment:

Translating a word problem into an equation is a core skill, and challenge, in algebra.  

The main problem with Connected Math is how easy it is.  In particular, rarely does it ask students to translate word problems into equations without guiding them through key steps.

If a problem is hard enough that students can't do it in their heads, then you don't need to ask them to explain how they found their answers.

Tuesday, April 22, 2008

Voting with my brain

In honor of Election Day here in Pennsylvania, it's time for some cool-headed, left-brained rationality & skepticism.

Always remember:
1. Across the political spectrum, there are decent, rational people.
2. Across the political spectrum, some of the people with whom I happen to agree are highly irrational, and/or untrustworthy, and/or ethically suspect.
3. The more cool-headed, rational, and skeptical I am, the easier it is for me to appreciate 1 and 2.
4. The more cool-headed, rational, and skeptical I am, the more likely I am to seek out, read/listen to, and accurately remember things that challenge my convictions.
5. The more cool-headed and rational I come across to others, the more likely they are to tell me when they disagree with me and flesh out opposing arguments.

In short, the more left-brained I am--or behave--the more sides of a charged argument I will truly appreciate. And the more informed a voter I will be.

Ultimately, of course, I can reduce no candidate to his/her political axioms & arguments, and so it's reasonable for me to vote as much by gut feeling as by rational deliberation.

Also, as Stephen Dubner and Steven Levitt discuss in their November, 2005 New York Times Magazine article Why Vote?, given the vanishingly small influence of individual votes on elections, it would be reasonable for me to opt out entirely.


Or to vote only for right-brained reasons: because it feels good, and/or because, socially speaking (see Dubner and Levitt's discussion of the Swiss voting experiment), it would look bad not to.

Gotta go!

Sunday, April 20, 2008

Left-brained epiphanies!

In an earlier post about two new movies featuring introverted eggheads who emerge from transformative experiences more passionate and socially engaged, I asked whether all epiphanies are right-brained.  How often, for example, does an outgoing musician learn that what makes him truly happy is retreating to his study to analyze business cycles and market equilibria?

In response, Dawn of Day by Day Discoveries posted an inspiring comment that she's given me permission to reprint here:

I've been a life-long dreamer who draws and sings and is (or so I thought) as right-brained as the day is long.

Then I started really looking at math while homeschooling my kids and engaging in demanding debates and suddenly I've had a blossoming of rational thought and a found a lot of joy in numbers. It has been wonderful.

I don't know what kind of thinker I am now though I suspect I hover in the middle.

Dawn went on to take an online test that assesses how systematizing (left-brained) and empathetic (right-brained) you are.

I scored in the average range at 34 on the systemizing quotient but above the male average of 30.

On the emotional quotient I was 52, slightly higher than the average for women.

There must be other unsung tales like Dawn's--ones that may not play in Hollywood but are nonetheless compelling.  And should be told.  

If you have one you'd like to share, please post it as a comment below. 

I'm hoping there will be enough to make today's feature a regular one.

Saturday, April 19, 2008

Fairy tale endings

Every once in a while, reading fairy tales with my daughter, I discover something the 100th time around that somehow escaped me on all 99 prior occasions. Like: how does the King react when the miller’s daughter, after he marries her for her gold-spinning skills (and after Rumpelstiltskin stops visiting), loses her apparent magic?

With the Hans Christian Andersen tales, what sometimes hits me on the 100th read is how they capture yet another human weakness. The Emperor’s New Clothes of my childhood memories is simply the story of a foolish emperor to whom only an innocent boy dares speak the truth. In my college and grad school years, the story morphed—as it does for so many of us--into a parable for how easily people fall for empty, ego-promoting jargon and rhetoric in order to avoid feeling--or looking--“stupid.”

But only since my crash courses in the higher superstitions of certain autism and education experts has Andersen’s second reason for claiming to see invisible clothes occurred to me: not wanting colleagues to judge you “unfit for your office.” For, of course, nowadays (always?) success in today’s education and academic establishments (among many others) means adhering to certain dogmas, however ridiculous they strike you, or risk your fitness, as assessed by peers and superiors, for anything from publication, grants, tenure, and promotion, to simple respect.

Then there’s The Ugly Ducking. In my childhood memories, it simply tells how someone who looks uglier than his more conventional peers can ultimately come into a beauty of his own. But then last night, as my daughter and I discussed how our hero wouldn’t seem ugly to swans, I suddenly saw his story as allegorizing the negative impressions wrought by narrow socio-cultural standards.

Imagine a world whose norms aren’t sleek yellow duckling feathers and a duck-like waddle, but “appropriate” interactions with peers and eager participation in group projects and class discussions. Such norms convert the quiet, disengaged introvert into an uncooperative, undeserving freak.

Until, we hope, she finds birds of a feather who appreciate all she is.

And until, we hope further, her right-brained peers revise their views of her from ugly “normal” person to beautifully eccentric left-brainer.

Thursday, April 17, 2008

Subjective writing standards: bad news for good writers

Today's elementary school writing assessments, just like those for math and science, have opened the floodgates of subjectivity.

Restricting ourselves to what's visible online, let's consider the Delaware State Writing Rubric as used, for example, by the Christina School District. Its standards for top quality elementary school prose include: "sufficient, specific, and relevant details that are fully elaborated," "smooth transitions," "effective introduction and closing," "appropriate" sentence variety, and "vivid word choice."

Turning to the California writing standards as used, for example, by the k-6 Museum School in San Diego, we see "writes with confidence," "well-chosen details," "descriptive words," "concrete, sensory details," "provides insight into why the selected incident is memorable," "self-motivated," "shows originality," "developed voice, sense of style, purpose," and "develops plot and character."

Of course, much of this has long governed middle and high school English assessments. Here, most teachers have majored or minored in English and, I would hope, know something about good writing.  

But such subjectivity is relatively new to elementary schools, where teachers once judged students primarily on penmanship, spelling, and grammar.

How qualified are these teachers, whose writing backgrounds may have ended with freshman English, and who often score in the bottom third on their SATs, to judge details, word choice, transitions, voice, and character development?  

How well they assess other people's writing must have something to do with how well they write themselves.

Howling over exceptionally bad teacher prose has become a blood sport among education bashers. Let's rise above this and consider, instead, a sample that reflects the writing skills, not of your average elementary school teacher, but of the leaders of today's Language Arts establishment. Here's the opening sentence of the overview by the National Council of Teachers of English of their Standards for the English Language Arts:

The vision guiding these standards is that all students must have the opportunities and resources to develop the language skills they need to pursue life's goals and to participate fully as informed, productive members of society.

There's not much here in the way of detail or vivid words, but in its four-fold embedding of clauses, this sample is nothing if not "elaborate." Eliminating the wordiness, repetition, and awkward, passive phrasing (as in "the vision.. is that") reduces the 36 words down to 25:

These standards seek to enable all students to acquire the language skills they need to pursue their goals and participate knowledgeably and productively in society.

But some teachers, especially elementary teachers, may be so poorly trained in writing, and so indoctrinated in "fully elaborated detail," that they are unable to distinguish useful detail from excessive verbiage. And, I expect, would rate my revision as inferior to NCTE's original.

The problem with subjective standards is that when under-skilled people use them, they often end up favoring inferior products.  

Until our grade school teachers' writing skills match their indoctrination in today's writing standards, the latter have no place in elementary school writing assessments.

Wednesday, April 16, 2008

Math problem of the week: Investigations vs. Singapore Math

1. Grade 1 Investigations, Number Games and Story Problems, Investigation 2, Session 3, Student Sheet 12.

Collect 25¢

Note to families: Your child may play this game with only pennies.  Other coins are optional. If you don't have a dot cube (or die), use Number Cards or write the numbers 1-6 on slips of paper, turn them facedown, and draw them from a pool.

Materials: One dot cub
Coins (about 30 pennies, 6 nickels, 4 dimes, and 1 quarter)

Players: 2
Object: With a partner, collect 25¢ in coins.
How to Play
1. To start, one player rolls the dot cube. What amount did you roll?  Take that amount in coins.
2. Take turns rolling the dot cube.  Take that amount in coins and add them to the collection. You may use all pennies, or you may trade coins at any time (for example, 1 nickel for 5 pennies).
3. After each turn, check your total amount.  The game ends when you have 25¢


2. Grade 1 Singapore MathFrom a similar point in the curriculum, Primary Mathematics 1B Workbook, p. 151


40 - 30 = 60 - 20 = 
49 - 30 = 62 - 20 =
30 - 10 = 80 - 40 =
36 - 10 = 83 - 40 =
50 - 40 = 90 - 80 =
57 - 40 = 95 - 80 =


OILF exercise for parents:  Enumerate the math skills that each assignment involves.

Extra Credit: For each assignment, from start (reading the directions) to finish (putting everything away), estimate the ratio of learning to time expenditure.

Tuesday, April 15, 2008

Inquiry-based science: more right-brained epiphanies

The new National Science Education Standards content guidelines unwittingly summarize everything that is wrong with today's grade school science programs:


Knowing scientific facts Understanding scientific concepts and developing
and information abilities of inquiry

Studying subject matter disciplines Learning subject matter disciplines in the context
(physical, life, earth sciences) for of inquiry,technology, science in personal and
their own sake social perspectives, and history
and nature of science

Separating science knowledge Integrating all aspects of science content
and science process

Covering many science topics Studying a few fundamental science concepts

Implementing inquiry as Implementing inquiry as instructional
a set of processes strategies, abilities, and ideas to be learned

Activities that demonstrate Activities that investigate and analyze science
and verify science content questions

Investigations confined to Investigations over extended periods of time
one class period

Process skills out of context Process skills in context

Emphasis on individual process skills Using multiple process skills--
such as observation or inference manipulation, cognitive, procedural

Getting an answer Using evidence and strategies for developing or
revising an explanation

Science as exploration and experiment Science as argument and explanation

Providing answers to questions Communicating science explanations
about science content

Individuals and groups of students Groups of students often analyzing and
analyzing and synthesizing data synthesizing data after defending conclusions
without defending a conclusion

Doing few investigations in order to Doing more investigations in order to develop
leave time to cover large understanding, ability, values of inquiry and
amounts of content knowledge of science content

Concluding inquiries with the result Applying the results of experiments to scientific
of the experiment arguments and explanations

Management of materials and equipment Management of ideas and information

Private communication of student ideas Public communication of student ideas
and conclusions to teacher and work to classmates

A sea change from scientific knowledge to vague entities like "concepts," "abilities of inquiry," "process," "procedure," "manipulation," and "investigations." A seismic shift from scientific breadth and depth to the personal, social, technological, and historical.  A quantum leap to the conclusion that scientific disciplines (physical, life, and earth sciences) aren't interesting "for their own sake."

What about those students who like learning physics, biology, or geology?

Other changes, shifts, & leaps: From solo work to group work. From experimentation, observation, and inference, to "argumentation" and "communicating explanations." From communicating answers privately to teachers to sharing them publicly with classmates.

What about shy students and others who don't work well in groups?

What about those who excel at empirical observations and scientific inference, but have trouble using enough Language Arts to explain things to their teacher's satisfaction?

Many people have asked why ever fewer Americans are pursuing careers in science. Blaming everything from Creationism to anti-nerd stereotypes (more on this in a later post), they've overlooked that which has gone furthest out of its way to repel our most promising future scientists.

Monday, April 14, 2008

Are all epiphanies right-brained?

Two movies released this week feature introverted, soul-deadened, betweeded professors who are as academically eggheaded, i.e., as left-brained, as they are clueless about all that right-brained stuff: human relationships and spirituality.

In Smart People, literature professor Dennis Quaid is upended by a visit from his brother and a fling with Sarah Jessica Parker. In The Visitor, a different sort of visit, along with romance and African drumming, transforms a professor of economics.

Naturally, both professors emerge from their shells more engaged with people, passionate about life, spiritually awakened, and happy.

In the stories we like best, it always seems to be a right-brained epiphany.

Imagine, instead, an outgoing African drummer whose life turns topsy-turvy when he takes an economics class and learns that what makes him truly happy is retreating to his study to analyze business cycles and market equilibria.

Saturday, April 12, 2008

The science we thneed

I've just received the Oral Presentation Rubric for my autistic son's 5th grade science assignment. For a top grade of 4, he must score high on Preparedness, Comprehension and Vocabulary, and:

--"Listen attentively" to other presentations and "not make distracting noises or movements."
--"Use several props (could include costume) that show considerable work/creativity and which make the presentation better."

The stakes are high: in the other Earth Day Unit assignments, he hasn't done so well. The one book they read, for example, completely flummoxed him. Here are his answers to the reading response questions:

1. Why did the Once-ler cut down the Truffula Trees?
To make thneeds which he... [illegible]
2. Why do the Brown Bar-ba-loots have to leave?
The Brown Bar-ba-loots are to leave because they didn't have food.
3. What kind of problems does the Thneed factory cause for the environment?  Name at least three.
The gunk he produced gets... [illegible]
4. What happens to the Onceler when there are no more Truffula trees?
He died.
5. What happened to the Lorax?
He [illegible]
6. What do YOU think the Lorax' message "UNLESS" means?
7. What could the Once-ler have done to minimize his factory's effect on the environment?
8. Is bigger always better?  Give an example to back up your opinion.
9.  A "Thneed" is defined as a fine thing that everyone THINKS they need (but probably really don't).  What are some other examples of thneeds - things we THINK we need but could do quite well without?
[5th grade science class, anyone?]

Fiction, character motivation, and open-ended questions aren't my autistic son's forte. Neither are vocabulary, communicating with classmates, listening attentively, and costume design.

But what about ecosystems, sustainability, photosynthesis/carbon absorption, greenhouse gases, and weather patterns?

Just possibly, science classes that favor such topics over fiction and creative performance would benefit not only autistic students and other left-brainers, but also--by raising the environmental awareness of all children--the very Earth we are soon to celebrate, and eventually to leave them in charge of.

Friday, April 11, 2008

NSTA's science classrooms: havens for unscientific assessment

After this week's Lorax Debacle, I met with my autistic son's science teacher to discuss his grades. She was quick to cite the new standards of the NSTA (National Science Teachers Association).

So I checked these out, and came across the following assessment guidelines:


Assessing what is easily measured Assessing what is most highly valued

Assessing discrete knowledge Assessing rich, well-structured knowledge

Assessing scientific knowledge Assessing scientific understanding and reasoning

Assessing to learn what students Assessing to learn what students do
do not know understand

Assessing only achievement Assessing achievement and opportunity to learn

End of term assessments by teachers Students engaged in ongoing assessment of
their work and that of others

Development of external assessments by Teachers involved in the development
measurement experts alone of external assessments

Along every dimension, a shift from objective and easily measurable to subjective and ill-defined. From "measured" to "valued"; from "discrete" to "rich, well-structured"; from "know" to "understand"; from achievement pure and simple to achievement plus "opportunity to learn"; from assessment by teachers to assessment by students; from external tests developed by disinterested parties to tests that include input from those (teachers) who are anything but.

Factor in the notoriously poor content-area training of most science teachers, and the NSTA's emphasis on...

The ability to inquire.
The ability to use science to make personal decisions and to take positions on societal issues.
The ability to communicate effectively about science.

..and you have a recipe for grades that may have less to do with a student's scientific achievement than with how well he communicates, gets along with classmates, "takes positions" that his teacher agrees with, and demonstrates curiosity during activities that all too often contain little of scientific interest.

Wednesday, April 9, 2008

Math lesson of the week: two approaches to the Quadratic Formula

1. McDougal, Littell's Algebra 1, (a traditional algebra text), p. 732 

Investigating the Quadratic Formula

Consider a general quadratic equation

ax2 + bx + c = 0      where  a ≠ 0

Perform the following steps.  Then describe how your result is related to the quadratic formula.

1. Subtract c from each side of the equation ax2 + bx + c = 0
2. Divide each side by a.
3. Add the square of half the coefficient of x to each side.
4. Write the left side as a perfect square.
5. Use a common denominator to express the right side as a single fraction.
6. Find the square root of each side.  Include ± on the right side.
7. Solve for x by subtracting the same term from each side.
8. Use a common denominator to express the right side as a single fraction.


2. The University of Chicago School Mathematics Project (UCSMP) Algebra: Integrated Mathematics, (a sequel to Everyday Math), p. 574-5 

Quadratic Formula
If  ax2 + bx + c = 0 and a ≠ 0, then
The Quadratic Formula is one of the most famous formulas in all of mathematics.  You should memorize it today. [Italics not mine].

Applying the Quadratic Formula

Example 1
Solve -x2 + 2x + 27 = 0 [reference to an earlier word problem]


Recall that-x2= -1x2. So rewrite the equation as
-1x2 + 2x + 27 = 0
Apply the Quadratic Formula with a = -1, b = 2, and 2 = 27.

[A demonstration of this, followed by similar problems for students.]


In short:

Derivation vs. memorization and plug-in.

In comparison with traditional approaches, UCSMP's Reform Mathematical "discovery" approach gives students much less practice factoring and finding perfect squares and common denominators.  

But unless you've mastered these things, you probably won't be able derive the Quadratic Formula. 

To use Reform Math's lingo, instead of "discovering" it, "authorities" must "spoon feed" it to you--a tactic that Reform Math's supporters purport to abhor.

Tuesday, April 8, 2008

Sugar brings out our inner left-brainer

A recent Economist article reports on a study concluding that sugar aids rational thought. Subjects who drank sugar-sweetened lemonade after a mentally taxing activity were more likely, once the glucose reached their blood, to make choices based on reason. Those who instead drank artificially sweetened lemonade more frequently succumbed to faulty intuitions.

Perhaps all that sugar our children ingest isn't for naught.  If only our schools were better at harnessing it.

Monday, April 7, 2008

What do you speak for?: Language Arts invades Science

My autistic son, who for years has been conducting his own experiments and figuring out how things work, and who can rig the internal lever in our new thermostat to trick the furnace into running perpetually, just brought home the following assignment for 5th grade science class:

The Lorax spoke for the trees, "for the trees have no tongues." What would you choose to speak for, and what would you say?  Write a one-minute talk on behalf of something, which cannot speak for itself.

The essay should include at least 5 important facts about the thing you are speaking for.  You need to write it as a persuasive essay so others will be persuaded to do something to help. 

The essay should be at least 5 paragraphs.

My son has difficulty writing 5 paragraph essays to persuade people of things. He also has difficulty speaking for other people, let alone for rain forests, spotted owls, and unspoiled beach fronts.  

Might this have something to do with why he's in danger of failing science?

Sunday, April 6, 2008

Why left-brainers depend more than others on high grades

You're a shy, introverted, and/or socially awkward child. You're also very analytical, excelling in math, science, foreign language, and expository writing. You are, in other words, a classic left-brainer. And so, as a grade school student, where are you going to shine?

Not in the schoolyard, where you shy away from peers or are teased by them.

Not in the classroom, where you're intimidated by discussions or have trouble following them and figuring out when to join in.

Not in your school-based extracurriculars, where leadership roles don't suit you, and many of which are uncomfortably social.

Rather, above all else, it's grades. More than for most of your classmates, your self-worth depends on grades that, as grades once did, reflect the extent of your strong analytical, mathematical, scientific, and linguistic skills.

Later on, there are your prospects for admission to colleges that challenge and nurture these skills. Increasingly, as Malcolm Gladwell observes in a recent New Yorker article, competitive colleges are placing less weight on SAT scores and more on school-based extracurriculars and leadership roles. Increasingly, they are shying away from those who seem shy or socially awkward when interviewed. 

In such a climate, your high grades are your trump card.

But, as I've explored in earlier posts (one, two, three), grade schools are increasingly reluctant to deal this hand, reserving top grades for the gregarious enthusiasts of today's right-brain classrooms.

Friday, April 4, 2008

Why should grades favor left-brain students?

As I've discussed in three earlier posts (one, two, three), low academic standards combined with high standards for sociability and for "normal" development (in penmanship, organization, and paying attention) cause today's left-brainers to get lower grades than many of their less academically skilled classmates.

So what? Experts like Howard Gardner (of Multiple Intelligences fame) say schools have long been biased towards such left-brain skills as math, logic, and verbal reasoning. Perhaps now they're finally righting the balance.


1. School should be about skills that require explicit teaching--e.g., how to write coherently on the causes of the Civil War. It shouldn't be about skills that children normally develop on their own, at their own rates, without classroom intervention. Except for those who are way behind schedule, there's little reason for classroom instruction and classroom activities on how to interact appropriately, listen attentively, transition between tasks, and stay organized.

2. School should be about skills that children don't have better opportunities to learn elsewhere: e.g. how to read and multiply, as opposed to how to get along with people and dance to music. Reading, multiplying, and academics in general just happen to involve lots of left-brain thinking.

3. Teachers shouldn't grade students on developmental skills like cooperating in groups, but on what they've learn specifically because of going to school.

4. All this means that grades should mostly assess left-brain skills and, as a result, end up favoring left-brain learners.

5. Finally, as we'll see in my next post, left-brainers depend much more on good grades than other students do.

Wednesday, April 2, 2008

Math problems of the week: grade 2 Trailblazers vs. Singapore Math

Today we'll see how another Reform Math program, Trailblazers, stacks up with Singapore Math.

A.  An exercise from half-way through Trailblazers Grade 2 (Student Guide, Book 2, p. 293).

Which Answer Makes Sense
Look at each problem.  Decide which number is the best estimate of the correct answer. Explain why you think so in the space below the problem.

 60 45 30 50
 50 30 40 15
  50 30 70 20
 94 50 40 75
 71 40 50 60

B. An exercise from halfway through Singapore Math Grade 2 (Workbook 2B, p. 9)

Write the missing numbers.
(a) 99 + __ = 100 (b) 95 + __ = 100
(c) 96 + __ = 100 (d) 91 + __ = 100
(e) 80 + __ = 100 (f) 35 + __ = 100
(g) 84 + __ = 100 (h) 63 + __ = 100
(i) 42 + __ = 100 (j) 58 + __ = 100
(k) 6 + __  = 100 (l)  9 + __ = 100

In short, estimation and verbal explanation vs. systematic practice calculating precise answers to more difficult problems.

Tuesday, April 1, 2008

Why Left-Brainers Don't Get High Grades, Part III: Motivation, Sociability, and Graphic Arts

As discussed in yesterday's post, today's grade school students are graded less for their skill level, and more on how well they work in class, their oral responses, their journal entries, and the work they produce.  

New report card guidelines encourage teachers to de-emphasize tests, and to forgo short-answer tests altogether, even for math. In the words of the Christina School District guidelines:

It is not recommended that the math teacher rely on multiple choice, fill-in-the-blank tests or tests with bare math facts. Tests that are better indicators of student learning may include:
• Descriptions (Describe the problem and the process used to solve it.) 
• Explanations (Explain how or why a process works to solve a problem.) 
• Graphing (Make a bar or line graph to show data collected.) 
• Drawing and Labeling (Draw a solution and label the parts.) 
• Performance Assessments (Demonstrate how you would set up an investigation, gather data, graph and analyze data to arrive at a reasonable and accurate conclusion.)

What about homework? Some report card guidelines recommend against factoring into grades those assignments where parents might have helped out. In the words of the Christina Guidelines: “Projects that are done at home with the help of parents should not be given a score that is reflected on the report card.” 

All this makes grades primarily a function of what students produce during class time, and how they behave during classroom activities. Which, in turn, is how many smart left-brainers, even if they do their best to play by the new rules, fall short of top grades.  

First, shyness or lack of sociability can make it hard for them to jump into class discussions and cooperate "appropriately" in group activities.

Second, the swift, subconscious calculations and sophisticated mathematical strategies of the left-brain math buff, however strong his or her verbal skills, are difficult, if not impossible, to explain to others.

Third, many of them have skewed developmental profiles, with analytical skills far exceeding not just social skills, but the penmanship, organization, and graphic arts skills that figure prominently in today's classroom activities and testing "rubrics" (see above).

Fourth, with their linear, one-thing-at-a-time minds, many of them find the sensory clutter, multidisciplinary breadth, and multi-step directions of today's hands-on activities confusing, overwhelming, and disengaging. They have trouble maintaining attention, making transitions, and motivating themselves to be motivated.  Without attention, motivation, and the ability to move on, their performance falters.

Finally, overwhelmed and under-motivated, even the most diligent of them struggle to go the extra mile that top grades now require: e.g., in the words of the Christina Guidelines, to "consistently extend mathematical concepts and make new connections beyond grade level expectations", and "integrat[e] connections,"  Or, in the words of the Philadelphia School District, to "demonstrate[...] superior understanding of concepts, skills, and strategies," and the "ability to apply concepts and extend learning."

Not that left-brainers aren't constantly extending concepts and making new connections.  It's just that they do so in highly abstract and analytical ways that most teachers don't recognize, particularly when adhering to today's "rubrics," "best practices," and "formative assessments."

Top grades, it would seem, are reserved for the eager-beaver who either relishes the right-brained classroom activities, or has enough social savvy to know how to act as if she does.