Math problems of the week: 3rd grade Investigations vs. Singapore Math

1. An assignment from 3rd grade (TERC) Investigations, given in late April:

2. From the middle of the first half of the 3rd grade Singapore Math curriculum:

3. Extra Credit: With reference to Investigations problem 3 above, comment on the role of parents in math instruction.

Academic well-roundedness, fiction, and autism

Recently, I've watched my 13-year-old autistic son struggle through a couple of 4th-grade-reading-level novels, getting almost nothing out of them.  This has made me wonder how reasonable it is to insist, as J's teachers do, that someone like him read fiction.

Here's a child who readily reads technical manuals, does pretty well with grade-level science texts and with 4th-grade-level history books, but who misses most of what matters in all but the most simple, basic, character-driven fiction.  

After all, he's autistic.  Empathy, perspective-taking, social reasoning, socio-cultural background knowledge--all of these to some extent elude him.  So do character's motivations, author's intent, theme, tone, and the emotional effects of literary devices.

On the one hand, carefully chosen fiction might help him with his social reasoning skills by giving him opportunities to see characters interacting.  To this end, however, I generally prefer movies and TV shows: audio-video venues capture many more of the cues of real-life social interaction than does printed text (though we keep this channel open as well, with captions turned on for extra feedback).  

But if the goal is the "well-rounded" liberal arts education that comes from appreciating literature, I'm not sure it's a realistic one when it comes to those children whose social deficits are as extreme as J's.  I'm all in favor of a well-rounded education for most kids, and wary of underestimating potential and prematurely shutting off opportunities for academic development.  But it seems to me that, for kids like my son, by the time they reach middle school, the benefits of eliminating literature from their academic portfolios may well outweigh the costs. In particular, it would free up time for activities specifically targeted to improve social skills.

GrammarTrainer demo debugged

An exception within an exception within an exception was causing problems with certain exercises involving optional articles when certain text inputting options are chosen. Fascinating stuff, I realize. (It reminds me of learning the endings for Russian numerals between 10 and 20 when they modify animate nouns in the prepositional case).

Now it's working:  here (it still takes a few moments to load)

I had to dig pretty deep to locate the cause; it took under 20 seconds to type the single line of code necessary to fix it. 

Here's where it's useful to be "perseverative," "detail-focused," and "insistent on sameness" (or, at least, on consistency)--all symptoms of Asperger's Syndrome. Or at least of extreme "left-brainedness."

The right-brained computer science fair

Dissatisfaction with the results of programming contest events has led to the development of a new type of competitive event, a computer science fair. The computer science fair, a combination art show/science fair, is designed to attract a wider diversity of participants, to encourage creativity, and to reinforce good project development practices. Students are asked to creatively use technology to express themselves, to design new inventions, and to solve problems by submitting projects in the following categories: 1) computers and the arts and humanities --- computer music, computer art, multimedia projects; 2) computer programs --- entertainment, education, scientific, business, modeling and simulation; and 3) computer or electronically controlled inventions.

--Sue Fitzgerald and Mary Lou Hines, The computer science fair: an alternative to the computer programming contest.

In the case of the Philadelphia Computer Science Fair, categories now include:

Graphic Design
Desktop Publishing
Multimedia

But what about the left-brained computer programmer, the linear coder who can do pointers and recursion but hasn't a clue about graphic design? Is there any way left the best of our true programmers--the kind who write the underlying code for all those "higher level" programs on which non-programmers depend--to distinguish themselves?

Or is the "dissatisfaction with the results of programming contest events" cited above one more instance of revenge against the nerds?

Math problems of the week: 5th grade Everyday Math vs. Singapore Math

Final graphing assignments:

Extra Credit: Based on what you know about Singaporean vs. American cultures, match these assignments to their respective cultures of origin.

No Child Left Behind(?) Achievement Tests: What about the brightest kid in the class?

Now that this year's Pennsylvania State, No Child Left Behind-mandated achievement testing has drawn to a close, it's time to assess the assessments. How has the standardized testing of grade schoolers changed over the years? And how do these changes affect the most academically gifted of our students?

A generation ago the standard test was the Iowa Achievement Test. With No Child Left Behind, the stakes are much higher... and the tests much easier. This is because No Child Left Behind requires states to create tests to measure student achievement and to impose consequences on schools whose aggregate scores fall shy of the state-determined standards. The easiest way for a state deal with these requirements is to create tests with low enough standards that few of its schools will fail to meet them.

This creates problems not just for students in underperforming schools, but for students in general, including the academically gifted. The pressure to make adequate yearly progress works its way down from states to individual schools and their principals and teachers, causing the latter to spend much of their class time teaching to the test. 

Teaching to low-level tests lowers the academic bar for everyone. What's the best route to adequate yearly progress? Minimizing the number of students who score "below basic" and maximizing the number of students who score "advanced." There's no time left, or any incentive, for addressing the needs of students whose skills already exceed the skills assessed on the tests.

Making matters worse, some school districts use individual test scores to screen applicants for admission to the best middle and high schools. In Philadelphia, for example, if you are below a certain percentile (85th or higher, depending on the school) you are automatically disqualified from special-admit schools, no matter your grades, teacher recommendations, or other qualifications. Since Philadelphia’s nonselective middle schools and high schools range from bad to terrible, the stakes are extremely high.

Here's where the replacement of the Iowa Achievement Test with the various lower-level state achievement tests becomes significant. Back in the days of Iowa, you could be in 7th grade and test at an 11th grade math or reading level. Now the best you can do is end up in the 99th percentile of your grade-level peers. 

Consider how all this affects the prospects of a particular sort of bright student: one who turns off his or her brain when faced with easy problems and doesn't check his or her work. Back in the days of Iowa, however many careless mistakes such a child might make on grade-level questions, he or she could more than make up for this with correct answers to much harder, above-grade-level questions. In the age of No Child Left Behind, this is no longer possible. Today's low-bar test don't allow this child any way to "redeem" careless mistakes on easy questions. The statistical “noise” created by such mistakes in a testing instrument that sets too low a ceiling is simply too great to properly distinguish the brightest students from their peers.

The result? Some of these students may find themselves not only unable to attend the most academically engaging of our public high schools, but, like altogether too many of their peers, faced with totally unacceptable alternatives.

Autism Diaries XVI: Letters to the principal

All profanities elided and identifiers removed.

Instigating event:

Dear school district of X,

Ms. X is a very terrible principal. She took the recess away for rest of a year. She's an a**hole. Her mother is a b****.

Apology letter, first draft:

Dear Ms. X,

I am sorry for calling you son of the b****. You can't just take away recess for rest of the year. It is terrible terrible terrible thing that you ever did. I'll tell the school district. If you don't let me have recess, I'll have recess myself. I will stop saying a word against other people's mother. You will be in trouble with the school district.

Second draft:

Dear Ms. X,

 I am sorry for calling you son of the b****. I will stop saying a word against other people's mother. I am worried that I might get suspended or expelled. I do not want any after school detention. I will stop writing an angry letter.

Final draft:

Dear Ms. X,

I am sorry for using bad words against you and your mother. I know that it's wrong to hurt other people with words. I will stop writing mean letters.

Math problems of the week: 3rd grade Investigations vs. Singapore Math

1. The third of the three investigations in the 3rd grade Investigations Landmarks in the Hundreds booklet:

2. The final exercise in the first chapter of the 3rd grade Singapore Math Primary Mathematics 3A workbook, "Numbers to 10,000":

3. Extra Credit: Estimate the ratio of effort to learning in each assignment.

Reading comprehension and Theory of Mind

An article in last week's New York Times reports that literary scholars are following on the heels of economists, historians, political scientists, and art critics, and venturing into cognitive science:

Literature, like other fields including history and political science, has looked to the technology of brain imaging and the principles of evolution to provide empirical evidence for unprovable theories.

Of particular interest to Lisa Zunshine, a professor of English at the University of Kentucky, whose specialty is 18th-century British literature, is Theory of Mind and reading comprehension. 

Ms. Zunshine is particularly interested in what cognitive scientists call the theory of mind, which involves one person’s ability to interpret another person’s mental state and to pinpoint the source of a particular piece of information in order to assess its validity.

Especially challenging in this department are the novels of Jane Austen:

Jane Austen’s novels are frequently constructed around mistaken interpretations. In “Emma” the eponymous heroine assumes Mr. Elton’s attentions signal a romantic interest in her friend Harriet, though he is actually intent on marrying Emma. She similarly misinterprets the behavior of Frank Churchill and Mr. Knightly, and misses the true objects of their affections.

Humans can comfortably keep track of three different mental states at a time, Ms. Zunshine said. For example, the proposition “Peter said that Paul believed that Mary liked chocolate” is not too hard to follow. Add a fourth level, though, and it’s suddenly more difficult. And experiments have shown that at the fifth level understanding drops off by 60 percent, Ms. Zunshine said. Modernist authors like Virginia Woolf are especially challenging because she asks readers to keep up with six different mental states, or what the scholars call levels of intentionality.

As for Ms. Zunshine's research:

Ms. Zunshine is part of a research team composed of literary scholars and cognitive psychologists who are using snapshots of the brain at work to explore the mechanics of reading. The project, funded by the Teagle Foundation and hosted by the Haskins Laboratory in New Haven, is aimed at improving college-level reading skills.

“We begin by assuming that there is a difference between the kind of reading that people do when they read Marcel Proust or Henry James and a newspaper, that there is a value added cognitively when we read complex literary texts,” said Michael Holquist, professor emeritus of comparative literature at Yale, who is leading the project.

The team spent nearly a year figuring how one might test for complexity. What they came up with was mind reading — or how well an individual is able to track multiple sources. The pilot study, which he hopes will start later this spring, will involve 12 subjects. “Each will be put into the magnet” — an M.R.I. machine — “and given a set of texts of graduated complexity depending on the difficulty of source monitoring and we’ll watch what happens in the brain,” Mr. Holquist explained.

It would be particularly interesting to look for individual differences in reading ability.  I've long suspected that those I call "left-brainers"--Simon Baron-Cohen's "systemizers"--will find books (or plays) that focus on the diverse perspectives of multiple interacting characters much more difficult to follow than books structured around a single argument (however complex), while for "right-brainers"--Simon Baron-Cohen's "empathizers," it's the other way around.

That's certainly true of me.  I can sail through (for example) Climbing Mount Improbable and How the Mind Works, while Arcadia and The Smartest Guys in the Room (most recently) have so totally overwhelmed me that I've had to put them down.

More media praise for hands-on science

A front-page article in this weekend's Philadelphia Inquirer Report Card on the Schools contains yet another rhapsody about the virtues of the hands-on science classroom:

In the new science classroom, teachers still offer instruction, but in smaller doses. Lab work and treks outdoors are now integral to elementary and middle school instruction.

"We're engaging kids more in the process of science, not just the content," said Paul Joyce, science supervisor in the West Chester district.

This is learning science by doing science, a change in instruction that is costing schools time and money, yet is fast gaining traction as educators heed warnings that the economic health of the region - and the nation - demands a science- and tech-savvy workforce.

Too bad no one is listening to Dan Willingham, a cognitive scientist specializing in education who argues that there's an important difference between the brains of experts (e.g., scientists) and novices (e.g., k12 students) and that k12 science education should focus not on creating and testing theories of natural phenomena, but on the comprehension of scientific knowledge.

In Willingham's words:

Does this mean we shouldn’t ask students to... conduct a scientific experiment? Of course not. But we should understand the difference between the thought processes of experts and novices.

Instead of heeding Willingham's advice--advice that is backed by scientific experiments constructed by experts--more and more schools are sacrificing the comprehension of knowledge imparted by experts for science experiments constructed by novices.

The Inquirer article goes on to state that schools are giving high priority to math education:

At the high school level, nearly a third of the 160 public high schools in South Jersey and suburban Philadelphia now require four years of college-prep science or math - or both - for graduation. That's one year more than the states mandate.

And large percentages of students in many schools are entering ninth grade already having taken Algebra I.

Unfortunately, the predominance of Cookbook Math (with all its guessing and checking and plugging in numbers rather than understanding concepts, manipulating symbols, and proving theorems) and Reform Math (with its watered-down, feel-good curriculum) means that math in general, and Algebra I in particular, are no longer what they once were. Regardless of the names we assign to the courses students take, our average high school graduate has learned far less actual math than her counterpart a generation ago.

Indeed, lurking in the Inquirer's fine print is a less rosy picture:

While there is progress, there is far to go: Results of the 2009 science assessments in Pennsylvania and New Jersey show high achievement in fourth grade but numerous low scorers in eighth and 11th grades, even in some of the region's top-performing schools.

What do the 4th grade results matter if the gains are lost by 11th grade?

One could argue that this latest generation of budding, hands-on "scientists" hasn't yet reached the 8th or 11th grades, when they will surely outperform today's middle and high school students.  But consider Philadelphia's flagship high school for hands-on science.  The Science and Leadership Academy is a project-oriented school affiliated with the Franklin Institute (erstwhile The Franklin Institute Science Museum) that gives every student a laptop computer and attracts some of the brightest students in the district. Last year there were about 10 applicants for every slot.  

And according to the Report Card on the Schools, the Science and Leadership Academy's 11th grade science scores place it in the 3rd quintile of Philadelphia public high schools, with only 15% scoring "advanced,"  and 59% "below basic."

Robotics documentary

Filmmaker Joey Daoud recently wrote me to tell me about a documentary he's been working on that follows high school robotics teams building combat robots for the National BotsIQ Championship. He is currently trying to raise funds to film the championship through a Kickstarter project.

His goal is "to be entertaining, yet motivational to students to get them involved in science and engineering (STEM) programs."

Mr. Daoud is also working on an educational video on getting students (especially girls) involved in STEM programs. "I'd love to get input on from teachers to shape it so it works best for their needs," he writes.

More information on these projects (and how to contribute to them) can be found here.

Math problems of the week: traditional algebra vs. CPM

1. The final factoring problems in McDougel Litell's Algebra I, a contemporary, traditional algebra text (p 773):

x² + 6x + 8

x² - 24x - 112

3x² + 17x - 6

4x² + 12x + 9

x² + 10x + 25

x² - 14x + 49

2. The only factoring problems in all of College Preparatory Math's Foundations for Algebra Year 2, a Reform Math algebra text (p. 159; p. 195):

2x + 20

5x + 35

3x - 15

5x - 60

8x + 12

9x + 6

-2x - 4

3. Extra Credit:

Why is factoring second degree expressions more traditional than only factoring first degree expressions?

New GrammarTrainer website: grammar software for autistic spectrum children

I'm officially divorced! From my erstwhile software vendor, that is.  And am now selling my grammar software directly from my website. Like a fresh divorcée, I'm sporting a whole new look.  Check it out at:

Autism Language Therapies

For the uninitiated, here's a quick description of GrammarTrainer:  

GrammarTrainer, in contrast to all other language teaching software, has users actively constructing phrases and sentences via interactive linguistic feedback.  It consists of four levels and over 100 lessons. The curriculum begins with simple phrases (e.g., "a circle," "under the rectangle"), gradually progressing to increasingly complex sentences (e.g., "the girls are giving the cookies to the boys";"the lower the sun is, the longer the shadow is"; "He said that you need to help her tie her shoes"). Topics include simple and complex verb tenses, a variety of subordinate clauses, and the pragmatics of pronouns.

Demos are available here (allow a few moments to load):

Level I

Level II

Level III

Level IV

Parent testimonials and preliminary data suggest that GrammarTrainer may significantly improve the basic grammar skills of language-delayed children on the autistic spectrum.  An efficacy study is currently in progress.

April 1st Times article on The Blue School

Last week, an April 1st front page New York Times article showcased an alternative elementary school in Manhattan's East Village that now requires an intelligence test (the E.R.B.) as part of the application test, charges $28,000 tuition for kindergarten, and has six families applying for each available slot in the school's preschool.  The school, reports the Times, is led by "the founders and spouses of the Blue Man Group, an alternative theater troupe" who

were attracted to the Reggio Emilia philosophy of learning, which integrates what children want to learn with what teachers and parents want to impart. To that end, [the school] has two “provocateurs” on the staff whose job is to inspire different “threads of learning,” planting ideas and building areas of study around the ones that take.

Tuition and intelligence testing aside, the school, reports the Times

has remained true to its progressive roots, with “imagination stations” and “glow time.” Children help direct the curriculum, and social and emotional skills are given equal weight to reading and math.

...

After a teaching candidate read “The Great Kapok Tree” to a class of first graders, they took interest in the rainforest. A provocateur built the classroom into a rainforest, replete with a kapok tree whose (plastic) leaves cover the ceiling. The children have studied the animals that live in the rainforest and are now exploring whether the Littles, characters from another story they read, might live in the kapok tree. They write letters to the Littles and even create math tests for them.

Once a week, 4-, 5- and 6-year-olds mix together at imagination stations grouped around subjects like language arts, music and movement. On a recent morning, one group was stationed in the hallway singing “Frère Jacques” in Mandarin; in a classroom, another group was imitating the sound of rain on a giant drum (it was pouring outside). When the children got overly excited by their Blue Man Group-style drumming, they calmed down with breathing exercises.

Another Blue Man inspiration is glow time, when natural or incandescent light is replaced by black light and children transform their environment using props like shaving cream, play dough or glow-in-the-dark blocks to study things like light, shadow and outer space.

Provocateurs and glow stations. Letters and math tests for The Littles. Anyone who spends any time visiting or reading about elite and model elementary schools should be forgiven for mistaking this article for an April Fool's day parody of current practices. 

But what's really striking is how feebly it justifies the new admissions criteria, in contrast to its justification for the tuition hikes (the burdens of Manhattan real estate). The justification for requiring an intelligence as part of admissions?

The founders said that since many of their applicants were also trying to get into schools that required the E.R.B. test, it did not seem like too much of an imposition, because the test result can be sent to multiple schools. 

and

The school, like many others, says that it does not require a minimum E.R.B. score, and that the test is one of many tools used in admissions.

and

“It gives us one piece of the child, which is valuable,” said Renee Rolleri, another founder, “and we can better respond to their academic strengths and areas to work on.”

Hmm... Isn't intelligence testing counter to progressive education in general, and The Blue School's philosophy in particular? Ms. Rolleri's implication, of course, is that the E.R.B. scores are being used to inform teaching, not to screen applicants.  But if this were really the case, wouldn't she say so more explicitly? And wouldn't the test then be administered once students start school, rather than the year before?

The other striking thing is how eager families are to apply to this school, and how unconcerned they are about its current lack of track record:

Despite the fact that it has no permanent real estate, limited financial aid and no track record of placing students in top schools, six families applied for each available slot in next fall’s preschool class.
...

Many parents are clearly unfazed by the school’s requirements and absence of a track record.

“This school was our top choice; we wanted to be a part of it,” said Marah Anderson, whose son Finnegan is in kindergarten. Ms. Anderson said that her son was accepted to two other schools, but that she preferred the holistic approach of the Blue School. “There is a lot of research that supports this approach to learning,” she said. “It’s just not as widely implemented.”

But perhaps these two striking facts are related.  There isn't a lot of research that supports this approach to teaching, but there is a lot of research showing that standardized test scores predict academic success.  If the school is going to avoid the most effective teaching strategies, then, to attract parents like Ms. Anderson, it had better make sure the students it admits are likely to succeed academically in spite of this.

Phonics vs. "Balanced Literacy": the Hebrew experiment

Learning how to break the code of written language is one area in which all the evidence shows that the left-brained/analytic/phonics approach trumps the right-brain/holistic/Balanced Literacy approach--for all types of students.

The most recent example of how ineffective the Balanced Literacy approach is can be seen on the following Youtube video, which was recently posted on kitchentablemath:

In the ensuing discussion thread, I had the following idea for an experiment.

The idea is to take American adults and turn them back into novice readers via a not-too-complicated phonetic writing system consisting of sound-to-symbol correspondences with which they are totally unfamiliar. The Arabic and Hindu writing systems strike me as too complicated; the Greek and Cyrillic systems involve too many familiar symbols and sound-symbol correspondences.

But you'd have to do this without introducing the added challenge of learning another language.

So what you do is you transliterate a bunch of simple English words--of the sort you'd start beginning readers on ("cat", "mat", "can", "pan", "and", "in", "on", etc.)--into Hebrew letters.

Then you teach American adults who don't already know how to read written Hebrew how to read this list of transliterated words. Group A learns via X hours of phonics-based lessons; Group B via X hours of balanced literacy. Then everyone gets a post-test in which they have to read a list of Hebrew transliterations of the words in question.

Of course, these adult subjects aren't completely comparable to American children learning to read for the first time. On the one hand, they already know how to connect phonemes together into syllables (p-e-n -> pen). On the other hand, they haven't had the kind of prior exposure to the Hebrew alphabet that beginning readers usually have had to their native alphabets by the time they start to learn to read. On the third hand they are adults, not children.

However, I still think that this experiment would be quite revealing, both in the likely results it would produce, and in how it would remind those who need reminding of (1) what it's like to learn how to read, and (2) how impractical it is to memorize written words as graphical wholes.

Math problems of the week: 3rd grade assessment, Pennsylvania vs. Singapore

In honor of this year's pending Pennsylvania State Assessment Tests (to be administered this month): I. The sample measurement subtest in Pennsylvania Test Prep Grade 3, proclaimed as "aligned to state and national standards" (p. 110):

2. The Singapore Math Placement Test for students at the end of 3rd grade:

3. Extra Credit: Below are the Mathematical Reasoning and Connections standards from the Pennsylvania State Standards, excerpted from Pennsylvania Test Prep Grade 3 (p. 107). Based on the Singapore Placement test above, extrapolate the corresponding Standards for Singaporean 3rd graders.