One of my areas of educational interest is talent in math and science--a common characteristic of many of those I call “left brainers.” This interest has me lurking on several gifted education lists. One of these lists has been dominated in recent weeks by parents arguing over what “giftedness” means. I’m guessing that many people would prefer whatever definition includes their own kids and excludes as many others as possible. Perhaps there are some philosophical issues as well.

Curiosity piqued, I spend some time on Google Scholar and consulted with my extremely intelligent and reliable collaborator, who holds a Ph.D. in cognitive psychology, and learned that there is no well-defined construct for giftedness. Instead all we have are:

1. Arbitrary IQ cutoffs (the legal definition in many states)

2. Arbitrary percentage cutoffs (typically, somewhere from 3 to 5% of the school-aged population)

3. “Divergent thinking” tests

4. Arbitrary definitions by supposed experts

Criterion 1 rules out anyone with uneven intellectual skills--the great writer with dyscalculia; the Aspie math whizz with working memory problems or below-average verbal skills.

Criterion 2 rules out the possibility that a much larger percentage of kids are gifted in one way or another and/or are significantly under-challenged by today’s dumbed down curricula.

As for Criterion 3, “divergent thinking tests”, these often involve open-ended tasks that are difficult to score objectively, and that typically are verbal/informational in nature (“name as many uses for a fork as possible”), or are visual in nature, and thus unlikely to correlate with, say, mathematical or musical creativity. In fact, my expert psychologist tells me that the only thing that divergent thinking tests reliably predict is how well you’ll do on similar divergent thinking tests.

Perhaps most insidious is Criterion 4, arbitrary definitions by supposed experts. I’ve heard some people cite these as definitive, and others breeze through the often fuzzy but mystical-sounding criteria to the breathless conclusion that their children must be gifted.

Consider, for example these oft-cited criteria, reported, for example, at Education.com:

A bright child: Knows the answers

A gifted child: Asks the questions

A bright child: Is interested

A gifted child: Is very curious

A bright child: Pays attention

A gifted child: Gets involved mentally and physically

A bright child: Works hard

A gifted child: Can be inattentive and still get good grades and test scores

A bright child: Answers the questions

A gifted child: Questions the answers

A bright child: Enjoys same-age peers

A gifted child: Prefers adults and older children

A bright child: Learns easily

A gifted child: Often already knows the answers

A bright child: Is self-satisfied (when gets right answer)

A gifted child: Is highly self-critical (perfectionists)

A bright child: Is good at memorizing

A gifted child: Is good at guessing

I asked my psychology expert, and there is no rigorous test out there that measures “curiosity.” Beyond that, these highly subjective standards lend themselves quite readily to being one more way to shortchange the shy, diligent, introverted, and/or often bored-because-under-challenged left-brainer or child with Asperger’s in favor of his or her more extraverted, teacher-pleasing counterparts.

What, then, is the best way to measure giftedness? Recognize that it is domain specific, and look at a child’s performance in particular domains. And as for what to do with that information, the only thing relevant to K12 education is *to make sure that everyone is appropriately challenged in all subjects*.

## Monday, April 30, 2012

### Red herring du jour: defining giftedness

## Saturday, April 28, 2012

### Letter from Huck Finn: The Barber and the Grandpa

*Out in Left Field proudly presents the eighth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.*

**The Barber and the Grandpa**

One thing I imagined I would do as a student teacher would be to ask my students “Who can show this problem who’s boss?” whenever I wrote a problem on the board. Students would of course rise to the occasion. Nothing remotely like that ever happened. What did happen varied, but one event stands out. It occurred the Friday of the second week that my teacher was gone.

We had just covered squares, square roots and irrational numbers. I had shown with the aid of a spreadsheet how we can close in on the square root of 2 through successive approximations. I tried to get across that some numbers can be expressed as a ratio of two integers and others cannot. That idea remained an abstraction. My students could say that square root of 5 is irrational because it isn’t a perfect square, but that was it. (I myself didn’t really grasp this until high school.)

I ended the topic in my honors pre-algebra class saying “I know that the concept of rational and irrational numbers is abstract, but it’s a foundation of what we call the ‘real numbers’. If you study mathematics, you will see this concept many times.”

To which a chubby, jovial boy named Arturo replied: “I’m going to be a barber.”

I didn’t know what to say, so I said nothing. This was at the end of the second week, as I say. A long week. On Monday of that week I announced the news to the class that their teacher’s father passed away over the weekend. I said she would be gone all week, but back next Monday. This was also parent teacher conference week. Our classes were shortened, so students only had half a day. The afternoon was devoted to the conferences. “I will be available to meet with your parents, and Mr. Ortega will be with me as well.” (Jaime Ortega was my sub while my teacher was gone.)

In all my classes, I circulated a sympathy card in which I had placed an accordion style insert of extra paper so everyone could sign. Marta in first period told me “That’s too bad her father died. I lost my grandpa last year; I was very sad.”

“I’m sorry to hear that,” I said.

Rebecca, who sat next to Marta said “My grandpa died too. A few months ago.”

“I’m sorry,” I said. She nodded.

Given these conversations, I was surprised when the first person to talk to me at the parent teacher conference introduced himself with the words: “I am the grandpa.” He was a distinguished looking man, about my age. He was with his granddaughter, Gabby, a girl in the honors pre-algebra class. She had her two year old brother in her arms. “This is Louie,” she said.

I explained to the grandpa the situation with my teacher but that I could talk to him about his daughter. He spoke in Spanish from then on, and Jaime translated. He wanted to know how Gabby was doing. I looked up her class grade; she was doing fine. “Does she talk a lot?” he asked. I told him that the people she sat with were often noisy and she sometimes joined in. All in all, she was fine. In fact, she was a delight. He looked stern and said something to her in Spanish and she nodded. He thanked me, shook my hand, and she said “Goodbye, Mr. Finn!”

The conferences were set up in the cafeteria. We were stationed at long tables with name cards. Families, often with little ones in tow, roamed around the room as if it were a train station. Not many parents came by. Those who did were concerned about their child’s grades. I told them I would alert Mrs. Stevens when she returned, but also said that Mrs. Stevens and I were always available during lunch to tutor. And on Wednesdays after school, Mrs. Stevens also tutors—there are late buses. Very few students took advantage of this.

After I had talked with Gabby’s grandfather Jaime told me “A lot of times the grandparents or other relatives are taking care of the kids.”

“The parents are working in the fields?” I asked.

“Many of them, yes, but you have some kids who have a father in jail and sometimes the mother leaves. So they live with a relative or a guardian.”

By the end of the week, I had found out that Jaime’s parents worked in the fields, that he lived with them, that he had been a substitute for two years and there were few openings in the schools. He worked weekends at a pizzeria. In the summers, he helped his parents in the fields. And on Friday, it was his birthday.

“Happy birthday, Jaime!”

“Yes. I woke up today and looked in the mirror and said ‘today I’m thirty’.”

“Pretty young by my standards,” I said.

“Yeah, maybe. For Mexicans it is not. In Mexico, if you’re 30, you’re married with kids.”

The room was getting quieter; there were fewer families on that last day.

“I think people take a lot for granted here,” he said. “A lot of kids think here think because they’re not in Mexico it’s going to be easier. I tell the students, it’s about choices. You can work hard and try to do something with your life, or you can work in the fields all your life.”

“Or you can be a barber,” I said.

“Yes,” he said and smiled. “You can be a barber.” He looked at his watch. “Well, there’s not much going on, so I’m going to get something to eat.” I nodded.

“I guess I won’t see you anymore after this,” I said. “I enjoyed working with you.”

“Same here. I sub a lot here so you’ll see me.”

“I hope so,” I said and we shook hands.

I left shortly after. On the drive home I passed by many strawberry fields and wondered which ones Jaime had worked in, or my students’ parents. When there were no more strawberry fields to see, I started wondering what it would be like to have Tina back in the classroom once again.

## Thursday, April 26, 2012

### Math problems of the week: 3rd grade 1920s math vs. Trailblazers

**I. The second to last page of the 3rd grade section of Hamilton's Essentials of Arithmetic, First Book**, p. 123 (published in 1923) [click to enlarge]:

**II. The second to last page of the 3rd grade**, p. 326 (published in 1997) [click to enlarge]:

*Math Trailblazers Student Guide***III. Extra Credit:**

In light of all the many claims made by educaction professors like Sherry Fraser (see Barry Garelick's first comment in my previous post), it's easy to forget that many children

*like*sitting in rows facing the blackboard and doing drills. I see this regularly on Thursday afternoons, when I teach (as I will in a few hours) in an after school math remediation program at one of our underperforming local schools. Here a group of 7 to 10-year-old boys regularly opts to join me at the blackboard (organizing their chairs in a

*row*) to do multiplication drills. To the extent that they are distracted and restless, it's not out of boredom, but mostly because they're arguing over who gets to solve the next problem and trying not to blurt out the answers when it's not their turn.

But just because children

*enjoy*drills doesn't mean we should indulge them. Comment on the relative merits of the final problem sets in the 1920s vs. the Trailblazers 3rd grade curricula.

## Tuesday, April 24, 2012

### Speaking of data

An anonymous commenter on one of my recent posts accused me of not providing data backing up my criticisms of Reform Math. I explained that I see my main contribution to the debate about Reform Math as being in my side-by-side comparisons of Reform Math with Singapore Math and pre-1960's U.S. math.

Detailed comparisons of particular problems sets and elicited skill sets also have their role, and help to ground such concepts as "higher level thinking," "conceptual understanding," and "mathematical challenge."I also noted that, "for a combination of practical, ethical, and ideological reasons," the kind of randomized study Anonymous was requesting are not as not as abundant as one might wish.

Meanwhile, my new friend Leigh Lieberman has been digesting and disseminating tons of what data there is. For example, she's shared with me a November, 2007 Washington Post article on the "unusual approach" of performance-based grouping, as conducted in Rock View Elementary, a low-income school with limited resources in Montgomery County MD. From the article:

The Kensington school's 497 students are grouped into classrooms according to reading and math ability for more than half of the instructional day.There groupings, the article notes, are "fluid and temporary:"

Students are tested regularly in multiple areas and are promoted to more challenging course work as their skills improve. No one is ever demoted.The results:

While some other Montgomery County schools serving low-income populations have posted higher test scores, few have shown such improvement or consistency across socioeconomic and racial lines.Leigh writes:

What I find most exciting about this case, besides the dramatic speed and degree of success in just a few years of use, is what happened the year when it was banned and then subsequently restored.From the article:

In the 2005-06 academic year, Roberson [Rock View's principal] was instructed to halt performance-based grouping, for at least one year, "to see if it really had an impact on student performance." Students returned to mixed-ability classrooms. Test scores fell. The next fall, performance-based grouping resumed. Scores rebounded to all-time highs.Leigh also notes that "Many of the reform movement's so-called 'successes' cannot be readily reproduced elsewhere (very few schools fall in their economic and professional parent range)."

I'll be sharing more of Leigh's data/links here on this blog. To start us out, here is a study she's forwarded me that examines how Michigan State students who used the Reform Math Core-Plus program in high school fared in comparison with those who didn't:

A Study of Core-Plus Students Attending Michigan State University (Richard Hill and Thomas Parker).

## Sunday, April 22, 2012

### TEDx Enola: Helping the World By Helping Our STEM Kids

As even the latest Education Week notes, arts education is alive and well. The story for STEM education is quite different, as I've noted here and there on this blog, and also in a TEDx talk I gave this past February, which just went up on Youtube:

## Friday, April 20, 2012

### Math problems of the week: 6th grade Everyday Math vs. Singapore Math

**I. The final angle deduction problems in the 6th Grade Everyday Math curriculum** (Student Math Journal I, p. 201) [click to enlarge]:

**II. The first angle deduction problems in the 6th Grade Singapore Math Primary curriculum**(Primary Mathematics Workbook 6B, p. 85) [click to enlarge]:

**III. Extra Credit**

Which problem set better prepares students for formal proofs in geometry?

Given that U.S. Reform Math geometry programs have drastically reduced or eliminated formal proofs, does this matter?

## Wednesday, April 18, 2012

### Buzzwords for Jobs

2*1st century skills; collaboration; interdisciplinary; hands-on, real-world; portfolios; risk taking*-- the Edworld's buzzwords have penetrated so deeply into our culture that they've even seeped into the Wall Street Journal. Consider an article from the front section of this past weekend's Weekend Edition, written by Tony Wagner, a former high-school teacher and the Innovation Education Fellow at the Technology and Entrepreneurship Center at Harvard. Entitled "Educating the Next Steve Jobs," Wagner's piece argues that:

Though few young people will become brilliant innovators like Steve Jobs, most can be taught the skills needed to become more innovative in whatever they do. A handful of high schools, colleges and graduate schools are teaching young people these skills—places like High Tech High in San Diego, the New Tech high schools (a network of 86 schools in 16 states), Olin College in Massachusetts, the Institute of Design (d.school) at Stanford and the MIT Media Lab. The culture of learning in these programs is radically at odds with the culture of schooling in most classrooms.Wagner cites no evidence showing that these programs are actually succeeding in teaching innovation; the only evidence he cites for any of his claims comes from interviews of "scores of innovators and their parents, teachers and employers." Since it's too soon to say whether the students in these programs qualify as innovators, Wagner's evidence, such as it is, doesn't apply here.

Wagner then tendentiously invokes what's supposedly happening everywhere else:

In most high-school and college classes, failure is penalized. But without trial and error, there is no innovation.Wagner seems to think that most classes are about innovation. But in systematic subjects like math and science, trial and error (Reform Math's guess and check) is not necessarily something to encourage. Even when trial and error is the best route, if the final product still contains errors, shouldn't assessments reflect this--as they do in the real world?

Wagner, like so many others who haven't visited enough actual classrooms, seems to be laboring under an antiquated notion of public schools in which students primarily lose points for wrong answers, as opposed to losing points for not explaining those answers in words or pictures, and not being sufficiently "colorful" and "creative."

When it comes to universities in particular, Wagner is right about one thing:

The university system today demands and rewards specialization. Professors earn tenure based on research in narrow academic fields, and students are required to declare a major in a subject area. Though expertise is important, Google's director of talent, Judy Gilbert, told me that the most important thing educators can do to prepare students for work in companies like hers is to teach them that problems can never be understood or solved in the context of a single academic discipline.But is the answer to make

*all*courses interdisciplinary (as opposed to simply encouraging coursework in different disciplines)?

At Stanford's d.school and MIT's Media Lab, all courses are interdisciplinary and based on the exploration of a problem or new opportunity.Like other STEM enthusiasts, Wagner implicitly biases his recommendations towards engineering, especially the business side of engineering, forgetting about math and science, where advancements depend less on hands-on, interdisciplinary activity and corporate marketing, and more on ever scarcer analytical reasoning skills:

At Olin College, seniors take part in a yearlong project in which students work in teams on a real engineering problem supplied by one of the college's corporate partners. The teams present their plans to a panel of business leaders who assess their work.

...

At High Tech High, ninth graders must develop a new business concept—imagining a new product or service, writing a business and marketing plan, and developing a budget.

What Wagner thinks of as his biggest conclusion--based presumably on those "scores of interviews"--is total Edworld boilerplate:

In conventional schools, students learn so that they can get good grades. My most important research finding is that young innovators are intrinsically motivated. The culture of learning in programs that excel at educating for innovation emphasize what I call the three P's—play, passion and purpose. The play is discovery-based learning that leads young people to find and pursue a passion, which evolves, over time, into a deeper sense of purpose.So are his concluding remarks:

The solution requires a new way of evaluating student performance and investing in education. Students should have digital portfolios that demonstrate progressive mastery of the skills needed to innovate. Teachers need professional development to learn how to create hands-on, project-based, interdisciplinary courses. Larger school districts and states should establish new charter-like laboratory schools of choice that pioneer these new approaches.The larger question is whether

*any*Steve Jobs, Bill Gates, etc., is ever the product of classrooms, regardless of how "innovative" they are.

## Monday, April 16, 2012

### Shakespeare, or Life Skills

For the last four months I’ve been teaching a class on language-related disabilities to Teach for America students. Soon they’ll be finishing up their program and looking around for jobs. We chat about their career goals from time to time, and it turns out that not one of them has teaching as their first choice. The two years they’ve spent in Philadelphia’s special ed classrooms have totally turned them off to what they once thought they wanted to do.

It’s not just the workload, the paper work, the often challenging students, the often unreliable parents, the underuse of crucial assistive technology (hearing aids and eye glasses, needed but not yet acquired, or simply left at home) and/or facilitating technology (computers and iPads are no longer allowed in one of my students' classrooms because her students, severely emotionally disturbed, too often destroy them).

More than anything, it’s the inflexibility of the curriculum and the resulting hopelessness of the job they being are asked to do. Their students, regardless of their reading levels, are all supposed to be reading on-grade-level texts. Even those entering 1st grade already significantly delayed in language and other pre-reading skills (many didn’t attend kindergarten, which isn’t yet mandatory in Pennsylvania) are forced to read books they’re incapable of reading. From there, the gap between reading ability and reading task only widens--such that the teachers, unless they manage to sneak in more appropriate assignments, often end up resorting to audio tapes and movies.

So, forget all we know about scaffolding and Zone of Proximal Development: all “included” children must be reading grade level texts from day one. The only alternative, my TFA students tell me, is placement in Life Skills classes. What our schools are offering then, even to special ed students, is a one-size-fits-all, all-or-nothing curriculum. That's what No Child Left Behind has come to mean.

I see this at my end with J. Because he’s not in a life skills program, but, thankfully, attends regular classes in a science and engineering magnet school that mostly match his talents, this child, who reads at around a 6th to 7th grade level, is currently “reading” Romeo and Juliet and “learning” Shakespearian vocabulary. Perhaps he’s also basking in the beauty of great literature, and sensitizing himself to the tragedy of star-crossed love. Yup. But this sure as heck beats the alternative.

## Saturday, April 14, 2012

### Letter from Huck Finn: Trying to Hold On and Another Guest on the Raft

*Out in Left Field proudly presents the seventh in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names (with the exception of Miss Katharine's) have been changed to protect privacy.*

**Trying to Hold On and Another Guest on the Raft**

I was thinking of not writing any more letters since it seems no one is reading them. Miss Katharine just about had a heart attack when I told her this. She said a lot of people are reading them and if I am going to quit writing them, why would I leave the readers hanging after telling them my teacher left for a family emergency and I was on my own? So, I’m not quitting. And I'll tell you know that I got through it all just fine. My teacher’s father would pass away and she would be gone for two weeks during which time I was pretty much on my own.

I did get some help the very next day. With five minutes to go before the first period bell, and no substitute, I had given up on anyone showing. But then I heard a key in the door and eagerly went to open it. This caused a problem. “I can’t get the key out,” the sub said. “Close the door.” I did so; he got the key out and opened the door. “My name is Jaime Ortega." he said. "Sorry I’m late; there was traffic. Is there a seating chart?”

I handed him the chart. "Yes," he said like a doctor looking at an X-ray. “I know a lot of these kids,” he told me. "Some of them I know from when I subbed at the elementary schools. And I know some of their brother and sisters.” He was a former student teacher of Tina’s as well, and had subbed in the school many times. He was also from the area. “I’m hungry,” he said. “I worked out this morning and didn’t get a chance to eat breakfast. We get a break after first period?” he asked.

For the next two weeks our routine began with Jaime telling me he was hungry and taking attendance. I would check in homework. After that I would do the warm-up problems, go over yesterday’s homework, and start the day’s lesson. Jaime would help answer students' questions and work with students who had trouble focusing.

That first day, we continued to work with exponents—this time presenting the rules for multiplying and dividing powers of the same base, like 5

^{6}/5

^{2}= 5

^{4}. Which actually should have been the lesson for the day before so as to better explain negative and zero exponents.

After first period would come a break, and Jaime's daily routine was to go to the teacher's lounge and eat whatever snack he happened to have on hand.

Second period was the discovery-based algebra class. I checked in homework, Jaime would go over the answers to the homework; then I would start the day's lesson. It was about functions and graphing. From the second week of class they were “finding the pattern” of growth in a variety of problems. For example, a person buys a tree seedling that is 3 feet tall and plants it. It then grows at a rate of 2 feet per month. Students are asked how tall the tree is after 2 months, 5 months, and finally x months. They are supposed to figure out that 3 feet is the starting point, so they will end up with the equation y = 2x + 3 which they then graph and notice that the starting point of 3 crosses the y axis.

CPM wants students, to "make connections” between the patterns of growth, the equation for such pattern, table of values, graph of the equation and word description of the patterns. After all this, the book finally provides a definition of y intercept and slope and the standard form of the equation of the line (y = mx + b). And then students get to “put it all together”. Except for figuring out slope from any two points; that was still three chapters away.

The authors believe that “simply memorizing what to do in a specific situation without an understanding of the reasons why the method works too often leads to quick forgetting and no real long-term learning.” To rectify this sad situation, students are “asked to solve problems designed to develop the method.” In other words, teaching procedures directly never leads to understanding--or connections.

Rudy was a very bright student, and despite earlier confusions in which he referred to 2x + 5 as “paste," he was definitely the “go to” person within his group of four. His group liked me because of my unusual habit of answering questions. He asked me about a problem that asked for the equation of the graph of a line. “Do you know what the y intercept is?” I asked.

“Where it crosses the y axis?” he asked.

“Yes. Do you know the equation y = mx + b?” I asked pointing to the equation I had written on the board earlier. He nodded. “So what is the y intercept in that equation?”

I reminded him of his past "connections": How to plug in the intercept value for ‘b’, how to compute slope, how ‘m’ means ‘slope’ in that equation. His eyes got big and he began speaking in Spanish to his group. Had Tina been there she would have told me I was giving them the answer. And I was—sort of. The problem-based "connections" approach wasn’t leading to "aha" moments for many students. I noticed Jaime giving similarly explicit answers to the other groups.

Later in the day during 4th period prep, I went to the teacher’s lounge. Jaime was there, eating salsa chips that he sprinked with Tapatio sauce.

“I thought you had left for the day,” I said.

“I was hungry,” he said. “Chip?”

“Too hot for me,” I said.

He appeared to be thinking about something. “You know, I am tutoring some students in calculus at the high school here and they use the CPM book for that. I don't think it’s very good. You have to be really smart to see what's going on.”

“I’m with you.”

“Any word on when Tina is coming back?” he asked. I said no one knew anything. In the meantime, though, it was nice to have somebody with me on the raft--someone who was not afraid to say the sun was not rising in the south.

## Tuesday, April 10, 2012

### A case for textbooks and survey courses

In my previous post (below), I argue that for many students, not just those on the autistic spectrum, the best way to present material is in a systematic, structured fashion.

For many students, information is not only best absorbed this way, but also, I suspect, best remembered. For the stability of our long term memories may depend in part on how well-organized they are in the first place--i.e., on how incoming memories are initially “slotted” with respect to existing and future memories--and on whether we end up having to shuffle our existing memories around later on (creating interference-like effects?) in order to organize them better with respect to future information.

Optimizing such slotting, especially vis a vis future memories, depends not only on structuring incoming information with respect to previous information in a way that is explicitly clear (i.e., building logically upon what’s come before), but on conveying ahead of time what the big picture, or the overall organization, looks like (especially in subjects where this is less predictable--for example history as opposed to math).

In practice, this is an argument not only for textbooks (ones with well-organized, linearly structured material), but for broad outlines and survey courses. That is, for starting a subject--say history--with structured, generalized information that allows you to internalize the big picture and create slots for incrementally more specific information that, in turn, is itself presented initially in outline or survey form. It’s a kind of breadth-first tree-creation process, where the biggest branches--those closest to the root--are created first, and then each “subtree” is begun, recursively, in the same fashion.

The only problem is that, just like the use of the structured information of textbooks, and the emphasis on richly interconnected networks of facts in the first place, introducing subjects via survey courses is completely out of fashion.

## Sunday, April 8, 2012

### Autism and complex information processing

I’ve written a couple of posts (here, and here) on the common misconception that autistic individuals are deficient in abstract thinking. Some of what I’ve written may put me at odds with a highly-reputed autism researcher, Nancy Minshew at the University of Pittsburgh, who has run a number of studies suggesting that autism is a disorder of what she calls “higher-level thinking” skills--or, more precisely, of complex information processing.

More precisely, Minshew finds that autistic individuals with normal IQ scores are worse than typical control subjects at tasks that require people to impose an organizational structure on highly complex information. For example, normal IQ autistics, playing a Twenty Questions-like game, are more likely to guess at random rather than organizing the possibilities into a systematic hierarchy that yields questions that increasingly narrow down the possible answers. Normal IQ autistics are also worse than typical control subjects at remembering complex figures, where it would help to chunk the abstract lines, squiggles and shapes into some sort of hierarchical organization. And they are perform worse when asked to come up with their own criteria for sorting cards that vary along multiple possible dimensions.

I’m not sure what to make of these results. The underlying studies appear to be rigorous, and they appear strongly supportive of Minshew’s conclusions. But a recent study by Nilli Lavie at University College in London (reported on by Jonah Lehrer in this past weekend's Wall Street Journal) finds that handling complex information is an area of strength in autism. And Minshew’s results don’t ring true for the autistic individual I know best. I’ve played 20 questions with J; he’s highly systematic. He’s also good at Set, which involves rapidly sorting cards by various criteria--trying different criteria on your own. I know anecdotally that numerous kids are excluded from Minshew’s studies because their IQ scores are too high, too low, or perhaps too much all over the map. Perhaps kids like J don’t make her cut? But if so, she’s missing out on whole subgroups of high functioning autistics who collectively might wash out her results.

On the other hand, there’s one underlying theme in Minshew’s work that does rings true to me--a theme picked up by other autism researchers, for example, Tony Attwood--and that is the idea that autistic children, perhaps more than others, don’t do well when presented with lots of information at a time whose organization is only implicit. More than others, they depend on things being presented in an incremental, explicitly organized fashion. I see this all the time with J: he flails when material is distributed across many different sources, and thrives when it’s all contained within one structured textbook. He flails when asked to come up with an organizing principle for an open-ended project, and does much better with a series of discrete tasks that someone else comes up with.

But, speaking from personal experience, I don’t think you have to be autistic to have these cognitive preferences/deficits. In my book I propose that they characterize what I call “left brainers” in general. Nor do I think they are particularly debilitating, especially because they seem often to be accompanied by significant strengths in the more systematic of subjects.

Whether or not Minshew is on to something, I’m concerned that her use of “higher level” thinking skills will further entrench the already widespread misconception that children with autism are deficient in abstract thinking--when, in fact, this is for many of them an area of great potential.

## Friday, April 6, 2012

### Math problems of the week: 2nd grade Investigations vs. Singapore Math

**Collecting data vs. interpreting it.**

**I. From the 2nd grade TERC/Investigations curriculum** [click to enlarge]:

**II. From the 2nd grade Singapore Math curriculum**[click to enlarge]:

**III. Extra Credit**

What do Singapore 2nd graders miss out on by skipping the data-collection step?

## Wednesday, April 4, 2012

### Letter from Huck Finn: My Teacher Leaves, and I Become a Gunslinger

*Out in Left Field proudly presents the sixth in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.*

**Letter from Huck Finn: My Teacher Leaves, and I Become a Gunslinger**

Things were looking pretty rosy by the third week in October. I was at the halfway point in my student teaching and had made it through two evaluations and was getting the hang of things. But then my trip down the river took an unexpected turn. I was getting the first period class through the morning routines on a Monday when Tina’s cell phone rang and she went outside to answer it. I could hear her voice through the door; it sounded like she was crying. She came in and told me she had to leave. “It’s an emergency. My father’s in the hospital. Can you handle the class?”

“I’ve got it,” I said.

“They’ll try to get you a sub. I’ve got to go.” She grabbed her purse and ran out the door. I found myself without my teacher, facing 30 or so students waiting for me to do something. I now had to explain zero and negative exponents and get my first period class to follow along with the explanation the text book had presented. This was clearly not going to be easy.

The way I learned was based on the rule for dividing powers: a

^{m}/a

^{n}=

*a*

^{m-n}. Suppose you have a

^{3}/a

^{3}with a not equal to 0. Then it equals 1, because any number divided by itself is 1. But using the rule for dividing powers, it also equals a

^{3-3}which is a

^{0}. So a

^{0}= 1. Similarly, for negative exponents, something like a

^{2}/a

^{3}is easily shown to be (a·a)/(a·a·a). By cancelling a’s, you’re left with 1/a, and by the rule for dividing powers, a

^{2-3}is a

^{-1}which equals 1/a. This all made perfect sense to me and I liked the idea that the application of one rule led to another. But the textbook I was using for teaching took a different approach. The authors seemed to think that deductive reasoning didn’t teach the right “habits of mind” given the prevailing belief that “math is about patterns”.

The explanation amounted to using a place value chart (thousands, hundreds, tens, ones, tenths, hundredths, etc). I first showed that the thousands column can be written as 10 x 10 x 10 or 10

^{3}. “Moving to the right of the thousand’s place, what’s next?” I asked. A few students responded: “One hundred”. I wrote 10 x 10 = 10

^{2}, and pointed out that this was 1,000 divided by 10. “And the number next to 100?” I asked. “Ten!” a few more said. “And look at our exponents. We had 10

^{3}, and 10

^{2}, what’s this next one going to be?” “Ten to the one” they said. I then asked what the number next to the ten’s place would be. “One,” they said. I pointed to the pattern of exponents, and said “Three, two, one…what’s the next exponent?” Someone said “Zero.” And there you have it: by the pattern, 10

^{0}is 1.

Continuing in that fashion, I showed how when we move further right we get into negative exponents: 10

^{-1}is 1/10, 10

^{-2}is 1/10

^{2}or 1/100 and so on. From this, students were to take it on faith that this pattern applied to powers other than ten, like 5

^{-2}is 1/5

^{2}. I was fairly certain the explanation wasn’t going to stick with them for more than two minutes—if that. So I had them write down the rules and relied on what can either be called an axiomatic definition or what the authors probably felt they were avoiding: rote.

My next challenge came when I saw a paper airplane fly across the aisle. Judging by where it landed, I had it narrowed down to one of two boys: Eliseo or Cesar. The class was silent, watching to see what I would do. I knew I had to do something. Up until now I had never made anyone “do a card”—a punishment in which a student had to copy what was written on a colored card kept in a folder at the side of the room: a treatise about the value of education. Increasing amounts of card punishments could get students a referral to the office, a parent-teacher conference, or even a suspension.

I’m not sure what prompted me to do this, but I walked down the aisle between Eliseo and Cesar in the manner of a gunslinger—slow, sure, and looking for danger. The closer I came to Cesar, the more nervous and fidgety he became. I made my choice. “Cesar, do a card,” I said. He fairly jumped out of his seat and took to the task; he even looked relieved. The class seemed impressed with my feat. They seemed much quieter after that.

The day moved on. By third period—my last one—I was still on my own. The class followed the exponent lesson, but Manuelo, one of the brightest students in that class, asked the question I had asked long ago when I first saw zero powers: “How can anything be raised to the zero power? And why would it equal one?”

The class looked at me like my first period class did when Cesar threw the paper airplane. “Well,” I said, “sometimes in math we just say something is a certain way because it fits the pattern.” “OK, but I still don’t get it,” Manuelo said. The axiomatic approach was not sitting well with him. Nevertheless, he worked with the concepts of zero and negative exponents with the rest of the class.

During the “prep” period (fourth period), the phone rang. It was Tina.

“How are you doing?” I asked.

“I’m OK. My father’s had a heart attack. It’s extremely serious.”

“I’m very sorry,” I said.

“So I may be out for at least another week. I talked to the principal; you’ll get a sub tomorrow. What did you do today?”

I told her about the exponent lesson. “Oh, we never do it the way the book does it. We teach them the rules for multiplying and dividing powers; you know--3

^{2}/3

^{2}equals 1 and also 3

^{0}?”

“I’m glad we’re having this little chat,” I said.

“Sorry,” she said. “There wasn’t time.”

“I know, I know. You don’t have to worry. I’m doing fine.”

“I have full faith in you,” she said. “I know you can do it.”

We both knew we were lying to each other, but sometimes that’s all you’ve got to get you through the day and night.

## Monday, April 2, 2012

### Explaining your answers when there's nothing to say

"How is a teacher, who uses formative assessments, supposed to know what a child truly understands or is grappling to understand, if they don’t have a window into their thinking?"

"By giving them truly challenging problems and seeing if they consistently get the right answers!"

*From a recent exchange I had on an Education Next thread on whether students should be required to explain their answers.*

It strikes me that there are two things about Reform Math assignments that cause today’s educators to be so insistent that kids explain their answers:

1. Reform Math problems, in comparison with traditional math (and overseas math, including Singapore Math) are often so easy that there’s no work to show. The calculations are relatively simple, and the word problems usually don’t involve more than one or two steps. Many students, unless you instruct them otherwise, have no reason to write down anything beyond the bare answers. Teachers, meanwhile, remembering how important it was to show their work back when they were in school, may conflate showing your work with explaining your answers.

2. Reform math assignments often involve no more than a half dozen problems, again with relatively easy calculations, where traditional (and overseas) assignments involve dozens of problems with much more complex calculations. It’s therefore conceivable for a student to get every problem on a Reform Math sheet correct by chance rather than by understanding. Getting all correct answers for several dozen complex calculations without understanding what you’re doing, in contrast, is so unlikely that, when students accomplish this, then it’s reasonable to assume that (unless they were cheating) they know what they are doing.

Two additional factors underlie Reform math’s insistence on explained answers. One is the notion that, regardless of how much your math skills exceed your verbal skills, if you can’t communicate your mathematical thinking then that thinking must be deficient. The other is that “mere calculation” has little to do with “deep understanding.”

These notions, of course, also help justify the educational malpractice of restricting students to problem sets in which the calculations are a fraction of the frequency and difficulty they once were and still are nearly everywhere else in the developed world.

**Postscript:**

I recently showed the Singapore Math curriculum to a student visiting from Mainland China, and the first thing she observed was that the 4th grade Singapore Math problems were problems that she and her classmates routinely did in

*first grade*.

"You must have gone to one of the top elementary schools," I said.

"Not one of the top schools; just a pretty good school" was her reply.