Monday, June 30, 2014

Pain means gain

Catching up a bit on some old, blog-worthy articles, I’ve come across one from a September, 2013 issue of the Wall Street Journal. Written by Joanne Lipman, co-author, with Melanie Kupchynsky, of "Strings Attached: One Tough Teacher and the Gift of Great Expectations," it adduces some empirical evidence for a number of points that have long rung true to me. Some of it is stuff that I’ve already blogged about ad nausaum--the benefits of drill for higher level thinking; the power of grit—but the bulk of it, even nine months later, still feels fresh. For example:

A little pain is good for you.
…Psychologist K. Anders Ericsson gained fame for his research showing that true expertise requires about 10,000 hours of practice, a notion popularized by Malcolm Gladwell in his book "Outliers." But an often-overlooked finding from the same study is equally important: True expertise requires teachers who give "constructive, even painful, feedback," as Dr. Ericsson put it in a 2007 Harvard Business Review article. He assessed research on top performers in fields ranging from violin performance to surgery to computer programming to chess. And he found that all of them "deliberately picked unsentimental coaches who would challenge them and drive them to higher levels of performance."
Or:
Failure is an option.  
…In a 2006 study, a Bowling Green State University graduate student followed 31 Ohio band students who were required to audition for placement and found that even students who placed lowest "did not decrease in their motivation and self-esteem in the long term." The study concluded that educators need "not be as concerned about the negative effects" of picking winners and losers.
I’m less certain about the psychological benefits of kids being sorting their peers into winners and losers: as happens, say, when students pick team members in gym class.
Strict is better than nice.
…What makes a teacher successful? To find out, starting in 2005 a team of researchers led by Claremont Graduate University education professor Mary Poplin spent five years observing 31 of the most highly effective teachers (measured by student test scores) in the worst schools of Los Angeles, in neighborhoods like South Central and Watts. Their No. 1 finding: "They were strict," she says. "None of us expected that."
The researchers had assumed that the most effective teachers would lead students to knowledge through collaborative learning and discussion. Instead, they found disciplinarians who relied on traditional methods of explicit instruction, like lectures. "The core belief of these teachers was, 'Every student in my room is underperforming based on their potential, and it's my job to do something about it—and I can do something about it,'" says Prof. Poplin.
“It's my job to do something about it—and I can do something about it"--that’s also refreshing. So many educators these days (at least among the ones whose voices one hears on the Internet) seem so ready to use factors beyond their control, like poverty or uninvolved parents, as excuses rather than as motivators.
Creativity can be learned.
…Temple University psychology professor Robert W. Weisberg… has studied creative geniuses including Thomas Edison, Frank Lloyd Wright and Picasso—and has concluded that there is no such thing as a born genius. Most creative giants work ferociously hard and, through a series of incremental steps, achieve things that appear (to the outside world) like epiphanies and breakthroughs.
That’s certainly true of the most creative people I know. Very little of the creative process is sudden, out-of-nowhere inspiration, even if a spark of that helps sets things in motion.’
… The bottom line, Prof. Weisberg told me, is that creativity goes back in many ways to the basics. "You have to immerse yourself in a discipline before you create in that discipline."
Then there's:
Praise makes you weak…
...Stanford psychology professor Carol Dweck has found that 10-year-olds praised for being "smart" became less confident. But kids told that they were "hard workers" became more confident and better performers.
I’ve seen this play out in my martial arts class: priase make me certain that my next kick will be weaker or will miss the target, as it almost always does. As a classmate of mine put it, compliments are the kiss of death.
.…while stress makes you strong.
...A 2011 University at Buffalo study found that a moderate amount of stress in childhood promotes resilience. Psychology professor Mark D. Seery gave healthy undergraduates a stress assessment based on their exposure to 37 different kinds of significant negative events, such as death or illness of a family member. Then he plunged their hands into ice water. The students who had experienced a moderate number of stressful events actually felt less pain than those who had experienced no stress at all.
"Having this history of dealing with these negative things leads people to be more likely to have a propensity for general resilience," Prof. Seery told me. "They are better equipped to deal with even mundane, everyday stressors."
This is in contrast more extreme levels of so-called “toxic stress”—related less to routine stressors and more to high levels of domestic instability--violence, abuse, and neglect, extreme poverty/economic insecurity—the debilitating effects of which have been the subject of recent research.

Within reason, pain leads to gain—something that, like so much of this, we readily accept in our athletic fields, but not in our classrooms.

Saturday, June 28, 2014

Conversations on the Rifle Range 2: Negative Numbers, Back-to-School Night, a Tattooed Man and a Mysterious Stranger

Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number two:



My back-to-school night was held on a Thursday evening during the first full week of school. Like most back-to-school nights, it was designed to give parents a peek at what goes on in their child’s school-day. And like most back-to-school nights that I’ve been to, parents shuffled from class to class, following their child’s schedule with somnambulistic fervor—each class lasting 10 minutes.

Of course, it wasn’t an exact replica of a school day: the school used block schedules, with hour and fifty minute classes and odd and even-period classes alternating every other day. I had three classes held during second, fourth and sixth periods—which meant that I taught every other day.

This was my first ever back-to-school night as a teacher. Parents from my three classes showed up, though attendance was fairly sparse. I assured all that I was certified to teach math, and that I would follow the teacher’s lessons and grading procedures. I had a list of topics that I would be teaching in my algebra classes and pointed to them. People nodded vaguely. I then said “I teach by providing instruction, worked examples, and lots of problems.” People nodded vaguely again. So far so good.

I was teaching the two-year sequence of Algebra 1. It is designed for students who are having difficulty in math. My high school, being very small, only offered the 2-year sequence. Students, who for whatever reason did not take Algebra 1 in 8th grade, unless they went to summer school, were therefore stuck with the two–year sequence of algebra, regardless of their ability to handle the one-year course in 9th grade. Two of my classes were the first year of the two-year sequence, and one was the second year.

I was most curious about the parents who showed up for my sixth period class “first year” algebra 1 class) since the students in that class were the most difficult. Out of 25 students, perhaps four were actually intent on learning anything. In fact the parents of a girl named Laura—one of the good students—showed up. She had two sets of parents. Her biological father was there; he bore tattoos on his neck including one of a poorly drawn heart with a number inside it. The step-dad and mother were there as well, along with Laura’s little sister—everyone but my student. And then there was a man who arrived late and sat in the back, looking somehow familiar, with a bored look on his face, slouched at a desk.

It had been hot earlier in the day, so that by sixth period the classroom was stifling. Many of the students were sophomores and some juniors who had taken the class before. My lesson that day had been on negative numbers and I was intent on making sure they had the basics down. In both of my first-year classes I asked, “How many of you know how to operate with negative numbers?” A few hands went up. “Let me ask you this then. What is 10 minus negative 5?” Immediately I heard wrong answers: five, negative five.

I then asked a question designed to get the attention of the many junior varsity football players in the class. I would guess that between my three classes, I had half the members of that team. I pointed to a football player. "Let me ask the question in another way. Suppose in a football game your team gets the ball and on the first play you lose 5 yards. The coach says you have to make enough yardage on the next play to get a first down. How many yards do you have to run on that next play?”

The answer came with no hesitation: “15”.

“Excellent; how did you do that?”

The boy explained that you need to make up the 5 yards you lost and then add 10 to that for a total of 15.

"You solved 10 minus negative 5. In two seconds.” I went on to say that they work with negative numbers every day but don't realize it, and then went into the day's business. While I had been successful in my 4th period class, I didn’t advance very far in 6th period. A few minutes into the lesson, a girl named Jeanne suddenly shouted “We learned this last year. Why are we learning it again?”

Suddenly the class which earlier had admitted to being weak with negative numbers was riled up at having to learn about them again. I suppose her question had deeper meaning for those taking the class for the second time—yes, why were they learning this again? But the disruption she set off in a hot room full of irritable students was irreversible and immune to any explanation I could offer. I offered one anyway.

“This is a review,” I said.

“I thought this was supposed to be algebra!” someone shouted.

I then resorted to the time-honored blame-laying approach. “Not everyone knows all this stuff like Jeanne claims to know…” I was drowned out by a chorus of accusations that I was putting words in her mouth. They knew I had lost control.

I walked over to Jeanne and told her to knock off the attitude. “This isn’t attitude—it’s natural” she said. “I’m Italian.” I admit I found this rather funny but I now had to follow through. “Would you like a referral to the office?” I asked. She became quiet and the class began to quiet down. Patrick, a football player who had failed the class last year and provided steady commentary to his neighbors about my teaching skills chimed in: “Are you losing control of the class, Mr. G?”

I walked over to Patrick. “Would you like to try to control this class?” I asked him.

“No,” he said.

“Would you like a referral?”

“No.”

The class quieted down enough that I could finish my lesson.

Now, several hours later in a considerably cooler and quieter room I projected the same aura of confidence that a trapeze artist displays even after he almost misses grabbing the bar. I asked if there were any questions. There were none, not even any about Common Core.

I suspected that the man who was slouched in his seat in the back was Patrick’s father. But when the dismissal bell rang, he was out the door before I could ask who he was.

Thursday, June 26, 2014

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

End-of 5th-grade poultry problems:

I. From the Everyday Math Student Math Journal, volume 2, final problem set, p. 441:



II. From Hamilton's Essentials of Arithmetic, final "Everyday Use of Numbers" problems, p. 363

Tuesday, June 24, 2014

End-of-year update for home school

I’ve just turned in my daughter’s homeschooling portfolio to the city of Philadelphia, and so now seems a good time to share what we’ve done this year.

Her language Arts readings included Ms. Peregrin’s Home for Peculiar Children, The Umbrella Man and Other Stories, If You Come Softly, A Wish After Midnight, The Book Thief (all these books were for the monthly Tween Book Club she belongs to); The Diary of Anne Frank, The Witch of Blackbird Pond, Tom Sawyer, Huck Finn, Peter Pan; D’aulaire’s Greek Myths, selections from the Old Testament; Nathaniel Hawthorne’s The Golden Book and Tanglewood Tales; selected poetry, including T.S. Eliott’s Cats poems and parts of The Song of Hiawatha; Romeo and Juliet (which we read out loud and later saw performed by the Philadelphia Shakespeare Company); and, finally, selections from the last volumes of My Bookhouse series, including pieces on Robert the Bruce, Punch and Judy, P.T. Barnum, Daniel Boone, retellings of Nordic and Germanic myths and of Robin Hood and Beowulf, tales from the Arabian Nights, excerpts from Don Quixote, and, lastly but (as it were) not least, Gulliver’s Travels to Lilliput.

You’ll notice that these readings span a range of levels; we went through the hard stuff quite slowly and she created flashcards for the words she had to look up.

For writing, she wrote daily reading summaries, as well as short stories and poetry for a creative writing group.

In history she completed the final volume of The Story of the World series, first outlining each chapter, then taking the chapter tests, and finally, choosing a particular country (she chose Russia) and synthesizing all the information on it into a multi-page essay.

For science, she did a combination of earth/physical science, basic electricity, and animal taxonomy. She continued through McDougal Littell’s Earth Science, worked through an instructional video series (which I blogged about here), and did a few at-home chemistry experiments. She also a series of electricity kit projects. And she used some great Internet-based resources on animal taxonomy kindly provided by one of commenters on this very blog.

For math, she continued to work through Wentworth’s New School Algebra. This year she’s covered polynomial long division, factoring of polynomials, rational expressions and equations, simultaneous linear equations, graphs of linear equations, and word problems involving linear rates, with boats and currents; one vehicle overtaking another vehicle; pipes filling pools; and painters working separately vs. simultaneously (i.e., word problems of the sort that used to abound and now, under Reform Math, are practically extinct).

For French, she’s continued with French in Action video tapes (created by the brilliant Pierre Capretz, who died on April 1st), the ALM textbook series, and various story books written for French children, including some fairy tales, several volumes from the Lu Lu series and Tin Tin.

Her music training includes classical music CDs, ear training, and piano, organ, and violin lessons, as well as ensemble activities with her peers. Musical highlights include Bach’s Little Fugue in G-minor, Bach’s French Suite in G, and Bach’s violin concerto in A minor and double concerto in D minor; Mozart’s piano concerto in F major; Beethoven sonatas from Opus 10; and the Passepied from Debussy’s Suite Bergamesque. She also took a drawing class and a pottery class at the neighborhood after-school art center and had field trips to the Art Museum. Finally, for “gym”, in addition to regular hikes and bike rides, she had horseback riding lessons, and, most recently, tennis lessons with her oldest brother.

Sunday, June 22, 2014

Should kids have homework? Should parents help?

In an opinion piece in this week's NYTimes Sunday Review, writer Judith Newman is skeptical.

Newman opens with a story about her attempt to improve her 12-year-old son's essay on To Kill a Mocking Bird. Even though she has "a master in literature from an Ivy League school" and "write[s] for all the major magazines," the end result of her edits was a score of 73. As she puts it:

He got a 73.
I got a 73.
Generalizing from her experience, she cites research by two sociologists that supposedly shows (but see the critical reviews) that parental help with schoolwork is mostly ineffective, and often detrimental.

I'm drawn towards a different conclusion. If someone with Newman's qualifications got a 73 on a 7th grade literature paper, there was probably something warped about the grading rubric. Perhaps the essay lacked a sufficient number of "text-to-self" references/personal connections--things that don't earn you points outside the Bizarro World of K12 classrooms. Perhaps it wasn't "creative" enough (see below).

Indeed, if parental help is ineffective, this may say more about the kinds of expectations that our K12 schools impose on kids than it does about parental help. Consider another of Newman's anecdotes:
“I think of myself as an intelligent, functioning adult,” says the writer Julie Klam, who has a daughter who just finished fifth grade. “But my God. Do you know what a ‘math lattice’ is? No, you do not. The way basic math is taught now, it’s not like A plus B equals C. It’s more like A plus B, and then you run out for oranges, and then you take the subway. My daughter’s recent assignment was like a buffet of confusion.”
Or this observation:
Further complicating the homework is the increasing fashion for making it “creative” — which often renders it unnecessarily complicated, at least for the age and dexterity of many younger children.
The age-inappropriateness of such assignments is often what prompts parents to want to help--at the same time that they may find themselves unable to do so:
“I used to be very involved in my kids’ homework until my second grader came back with an assignment to recreate New York City’s waterways using a baking sheet, mounds of paper towels, tin foil and rivers of water poured from a pitcher,” says Marjorie Ingall, a Manhattan public school mother. “First of all, I don’t care about New York’s waterways as long as the water that comes out of the tap does not catch fire. But that aside — this is an assignment for me, not for an 8-year-old. There was just so much crying at my house.”
Even when parents can and do help with such assignments, Newman points out, there's a huge downside:
First, you are conveying to your kids that they can’t make it without Mom and Dad’s help... But more important, you are sending the unmistakable and not so subtle message that it’s better to be right than smart.
Or, at least, that it's better to be rubric-compliant (and "creative") than smart.

But does this mean that schools should assign little or no homework and that parents should offer little or no help? The fact that silly, time-consuming assignments don't make kids smarter or more knowledgeable (or more self-reliant) doesn't mean that all homework is pointless. And the fact that parental help is ineffective when channeled into these assignments doesn't mean that all parental help is worthless. Parental help with math (not Reform Math; not Common Core Math) and parental help with writing (not rubric-compliance) can be extremely helpful in preparing kids for the broader world beyond the Bizarro World of K12 classrooms.

Newman cites Finland and Denmark, "countries with some of the highest levels of academic achievement," as having "little or no homework." This is misleading (see here and here): particularly in the upper grades, both countries do assign homework. It's just that it's a lot less time consuming than ours is.

What American students need more of is what students around the rest of the world get regularly: homework with a much lower ratio of time-expenditure to learning; homework that respects that their time outside classrooms is at least as valuable as their time inside--and should be imposed upon only for very good reasons.

Friday, June 20, 2014

Math problems of the week: 4th grade Everyday Math vs. 1920s math

Final rates problems for 4th grade:

1. From the 4th grade Everyday Math Student Math Journal, Volume 2:



II. From Hamilton's Essentials of Arithemtic for years 1-4, final chapter:


Wednesday, June 18, 2014

Squandering STEM by broadening its appeal

Two days ago, an article appeared in the Science Times STEM series entitled Efforts to Inspire Students Have Born Little Fruit. The article cites the fading, hodgepodge efforts of the companies and nonprofits and private-public partnerships that made up President Obama’s Educate to Innovate initiative. It also cites the lack of improvement by American students on the Program of International Student Assessment (PISA) tests, last given in 2012. And, finally, it cites the lack of follow-up investigations of what’s happening in classrooms, and the fact that, since “public education is run at the state and local levels… any presidential education initiative is inevitably indirect”.

I’d put the problem differently. As I’ve noted here, here, and here, our schools, and our society more generally, are no longer encouraging and educating the kind of student who is most likely to persevere in STEM careers. These are the left-brained math and science types, more and more of whom face a dumbed-down, language-arts intensive Reform Math curriculum, and a science curriculum that increasingly emphasizes projects over the core knowledge and quantitative skills needed to succeed in college level science courses.

At the expense of encouraging this type of student, K12 schools are trying to broaden the appeal of math and science—by making them even less mathematical and scientific. And so we have algebra taught as dance, fraction murals, photosynthesis as dance, and science festivals featuring showy displays of gadgetry as well as theater, art, and music.

The thing is, the kind of student who finds these approaches engaging and enlightening, as opposed to trivializing of what’s interesting about math and science, probably isn’t the kind of student who ever would persevere through a STEM major.

Meanwhile, the kind of person who potentially would is getting less and less of the training s/he needs to compete with growing number of international peers in college.

Monday, June 16, 2014

Teachers, tear down those wall hangings!

Even as preschools compete to become more “academic,” kindergarten still marks, for most kids, a time of unprecedented demands on their attention spans. All of a sudden, they find themselves having to spend most of their class time focused on activities that their teachers choose for them. With the recent expansion of kindergarten into full, 6 ½-hour-a-day affairs, these expectations are unprecedented not just personally, but also historically. 

Also unprecedented—so far as I can tell--is the amount of time kids have to spend listening to one another during Circle Time. Extended Circle-Time peer-talk, after all, is one of the apotheoses of cooperative, child-centered learning. It also happens to be, as far as maintaining focus is concerned, one of the most trying tasks of all—at least for 5 and 6 year olds. For here, unless it happens to be your turn to speak, you have to sit still, criss-cross apple sauce style, with nothing to lean on, and quietly look and listen as one after another of your not-so-articulate classmates (they are, after all, only 5 and 6 years old) eventually gets their point across.

Should your attention wander away, to, say, the objects in the room or posters on the wall, your teacher may interrupt the discussion to rebuke you. J got a special dispensation (for being autistic), but my daughter did not, and I can’t tell you how many times she (and I) were told she was disrespectful and/or unkind for fidgeting and not attending to peers. (Countless others, of course, are judged not as morally deficient, but at attentionally so—and medicated accordingly).

Now a recent study reported in last week’s Science Times confirms what I (and many others) have long suspected. If anyone is being--however unwittingly—inattentive or inconsiderate, it’s the teachers of their students:

A large body of evidence supports the importance of focused attention for encoding and task performance. Yet young children with immature regulation of focused attention are often placed in elementary-school classrooms containing many displays that are not relevant to ongoing instruction. We investigated whether such displays can affect children’s ability to maintain focused attention during instruction and to learn the lesson content. We placed kindergarten children in a laboratory classroom for six introductory science lessons, and we experimentally manipulated the visual environment in the classroom. Children were more distracted by the visual environment, spent more time off task, and demonstrated smaller learning gains when the walls were highly decorated than when the decorations were removed.
Teachers of the world, before you assume it’s the kids’ fault, cut down on Circle Time, and tear down those wall hangings!

Saturday, June 14, 2014

Conversations on the Rifle Range, I: Not Your Mother’s Algebra 1 and the Guy Who Really Knows



Barry Garelick, who wrote various letters under the name Huck Finn and which were published here is at work writing what will become "Conversations on the Rifle Range".  This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents the first episode:

Those familiar with my writing on math education know me from my previous incarnations as John Dewey and Huck Finn, whose adventures I recounted in a book called “Letters from John Dewey/Letters from Huck Finn”. I am in a second career which for lack of a better title is known as “trying to obtain a permanent math teaching position in a desirable area of California.” I retired a few years ago and obtained a math teaching credential. Although I have applied for various math teaching jobs, I have only managed to get two interviews, so I’ve had to make do by being a substitute teacher. This situation may be due to age, or perhaps my views on math education are becoming known, or both.

In the course of the 2013-14 school year, however, I took on two long-term substitute assignments. The first one was for six weeks at a high school which started at the beginning of the school year. The second was for an entire semester at a middle school, starting in January and ending in June.

Both assignments took place amidst the media hype that focused on the 50th anniversary of events occurring in 1963 and 64 including but not limited to the Kennedy assassination, the Beatles’ arrival in the US and performance on the Ed Sullivan Show. Not mentioned by the press but every bit as important is the fact that it was also the 50th anniversary of my taking Algebra 1. And while I am not an outright proponent of the philosophy that “If you want something done right, you have to live in the past”, when it comes to how to teach math there are worse philosophies to embrace.

As if to keep me from delving too far into my past, my teaching assignments occurred during a year of transition to the Common Core standards. In both assignments, I came to know the person from the District office, who I shall call Sally, whose role was to get the teachers—as part of the transition effort— to try various Common Core type activities with their students. I met her for the first time on the teacher workday held before the first day of school.

Sally started out the meeting by telling us that she had been meeting with the person in charge of putting together the California "Framework" for Common Core. “So he REALLY KNOWS what's going on,” she said. This stated, she then talked about this in-the-know person’s view of Common Core’s Standards for Mathematical Practice (SMPs).

For those who may not know what these are, the SMPs are eight practices that 1) supposedly embody the work habits and general mode of thought of mathematicians, 2) were defined largely by non-mathematicians, and 3) which most real mathematicians believe are nonsense. Yes, criticizing and analyzing the reasoning of others (one of the SMPs) is what mathematicians may do, but it is something learned through accumulation of expertise in the subject area. But distinctions between novices and experts have never bothered the non-experts who write this stuff and even some mathematicians are swayed by the “wouldn’t it be nice if students could do this” quality of such daydreams.

Sally was therefore quite excited to tell us about the person who REALLY KNOWS’ view of the role of the SMP’s. “Up to now everyone thought the Standards for Mathematical Practice were the instructional methodology for teaching the content,” she said and then quickly added that the guy who REALLY KNOWS says, no, that's wrong.

I was getting hopeful here.

“He says it’s the other way around: the content is there for students to learn the Standards for Mathematical Practice,” Sally announced triumphantly. While this view can be interpreted to mean that math procedures and content lead to understanding I don't think that's what the guy who really knows meant. In all likelihood he meant that the Common Core content standards require students to work in groups, discuss, conjecture, critique each other’s arguments and that teachers are to be (in Sally’s words) "the guides on the side".

She talked about how the District was phasing out the “accelerated math” in which students in 8th and even 7th grade could take Algebra 1. She then put up a slide from a Power Point presentation titled “Not Your Mother’s Algebra 1” referring to Common Core’s approach to algebra. The whole idea being that Common Core gets into “deeper learning”. Which means that students will now get a smattering of algebra in 8th grade, and the rest of it in 9th, thus taking two years to do what used to be done in one—with some topics left out. She did say they are working on pathways for those students who may “really truly” be gifted and for which algebra in 7th or 8th grade may be appropriate. She also alluded that they are working on a pathway for those who did not qualify. I would in fact find out what the various math pathways would be and how one qualified to take a normal class in algebra. But that would be when I started my second assignment.

This was likely not going to sit well with some parents, she said and that for "back to school night" expect some questions on Common Core. “There's been a lot of parent pushback,” she said. She put some talking points up on the screen for us and said to feel free to use them. One of them was "writing and deep thinking" which I decided would not appear on my board on back to school night or any other night.

That evening I decided to look at the website for information about the “guy who really knows”. I found nothing. But I did find some information about the Superintendent of my school district. In particular, he wrote a piece on his philosophy of teaching and even made it available for downloading. I saw this:

“I believe students in the 21st century are different. They are digital natives and live in a world where “any knowledge” can be found immediately on Google. Therefore, why regurgitate knowledge (like an “academic rationalist”) when it is far more reasonable to expect a student to apply this knowledge and to make new meaning from this knowledge. Relevance is critical among this generation of students in order to motivate them to move beyond what I see as low-level thinking.”

I pretended I never saw it and decided to just stay invisible and teach math.

Thursday, June 12, 2014

Math problems of the week: rates problems in Chicago vs. 1920s algebra

I. The only rates problems in The University of Chicago Math Project Algebra:




II. A few of the rates problems in Wentworth's New School Algebra:



III. Extra Credit:

What's missing from Chicago Math are problems like the Wentworth problems above, which require you to set up a rate problem from scratch, with only a worked example to guide you, and with three different entities (time, speed, and some sort of change of state) to interrelate from multiple perspectives. When was the last time you saw a problem like this?

Tuesday, June 10, 2014

Autism Diaries: a practice SAT essay

ESSAY

Time -- 25 minutes

 
Think carefully about the issue presented in the following excerpt and assignment below.
Sometimes it is necessary to challenge what people in authority claim to be true. Although some respect for authority is, no doubt, necessary in order for any group or organization to function, questioning the people in charge- even if they are experts or leaders in their fields- makes us better thinkers. It forces all concerned to defend old ideas and decisions and to consider new ones. Sometimes it can even correct old errors in thought and put an end to wrong actions.
Assignment:
Is it important to question the ideas and decisions of people in authority?
Begin your essay on this page. If you need more space, continue on the next page.

People can be wrong sometimes. It is sometimes important to correct them, and sometimes it's not important. Sometimes, people come up with an idea which might be a bad idea. Sometimes, there are hidden reasons why it's not really a bad idea. So it's good to explain why you think it's a bad idea, instead of just saying it's a bad idea.

One example is when people are installing the ceiling fan, and it looks like it's a bad idea to install it that way, there could be something hidden that keeps the fan from falling down and getting a person killed.

Another example is when you see people microwaving a food while it's still in a box. A box might look like it's cardboard, but it's made to be cooked in microwave. So it's good to just say that it's cooking while in a box instead of just turning off a microwave and make people very upset.

Another example is that when you try to put eggs on chandelier because it's easter, and people says it's a bad idea and says to stop. Instead of just arguing and saying nothing's wrong with eggs on chandelier, you can just ask why they think it's a bad idea, and they can say that the light can melt the egg and make a smoke.

So instead of just argue you can just ask a question why they think it's a good or bad idea, and they can explain why, in a very calm way. This way, you don't end up getting into fight and ending up in the hospital.

[Posted with permission from author.]

Sunday, June 8, 2014

What students lose when schools abandon penmanship

This past week's Science Times showcases yet another cases where classroom practices are diverging from recent findings in cognitive science (see also "critical thinking"; novices vs. experts; learning styles; and real-life connections). The latest findings are about penmanship, increasingly downplayed in classrooms, and, as the article points out, by the Common Core:

The Common Core standards... call for teaching legible writing, but only in kindergarten and first grade. After that, the emphasis quickly shifts to proficiency on the keyboard.
The article cites a 2012 study by Karin James, a psychologist at Indiana University in which:
Children who had not yet learned to read and write were presented with a letter or a shape on an index card and asked to reproduce it in one of three ways: trace the image on a page with a dotted outline, draw it on a blank white sheet, or type it on a computer.
The kids who drew the letters freehand--i.e., on blank pages--exhibited increased activity in the same areas of the brain that are "activated in adults when they read and write." The other kids didn't. Freeform writing, James speculates, requires that "we first plan and execute the action in a way that is not required when we have a traceable outline."

Freeform writing, furthermore, produces "highly variable" results. No two handwritten letters look exactly alike--particularly when drawn by novice writers. This variability, James suggests, may aid kids in learning letters--presumably by requiring them to abstract the signal (the letters' essential features) from the noise (all that variability).

Another of James' studies, which compares "children who physically form letters with those who only watch others doing it," suggests that:
it is only the actual effort that engages the brain’s motor pathways and delivers the learning benefits of handwriting.
This is consistent with other findings on the benefits of active over passive learning: e.g., of retrieval practice and productive practice in second language learning. (And it is one of the guiding principle behind my linguistic software for autism.)

Beyond form, there's content. In a study that followed children in grades two through five, psychologist Virginia Berninger finds that:
When the children composed text by hand, they not only consistently produced more words more quickly than they did on a keyboard, but expressed more ideas. And brain imaging in the oldest subjects suggested that the connection between writing and idea generation went even further. When these children were asked to come up with ideas for a composition, the ones with better handwriting exhibited greater neural activation in areas associated with working memory — and increased overall activation in the reading and writing networks.
Then there's note-taking and recall:
Two psychologists, Pam A. Mueller of Princeton and Daniel M. Oppenheimer of the University of California, Los Angeles, have reported that in both laboratory settings and real-world classrooms, students learn better when they take notes by hand than when they type on a keyboard. Contrary to earlier studies attributing the difference to the distracting effects of computers, the new research suggests that writing by hand allows the student to process a lecture’s contents and reframe it — a process of reflection and manipulation that can lead to better understanding and memory encoding.
For experienced keyboardists, typing is generally much faster than writing. Handwritten notes, then, require more triaging of content, and, therefore, more active attention to what's most important.

Of course, typing's advantage in speed brings plenty of plusses. That's why I made fluent typing one of this year's first home-schooling priorities. My daughter now types super-fast. But she still writes about half of her assignments by hand.

Friday, June 6, 2014

Math problems of the week: 4th grade Investigations vs. Singapore Math

1. The final fractions problem set in the 4th grade TERC/Investigations Student Activity Book, "Fraction Cards and Decimal Squares" Unit (the final unit on fractions) [click to enlarge]:



2. The final fractions problem set in the 4th grade Singapore Math Primary Mathematics 4B Workbook, "Fractions" Unit (the final unit on fractions) [click to enlarge]:



Wednesday, June 4, 2014

Autism Diaries: "We don't want another Hitler!"

J turned 18 yesterday, and, for the first time in a very long time, awoke before we did. Instead of having to be dragged out of bed as usual, he showed up in our room with a huge smile on his face, fully dressed and ready to go to school--and ready to appreciate, among other things, his newly-acquired political powers.

It used to be the prospect of driving that excited him (and horrified us); now it's voting. He can't wait for November, all the time realizing that, statistically, his one small vote counts for little. His one big motivator--and his recurring refrain: "We don't want another Hitler!"

"Is that why we need to learn history?" is his abiding follow-up. Thus does preventing another Hitler motivate, in his eyes, the five mornings per week that we sit on the train together on the way to school and read his history textbooks--world history last year; American history now.

(The Holocaust, interestingly, is covered in both of these Pennsylvania-approved textbooks--in spite of a recent report showing that graduates of Pennsylvania high schools know woefully little about it and suggesting that the answer is mandated, Holocaust-specific instruction.)

Interestingly, J hasn't brought up driving in a while; the prospect of him voting, in contrast to this, horrifies me not one whit.

Monday, June 2, 2014

Autism Diaries: Irony and Jokes

"Yeah, right."

This expression so routinely expresses ironic skepticism that linguists have joked that, in addition to having a double negative, English also has a double positive.

Perhaps because I wrote my dissertation on irony, I'm prone to use irony even with J. And, increasingly, he gets it, as we see in this recent text message exchange:

J: I'm stuck in traffic and am going to be home really late.

Me: Yeah right.
J: No, I'm telling the truth.
J is as aware of the irony in "yeah, right" as I am of the fact that whenever he tells me he's going to be late, he's actually going to be on time. He's aware of my irony even though irony is supposed to be particularly difficult for people with autism.

Of course, "yeah, right" is the kind of conventionalized irony that, as with the conventional metaphors I discussed earlier, can be learned and memorized in the same way that vocabulary words are. But learning that "yeah, right" can mean something like the opposite of its apparent meaning can help you realize something much broader: namely, that you should always be on the alert for other instances in which people may mean the opposite of what their words appear to imply. And I'm guessing there's nothing inherent to autism that should prevents people from being alert to this. 

J certainly is. Recently, for example, my Monday evening class was canceled, as it often is, and J asked me, as he often does, "Why aren't you at class"? He prefers for me to be at class, since while I'm there, I can't bug him about finishing his homework.

This time, instead of giving him the response he was expecting, I said "I'm staying home because I love you very much and want to spend more time with you."

"You're joking!" was his immediate reply, a strange smile on his face.

"But I do love you very much!"

"You're joking about wanting to spend more time with me!"

He, too, has been playing around with verbal jokes lately--another supposed deficiency of people on the autistic spectrum. Here's another recent text exchange:
Me: Put on your shoes and docks.

J: What are docks?

Me: Can't you read typos?

J: Can't you read jokes?
Part of what's going on here is ego: J's been realizing that conversations can be competitive. Newly aware of this thing called having the last word, he wants it to be his.