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

I. A 2nd grade (TERC) Investigations subtraction assignment, assigned in mid-December:

II. From a similar point in the 2nd grade Singapore Math curriculum (Primary Mathematics 2B, pp. 20-21):

III. Extra Credit:

What kinds of answers can Investigations parents expect from their children when they ask (as per the Family Connection above) "Is it helpful to have the counting-by-5's and county-by-10's numbers shaded on the 100 chart? Why?"

Should parents worry if their child is one of those who "counts up or back by 1s" rather than "by 10's and then 5's," or "by 10's and then 1's"?

Will geeks inherit the earth?

Is it a left-brain world with right-brainers far afield or a right brain world with left-brainers far afield? Will right-brainers rule the future, or will geeks inherit the earth? The publishing world abounds with contradictions (some of which I address here).

Its latest contribution, Geeks will Inherit the World, argues that those who were outsiders in high school will ultimately prevail. How does this mesh with my claim that the world is fraught with right-brain bias?

For one thing, most of author Alexandra Robbins’ so-called “geeks” aren’t socially awkward, analytically-oriented left-brainers. Her outsiders include a “popular bitch,” a victim of racism, and a social under-achiever who is openly gay. They're certainly ostracized, at least by some, but not because they’re geeks.

Furthermore, much as they might deserve to, it’s not clear whether any of the particular individuals Robbins profiles will actually inherit the earth—or, at least, more of the earth than their more social counterparts. First, we only see these characters during a year of high school, and not years, let alone decades, into the future.

Second, what does it mean to inherit the earth, anyway? It's easy to proclaim that non-conformists are the true heroes and original thinkers, but how do their lives actually turn out? Many may excel in their chosen careers, but do they really come out on top overall? How many truly non-conformist heroes become leaders? How many truly original thinkers succeed in broadly publicizing their thoughts? In general, no matter how heroic you are in private, and no matter how original your ideas are, conformity, social connections, and social charisma play an enormous role in public success—and even, according to some, in success within your chosen field.

Third, the notion that geeks do better later on in life may conflate two different senses of “doing better later.” Doing better later may mean doing better in the future than other types of people do, or it may merely mean doing better than you did earlier. Many high school outsiders, even those who are properly called “geeks,” are simply late social bloomers, and become, for better or for worse, much more sociable later on--eventually enjoying many of the same perks as their more socially precocious peers.

None of this subtracts from my celebration throughout this blog of left-brainers and of left-brain talents. But, whether the arena is the k12 classroom, Washington politics, corporate America, popular entertainment, mainstream publishing, or even academia (think teaching ratings, collegiality ratings, and the role of connections in successful grant applications), a world that truly celebrated left-brain talents would be look quite different from the world in which we currently live.

Self-esteem that makes a difference

Throughout this blog I've critiqued self-esteem boosting strategies as either backfiring on unsocial children, and/or coming at the expense of academics--such that, for example, American school children have higher self-esteem about their math skills than their Japanese counterparts do, even though they perform significantly significantly worse on international math tests.

But I don't intend to rule out meaningful ways to boost self-esteem that actually raise academic performance. A recent article in the Philadelphia Inquirer reports on one highly successful strategy--one that addresses the "stereotype vulnerability" experienced by members of certain minority groups:

College freshmen read the results of what they were told was a survey of upperclassmen, together with ostensible first-hand reports of navigating college life. The stories detailed how, at first, the juniors and seniors had felt snubbed by their fellow students and intimidated by their professors, but their situation had improved as they gained self-confidence. The freshmen were asked to write essays explaining how their own experiences dovetailed with those of the upperclassmen; they then crafted short speeches that were videotaped, supposedly to be shown to the next generation of undergraduates. The exercise took about an hour. Meanwhile, a control group was reading and writing about an unrelated topic.

This simple experience didn't affect how well white students in the study performed academically. That's not surprising, because whites aren't hostage to stereotypes of inferiority. But it appeared to change the arc of the minority students' college lives. Over the next three years, their grade-point averages steadily rose, compared with the GPAs of a similar group of black undergraduates: the control group that didn't participate in the "social belonging" exercise.

At graduation, the grades of the students in the experiment were a third of a point higher than the grades of the students in the control group; that's the difference between a B-plus and an A-minus average. Twenty-two percent of the minority participants, but only 5 percent of the control group, were in the top quarter of their class; only a third of them, compared with half of the control group, wound up in the bottom quarter. What's more, they were substantially less likely to have become sick, and more likely to report being happy, during their undergraduate years than the other minority students.

The effects of this strategy are so heartening, and so powerful, that it should be tried at schools around the country. But notice how much more meaningful this sort of self-esteem boosting is--dramatizing, as it does, the power people have to overcome life's adversities--than are the more typical self-esteem boosting tactics of vacuous praise, "cooperation" over competition, and me-focused assignments.

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

I. The first decimal-to-fraction conversion exercise in the 4th grade Singapore Math Primary Mathematics 4B workbook, p. 32 [click to enlarge]:

II. From a similar point in the 4th grade Investigations curriculum, where "you may use your calculator" [click to enlarge]:

III. Extra Credit What fraction of Investigations vs. Singapore Math students will need to bring calculators to restaurants?

The virtues of explaining your answers

Throughout this blog I've complained about requirements that students explain their answers to easy Reform Math problems. What is gained by making children put into words that which they can easily do in their heads? Why should those who repeatedly get the right answers lose points for not explaining how they got them?

Does that mean that there's no virtue to explaining your solutions? Not when it comes to harder problems, e.g. multi-step problems in algebra and beyond, where there's actually some work to show. Furthermore, although I believe it's quite possible to have a perfectly adequate mathematical understanding without being able to verbalize what you're doing (consider all those language-impaired, mathematically-gifted children on the autistic spectrum for example), I also believe that there are different levels of understanding, and that the ability to verbalize things indicates something about how deeply you get it. When it comes to math in particular, it strikes me that there are at least two major dimensions to understanding: one is how abstract you can go; another is how well you can explain it to others (a third might be how well you can visualize it).

Throughout my middle school and high school years, the resident student math genius would constantly pester me about why I bothered taking math classes when it was clear to him that I didn't really understand what was going on. He couldn't appreciate that I understood things at a level that was meaningful to me, if not to him.

Weak though I am compared to math buffs, I think my relative strength is in being able to explain how things work to others (again, not in that inane "explain your answer" sense, but more in terms of verbalizing math concepts; of verbalizing what's going on logically, as I try to do when I teach math to my kids). Driving this home in a very odd way have been math classes I've taken with classmates who outperformed me on tests, but would turn to me for verbal explanations of what was going on. I helped them out, and then did ultimately did worse than them overall.

Given a choice, I actually prefer to have strengths in the ability to verbalize rather than in the ability to solve really hard problems. I like being conscious of what's going on--and, as a linguist, I've come to believe that, for those of us who aren't language impaired, there's a very strong connection between verbalizing things and being fully conscious of them.

I wonder whether those who can solve really hard problems but can't put into words what they're doing are truly conscious of what's going on. Again, I mean something different here from the idea that if you can't explain your answer you don't know what you're doing. If you can solve hard problems, at some level you must know what you're doing; you just may not be fully conscious of it, and that lack of full consciousness doesn't really matter unless you happen to be a teacher and it impairs your ability to teach others.

When "relevance" is effective

Throughout this blog I've repeatedly lamented the relentless attempt by educators to make things relevant to students' lives and to the real world. In reading assignments, this deprives children of windows into exotic worlds. In writing assignments, it deprives them of imaginative ventures in fantasy. In math, it limits their engagement with rigorous, abstract math problems.

But there is a way to move beyond All About Me and still keep things relevant--in a good way. Indeed, this kind of relevance is at the heart of good teaching. Rather than making things relevant by keeping them close to home, why not make things relevant by taking children there? In other words, rather than only asking "How is this like your life?", instead say "Imagine if this life were your life." What would it be like to live as a nomad in Arabia, or as a child-prince(ss) in ancient Japan, or as a monk in England, or as a knight?

Tellingly, the best way to facilitate this kind of personal connection isn't to make the material as similar to the child's life as possible, but to make it as vividly detailed as possible--especially in all the exotic ways it differs from the here and now. And isn't this, after all--this imagining of other worlds--what a good education is all about?

The harmful effects of uttering "think" and "know" (at least when you're under 4)?

Some people argue that it's harmful for children to use the standard algorithms of arithmetic before they understand how and why they work. Supposedly "premature" use blocks future understanding.

By extension young children are harming themselves all the time--specifically when they use words like "think" and "know." That's because, as it turns out, children younger than 4 don't understand the meanings of these words--and, in particular, what distinguishes "think" from "know"--even though they use these words frequently.

I'm reminded of this phenomenon every time I teach my Autism, Language and Reasoning class and revisit the following passage from Katherine Nelson's "Language Pathways into the Community of Minds" (from Why Language Matters for Theory of Mind):

At 3 years, most children studied produce the focal cognitive terms think and know in the course of everyday conversations, at least occasionally...It is not until about 4 years that children appear to use these terms to indicate specific mental states, distinguishing between the meanings of think and know on the basis of certainty...It is not until the early school years that tests of comprehension show clear discrimination among the terms think, know , and guess. And even in the late elementary years, children do not demonstrate understanding of the full range of distinctive meanings of know.

As Nelson explains, the earliest uses of these terms can be described as "without meaning." Does such meaningless use interfere with later understanding? Quite the contrary:

The acquisition of meaning of abstract terms such as mental-state words is best conceptualized in terms of acquiring meaning from use.

Perhaps language isn't the only place where meaningless use is productive rather than harmful; where, in Nelson's words, one can acquire meaning from use. And perhaps if we substitute "acquire" with "construct," even a hard-core Constructivist might consider this possibility.  

Or, at least, isn't it pretty to think so?

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

I. A subtraction problem set from about 3/4 of the way through the 2nd grade (TERC) Investigations curriculum:

II. The second half of subtraction problem set from about 1/8 of the way into the 2nd grade Singapore Math curriculum:

III. Extra Credit

Defenders of Investigations argue that explaining your answers to a smaller number of problems fosters a greater depth of understanding than doing multiple problems using the standard algorithms. How might they explain their answer to America's math woes, and how did they check their answer?

The virtues of creativity

In this blog I've repeatedly complained about wrong-headed priorities involving creativity, explaining answers, learning styles, self-esteem, and relevance. In the next few blog posts, I'm going to argue that each of these things should still play some role K12 education. I begin, today, with creativity.

As regular readers of the blog know, I've repeatedly criticized the practice of grading students on creativity when there is no established protocol for teaching it or for measuring it objectively (beyond the crude, reductionist, and pointless tallying of things like "illustrations per page" and "colors per illustration"). Does that mean that creativity should play no role in K12 education? Not at all. Teachers can, and should, attempt to inspire creativity in their students, and to be creative themselves in their teaching.

But inspiring creativity does not mean simply exhorting students to "be creative." Far more promising is helping students appreciate the creative works of others--in anything from poems, to stories, to essays, to paintings, to clever solutions to math problems. Or giving them inspiring assignments or prompts--e.g., extend such and such a bizarre opening sentence into the first of a story; or extend such and such a random squiggle into a page-sized illustration. Or taking them on exotic outings--topiary gardens; whirligig exhibits; kitchen instrument concerts; or La Compagnie Transe Express.

Teachers themselves can be creative, too, even in the driest of subjects, and without reducing the subject matter to meaningless mush. A friend of mine who teaches at St Ann's in Brooklyn--a school whose combination of a classical education and a creative teaching staff is perhaps unparalleled--dresses up every Wednesday like a puritanical schoolmarm, topped with a puritan cap and equipped with a dunce cap, teaches her 5th graders the fundamentals of grammar and style. They love it, and they learn it.

Ironically, at the same time that students are increasingly graded on "creativity," schools are inspiring it less than ever. They're increasingly more likely to punish than to reward teachers who don't adhere to their scripted lines. And, their language-policed, all-about-me curriculum, which prefers mirrors to windows, provides way too few windows, in particular, into creativity at its most inspiring.

Perhaps that's why America is less and less a land of innovation--except for first-generation immigrants, assuming they don't go back home.

Cooperative learning in online classes

So obsessed is the education establishment with cooperative learning that even online classes attempt to have students working in groups, even though the greatest potential benefit of group work, namely brainstorming or the spontanous bouncing of ideas back and forth, is not really possible over email, file exchange, and discussion board venues that online classes provide. Furthermore, when you never actually meet the people you're working with, there's less social pressure to fully cooperate with them.

What I've seen as a result is almost no collaboration and a great deal of frustration. Students typically divide up the work immediately and do their separate parts, spending little or no time giving one another feedback once they're done--perhaps because of that universal human tendency to finish things at the last minute. If one person's part was done poorly, lowering the group grade, the other group members are understandably demoralized.

If I alter things so that each person gets a separate grade for their part, I take away any incentive for collaboration, unless I try to set things up such that part of what each student does builds on something that one of their group members has done. But this falls apart whenever any group member fails to finish their part in time for their partners to add their pieces.

There are already reasons aplenty to question our obsession with groups in actual classrooms with actual classmates; when everything is virtual, what arguments remain?

Math problems of the week: 8th Connected Math vs. enrichment math

For years I have turned a blind eye to what happens inside J's math class--so long as I get to decide what we do at home. Just recently I've learned what I've been missing. Without my lifting a finger or wagging a tongue, a minor miracle has occurred. The interim math teacher who took over mid-year appears to have been giving J some above grade-level classwork. Given what the alternative would look like (the school uses Connected Math), I'm all the more appreciative. 

Consider the final algebra problems in 8th grade Connected Math vs. the worksheet that J brought home three days ago:

I. The final four problems in the final algebra chapter in 8th grade Connected Math: "Frogs, Fleas, and Painted Cubes," p. 87:

1. What patterns would you expect for a quadratic function

a. in tables of (x, y) value pairs?

b. in graphs of (x, y) value pairs?

c. in equations relating x and y?

2. How are equations, tables, and graphs for quadratic relations different from those for

a. linear relations?

b. exponential relations?

3. For a quadratic relation in the form ax² + bx + c, how can you tell whether the graph will have a maximum or minimum point?

4. What strategies can be used to solve quadratic equations such as 3x² - 5x -3 and x² + 4x = 7 using

a. tables of values of a quadratic function?

b. graphs of a quadratic function?

II. The last four problems in J's sheet: (source unknown)

Write the following functions in the form y = a(x + h)² + k. Find the vertex.

y = x² + 4x -9

y = x² - 6x + 16

y = x² - 8x - 1

y = x² + x + 2

III. Extra Credit

Consider the likely performance of a language-delayed, high functioning autistic child on each problem set. How might estimates of mathematical ability in autistic children depend on whether the instrument involves Reform Math problems?

Everything but the curriculum

Recent places where the word "curriculum" is conspicuously absent:

Waiting for a School Miracle (Diane Ravitch)

What do Philadelphia Schools Need Most? (Philadelphia Inquirer poll)

GrammarTrainer: now in an Ipad incarnation

Thanks to a team of Drexel Digital Media students who made it their senior project, GrammarTrainer is now a (very preliminary) Ipad App.

(The main GrammarTrainer site is as before).

Honors classes for all?

In his most recent Washington Post column, Jay Matthews proposes a new way to de-track schools. Instead of eliminating the honors track (the typical strategy), why not eliminate the lowest track instead and allow regular students into honors classes? In support of this, he cites an anecdote from Alexandria Virginia:

Jack Esformes of T.C. Williams High School in Alexandria mixed seven AP students with 21 regular students in each of the five government course sections he taught each year. Nothing was dumbed down for the AP students. The regular students received less homework, but once they discovered they were often as clever in class as the alleged smart kids, some of them switched to AP. Many of them told me they liked the challenge of being taught at such a high level.

Noting that only five county high schools in Fairfax County still have three tracks and that "those will disappear next school year," Matthews asks:

Why not show that Fairfax can do even better than other systems? The county should keep honors, eliminate the basic course, and give everyone’s kid a chance at a head start in college, or life.

But as Barry Garelick of the Coalition for World Class Mathematics points out:

The fact that non-honors courses are watered down and unchallenging is unfortunate, but is an artifact of the poor preparation many students receive in K-8. The elimination of formal "tracking" (into programs such as vocational, general, and college prep) resulted also in the elimination of ability grouping. This translated into full inclusion, and brought about the so-called differentiated instruction, in which instruction is 1) not differentiated and 2) not very often given. Although Mr. Mathews has maintained that the theories of teaching taught in ed schools stays in ed schools and that they don't make their way into real classrooms, he is tragically mistaken. All he need to is read the accounts of parents in newspapers to see that the "math wars" and other battles of the education wars are more than "two groups of smart people calling each other names" as he once intoned to me.

Students who do not have access to outside help (parents, tutors, or learning centers like Sylvan) are held hostage to poor educational practices and are thus bound for the non-honors track in high school. The non-honors track could be much more challenging than it is, and there can still be an even more challenging honors track. Mathews is quite rightly longing for more challenging courses in high school, but is not acknowledging some of the major reasons why they are offered only in the honors track. 

For more on this, see Garelick's most recent piece for Education News.

Crowds vs. herds

In my book I draw a distinction between "cooperation" and "collaboration," defining the former as people working while interacting, and the latter as people working on joint projects, but not necessarily in one another's presence or with much productive interaction. In collaborations, after the work is divvied up, participants might spend the majority of their time working independently. 

I argue, furthermore, that this is what typifies most successful real-world collaborations. Except for those of us working on construction sites or film sets, we tend to get most of our work done at desks in private offices or cubicles; not at conference tables.

It turns out that there is a good reason for this. In an article in last weekend's Wall Street Journal, Jonah Lehrer reports that:

The good news is that the wisdom of crowds exists. When groups of people are asked a difficult question—say, to estimate the number of marbles in a jar, or the murder rate of New York City—their mistakes tend to cancel each other out. As a result, the average answer is often surprisingly accurate.

But here's the bad news: The wisdom of crowds turns out to be an incredibly fragile phenomenon. It doesn't take much for the smart group to become a dumb herd. Worse, a new study by Swiss scientists suggests that the interconnectedness of modern life might be making it even harder to benefit from our collective intelligence.

The experiment was straightforward. The researchers gathered 144 Swiss college students, sat them in isolated cubicles, and then asked them to answer various questions, such as the number of new immigrants living in Zurich. In many instances, the crowd proved correct. When asked about those immigrants, for instance, the median guess of the students was 10,000. The answer was 10,067.

The scientists then gave their subjects access to the guesses of the other members of the group. As a result, they were able to adjust their subsequent estimates based on the feedback of the crowd. The results were depressing. All of a sudden, the range of guesses dramatically narrowed; people were mindlessly imitating each other. Instead of canceling out their errors, they ended up magnifying their biases, which is why each round led to worse guesses. Although these subjects were far more confident that they were right—it's reassuring to know what other people think—this confidence was misplaced.

The scientists refer to this as the "social influence effect." In their paper, they argue that the effect has grown more pervasive in recent years. We live, after all, in an age of opinion polls and Facebook, cable news and Twitter. We are constantly being confronted with the beliefs of others, as the crowd tells itself what to think.

...

This research reveals the downside of our hyperconnected lives. So many essential institutions depend on the ability of citizens to think for themselves, to resist the latest trend or bubble. That's why it is important, as the Founding Fathers realized, to cultivate a raucous free press, full of divergent viewpoints.

The ideal, then, isn't group think, but independent thinking followed by a compilation of people's thoughts. 

Jonah Lehrer, however, neglects to mention one reason why the social influence effect has grown in recent years:  all the time that today's students are forced to work in groups in K12 classrooms, and, increasingly, in college classrooms as well.  In this case it's not the hyperconnectedness of our wired and wireless lives that's responsible, but the group think of the education world, with its systematic confusion of "cooperation" with "collaboration."

Math problems of the week: grade 3 Investigations vs. French Math

I. From Landmarks in the Hundreds, a student activity public for grade 3 Investigations (TERC):

Calculator Skip Counting

Choose a number to count by. Pick one you think will land exactly on 300.
Skip count by this number on your calculator.
Does it work? If so, write how many of your number it takes to get to 300.

Numbers               Did you land on         If it worked:
we tried 300         exactly?                       How many in 300
--------------------------------------------------------------------------------------------------

______________ Yes No ________________
______________ Yes No ________________
______________ Yes No ________________
______________ Yes No ________________
______________ Yes No ________________


... (18 iterations in all)

--------------

II. From Cahier d'activités mathématiques, CE2 (3rd grade), translated from the French:

Goal: Calculate in your head.

Observe:

32 x 4 = 32 x 2 x 2
32 x 4 = 64 x 2
32 x 4 = 128
32 x 40 = 1280

5 is half of 10
12 x 5 is half of 12 x 10
12 x 10 is 120
12 x 5 is 60

50 is half of 100
12 x 50 is half of 12 x 100
12 x 100 is 1200
12 x 50 is 600

1. Solve in your head:

43 x 2 = _____ 82 x 2 = _____ 16 x 10 = _____ 35 x 10 = _____
43 x 4 = _____ 82 x 4 = _____ 16 x 5 = _____ 35 x 5 = _____

2. Solve in your head:

24 x 4 = _____ 43 x 4 = _____ 23 x 5 = _____ 74 x 5 = _____
24 x 40 = _____43 x 40 = _____23 x 50 = _____74 x 50 = _____

3. Solve in your head:

33 x 4 = _____ 13 x 40 = _____ 26 x 5 = _____ 14 x 50 = _____
120 x 4 = _____ 21 x 40 = _____ 31 x 5 = _____ 15 x 50 = _____

---------

III. Extra Credit

A recent commenter on my review of Diane Ravitch's latest book writes:

We keep wondering why the USA is totally opposed to looking at successful
countries. Good old American exceptionalism concludes that right wing solutions
are required even though none of the successful nations stress testing,
privatization, teacher bashing, merit pay, or any of the other conservative
solutions.

What key factor does our commenter omit?

---

Note: I've recycled this math comparison from a much earlier post. I thought now was a good time to recall that Singapore is only one of many countries whose curriculum excels over ours; it's also a good time for me, personally, as I've literally run out of time for blog-related activities this week.