Thursday, September 18, 2014

Math problems of the week: more Common Core-inspired math problems

A continuing series...

From a site called the Mathematics Common Core Toolbox, which advertises itself as

a resource designed to support districts working to meet the challenge and the opportunity of the new standards. Here you will find tools and instructional materials that help you to better understand and to implement the CCSSM.
Here are two (part A and part B) of its 7th grade tasks, designed to align with the following math Standards:
MP.3    Construct viable arguments and critique the reasoning of others.
7.RP    Analyze proportional relationships and use them to solve real-world and mathematical problems.
7.EE    Use properties of operations to generate equivalent expressions.

According to the site's explanation:
This two-part task integrates content from across the grade level and calls for students to apply proportional reasoning and properties of operations as they critique others’ reasoning. The idea of critiquing the reasoning of others (MP.3) is a new demand in the CCSSM not typically addressed on most current state assessments. This is an important feature of the task, thus making it a “practice forward” task.
According to the CCSSM, students in grade 7 use their understanding of ratios and proportionality to solve a wide variety of percent problems and use properties of operations to generate equivalent expressions. In this task, students must also evaluate two different forms of reasoning and justify their analysis.
In Part a, students engage in a multistep percent scenario; they must deal with both discount and sales tax. This fairly traditional assessment item provides an entry point to set up the reasoning and understanding required in Part b, in which students are presented with two lines of reasoning and must evaluate whether the two are equivalent. The crux of the reasoning rests in their ability to apply properties of operations to algebraic equations.

Tuesday, September 16, 2014

Is book reading really trending upwards?

This past week, headlines about millenials screamed from the LA Times: Millenials Read More Books Than Their Elders, Study Finds; and from the Atlantic: Millennials are reading more books than the over-30 crowd, Pew found in a survey of more than 6,000 Americans.

And, from NPR:

As it turns out, the generation that has grown up in the age of technology has a fondness for a very old-fashioned habit - reading. According to a new Pew Research Center report, those under 30 were more likely to have read a book in the last year than those over the age of 30. And they're more likely to use the library as well.
Perhaps our schools aren't doing such a bad job fostering enthusiasm for the printed word.

Before leaping to this conclusion, however, let's read the printed word a bit more carefully, starting with the Pew's own website:
Millennials are quite similar to their elders when it comes to the amount of book reading they do, but young adults are more likely to have read a book in the past 12 months. Some 43% report reading a book—in any format—on a daily basis, a rate similar to older adults. Overall, 88% of Americans under 30 read a book in the past year, compared with 79% of those age 30 and older. Young adults have caught up to those in their thirties and forties in e-reading, with 37% of adults ages 18-29 reporting that they have read an e-book in the past year.
There's a rather simple explanation for the slight edge that millennials have over older adults in their reading and library habits: school.
Our surveys have found that older teens (ages 16-17) are more likely to read (particularly print books), more likely to read for work or school, and more likely to use the library for books and research than older age groups.
As far as reading that isn't for work and school goes:
[R]espondents ages 16-29 are somewhat less likely than adults 30 and older to say they read for pleasure; 76% of those younger than 30 read for this reason, compared with 81% of adults 30 and older.
As for library use, if you factor out those millennials who are most likely to be in school, the group differences are hardly surprising:
...[M]any of the library habits and views of adults in their late twenties (ages 25-29) are often more similar to members of older age groups than their younger counterparts. They are less likely than college-aged adults to have read a book in the past year, but are more likely to keep up with the news. In addition, a large proportion (42%) are parents, a group with particularly high rates of library usage. Additionally, library users in this group are less likely than younger patrons to say their library use has decreased, and they are much more likely to say that various library services are very important to them and their family.
Why is any of this headline news? To the extent that there are any significant differences, it's a huge leap to say that they indicate changes in trends. Perhaps 16-17-year olds reading more than older people dates back to the onset of universal high school.

These sorts of snapshot-based measurements of generational differences can lead to all sorts of bizarre conclusions. For example:

1. Millennials are healthier than baby boomers. (After all, they're decades younger).
2. Millenials have lower life expectancy than baby boomers. (The life expectancy today of an American 16-year-old is about 79; that of a 60-year-old is 83)
3. Life expectancy is negatively correlated with health.

How's that for a screaming headline?

Sunday, September 14, 2014

Making things interesting in the Age of Narcissism

Liking Work Really Matters, proclaims the Gray Matter column in this past weekend's New York Times. As author Paul O'Keefe, assistant professor of psychology at Yale-NUS College in Singapore explains:

Some of us are lucky enough to find our goals interesting — but that won’t always be the case. If interest is critical, how can we take a boring activity and make it interesting?
The answer, as O'Keefe presents it, turns out to be a complete validation of two things education experts have long been promoting: personal connections and group work:
Research by the psychologists Chris S. Hulleman of the University of Virginia and Judith Harackiewicz of the University of Wisconsin suggests that for most of us, whether we find something interesting is largely a matter of whether we find it personally valuable. For many students, science is boring because they don’t think it’s relevant to their lives.
With this in mind, the researchers asked high school science students to periodically do some writing over the course of a semester. They randomly selected half of them to summarize what they had learned in their class. The other half wrote about the usefulness of science in their own lives, thereby making it personally relevant and valuable.
At the end of the semester, the researchers found that, compared with those who simply summarized the material, the ones who reflected on its personal relevance reported more interest in science — as well as significantly higher grades, on average by almost a full grade point. This was particularly true for those with the lowest expectations for performing well in their class.
Research also shows that social engagement in activities can foster interest. In a study I co-wrote in the Journal of Educational Psychology, we had middle school students play a math-focused video game either alone, in competition with another student or in collaboration with another student. Compared with those who played alone, those playing with a partner reported greater interest in the game and a stronger desire to master it. [Emphasis added]
There are several problems here. First, there's the depressing (or cynical) assumption that people only find personal value in that which is relevant to their lives. That would appear to rule out entire fields--for example, ancient history, esoteric languages, or abstract mathematics, i.e., pretty much everything removed or abstracted from the here and now.

Those who trumpet personal relevance as the only route to interesting ignore the possibilities that:

a. not everyone is a narcissist, and
b. the features that make something interesting often stem from their irrelevance to our personal lives.

Indeed, I suspect that the least narcissistic and more intellectually curious of students find that these personal connections assignments detract from what's actually interesting. So is making everyone write personal connections (as many teachers, in fact, already do) really the best strategy? Why not instead figure out how to generate an outward-looking intellectual curiosity in all students? We might begin with a curriculum that is inherently interesting (i.e., appropriately challenging, tailored to different ability levels, not white-washed by language police, and not based on standardized testing); a curriculum taught by Sages on Stages who themselves are intellectually curious and who know how to make things interesting to others.

Moving from Relevance to Cooperative Learning, it's worth noting how O'Keefe's social engagement study looked only at games (math games), and that, in discussing its conclusions, O'Keefe fails to distinguish between the competition vs. cooperation results. This failure is rather surprising: competitive learning is more or less the opposite of cooperative learning, and it's only the latter that our schools have taken to extremes.

Interestingly, O'Keefe doesn't link to this second study; I tracked it down the abstract here:
The present research examined how mode of play in an educational mathematics video game impacts learning, performance, and motivation. The game was designed for the practice and automation of arithmetic skills to increase fluency and was adapted to allow for individual, competitive, or collaborative game play. Participants (N = 58) from urban middle schools were randomly assigned to each experimental condition. Results suggested that, in comparison to individual play, competition increased in-game learning, whereas collaboration decreased performance during the experimental play session. Although out-of-game math fluency improved overall, it did not vary by condition. Furthermore, competition and collaboration elicited greater situational interest and enjoyment and invoked a stronger mastery goal orientation. Additionally, collaboration resulted in stronger intentions to play the game again and to recommend it to others. [Emphasis added]
Note that collaboration actually worsened game-time performance, and that its only benefits related to how positively the participants felt about the experience (including their reported interest!). I wonder why the Gray Matter column left this out. Perhaps the Times didn't consider it fit to print--perhaps because it isn't relevant to the personal lives, values, and agendas of its intended readers.

Friday, September 12, 2014

Math problems of the week: a Common Core-inspired multi-step word problem

A continuing series...

Today's problem comes from ixl, which promotes itself as "the Web's most comprehensive K-12 practice site" and which, on its math page, explains that

IXL's math skills are aligned to the Common Core State Standards, providing comprehensive coverage of math concepts and applications. With IXL's state standards alignments, you can easily find unlimited practice problems specifically tailored to each required standard.
Here's of its multi-step word problems for 8th grade:

This page provides a randomized series of problems of similar difficulty. What they all have in common is that you can pretty much set things up as you read along. In this problem, for example, each sentence corresponds to a clear-cut mathematical step. Sentence 1: do nothing. Sentence two: we clearly have a bunch of costs to add up. The only complication is doubling that of the lemonade--no big deal for 8th grade. 4.40 plus 4.25--you can do that in your head; add 1.40, also in your head. Total: 10.05 Add the tax (already calculated for us)--easy: 11.05. Then we have the $15 given to the waiter: it's clear before you even get to the last sentence that you're going to be subtracting the 11.05 from the 15.00--again, a "friendly number" computation easily done in your head.

Compare this to one of the challenging word problems for 6th grade Singapore Math:
Each pencil cost $0.30 less than each ruler and each ruler cost $0.40 less than each pen. Weimin bought 2 pencils, 2 rulers and 5 pens and paid $5.45 altogether. How much did each ruler cost?
Though the numbers here are no less "friendly" than those in the Common Core-inspired problem, this problem does not reduce to a series of simple arithmetic computations that you can do as you read it through sentence by sentence. "Each pencil cost $0.30 less than each ruler"--what do you do with that? You could either do it algebraically, defining variables, expressing quantities, and setting up equations, or you could do what Singaporeans do, and draw and label some bar diagrams--once again deciding how best to represent things. In either case, it's helpful to read the whole problem first to see where it's heading. Finally, you have to pay careful attention to which quantity the problem is asking you for, and what this quantity corresponds to in your chosen setup.

Another sort of problem that we see less and less of in contemporary American math are those with multiple simultaneous constraints-- of the sort that require you to set up multiple simultaneous equations with multiple unknowns. Here, once again, you have to read through the entire problem before you set it up, and how best to set it up can take a fair amount of all those things to which today's education experts pay such enthusiastic lip service: grappling, grit, and higher-level thinking.

Most contemporary American math problems strike me as the equivalent of long sentences that consist of simple sentences strung together on a single strand. Their more traditional or Singaporean counterparts are instead like complex sentences with lots of subordination. It's the difference between:
Ten people got on the bus and then two people got on and then 1 person got on and 2 people got off and then 5 people got on and 1 person got off and then 3 people got on and 10 people got off and then 1 more person got on....
Of all the people who got on the bus that morning, the one that made the greatest impression on those passengers who were paying attention--for, given how early in the morning it was, few people were doing so--was...
Even if each sentence boils down to "What was the name of the bus driver?" there's a huge difference in the amount of work you have to do to make sense of it all.

Wednesday, September 10, 2014

Conversations on the Rifle Range 9: Sad Girl, Angry Boy

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 nine:

My six week assignment was coming to a close and in one of the last classes held before the final one, I was explaining how the distributive property can be used to multiply numbers quickly. For example, multiplying 8 x 32 can be thought of as 8 x (30 + 2) = 8 x 30 + 8 x 16, which becomes 240 + 16 = 256. After I had demonstrated this, Steve, a junior in my sixth period class, asked how you would use this technique to multiply 9 x 62. I was pleased at what I thought was genuine curiosity until I realized he was only asking because it was one of the problems assigned in the homework he was working on. Nevertheless, it was better than the usual attitude he gave me in front of his peers so I addressed his question.

“How would you break up 62 into two numbers based on what I’ve shown so far?” I asked.

He thought a moment and offered 60 + 2. “That’s good,” I said and wrote 9 x (60 + 2). “Now let’s multiply 9 by 60 and 2. First, what is 60 x 9?”

“I don’t know,” he said. “I don’t know my nines.”

Steve’s lack of knowledge of multiplication facts was not unusual. Nor was his transfer out of the high school a few days later to move back to Bakersfield. Students were transferring in and out of my classroom since I started. From what I've heard, this pattern would continue almost throughout the whole school year.

On my last day, I gave the unit test in all my classes: Chapter 6 for second period, and Chapter 1 for fourth and sixth period. As was her custom, Elisa came in 15 minutes before class and began drawing one of her dog pictures. Also as was her custom lately, she told me some of her observations of the class. “You know that Patrick is the one starting the paper wad fights, don’t you?” she said. I said I did, though I really did not.

“I think he thinks because you’re just a sub, he can do anything he wants,” she said.

“Maybe so.”

“I don’t understand why you don’t give him a referral.”

For that matter, neither did I. I felt I had to catch him in the act, even though he smart-mouthed me regularly, which was grounds for a referral by itself. But I was new to all this and in a school where many of the students had discipline problems. I figured that by limiting referrals, when I did refer a student it would have more weight—not that any referrals were ever questioned.

The class filed in noisily. Some students asked if we would have a party since it was my last day. I told them no, as I had for the past few days whenever they suggested a party, and got the test underway. I expected them to zip through the test not because they would find it easy, but because many were unprepared and simply gave up—putting down any answer they felt was reasonable.

During the test, Patrick called me over. “Is this right?” he asked pointing to the question “1/0 is _______”. He had written “undefined,” which was correct but this was a test—I couldn’t tell him whether it was right or not.

“I just wanted to know whether I spelled ‘undefined’ correctly,” he said in a loud voice, which produced snickers across the room and rapid writing down of answers among some students.

“Why don’t you just say it louder?” I said. “I don’t think all of the class heard you.” He smirked at this. He stood up to hand in his test, and in doing so, let one of the pages dangle in full view of the girl who sat behind him. “Very clever,” I said grabbing his test and putting it in the completed pile.

As I expected, students finished the test quickly. I knew that things would get out of control rapidly since I had nothing else planned for that day. When about 40 minutes remained, I excused students to go to the library if they wanted, to pass the remaining time. All of the class left except for Elisa, who asked me for a piece of computer paper. “I’m doing a special drawing,” she told me. She had told me once that she had two dogs and had learned how to draw dogs and enjoyed drawing in general. She lived on her aunt’s farm, she once told me. It wasn’t much of a farm, she said; just a little land and a bunch of chickens and some pigs. I recalled a sadness about her when she had told me this. I sensed the same sadness as she sat drawing on my last day.

During the remaining time I worked on my notes for my teacher, Mrs. Lassen, when she returned the following week. I thought of Patrick saying “undefined” loudly and his dangling of the test paper. I could have elected to give him a zero on the test for cheating, but decided to put the incident in my notes. Mrs. Lassen had said she wanted to do the grading of the unit tests. I had mixed feelings about doing this. As fate would have it, at just that moment Patrick and three others came back into the room.

“I thought you were in the library,” I said.

“We were,” Patrick said. “We got bored.”

He sat in the back of the room, slouched down in exactly the same way as the man I saw on back-to-school night. On a hunch, I asked him “Was your father here on back-to-school night by any chance?”

“Yeah, my step-dad was,” he said. “One of the few times he actually did something on his own.” He went on complaining bitterly about his step-father and then became silent. After a moment, still slumped in his chair, he said “Looks like things’ll be different in class when Mrs. Lassen comes back next week.” She was known for her strictness. He added “Thank you for not giving me referrals.” I never expected any kind of gratitude from Patrick and though perhaps I would have served him better by giving him referrals, I silently accepted his thanks. I could see he was an angry boy like so many others at that school.

The bell rang, and I wished the remaining students luck. Elisa came by and presented me with her drawing. “I usually do a drawing of a wolf for teachers who help me out,” she said. “Thanks for answering all my millions of questions. Some math teachers tell me I ask too many.”

“You’ll do fine, Elisa; just keep asking questions and working hard.” On the drawing was the message “Farewell Mr. Garelick”

For the next few weeks I felt that my assignment was a large failure. Over time, however, I’ve summed up my accomplishments as follows: I showed up for every class period, taught to the best of my ability, and tried to be consistent.

For some students, my inconsistent consistency was probably more than they had anywhere else.

Monday, September 8, 2014

All the news that's innovative

More of what's fit to print from last week's New York Times... First we have, from American Historical Association director James R. Grossman, an Op-Ed claiming enthusiastically that:

Fewer and fewer college professors are teaching the United States history our grandparents learned — memorizing a litany of names, dates and facts — and this upsets some people. “College-level work” now requires attention to context, and change over time; includes greater use of primary sources; and reassesses traditional narratives.
Sound familiar? Once again, it seems that our grandparents spent their school days memorizing litanies, whether in history or math. I guess that's why they were such mindless people who only knew how to spew boring, context-free facts, and who lacked the depth of historical understanding so common in today's young adults.

And who, unlike today's highly attentive kids, didn't know how to pay attention to change over time or to reassess traditional narratives.

And who missed out even more by not spending time on the primary sources that today's "little historians" are so well qualified, with the depth and breadth of their historical understanding, to assess and contextualize.

Then we have an article rhapsodizing about "the trend away from classes based on reading and listening passively to lectures, and toward a more active role for students" and explaining how, according to a new study, this "active learning raised average test scores more than 3 percentage points, and significantly reduced the number of students who failed the exams."

[Whoever said that listening to lectures is passive? Of course, the trend in question, so characterized, sounds so much more positive than "the trend away from classes based on reading and listening to lectures.]

It turns out this study involved just one class--an introductory biology class at the University of North Carolina at Chapel Hill--three sections of which "took a more traditional, lecture-based approach, and three demanded more participation by students." These sections were taught by one of the study's authors, Kelly A. Hogan, director of instructional innovation for UNC's College of Arts and Sciences. Hmm, I'm wondering whether Dr. Hogan is more likely to publish a study that supports traditional teaching, or one that supports... innovation? But let's not worry about objectivity. After all, everything seems pretty clear cut:
The more active approach gave students more in-class activities, often done in teams, including sets of online exercises. There were similar online exercises assigned to be completed before class along with textbook reading, intended to force students to think about the material rather than just memorize it, and still others for review after a lesson. Many of the exercises were ungraded, but the instructor could tell whether students had done them.
Particularly unassailable are Dr. Hogan's conclusions:
“In a traditional lecture course, [students are] not held accountable for being prepared for class, and they really don’t need to be, because an instructor is going to tell them everything he or she wants them to know,” Dr. Hogan said. “Would you read a report for a meeting if you knew your boss was going to spend 15 minutes summarizing it for you? I know I wouldn’t.”
Well, I would, if it was something I actually wanted to learn and remember. That's why I read my college history texts in addition to listening (actively) to the lectures. But clearly Dr. Hogan's students are different:
Surveys of students who had taken the class showed that those who had the more active approach were far more likely to have done the reading, and they spent more hours on the work, participated more in class and were more likely to view the class as a community.
But is it really newsworthy that many of today's students won't do the reading unless they are held accountable for it? Or that students are more likely to participate in class if they are given in-class activities to do?

Of course, there are long established ways to hold students accountable for doing the reading--ways that don't involve Hogan's innovations. But these ways are so much less... innovative.

And so much less fit to print.

Sunday, September 7, 2014

So Katharine Beals Has This Idea for Language Arts Class...

I admit it; the title of this blog post is pretty obviously inspired by an article in this weekend's New York Times Magazine. It's entitled So Bill Gates Has This Idea for a History Class..., and actually, I've been inspired by more than just the article's title. Consider this:

As Gates was working his way through the [Great Courses] series, he stumbled upon a set of DVDs titled “Big History” — an unusual college course taught by a jovial, gesticulating professor from Australia named David Christian. Unlike the previous DVDs, “Big History” did not confine itself to any particular topic, or even to a single academic discipline. Instead, it put forward a synthesis of history, biology, chemistry, astronomy and other disparate fields, which Christian wove together into nothing less than a unifying narrative of life on earth. Standing inside a small “Mr. Rogers"-style set, flanked by an imitation ivy-covered brick wall, Christian explained to the camera that he was influenced by the Annales School, a group of early-20th-century French historians who insisted that history be explored on multiple scales of time and space. Christian had subsequently divided the history of the world into eight separate “thresholds,” beginning with the Big Bang, 13 billion years ago (Threshold 1), moving through to the origin of Homo sapiens (Threshold 6), the appearance of agriculture (Threshold 7) and, finally, the forces that gave birth to our modern world (Threshold 8).
As Gates sweated away on his treadmill, he found himself marveling at the class’s ability to connect complex concepts. “I just loved it,” he said. “It was very clarifying for me. I thought, God, everybody should watch this thing!”
He told Christian that he wanted to introduce “Big History” as a course in high schools all across America. He was prepared to fund the project personally, outside his foundation, and he wanted to be personally involved.
This fall, the project will be offered free to more than 15,000 students in some 1,200 schools, from the Brooklyn School for Collaborative Studies in New York to Greenhills School in Ann Arbor, Mich., to Gates’s alma mater, Lakeside Upper School in Seattle. And if all goes well, the Big History Project will be introduced in hundreds of more classrooms by next year and hundreds, if not thousands, more the year after that... Last month, the University of California system announced that a version of the Big History Project course could be counted in place of a more traditional World History class, paving the way for the state’s 1,300 high schools to offer it.
Wow. It never occurred to me to to use my billions of dollars to get millions Americans to take the course I personally found most enlightening. That would be "The Syntactic Phenomena of English," originally taught by the late Jim McCawley of the University of Chicago Linguistics Department. Now, you might think this course is somewhat narrower in focus than "Big History." But language is the basis of everything human, and syntax is the basis of language. Without language, there simply is no history, of any size--small, medium, or big.

Indeed, back when I first sat through Jim's course and read through his two-volume The Syntactic Phenomena of English, with its grand tour through the smallest constituents of grammar, through those Deep Structure to Surface Structure movement rules, through to such phenomena as quantifier score, patches, and syntactic mimicry, I thought to myself, "God, everybody should take this course!"

Inspired by Gates' history initiative, I am prepared to fund this project personally, offering it free to cash-strapped schools from Portland, Maine to Portland, Oregon as a replacement for their traditional language arts curricula. And I'm confident that, as this happens, and as everyone sees how great this course is, the proliferation of "The Syntactic Phenomena of English" around the entire United States of America will reflect what principals and teachers know is best for our school children, and what those school children and their parents most desire from our country's public schools.

Indeed, what the New York Times Article says in conclusion about Bill Gates could also be said of me:
[A]ttempts to paint Bill Gates as a self-interested actor in his education projects don’t make much sense. Joel Klein, the former chancellor of the New York City Department of Education,... laughed off the idea that Gates had an ulterior fiscal motive. “The notion that he has an agenda other than trying to improve education is just embarrassing,” said Klein, describing how Gates continued to contribute — and even increased his contributions — to New York City public schools during Klein’s tenure. “I can’t think there is a malevolent bone in his body.”
Big History may one day become an heir to Western Civ or World History, but that didn’t seem to be Gates’s goal; it was more personal. Really, Big History just seems like a class that he wished he could have taken in high school. But he wasn’t a billionaire then. Now, a flash of inspiration on the treadmill might just lead to something very big.
There isn't a malevolent bone in my body, either, and I'm confident that the flash of inspiration I had when I first took "The Syntactic Phenomena of English" might just lead to something equally big.