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

A continuation of last week's problems-- the next pages in the chapters on circles sections of 6th grade Connected Math and Singapore Math:

I. First set: from Connected Math "Going Around in Circles" section:

II. Second set: from Singapore Math Primary Mathematics 6 Workbook "Circles" chapter

[click to enlarge]:

"Students will Plan an investigation to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object"

There's something deeply boring about the way the Common Core Standards are written--so much so that nearly every time I read them I start to space out. For skills-intensive fields like math and language arts, perhaps this is inevitable: it's hard to make generalized skills sound interesting. But the same vague tedium pervades the newly unveiled science standards. Here, excerpted from a recent Edweek, are some examples:

Energy: (Kindergarten)
• Make observations to determine the effect of sunlight on Earth’s surface.
• Use tools and materials to design and build a structure that will reduce the warming effect of sunlight on an area.

Biological Evolution: Unity and Diversity: (Grade 3)
• Analyze and interpret data from fossils to provide evidence of the organisms and environments in which they lived long ago.
• Construct an argument with evidence that in a particular habitat some organisms can survive well, some survive less well, and some cannot survive at all.

Motion and Stability: Forces and Interactions (Middle School)
• Apply Newton’s Third Law to design a solution to a problem involving the motion of two colliding objects.
• Plan an investigation to provide evidence that the change in an object’s motion depends on the sum of the forces on the object and the mass of the object.

Engineering Design (Middle School)
• Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.
• Develop a model to generate data for iterative testing and modifications of a proposed object, tool, or process such that an optimal design can be achieved.

Earth and Human Activity (High School)
• Evaluate competing design solutions for developing, managing, and utilizing energy and mineral resources based on cost-benefit ratios.
• Use a computational model to make an evidence-based forecast of the current rate of global or regional climate change and associated impacts on other Earth systems.

Science is one of those fields that should be inherently interesting to nearly everyone. But what is it that makes someone want to study, say, biology or earth science? Is it so they can learn how to construct arguments about habitats, or is it so they can learn about the organisms that make up a habitat?  Is so they can learn how to evaluate competing design solutions for developing mineral resources, or is it so they can learn about minerals and how people use them?

There are other problems with these content-poor standards. Make goals vague enough, as I've argued earlier, and nearly any strategy can justified as serving them. Which strategies then prevail isn't determined by the goals themselves, but by who's in power. In the highly problematic world of education, this dynamic makes the Common Core part of the problem rather than the solution.

The other downside of vague standards is that it's hard to know how to implement them. Ideally they give schools and teachers flexibility, but in an arena so pervaded by one particular ideology, educators must constantly second-guess what is the "right" way to, say, teach kids how to "construct an argument with evidence that in a particular habitat some organisms can survive well, some survive less well, and some cannot survive at all."

Americans tend to be highly critical of the centralized curricula of countries like France, in which all schools go through the same textbooks on more or less the same schedules. But there's something to be said for specific, content-based guidelines. I'm guessing that many teachers would prefer being told exactly what content to cover, and being given the flexibility about how to go about teaching it, to being handed some vague, uninspiring goals and an ideological environment in which only certain strategies are acceptable--strategies that must be justified every day with Common Core-flavored subgoals submitted with every lesson plan and posted on every whiteboard--and that, furthermore, provide no guarantee that students will actually learn and retain anything of actual substance.

A piano students' lament

In his A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, Paul Lockhart opens with an allegory about a musician, who awakens from a nightmare in which the “curious black dots and lines” that “must constitute the ‘language of music’” become the center piece of what has become a universally mandated music curriculum. He proceeds to describe just how tedious this curriculum is for all concerned:

It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.

And then, of course, he famously proceeds to connect this musical nightmare to the way K12 mathematics is supposedly actually taught: all meaningless, mindless drill.

As Alfred North Whitehead writes back in 1911, however, mindlessness is often a virtue:

It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.

Furthermore, while no sane piano teacher would ever make "curious black dots and lines" the centerpiece of music instruction, some of the best ones give top priority to mind-and-soul-numbingly tedious muscle exercises. I was reminded of this reading an accomplished pianist's recent New Yorker memoir about his "Life in Piano Lessons." Here's an excerpt (the student/narrator is Jeremy Denk, and the teacher is William Leland):

Learning to play the piano is learning to reason with your muscles. One of the recurring story lines of my first years with Leland was learning how to cross my thumb smoothly under the rest of my hand in scales and arpeggios. He devised a symmetrical, synchronous, soul-destroying exercise for this, in which the right and left thumbs reached under the other fingers, crab-like, for ever more distant notes. Exercises like this are crucial and yet seem intended to quell any natural enthusiasm for music, or possibly even for life. As you deal with thumb-crossings, or fingerings for the F-shart-minor scale, or chromatic scales in double thirds, it is hard to accept that these will eventually allow you to probe eternity in the final movement of Beethoven's last sonata. Imagine that you are scrubbing the group in your bathroom and are told that removing every last particular of mildew will somehow enable you to deliver the Gettysburg Address.

Of course, a certain amount of grit and gruel also underlies good writing. The Gettysburg Address doesn't just happen, either.

A left-brained approach to getting along politically

It's become a truism that we live in a world increasingly segregated by viewpoint, rarely talking civilly and substantively about politics with those with whom we strongly disagree.

One of the few proposed "remedies" is the open-ended discussion session. Get groups of people with opposing viewpoints talking to one another about how they feel, and they will become more open and respectful towards those with widely differing opinions, moderating their own views along the way. It turns out, however, that such open-ended discussions have the opposite effect, with those on opposing sides digging in and becoming even more entrenched:

Rather than adopting a middle ground, continuing discussion and debate often result in more extreme positions. In such a condition, consensus building is difficult and temporary when it is achieved at all because individual group members tend to shift away from an average attitude rather than toward it.

A recent study finds that a more analytical "left-brained" approach to be more fruitful. The best way to get someone to question and moderate their views, as it turns out, is to ask them to explain in detail how it is that a policy that they either support or oppose actually works. In response to such questions:

They become more moderate in their political views — either in support of such policies or against them. In fact, not only do their attitudes change, but so does their behavior. In one of our experiments, for example, after attempting to explain how various policy ideas would actually work, people became less likely to donate to organizations that supported the positions they had initially favored.

Interestingly, asking people to justify their position — rather than asking them to explain the mechanisms by which a policy would work — doesn’t tend to soften their political views. When we asked participants to state the reasons they were for or against a policy position, their initial attitudes held firm. (Other researchers have found much the same thing: merely discussing an issue often makes people more extreme, not less.)

Why, then, does having to explain an opinion often end up changing it? The answer may have to do with a kind of revelatory trigger mechanism: asking people to “unpack” complex systems — getting them to articulate how something might work in real life — forces them to confront their lack of understanding.

Reviving our country’s civil discourse, in other words, means constantly asking one another for detailed explanations of "how.” Back in the pre-pc, pre-Constructivist Dark Ages, this was routine, particularly in school settings. In my experience, it was what distinguished the best teachers and classmates. With the decline in both the analytical essay and the multiple revisions-feedback loop, I wonder how often today's students--let alone today's adults--are ever asked to flesh out the practical ramifications of their opinions. Indeed, in an age in which even asking someone for a specific example of what they're talking about can totally derail a conversation, many people seem to find it downright rude when their personal opinions are met with anything other than reflexive, unconditional respect--however vacuous this often must be.

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

A continuation of last week's problems-- the next pages in the chapters on circles sections of 6th grade Connected Math and Singapore Math:

I. From Connected Math "Going Around in Circles" section [click to enlarge]:

II. From Singapore Math Primary Mathematics 6 Workbook "Circles" chapter [click to enlarge]:

Letter from Huck: I Guess I’m Just a Cheater

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

With all the scandals about teachers and administrators cheating to raise their students’ scores on standardized tests, Miss Katharine was a bit concerned over my letter of a few weeks ago in which I described how I handled an assessment I had to administer. I told her she didn’t have to worry: no one was out looking for me, and the only knock at my front door was from a young man who was selling magazine subscriptions that would serve some worthy cause which had benefitted him somehow and which slips my mind at the moment.

Recently I was back at the same school where I had administered that Common Core flavored exam and ran into one of the students from the 8th grade algebra class. They were the ones who had demanded to know what “explain your reasoning” meant.

“Oh, you’re the guy who gave us that explanation for how to do the problem that no one could do,” she said. Interesting that the explanation I gave took less than a minute but apparently was enough to get the point across; so much so that I’ve become somewhat of a legendary sub. Which perhaps suggests that explicit and direct instruction might not be the “rote learning” approach feared by otherwise sane and pleasant people.

The “understanding” and “connection” mantras are prevalent in the groupthink that makes up much of the education establishment’s view of math education. It was at play big time with the algebra program I had to use when I was student teaching—a program called CPM algebra. I’ve mentioned it before and it seems appropriate to mention it again what with Common Core surfacing and being interpreted along the ideologies of reform math.

With CPM algebra, students were taught “slope” in a series of discovery lessons that spanned many weeks. They had to make “connections” between tables of values, and equations, how both described the patterns of growth, how the y interecept value helped to draw a graph, how to draw a graph given an equation, and how to determine the slope and y intercept when looking at a graph. All well and good, but the point-slope form of finding an equation was not presented initially; students were left in the dark for quite a while about how to determine the equation of a line given the coordinates of any two points on the line.

I kept to the script of the algebra text as best I could. This turned out to be disturbingly easy. You just went over the previous problems, gave a short intro for the topic of the day to get them going and then assigned the problems in the book for that day. They then worked on them in groups and Tina and I circulated to answer questions. I could see how if a teacher were lazy (which I hasten to say Tina, my supervising teacher, was not), they wouldn’t have to do very much, and lesson plans were pretty much automatic. Tina bought into the program; she believed in it and worked hard to make it work. But it also seemed she was seeing what she wanted to see. There were times when, circulating around the classroom, she would say to me “They’re getting it! They’re making connections!” Yet, there were students who seemed quite confused and some of them knew that if they pushed me hard enough during my circulatory tour of the classroom, my hints (given while Tina was working with other students and out of earshot) would often tell them what they were supposed to discover. Maybe the connections she was seeing the students make were because of that.

The reason why CPM eschews procedures like the point-slope method of finding an equation is that it supposedly gets in the way of true understanding. I heard this recently from a teacher, in fact: “Kids buy into the slope formula, plug in numbers, do the calculations and yet they still do not understand what they are doing. They are simply memorizing yet another formula for some unknown reason.”

I don’t know. I just don’t find slope all that terribly difficult to understand. Similar triangles and proportion seem to explain why the slope of a straight line is always going to be the same for any two points you pick. But people seem to think that if a kid is doing procedures without “complete and true understanding” he's doomed to a life of failure. It is as if the moment a student stops doing all the intermediate steps/algorithms and fails to make the appropriate connections each time, then he or she is using a trick or rote memorization to jump to the end result and not using understanding or strategies to solve something.

I recall one time when student teaching, talking to a fellow math teacher. This was during the time that Tina was gone for two weeks when her father passed away. The teacher was telling me about the math teaching philosophy. “Tina always says we can teach them how, but what’s really important is that they understand ‘why’". As she told me that, she looked to me as if she wanted me to say something. I sensed that underneath it all, she felt the same way I did—but was afraid of being disloyal.

I think of that hallway conversation often. I think of it when I see the posters for the Standards for Mathematical Practice on the walls of the various classrooms in which I substitute. They make me feel as if I’m back as a student teacher, trying to figure out the best way through a ridiculous program. And despite my strong beliefs about what I talk about here, I still feel like I’m cheating when I teach the way I see fit, as if maybe 1) there's something wrong with me, or 2) I’m being disloyal. I’ve only met a few teachers who have told me they don’t like the trends I’ve been describing in math education. They’ve usually been teaching for over 30 years and are about to retire.

Autism Diaries XLVIII: Proud parenting moments

In honor of Mother's Day, it seems fitting to share some of my more memorable autism parenting moments--moments when I said or did things I once thought I'd never do or say to any child of mine. Namely:

1. Shamelessly manipulating my child through outright lies.

At one point, for example, just after an ill-fated trial with Ritalin, J acquired the disturbing tick of periodically licking his palms. Lick the right one, then lick the left one, then do a couple of hand flaps. He'd never had a tick before and, except for definitively ruling out further medication, we were at our wits end.

But then it occurred to me to tell him about a paralyzing disease, called polio, that lurks on the surfaces around him and that he could pick up by licking his exposed skin. It took just a few reminders and he stopped. I didn't even need to show him pictures of people in iron lungs.

2. Revenge fantasies.

Back in the Terrible Twos, which extended nonstop into the Terrible Fives or even the Terrible Sixes (it's hard to remember now when the trashing of rooms, the puncturing of air mattresses and bicycle inner tubes, the booby traps, the adulterated food, the pushing and grabbing and eye-poking of family members and classmates, and the constant running off into stores, crowds of people, and busy streets [looking over his shoulder with a grin and a glint in the eye], finally abated), I used to sit awake and night, listening to J's maniacal laughter in the bedroom above me as he reminisced about the day's malicious escapades, and dream up what it would take to not only quell the mischief, but also exact revenge.

How could I force him into the kind of Time Out that normal kids somehow consent to? (Bring in his car seat and strap him into it say, in front of an American Sign Language video tape... or down in the dark basement?).

How could we buy ourselves just a tiny bit of the rest and relaxation that the parents around us could take for granted at certain times and places--say at the neighborhood swimming pool? (Stop him from running at top speed, nonstop, around the edge of the pool, vexing the life guards and parents of unsteady toddlers and us as we chased after him, by inserting him into an inner tube and pushing him out into the middle of the pool--exploiting the fact that he didn't yet know how to swim or paddle?).

How could I instill in him a regret that would substitute for the guilt he was proving to be completely incapable of feeling? (Make him use the air mattress he'd permanently deflated with a sewing pin? Take away his cookie and give it to his pushed, grabbed and/or eye-poked brother to eat in front of him?).

3. Making fun of my child's handicap.

This--my most memorial moment of all--occurred after a 3 1/2 hour Ferry Ride from Bar Harbor to Nova Scotia in the wake of Hurricane Bill, when huge swells caused nearly the entire boat to get sick--except for J. He kept asking his green-faced Daddy for money to buy snacks from green-faced clerk at the snack bar, and, eventually, had to go to the bathroom to relieve himself of all the green lemonade he'd consumed.

It was there that he started to notice that people were getting sick--and, naturally, started chuckling at all the "throw uppers." Only after it was all over did it occur to him to ask why so many people fell ill and why he was spared. I explained that the same thing that made him deaf--the absence of hair cells in his inner ears--also kept him from getting those out-of-balance signals that sometimes cause motion sickness.

It was hard to say which revelation delighted him more: that he had a special advantage, or that everyone else in the ferry had suffered--particularly his parents and siblings.

"Ha ha ha, you threw up!" became his joyous refrain.

After a few iterations of this, I thought of the perfect retort--though I'm not sure I would have called it out if all the car windows hadn't been tightly closed up or if there had been anyone outside the immediate family in the car.

"Ha ha ha, you are deaf!"

I must add that I hastily followed this up with: "Which do you prefer: to be hearing but sometimes get motion sickness, or to be deaf and never get motion sickness?"

His response, immediate and enthusiastic, did not surprise me.