All for the good of the children--and their brave new century

The largest textbook company in the world has teamed up with the 16th largest school system in the US--and one of the highest performing ones--to boost students' futures even more by teaching them 21st century skills.

You can read all this partnership in a paper by an Associate Research Scientist at Pearson by the name of Emily Lai, entitled “Creating Curriculum-Embedded, Performance-Based Assessments for Measuring 21st Century Skills in K-5 Students” and published by the American Educational Research Association.

The partnership between Pearson Publishing and the Montgomery County (Maryland) Public Schools, notes Lai, began in 2010 “with the goal of developing Pearson Forward, a digital curriculum featuring embedded assessment and professional development resources,” centered around 21st century skills.

21st century skills, explains Lai, include the following:

Critical thinking (encompassing analysis, synthsis and evaluation), creativity (encompassing fluency, flexibility, originality, and elaboration), collaboration, metacognition, motivation, and intellectual risk taking.

These, she assures us, have been operationalized based on a thorough literature review, which appears, from Lai’s bibliography, to be based predominantly on publications by education journals.

So important are these skills, Lai explains, that they will be assessed every 9 weeks during a 3 week period, each one in the course of 1-2 class periods, through Pearson’s Performance Based Assessment, (a variety of “embedded assessment,” or assessment that is “seamless with instruction”). Assessment tools include “holistic” rubrics, checklists, and student self-ratings. The 8 assessment tasks (assigned every 9 weeks) include “open-ended or ill-structured tasks,” tasks embedded in “authentic, real-world contexts,” and strategies for “making student thinking and reasoning visible.” The goal of all this assessment? “Tracking progress to predict success in post-secondary education.”

As Lai notes:

Each of the skills targeted in the curriculum entails both cognitive and noncognitive or affective components… Cognitive components of these constructs include knowledge and strategies, whereas noncognitive components include attitudes, traits, and dispositions.

Different assessment tasks target different 21st century skills. A task assessing intellectual risk-taking might look at whether, when a student is given a particular reading task, he or she chooses a story that is already familiar to them, or one that isn’t, since:

choice of unfamiliar story [is] arguably more of an intellectual risk than the choice of a familiar story.

Tasks assessing motivation similarly vary by subject (after all, different kids are more or less motivated in different subjects). Pearson’s task for assessing motivation in reading is:

a task that paired a teacher observation tool designed to capture students’ use of strategic behaviors with a student self-rating tool designed to capture more affective aspects of motivation, such as the student’s interest, self-efficacy, and goal orientation.

Tasks assessing metacognition might include open tasks that

allow students to decide what relevant information to use or how to use the information to solve the problem,” as opposed to “closed” tasks that “are characterized by more teacher control and structure.

Such tasks should also make student thinking and reasoning visible, which is

typically accomplished by embedding some sort of informal teacher-student interview into the assessment.

For example, during a ramp-construction project:

Students were encouraged to share their thinking with teammates as they worked together. We provided a set of interview questions for the teachers to pose to individual students as they worked:

How is it going?

What are you doing right now?

Why did you decide to build the ramp this way?

What is working well about your ramp?

What would you change about your ramp?  

As Lai notes:

Using this tool, teachers could observe the extent to which students were able to share their thinking and explain their ideas to others, both key indicators of metacognition at the Kindergarten level.

A task that measures “creativity” could be a time-limited response to a prompt:

We point out aspects of tasks that should not be varied, such as the time provided to students to respond to prompts (when assessing the creativity indicator of fluency, for example)

The most important 21st century skill, of course, is collaboration, and Lai’s proposals here are commensurately elaborate. A task assessing collaboration might involve:

Ill-structured tasks that cannot be solved by a single, competent group member… Ill-structured problems are those with no clearly defined parameters, no clear solution strategies, and either more than one correct solution, or multiple ways of arriving at an acceptable solution.

Collaboration-assessing tasks might also involve time constraints that make it impossible for one person to complete the task:

For example, to assess collaboration in math, teams of 2nd -grade students were required to design and create a mosaic using multi-colored tiles and then to devise and implement a method for representing the data on tile color by creating a graphical depiction of it (e.g., a bar graph showing the number of tiles of each color used to create the mosaic). 

In rating students based on these collaboration tasks, teachers should consider:

the quality of the completed group work project… the student’s ability to work respectfully and productively with others, and the student’s self-reported collaboration skills and contribution to the group.

As well as:

factors related to students’ use of helping behaviors (e.g., communicating respectfully, soliciting diverse opinions).

Citing a 1995 article written by N. M. Webb and published in Educational Evaluation and Policy Analysis, Lai writes:

As Webb explains, assessments that occur in group contexts can fulfill several different purposes. For example, teachers may wish to determine how much a student can learn from collaborating with others, whether a group of students can complete a product together, or whether individual students can communicate respectfully with teammates. Group processes that support one goal may not support another goal. For example, if the goal is to measure a student’s ability to learn from collaboration, then group processes such as co-construction of ideas, identification of conflict, giving and receiving elaborated help, and equality of participation should all be encouraged. In contrast, if the goal of group assessment is to determine whether a group can successfully complete a task on time, then group processes that facilitate student learning, such as trying to ensure equal participation among all group members, may be counterproductive. In this case, it will be more efficient to use processes that maximize group productivity, even if they minimize learning opportunities. Such processes might include letting the most competent student in the group perform most of the work. 

All of this, of course, is for the sake of the children--as the emphases on assessment (as opposed to instruction and remediation) and on predicting (as opposed to influencing) who will be successful in post-secondary education make abundantly clear.

In other words, in no way is it about multi-million dollar backroom deals between powerful companies and school boards that shut out all meaningful input from parents and traffic in educational buzz words and double speak. And in no way is it about branding half-baked assessment tools as “Pearson Forward Performance Based Assessment” tools and then associating them with a famously high performing school district whose current reputation lends them credibility (however much their relentless deployment--every nine weeks over a three week period,1-2 class periods for each 21st century skill--might help diminish this reputation in the future).

The child-centered classroom vs. the child-friendly classroom, II

Another thing I’ve noticed in my classroom observations is how, especially with primary school students, the more child-centered the class discussions, the more children tune out.Young children are often long-winded, inarticulate, unfocused, and slow to get to the point. Many of them routinely raise their hands before they’ve formulated a response. Soon their classmates are fidgeting and talking amongst themselves, losing the current thread and, along with it, the ability to say something relevant when it’s their turn.

Countrary to some people’s intuitions, then, the way to keep young kids engaged in a whole-class activity is to minimize their airtime, to interrupt and redirect them frequently, and, generally, to control the conversation tightly enough to keep things on track and moving swiftly.

It is, of course, the youngest children whose combination of short attention spans and unfocused, long-winded responses require the tightest teacher control. But I saw a similar need while in college and grad school seminars. After all, over-eager students who think they have something important to say that all of us should listen to at great length aren’t specific to primary school. I’ll never forget how frustrated I was with one particular professor who exerted so little control over the graduate student windbags that by the end of the 10-week quarter we were two weeks behind where we should have been. He was a brilliant scholar, highly articulate and full of revelations, and it was him I had signed up to listen to, not my fellow classmates.

Were people to bother asking students what they prefer, I wonder what sorts of students would say they prefer student-centered discussions to teacher-centered ones.

The child-centered classroom vs. the child-friendly classroom

I’ve been spending a little more time in schools lately, and have formed a few more impressions of child-centered classrooms--particularly of those in which students sit in pods facing one another rather than the teacher. I’ve written earlier about how seating students in pods makes it hard for them--particularly those facing away from the teacher--to focus on the teacher or any material that is being presented to them in the front of the classroom. I’ve also written about how pods can be arenas for the sort of subtle bullying that is particularly difficult for teachers to detect and discourage. And I’ve also written about how when students often opt to arrange themselves in rows rather than pods.

What I’ve noticed most recently is how much more disruptive behavior results when students are facing one another rather than forward. In particular, the temptation to talk to, mouth words at, exchange glances with, and otherwise interact with the peers you’re sitting next to and across from, even when the teacher requests your attention, is extremely high. This not only disrupts learning; it can also lead to a lot more angry yelling by the more frustrated teachers--some of whom would perhaps prefer to arrange the desks differently, if only their principals would allow it.

When armchair education experts conflate child-centered with child-friendly, they’re failing to apply a child-centered approach to their thinking. In particular, they’re failing to imagine what it’s like to sit facing the side or the back of your classroom opposite classmates who distract or bother you all day long, “guided” by a teacher who yells much more frequently that he or she might have in the teacher-centered alternative that no longer appeals to the experts.

Math problems of the week: traditional vs. IMP trigonometry

The second in a series of posts comparing the introduction of trigonometry in traditional high school math vs. the Reform Math program Interactive Math Program.

I. The next two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.395-396 [click to enlarge]:

II. The next two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 6-7 [click to enlarge]:

Stacking, regrouping, and corrupting the children, II

Rumor has it that a principal at a local school that uses Investigations Math has recently been making surprise visits to classrooms and demanding who knows how to add and subtract numbers via "stacking." (Stacking, which Investigations mentions reluctantly in passing and resists teaching to mastery, is that old fashioned method of arranging numbers one on top of the other before adding, subtracting, or multiplying them, and then "borrowing" or "carrying"--aka "regrouping"--from one column to the next.)

Since this principal has, for years, been a stalwart defender of the use of Investigations at her school, this was a bit of a surprise not just for the students, but also to their parents--especially the many who dislike Investigations. Some--including one who has set up an after school math program to teach stacking and other things that Investigations fails to teach--are now hoping that the principal is having second thoughts about Investigations (in spite of what the school's math consultant has said against stacking).

The specific surprise visit I heard about involved a 3rd grade class. Here a particularly brave girl who'd attended the after school enrichment program and knew how to stack volunteered to go up to the board and do so. She proceeded to stack, subtract, and get the correct answer.

What's unclear is what the principal made of this--or, for that matter, what her intentions were in the first place. While it's possible she's been having second thoughts about the curriculum, it's also possible she was simply fishing for confirmation that her kids can stack. That would give her a ready response the next time someone claimed that Investigations doesn't teach this. 

In general, the proliferation of Investigations in the greater Philadelphia area has been a boon to those running after school math programs. (One told me she really should have named her program "Thank You, Investigations.") But the symbiosis between Investigations schools and after school math programs is more dynamic than it might first appear. The more parents resort to after school math remediation, the more students (in spite of Investigations) learn math, and the easier it is for Investigations proponents to claim that Investigations is working--further entrenching both Investigations and after school math remediation. 

A "left-brain" take on misbehavior and discomfort

In the largely right-brain world of pop psychology, harking back at least to Freud, most afflictions would seem to have a socio-emotional source. In particular, misbehavior results from anger, poor emotional self-regulation, social insecurity, or a craving for attention. Distress when routines change results from fear of novelty and uncertainty. A desire for black and white categories results from a discomfort with ambiguity. The remedies, too, are social and emotion-based. If you’re upset about something, talk it out and process it emotionally.

Too often people ignore the possibility of cognitive causes and remedies. Misbehavior, for example, is often the result of cognitive disengagement, aka boredom. That’s why it occurs disproportionately when kids are waiting: waiting in line, waiting for a transition to end and a new activity to begin, or waiting for a long-winded classmate to finish talking.

Irritation when routines change, as I’ve noted earlier, can likewise have a cognitive source. Whenever someone puts your salt and baking powder containers in the wrong place, or changes the user interface on Windows or Blogger, you’re forced to relearn the boring, tedious stuff you’d earlier been able to automate, your mind no longer free to wander to more interesting places.

When you suddenly discover that a system of categories is more complicated than you thought it was, your heart may sink not because you’re emotionally uncomfortable with ambiguity, but because messy categories are much more of a cognitive pain in the neck to learn.

When you read the latest reports about how a nonhuman species supposedly communicates via nouns, verbs, and productive syntax, your failure to embrace these conclusions may not be because you feel threatened by the notion that humanity isn't as unique and privileged as you thought it was, but because of what you know about language, cultural transmission, and common misunderstandings about grammar.

Even when the cause is emotional the remedy may be cognitive. If I find myself brooding or unproductively anxious, I’ll seek out an intellectually engaging book or article to distract me, finding greater solace in Malcolm Gladwell, Steven Pinker, Jared Diamond, Matt Ridley, or Nancy Minshew than I would hashing things out with a therapist.

Math problems of the week: traditional vs. IMP trigonometry

The first in a series of posts comparing the introduction of trigonometry in traditional high school math vs. the Reform Math program Interactive Math Program.

I. The first two pages of the first trigonometry chapter in A Second Course in Algebra (published in 1937), pp.393-394 [click to enlarge]:

II. The first two pages of the first trigonometry chapter in Interactive High School Mathematics Math Program Year 4, pp. 4-5 [click to enlarge]:

How to disempower the best teachers and students

All over the country, gifted programming and opportunities for acceleration have been vanishing, thanks not just to budget cuts, but to the edworld fantasy that it’s possible simultaneously to eliminate the achievement gap and meet the needs of all students through “differentiated instruction” in heterogeneous-ability classrooms. Here, everyone uses the same curriculum, but the brighter students supposedly receive greater challenge either by teaching their less capable peers, or because the curriculum is so “rich” and “deep” (red flags that I should have included in my examples of edworld doublespeak) that they can engage with it at all different levels.

The most striking recent example of this is in Montgomery County Maryland (one of the most highly reputed school districts in the country, and one whose recent changes I blogged about earlier). Here, the school district recently sold its good name to Pearson, one of our country’s biggest textbook publishers, to the tune of $4.5 million dollars, and has adopted a “rich,” “deep” math curriculum known as “Curriculum 2.0” (or “Montgomery County Math” or "Pearson Forword") that looks suspiciously like Pearson’s other abomination, TERC Investigations.

Having thus “enriched” and “deepened” its math curriculum, MontCo is eliminating the once common practice (known as Math Pathways) of allowing those who are mathematically capable to attend math classes in classrooms one to several grade levels ahead of their official grade level. A mathematically advanced second grader, for example, might take math with fifth graders.

One year into the new curriculum, parents and kids are increasingly frustrated. One parent describes her second grade daughter as coming home in tears about how boring her math class is. Another parent notes:

Kids in this curriculum are bored, losing interest and not being taught at an alarming rate. Teachers don't feel empowered to give kids what they agree with parents that kids need.

One of the scariest aspects of the latest reforms in education is the disempowerment of teachers--particularly those experienced enough to know what works best. Why haven’t the teachers’ unions made this particular variety of disempowerment of one their topmost concerns?

Letter from Huck: I Say Goodbye and Head Out for the Territories

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

I Say Goodbye and Head Out for the Territories

Tina's moods were an up-down affair. On one day she could be very pleasant; but the next day she would be super-critical and negative. I tried working around her moods, but it was difficult. I knew her moods were due to her father passing away. And in all fairness, her advice to me—though annoying—was generally pretty good.

“When you ask a question, you have to give them time to answer it,” she would tell me. “Otherwise, they know you’ll just tell them the answer, and you’ll never get a response from them.” This is valuable advice—wait time is critical. And in the pre-algebra classes I taught, we used a direct instruction technique which entailed asking questions—and waiting for answers. Critics of the direct instruction method will tell you that it is all about kids sitting in rows, facing front, while the teacher lectures and provides instruction by rote. Well, first of all, our students were seated in groups of four, though I would have preferred rows. Secondly, I tried to engage the students by asking questions—not just “lecturing”. Third, with the exception of the Pythagorean Theorem, there was nothing rote about how we taught.

In the discovery based algebra class, students were expected to work in groups and come up with an answer, much of the time without the benefit of prior instruction. The method they were to have discovered would often be revealed in the next section. No one ever seemed to catch on to this. Tina’s admonishment to me in that class was similar: “You’re telling them the answers; let them figure it out themselves.”

I tried to follow this advice while they were learning to solve systems of equations with two variables. They had learned about eliminating variables; that is, given two equations like x + y = 6 and 5x – 2y = 8, you can multiply the first equation by 2 to obtain 2x + 2y = 12. Adding the two equations result in the 2y’s dropping out. But suddenly the book presented these two equations: 4x - 3y = 1 and 3x – 4y = -1. The students were stumped. Even the brightest student, Jorge, didn’t know what to do. He asked for help. Tina was out of the room. That was the other thing. While she criticized me a lot, she also gave me time alone with the class.

I wanted to stay true to what Tina wanted, so I did the following, knowing that Jorge was quite bright and would catch on to the hint I was about to give him. “Remember when you were learning how to add fractions, what you did when you added 1/3 and ¼?”

“Yeah, what about it?”

“What did you do?”

“Found a common denominator,” he said, clearly not getting where I was going.

“Which was what?”

“Twelve.”

“How did you do that?

”Multiplied 3 by 4.”

“OK,” I said. “Look at your equation. You have a 3y up here and a 4y in the second equation.”

“OH!” he said “You multiply the top one by 3 and the bottom one by 4.”

Now I imagine Tina might have criticized that, but I felt I wasn’t giving it away and was using proper scaffolding techniques. I tried the same technique with other students, and they eventually saw where I was going.

I guess when push comes to shove, although I dislike the CPM program, I’ve seen worse. It covers what needs to be covered. This doesn’t let CPM off the hook. It could be done a lot more efficiently with a lot more practice problems. Going back to first principles like generic rectangles and guess and check leaves a lot to be desired and takes credit for the hard work teachers do in the pre-algebra courses.

I also think that Tina gave more direct instruction in that class than she was willing to admit. There were plenty of times when kids weren’t getting it that she would stop everything and offer instruction at the front board. All in all, I think Tina is a good teacher and relies more on direct instruction than she might think. She often advised me on things that sounded an awful lot like things I would say.

She gave me a nice farewell, and in the last week or so, told the kids that I would be leaving soon.  They seemed sad about this.  A girl named Gabriela in the algebra class asked me if I was leaving because I was moving away.  This seemed likely to her since students were used to seeing their friends leave because of “moving,  Some parents moved to another part of town, or out of town, sometimes because of rent, or losing homes, or finding new jobs.  I explained that I was finishing up the program; my teaching was an assignment, just like her algebra course was an assignment. I don’t know if she understood. Her question threw me off guard, though, and I became sad.  “Why are your eyes watering, Mr. Finn?” Gabriela asked.

I was ready for that. I was always ready for student questions. “Oh, you know I have terrible allergies and they’re mowing the lawns today.” They were in fact. “Whenever that happens, my eyes just itch and burn.” She seemed satisfied with the answer, but I don’t know if she believed me.

Although sometimes I couldn’t wait for student teaching to be done, I miss working with Tina and I think of my students often, with great fondness. I tried to teach them as if what I was teaching mattered. I have an idea of who it mattered to and who it didn’t. For most, though, I simply couldn’t tell.

Well, I guess I better wrap this up. I appreciate Miss Katharine letting me tell my story. I really don’t know what’s going to happen to me; teaching jobs are pretty hard to find. But thank you all for listening. I’ve said just about everything I had to say. For now, anyway.

Your pal,
Huck

Mindlessness for Mindfulness

It’s nearly breakfast time, and I’m about to go down and make waffles. I do this every few days when the batter runs out, for waffles are a routine, particularly with J and me.

Our waffles are routine in more ways than one. I follow the several-generations-old family recipe, as I’ve done for the last hundred upon hundreds of times I’ve made waffles. And this morning I’ll follow exact same procedure I have for years now. I’ll walk mindlessly over to the radio and mindlessly push the preset button for NPR. Then I’ll proceed to mindlessly open up drawers and cabinets, mindlessly take out equipment and ingredients, and mindlessly mix things together while I listen to Morning Edition.

A key ingredient of happiness, I’ve decided, is minimizing boredom, and over the years I’ve striven to make as much as possible of the boring stuff mindless enough that my mind can wander somewhere more interesting. The best route to mindlessness, of course, is unchanging routine. So when I make waffles, or fold laundry, or pay bills, or enter grades on Blackboard, I first create a script and then stick to it over and over and over again. If something forces me to deviate from it--perhaps someone has put something in a different cabinet or otherwise changed the user interface--I get irritated, not because I can’t stand change, but because suddenly I’m trying to figure out where the baking powder is instead of listening to the latest from Syria.

Educators hate scripts and routines precisely because they’re mindless. One of the biggest criticisms I’ve heard of Direct Instruction, for example, is that it involves highly-scripted exchanges repeated ad nauseam. But mindlessness in one place means mindfulness elsewhere, and the more you can script the boring stuff, the more both students and teachers can focus on what’s meaningful and new.

The Constructivist activities that proponents tout as far superior to Direct Instruction, elevating critical thinking over mindless rote procedures, are often much more mind numbing. This is because they tend to involve large amounts of low-ratio-of-effort-to-learning busy work of the sort that, non-rote as it is, is impossible to automate and draws mental energy away from what’s new and interesting. Instead of focusing, say, on Magellan’s adventures, you’re searching for a tissue with which to gingerly dab up the glue spillage on your diorama of his fleet so you won’t get points taken off for sloppiness.

The key ingredients, then, are engaging material and routine. Make sure there is something interesting to listen to, or an interesting place for your mind to wander, or an interesting concept or narrative or set of interconnected facts in the lesson at hand. And make what’s interesting maximally accessible by making everything else as thoroughly mindless as possible.

Math problems of the week: varation in traditional vs. reform algebra

First variations problem sets: 


I. From Wentworth's New School Algebra, published in 1898, pp. 320-321 [click to enlarge]:

II. From Integrated Math 2, published in 2002, pp. 79-80 [click to enlarge]:

III. Extra Credit:
Comparing the two problem sets, does time expenditure vary directly or inversely with mathematical egagement?

Free riding off of what develops naturally rather than teaching what doesn't

Besides teaching TFA students about language disabilities, I’ve just finished up my last (and final) quarter teaching a class on language development in typical children. This has long been a requirement for teacher certification, even though it’s long been known by psycholinguists that typical children develop language on their own without special intervention from teachers. The authors of the materials used in this class (which I didn’t design) are therefore at pains to show ways in which teachers can nonetheless make a positive contribution to their students’ language development.

The (unintended?) side effect is that teachers are being encouraged in yet another way to steer their classrooms away from academic skills that must be taught towards skills that are largely developmental and aren’t easy to teach. I’m thinking especially of emotional maturity, empathy, social skills, organizational skills, and public speaking ability, all of which significantly influence the grades students get for the growing numbers of assignments involving group activities, interdisciplinary projects and project presentations.

When it comes to language skills, my students tend to finish up my class convinced (by all those materials I didn’t choose) that they need to be infusing even math classes with more language activities--not questioning either whether this is necessary, or whether it might water down the math.

Like I said, I no longer teach this class. But classes on how math, science, social studies, and reading skills don’t develop naturally the way language skills do, and on how to teach these skills based on the latest actual cognitive science research--those I would find highly worthwhile: for myself, for my students, and, most of all, for the students of my students.

If these were the classes that ed schools offered.

Letter from Huck Finn: Not Quite Like Old Times

Out in Left Field proudly presents the ninth 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.

Not Quite Like Old Times

On the Monday that Tina returned, I was preparing for first period. I was always nervous before the first period bell rang, like an actor hearing the audience as he waits for the curtain to rise and the play to begin. I was particularly nervous today because there was a possibility that Tina might not show.

I heard the usual sounds outside my classroom—kids playing handball against the giant wall of the gym building that stood across from our module. As the school buses arrived, more kids poured into the courtyard. The lonely sound of handball was replaced with what sounded like 10,000 students milling around. I was relieved finally to hear the familiar clacking of high heels on the walkway to the classroom and the sound of the key in the door.

“Welcome back,” I said.

“Thank you,” Tina said, and headed for her desk. She saw the group sympathy card from the class and one from me and put both in her purse.

A math teacher came into the room, hugged Tina and gave her a small gift. The two chatted for a minute and then the teacher left. In the remaining few minutes, Tina tried to catch up on what was going on. “What are we doing in pre-algebra?” She looked tired and tense.

“Just starting Pythagorean Theorem,” I said.

 “And algebra?”

“Multiplying binomials using generic rectangles.”

She suddenly brightened and said “Don’t you just love generic rectangles? It’s such a great way to teach how to multiply binomials.” She went on about how much she liked the approach, and how CPM would connect it to factoring later. I was glad to see her perk up and talk excitedly about CPM, even if I didn't care for it.

Generic rectangles are a way to represent the multiplication of two binomials as the area of a rectangle. So x + 5 multiplied by y + 10, can be represented as a rectangle with those binomials as the lengths of the sides:

The students had experience computing areas within rectangles; now they could represent rectangles as above. They compute the areas of the four smaller rectangles, and add up the results to get xy + 5y + 10x + 50. The challenge I faced was not so much in teaching the students how to do it, but how to keep them from calculating (x+5)(y+10) using the procedure known as FOIL, which they had learned in 7th grade pre-algebra.

In fact, later that day in algebra class the issue of FOIL came up in a rather surprising way. I had been in charge of the class. I became concerned when I saw that Tina suddenly looked very tired; she sat down and put her head on the desk. I watched her out of the corner of my eye while I continued working with the students on the generic rectangle problems. One girl, Samantha said “We learned how to do this last year using this method. Why can’t we use it?” She held up her notebook. I began to answer. “Yes, that’s FOIL, but…”

Tina suddenly stood up and shouted: “No! Don’t let them do it!” She then addressed the class: “You are NOT to use the FOIL method yet. I know some of you know it, but you have to understand what it is you’re doing first.”

She was fiercely loyal to the philosophy of the CPM authors who believed that the connection between binomial multiplication and area representation provided the understanding that students need. The authors were also fiercely loyal to the theory that teaching a procedure will frequently skimp on understanding. Well that’s fine, but just use it to introduce the topic, then teach them how it’s done algebraically. Do you really have to spend two weeks working with generic rectangles to instill “connections”, “understanding” and other educational trendy ideas in order to teach them something they’ve already learned in pre-algebra?

In fact, the pre-algebra text had a pretty good explanation for how to multiply binomials prior to bringing in FOIL as a shortcut. They start with the distributive property: a(b + 3) = ab + 3a. Then if "a" is replaced by a binomial such as b + 4, a(b+3) becomes (b + 4)(b + 3). Substituting in the original equation, you get (b + 4)b + 3(b+4), or b² + 4b +3b + 12, which is b² + 7b + 12.

That’s another thing that made me wonder. How much of this discovery and connection that CPM is bragging about is because of pre-algebra courses that students have had?

All that aside, Tina was happy to see her students again, and they were happy to see her which I was glad to see. She continued on as if nothing had happened. Especially in the pre-algebra class. In that class, I had explained the Pythagorean Theorem, but I was a bit too slow for her, and forgot to give the formula, so she jumped in, just like the old days.

“Let me interrupt and point something out about the Pythagorean Theorem,” she said, “because it wasn’t clear.” These last two words she said looking at me out of the sides of her eyes. I knew she wasn’t pleased. She summarized it as “It’s a² + b² = c². Can you say that?” The class repeated it. She then showed them the trick for how to identify the hypotenuse which is the “c” side in the equation: pretend the triangle is a bow, and the arrow goes where the little right angle sign is. The arrow is pointing at the third side: the hypotenuse. Funny how she could be so explicit in the pre-algebra class and yet buy in to CPM’s philosophy in the algebra class, I remember thinking.

But it was good to see her again, and I told her so at the end of the day. “Good to see you too, Huck,” she said. Something didn’t feel right, though. It felt like we both were trying to get back to the way things were before she left—and that we both knew that would never happen.

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

I. The last page of the 3rd grade section of Hamilton's Essentials of Arithmetic, First Book, p. 124 (published in 1923) [click to enlarge]:

II. The last page of the 3rd grade Math Trailblazers Student Guide, p. 327 (published in 1997) [click to enlarge]:

III. Extra Credit: Which problem set involves a higher ratio of conceptual reasoning to busy work?

Recommend what you know, or risk embodying what you don't

The latest to join the chorus of American intellectuals who argue that schools should stop teaching "tedious" facts and instead teach "21st century skills," fast on the heels of former Harvard president and presidential economics adviser Larry Summers, is popular science guru Jonah Lehrer. Here is he in this past weekend's Wall Street Journal:

One useful model for the 21st-century university is preschool. As the economist James Heckman has demonstrated, successful preschools don't increase the intelligence of toddlers or endow them with new knowledge they take to kindergarten. Rather, their early education leads to long-term improvements in "noncognitive" skills, boosting character traits such as self-control and conscientiousness.

Such traits often predict success in the real world better than I.Q. scores. And yet, colleges don't even attempt to improve them. Students are never taught how to regulate their emotions or study for a test. They don't learn how to take criticism or cope with failure.

Take perseverance. According to a study by the College Board in the early 1980s, a trait known as "follow-through" was one of the best predictors of success in college and beyond. But the modern university teaches follow-through only by accident, forcing students to take tedious classes and then rewarding those who don't drop out. That's a mistake. It's time to give students underlying skills that are not forgotten.

Is it really a surprise that preschools best prepare three and four-year-olds for kindergarten by fostering self-control and conscientiousness as opposed to attempting to "increase intelligence" and convey "new knowledge"?

And is it really a surprise that self-control, perseverance, and conscientiousness predict real world success better than IQ scores do?

But neither of these "findings" imply that post-K schools should give up on teaching the fact-rich disciplines for which self-control, conscientiousness, and perseverance are the prerequisites.

Dismissing factual content as "the sort of information that can now be looked up on a phone," and assuming that one can teach students "how to learn to think about thinking" in the absence of particular facts, Lehrer ignores both recent cognitive science research on the domain-specificity of critical thinking, and the possibility that self-control, conscientiousness, and perseverance matter largely because they are the prerequisites for mastering the bodies of knowledge that (yes, more than IQ scores!) predict real-world success.

Ironically, when they venture out of their fields of expertise and start making recommendations to educators, Jonah Lehrer, Larry Summers, et al end up unwittingly illustrating the domain-specificity of critical thinking that they so consistently ignore. If they knew more about what cognitive science research has to say about education, they wouldn't be making such foolish remarks about how we can teach children critical thinking and "21st century skills" without teaching them factual content.