I. The final data graphing problem in the 5th grade TERC/Investigations Student Activity Workbook (Unit 8, "How Long Can You Stand on One Foot"). [click to enlarge]:
Thursday, July 31, 2014
I. The final data graphing problem in the 5th grade TERC/Investigations Student Activity Workbook (Unit 8, "How Long Can You Stand on One Foot"). [click to enlarge]:
Tuesday, July 29, 2014
Finally a New York Times Op-Ed piece critiquing an education fad. In "The Fallacy of ‘Balanced Literacy’," author Alexander Nazaryan writes:
Now the approach that so frustrated me and my students is once again about to become the norm in New York City, as the new schools chancellor, Carmen Fariña, has announced plans to reinstate a “balanced literacy” approach in English classrooms. The concept’s most vociferous champion is probably Lucy Calkins, a Columbia University scholar. In her 1985 book, “The Art of Teaching Writing,” she complained that most English teachers “don’t know what it is to read favorite passages aloud to a friend or to swap ideas about an author.” She sought a reimagination of the English teacher’s role: “Teaching writing must become more like coaching a sport and less like presenting information,” a joyful exploration unhindered by despotic traffic cops.(One has to wonder whether Calkins has ever been coached by a sports coach: since when is this a "joyful exploration" that doesn't involve despotism and "presenting information"?)
Ms. Calkins’s approach was tried by Mayor Michael R. Bloomberg, but abandoned when studies showed that students learned better with more instruction.It's striking that Nazaryan, who believes in "the pedagogical value of a deep dive into sentence structure," is not a native-born American:
My students craved instruction far more than freedom. Expecting children to independently discover the rules of written language is like expecting them to independently discover the rules of differential calculus.
I am somewhat prejudiced on this issue, for my acclimation to the English language had nothing balanced about it. Yanked out of the Soviet Union at 10, I landed in suburban Connecticut in the English-as-a-second-language classroom of Mrs. Cohen. She taught me the language in the most conventionally rigorous manner, acutely aware that I couldn’t do much until I knew the difference between a subject and a verb. Mrs. Cohen was unbalanced in the best possible way.Mrs. Cohen is, in fact, unusual; here in American, where we are more ignorant than anywhere else of the grammars of other languages, Grammar Denialism is the norm, and balanced literacy, to some extent, is as much a symptom as a cause.
But knowledge of grammar and sentence structure is, indeed, key to improving writing. Consider some excerpts of Calkin's latest book (Pathways to the Common Core: Accelerating Achievement):
We can regard the Common Core State Standards as the worst thing in the world. Frankly, it can be fun to gripe about them. Sometimes we say to the to the educators who convene at our Common Core conferences, "Right now make your fact into a curmudgeon's face. As a curmudgeon, think about those standards—the timing, the way they arrived on the scene, their effect on your school."One can improve this choppy paragraph using techniques found in books that teach the grammar writing in the way that Calkins renounces: for example, creating cohesive sentence topics via sentence combining and rearranging.
We can regard the Common Core State Standards as the worst thing in the world, and, frankly, it can be fun to gripe about them. At our Common Core conferences, we sometimes say to educators, "Right now make your fact into a curmudgeon's face. As a curmudgeon, think about those standards—the timing, the way they arrived on the scene, their effect on your school."Here's another paragraph from Calkins' book that could use some syntactically-informed revision:
The entire design of the standards is based on the argument that the purpose of K–12 education is to prepare K–12 students for college (the rhetoric touts preparation for career as well, but this is not reflected in the standards). Because the standards were written by taking the skills that college students need and distilling those down through every single grade, kindergarten children, for example, are expected to “use a com-bination of drawing, dictating and writing to compose opinion pieces in which they tell a reader the topic or the name of the book they are writing about and state an opinion or preference about the topic or book."These overly long sentences can be rearranged and broken up for much greater readability, maintaining cohesion, in part, via a wh-cleft structure:
Informing the design of the standards is the notion that K–12 education exists to prepare students for college. (True, the rhetoric also touts career preparation, but the standards don't reflect this). Essentially, what the designers did was take the skills that college students need and distill these down through every single grade. The resulting expectation for kindergartners, for example, is that they “use a combination of drawing, dictating and writing to compose opinion pieces in which they tell a reader the topic or the name of the book they are writing about and state an opinion or preference about the topic or book."A small subset of people have a natural ear for language and do enough attentive reading to learn these techniques, implicitly, on their own. For everyone else, including many published authors, improving prose requires deliberate instruction--of the sort found nowhere in the joyful explorations of Writer's Workshop.
Sunday, July 27, 2014
A few weeks after writing a post about the latest research on the downsides to abandoning penmanship, it occurs to me that there’s yet one more facet of penmanship that we’ve abandoned. Not only are schools (for the most part):
1. No longer teaching penmanship
2. No longer teaching cursive
3. Replacing handwriting with keyboarding
...they are also, increasingly, having kids graph functions via calculators rather than by hand, via pencil and graph paper. Instead of slowly plotting out the points of a particularly function, strategically choosing which numbers to plug in to most efficiently determine the general shape of a function, today's kids simply type in a function and see its graph immediately. It’s a much more passive learning process, which, I’m guessing, substantially reduces how deeply students conceptualize the geometry of functions.
A month ago, during my daughter’s last week of homeschool, I watched as she slowly plotted out the graphs of y=x, y=x+1, y=x+2, … y = x-1, y=x-2, … y=2x, y=3x, … y=-x, y=-2x, y=-3x, … y=1/2x, y=1/3x, … Every step of the way, numerous though those steps were, she was learning something—in the best sense of hands-on, child-centered, discovery learning! If she’d simply typed all these functions, one by one, into a graphing calculator, I’m not sure how much she’d still remember today.
Friday, July 25, 2014
I. A set of 1st grade word problems from Hamilton's Essentials of Arithmetic, published in 1919:
II. A 1st grade Common Core-inspired math test, circulated via Slate and the Washington Post:
Wednesday, July 23, 2014
I was recently chatting with my French about the French baccalaureate, which is the exam you must pass in order to receive your high school diploma and move on to university. “Le bac”, in fact, is the only thing that determines your admission to university. Not your grades, not your extracurriculars, not your teacher recommendations. Talk about high stakes testing!
Except that the French bac is not at all like America’s high stakes testing. It consists, mostly, of essay questions, lab work, oral exams, and a research project, in all major academic subjects. Examinations are spread out over multiple days and are assessed by multiple examiners, and, except for the oral components, are done so anonymously. The process is, at once, more comprehensive, holistic, challenging, and individualized than the American testing system is, whether we’re talking about the No Child Left Behind Tests, the Common Core tests, the SATs, or the Aps, even if you put all these tests together.
Other European countries are similar. Whether we’re talking about the A-Levels in Britain, or the Abitur in Germany, it’s one big, challenging, comprehensive exam that determines your prospects for university.
And, while its detractors are legion, might there be some real virtues to this system?
It does, of course, favor academic skills over resume-padding and so-called leadership skills. And while this might seem overly narrow to many of us Americans, we should keep in mind the breadth of academic skills covered. Un-aided writing (no ghost-written essays here), analytical thinking (both in writing and orally), problem solving (with some truly challenging problems in math and physics), research skills, laboratory skills, breadth and depth of knowledge in all academic subjects: aren’t these what determine how prepared one is to benefit from university-level courses?
The baccalaureate system is also far more resistant to the tinkerings of privileged families than America’s college admissions system is. Expensive test-prep won’t get you very far; the bac tests aren’t gameable the way the SATs are. You can’t hire people to write or edit your essays for you. Expensive resume stuffers--unpaid internships, private lessons, and expensive programs abroad—have no place in the European university-level admissions process. The only way to prepare for the bac is to read a lot, write a lot, do a lot of practice problems, and study a lot—which is identical to the best way to prepare for university.
There’s also something to be said for final examinations that ultimately trumps everything else—your grades, your broader academic portfolio, your day-to-day class participation. Yes, you might not be someone who tests well, either orally or in writing. Yes, you might being having an off day while being tested. Yes, it would be nice if you could retake the bac should you fail it, without being stigmatized for life (and, in fact, you can retake it). But (unlike the SATs) the bac is given over a number of days (with some of the subject exams given at the end of the junior year of high school), and it seems to me that simply being a good or bad test taker is less of a variable when we’re talking, not about a bunch of tricky, tightly-timed multiple choice questions, but about a breadth of essays, problems, labs, oral questions, and research projects.
Then there are the biases that the baccalaureate system bypasses. What if you’re not a teacher-pleaser, or a diligent and timely homework-completer, or a big participator in class, or an effective grade-grubber, and what if, nonetheless, you are able to prepare yourself, in your own way, to write excellent exam essays and solve tough math and science problems and effectively complete the lab work and research project and ace the orals, and, in short, ace the entire, comprehensive exam? Whose business is it, really, how you got there, as long as you found your path?
Isn’t this, in fact, a system that favors multiple strategies and learning styles—in the very best sense of those terms?
Monday, July 21, 2014
How do you accommodate a complex information processing disorder?
Here are some of neurologist Nancy Minshew’s findings about individuals on the autistic spectrum, including those with normal to above-normal IQs:
- In autistic individuals, there is “a problem with the brain’s fundamental mechanisms for processing complex information.”
- There are “deficits across multiple domains that selectively involved higher-order abilities that involve the processing of complex information”
- Particularly impaired: “higher-order language comprehension” and “mental inferencing.”
For 5th grade:
- CCSS.ELA-LITERACY.RL.5.5 Explain how a series of chapters, scenes, or stanzas fits together to provide the overall structure of a particular story, drama, or poem.
- CCSS.ELA-LITERACY.RL.8.3 Analyze how particular lines of dialogue or incidents in a story or drama propel the action, reveal aspects of a character, or provoke a decision.
- CCSS.ELA-LITERACY.RL.11-12.2 Determine two or more themes or central ideas of a text and analyze their development over the course of the text, including how they interact and build on one another to produce a complex account; provide an objective summary of the text.
- CCSS.ELA-LITERACY.RL.11-12.6 Analyze a case in which grasping a point of view requires distinguishing what is directly stated in a text from what is really meant (e.g., satire, sarcasm, irony, or understatement).
- CCSS.ELA-LITERACY.RL.11-12.7 Analyze multiple interpretations of a story, drama, or poem (e.g., recorded or live production of a play or recorded novel or poetry), evaluating how each version interprets the source text. (Include at least one play by Shakespeare and one play by an American dramatist.)
Saturday, July 19, 2014
While there will always be readers, and there will always be teachers who assign the classics, I wonder how many of today’s kids are still engaging on a regular basis with the archaic constructions that permeate the older classics. I’m speaking, not just of archaic vocabulary (relatively easy to look up), but of archaic syntax. Even if we restrict ourselves, as schools long have tended to, to “Modern English” (which dates back to the year 1550), there are still a number of syntactic constructions we no longer find in the majority of the texts that today’s young readers encounter. I wrote about some of these earlier, but have collected a few more since.
Some constructions may simply bog readers down and/or baffle them:
1.”of a” plus time expression to express habitual time:
“of a night” (at night); “of an evening” (in the evening); “of a Sunday morning” (Saturday morning)
“..she had her cap on, which he had never seen her in before when he came of an evening.” (Adam Bede)
2. “that” for “so that”:
“Let us die that we may live”
3. “as” for the relative conjunction “that”:
"those as sleep and think not on their sins." (The Merry Wives of Windsor)
4. “were” for “would be,” with “that” for “if:
“It were better for him that a millstone were hanged about his neck, and he cast into the sea, than that he should offend one of these little ones.” (King James Bible)
5. “but” for “that” plus “wouldn’t”:
“There is no good man in any line but I call to my standard” (My Book House retelling of Robinhood)
6. Inversions: of subject and verb; of object and verb; of adjectives and nouns:
“On her head sang its war-song wild”. (My Book House retelling of Beowulf)
“For them the gracious Duncan have I murther’d” (Macbeth)
7. Nonrestrictive relative clauses shifted away from the definite nouns that modify:
“My lair is empty that was full when this moon was new” (The Jungle Book)
Some instances of archaic syntax may not merely baffle today’s kids, but lead them astray:
8. “Should” for “would”:
“I should have asked you to lunch with me even if you hadn't upset the vase so clumsily.” (Screenplay to Rebecca)
Here, one might think that the speaker is expressing an obligation to have lunch.
“You want to know if I can suggest any motive as to why Mrs. de Winter should have taken her life?” (Screenplay to Rebecca)
Here, one might think that the question at hand is why there was an obligation for Mrs. de Winter to take her life.
9. “had” for “would have”:
“So had life ended for Beowulf.” (My Book House retelling of Beowulf)
One might think Beowolf actually died.
10. “Though” for “even if”:
“Though ye gave me a thousand pounds, yet would I never sign the lease” (My Book House retelling of Beowulf)
One might think that the speaker actually did receive a thousand pounds.
As I noted earlier, even now things are a-changin’: “before”, “beside,” and “about” are losing their spatial meanings (“in front of,” “next to,” “around”), making sentences like “She stood before the crowd of people about the grounds beside the lake” not as readily understood as they once were. Also falling out of favor are the locative meanings of "within" and "without," upstaged by counterparts in which the locative morpheme comes first, rather than last: "inside," "outside."
Whether these archaisms merely confuse contemporary readers, or actually lead them astray, cumulatively, they make pre-20th century classics (e.g., Shakespeare, Dickens, Dumas, and James) increasingly inaccessible.
No wonder so many English classes now accompany the written classics with the growing number of movie versions that Hollywood is so eagerly churning out. Although, as we see from the screenplay for Rebecca (1940), it's hard to completely escape archaisms unless one sticks to relatively recent movies. Which I'm guessing is pretty much what today's K12 English teachers are doing.
Thursday, July 17, 2014
Chart reading versus estimation: large numbers in Investigations vs. Singapore Math.
I. An early problem set involving large numbers in the 5th grade TERC/Investigations Student Activity Book, from Unit 3 of the book [click to enlarge]:
II. A continuation of the first first problem set involving addition and subtraction (and multiplication and division) of multiples of ten from large numbers in the 5th grade Singapore Math Primary Mathematics 5A Workbook, from Unit 1 of the book [click to enlarge]:
III. Extra Credit:Is chart reading a 21st century skill?
Tuesday, July 15, 2014
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 four:
When it comes to teaching math, I am a drill instructor. I say this without apology. I believe that practice is essential in mathematics; it results in automaticity which ultimately allows students to take on increasingly complex tasks. It is part and parcel to the much ballyhooed concept of “understanding”. Yes, I put this word in quotes.
I am also—again in terms of teaching math—the equivalent of a rifle range instructor. I’ve never served in the military but from what I’ve heard, in basic training the instructors on the rifle range are not the same drill instructors that wreak havoc on the recruits. According to a friend who served, on the rifle range the instructors were very patient, talked in quiet tones and gave the recruits advice and encouragement in learning how to shoot their rifle. It is in this spirit—one-on-one encouragement as opposed to the enmasse drilling techniques—that I use this term.
Out of necessity, I start out as drill instructor. Prior to the start of any class, when confronted with a sea of faces, people talking, not wanting to start class (particularly true of my 6th period class), my lesson plan disappears from my mind to be replaced with this solitary and recurring question: “What the hell am I going to do now?” I then revert to a technique I learned when I was student teaching. Clipboard in hand, I stand at the front of the room and shout: “In your seats and homework out, please!” This technique serves two purposes. It gives me the illusion that I am in control. Secondly, it tells the students what to do, and even if they don’t do it, they at least know what it is they should be doing.
My routine after that is something I’ve seen written up in articles that decry the traditional classroom (sometimes referred to as the “I do, we do, you do” technique) for its dull, staid, predictable and uninspiring drills. It is also something that I have heard people say will disappear once Common Core kicks in full tilt. Thus, I am a vanishing breed soon to be relegated to the world of buggy whips, slide rules, 8-track tape players, and well-written math textbooks. While I check in homework, they are to work on the warm up questions that I have on the screen. We go through the warm-ups, I put up the answers to the homework, answer any questions they may have on the homework and then embark on the lesson.
My rifle range instruction occurs mostly when I circulate around the room after I assign the problems they are to work on. The conversations can be personal as well as instructive, and it is a time for me to get to know them. My second period class is algebra, 1 second year. This means that the students have passed the first year of the 2-year sequence of algebra 1—most of them anyway—and are older and somewhat more mature and well behaved than the first year students.
In this class I had about six football players on the junior varsity team, almost all of them struggling with the material. I was working on solving systems of linear equations. The chapter started out by solving by graphing—always a frustrating topic because the main intent of the lesson, it seems, is to show how inaccurate and unreliable graphing is as a means of solving equations. I suppose it’s also to show that the solution of a system of equations exists at the point of intersection. And then there are the typical “real world” problems that describe two health club plans or two long distance plans, or any two plans that involve 1) a membership fee (which translates to the y-intercept) and 2) a monthly, weekly, or daily rate (which translates to the slope) and asks at what point (days, weeks, or months) will the plans cost the same. I have nothing against such problems but, like most problems in Holt Algebra, little is ever varied so that once you learn how to solve one such problem, you’ve solved them all. As it was, many of my students were struggling even with basic types of non-word problems.
During one of my rifle range tours, one girl in that class, worried about a quiz upcoming in the next week, asked “Will the quiz have any division on it?”
“You mean will you have to know how to divide? Yes, of course,” I said. She looked crestfallen. “I mean, it won’t have a lot but can you divide things like 24 by 8 and things like that?”
“Oh yes,” she said, looking relieved. “It’s just the double and triple digit division I have a hard time with.” For some reason I didn’t find this very assuring.
My fourth and sixth period classes were also a mixture of abilities. Some students asked more questions than others. In sixth period it was Elisa, the girl who told me she had trouble with math. I found out she lived with her aunt, and had just moved from Colorado. She had had a tough time with math in Colorado and said she developed a stomach ulcer because of her last math class. The teacher wouldn’t answer any of her questions. Maria, a Mexican girl, was another who asked many questions and, like Elisa, said her previous teachers in math didn’t answer her questions. I don’t know if their teachers were of the philosophy that less teaching means more learning but both expressed gratitude to me for answering their questions.
Although I would explain the concept behind the problem, in most cases it always came down to telling students the procedures. Maria, for example, asked me how to solve -7 -3. “Maria, if you lost 7 dollars and then lost 3 more how much have you lost?”
“Ten dollars” she said, counting on her fingers.
“OK. So what you’re really doing is adding two negatives. It’s really (-7) + (-3).” I showed this on a number line.
“So if you have two negative numbers, you just add them and put a minus sign in front?” she asked.
I found myself thinking “What the hell do I do now?”
Knowing she was right, but also knowing and not overly concerned that she didn’t understand the “why” of the procedure, I answered her question, guiltily confident in my belief that procedural fluency leads to understanding. “Yes,” I told her. “That’s what you do.”
And that’s what she did.
Sunday, July 13, 2014
A survey of more than 500 district superintendents and administrators from 48 states [conducted by American Association of School Administrators] shows that most of the local K-12 leaders are firmly behind the Common Core State Standards.
The AASA survey finds that 93 percent of the superintendents say the new standards are more rigorous and will better prepare students for success after high school. In all, 78 percent of those surveyed believe the education community specifically supports the standards.Among the other "notable findings" that Edweek highlights is this one:
73 percent of those surveyed by AASA believe that the fight between supporters and opponents is actually hindering implementation of the standards.I'm not exactly sure what makes this finding notable, but I'd say it's reason, not for concern, but for hope.
Friday, July 11, 2014
I. The second multiplication problem set in the 5th grade (TERC) Investigations Student Activity Book, Unit 1: "Number Puzzles and Multiplication Towers," p. 10 [click to enlarge]:
What grade should/will an Investigations student get if they leave their assignment blank because their parents saw to it that they learned all their multiplication "combinations" over the summer?
Wednesday, July 9, 2014
The latest New York Times article on the Common Core State Standards begins with a refreshing acknowledgment of the frustrations that the Common Core has been causing among those most directly affected by it. The article opens with a description of a mother who, because of "the methods that are being used for teaching math under the Common Core,” plans to home-school her four children:
Ms. Nelams said she did not recognize the approaches her children, ages 7 to 10, were being asked to use on math work sheets. They were frustrated by the pictures, dots and sheer number of steps needed to solve some problems. Her husband, who is a pipe designer for petroleum products at an engineering firm, once had to watch a YouTube video before he could help their fifth-grade son with his division homework.
“They say this is rigorous because it teaches them higher thinking,” Ms. Nelams said. “But it just looks tedious.”The article also mentions
viral postings online that ridicule math homework in which students are asked to critique a phantom child’s thinking or engage in numerous steps, along with mockery from comedians including Louis C. K. and Stephen Colbert.But then the Times proceeds to regurgitate the Common Core's tired rationale:
The new instructional approach in math seeks to help children understand and use it as a problem-solving tool instead of teaching them merely to repeat formulas over and over. They are also being asked to apply concepts to real-life situations and explain their reasoning.When did math students ever repeat formulas over and over again? And when did students ever not apply concepts to real-life situations? And when did “explain your reasoning,” ubiquitous to American Reform math and rare everywhere else, become the one, one-size-fits all path towards, and the one, one-size-fits measure of, conceptual understanding?
The Times also cites employers, who “are increasingly asking for workers who can think critically”:
Employers “want a generation of people who can think and reason and can construct an argument,” said Steven Leinwand, a researcher for the American Institutes of Research.But if there’s even one employer out there who (a) is looking for mathematically competent employees and (b) has taking a close look at a representative sample of Common Core-inspired math assignments (as compared with traditional math assignments), I have never seen him or her cited anywhere.
Citing global tests like the TIMSS and PISA, in which American children lag behind those in other developed countries, the Times also claims that “traditional ways of teaching math have yielded lackluster results.” The Times does not mention that many of America’s current students grew up, not with traditional math, but with Reform Math, and that the countries that outperform us in math have eschewed our types of reforms in favor of more traditional ways of teaching math.
Having carefully omitted these facts, the Times proceeds on to argument by authority:
The [Common Core] guidelines are based on research that shows that students taught conceptually retain the math they learn. And many longtime math teachers, including those in organizations like the National Council of Teachers of Mathematics and the National Council of Supervisors of Mathematics, have championed the standards. …
Math experts say learning different approaches helps students develop problem-solving skills beyond math.Do these math experts include (a) any actual mathematicians who have looked at representative samples of Common Core-inspired math assignments or (b) anyone who has conducted or reviewed studies that actually address this question empirically?
As for the not-at-all-surprising claim that “students taught conceptually retain the math they learn,” this does not support Common Core-inspired math in particular. Traditional math also teaches conceptually.
The Times cites one authority in particular, Linda M. Gojak, the former president of the National Council of Teachers of Math:
“I taught math very much like the Common Core for many years.When parents would question it, my response was ‘Just hang in there with me,’ and at the end of the year they would come and say this was the best year their kids had in math.”For what it’s worth, I’ve never met a single parent who has said that. But given how much Reform Math has watered down the actual math, it’s easy to imagine that some of the least mathematically inclined children experience their first year in Reform Math as their “best year in math.”
The Times ends up suggesting that most of the frustrations amount to growing pains—the pains of a transition to a whole new curriculum, or as one superintendent puts it, “a shift for an entire society.”
The article notes that, besides the issues of winning over skeptical parents,
textbooks and other materials have not yet caught up with the new standards, and educators unaccustomed to learning or teaching more conceptually are sometimes getting tongue-tied when explaining new methodologies.Frederick Hess, directory of education policy studies for the American Enterprise Institute, puts it better:
“It is incredibly easy for these new instructional approaches to look good on paper or to work well in pilot classrooms in the hands of highly skilled experts and then to turn into mushy, lazy confusing goop as it spreads out to classrooms and textbooks.”To its credit, the Times article, besides including Hess among its authorities, acknowledges that certain subpopulations may have particular problems with Common Core-inspired math:
Some educators said that with the Common Core’s focus on questioning lines of reasoning and explaining answers, the new methods were particularly challenging for students with learning disabilities, or those who struggle orally or with writing.
“To make a student feel like they’re not good at math because they can’t explain something that to them seems incredibly obvious clearly isn’t good for the student,” said W. Stephen Wilson, a math professor at Johns Hopkins University.(It's refreshing to see an actual math professor cited here.)
Besides the language impaired, there are the mathematically gifted:
Some parents of children who have typically excelled at math find the curriculum laboriously slow.
In Slidell, an affluent suburb of New Orleans, Jane Stenstrom is concerned that her daughter, who was assigned to a class for gifted students as a third grader last year, did not progress quickly enough.
“For the advanced classes, it’s restricting them from being able to move forward,” Ms. Stenstrom said one recent afternoon.
Her daughter, Anna Grace, 9, said she grew frustrated “having to draw all those little tiny dots.”
“Sometimes I had to draw 42 or 32 little dots, sometimes more,” she said, adding that being asked to provide multiple solutions to a problem could be confusing. “I wanted to know which way was right and which way was wrong.”Surely some people will see this child, however gifted, as overly rigid in her mathematical reasoning and problem solving skills. But here's my take: when it comes to mathematics, (or, for that matter, engineering, accounting, pharmaceuticals, surgery, piloting airplanes, operating machinery, or, dare I even say it, educating our children), "wanting to know which way is right" is a pretty reasonable desire--especially when it comes to one of the Common Core's main obsessions: all those "real-life situations."
Monday, July 7, 2014
Conversations on the Rifle Range 3: The Broom in the Store Room, Multiple Answers, and the Rituals of Groupthink
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 three:
I believe strongly in how math should be taught, and even more strongly in how math should not be taught. Nevertheless, when I am involved in teaching it as I believe it should be taught, I feel vaguely guilty, as if I am doing something against the rules and perhaps even wrong. That’s how groupthink works. It is an acculturation process.
I am reminded of a job I had as a nighttime janitor at the University of Michigan Medical School the summer between my sophomore and junior years. The janitors put up with the college kids who worked with them, but they also could give us a hard time. On my first day, the supervisor told me to get a broom from the store room. This was an initiation rite. No matter which broom I laid a hand on, someone piped up “That’s mine!” In fact all the brooms had been claimed except one which belonged to someone who was not there. That one was off limits as well, but the supervisor finally said with an air of reluctance, “Well you may as well use that one. He probably won’t be coming back.” And true enough, he never did and the broom was mine. Several weeks later, another “new guy” joined the ranks and he was told to find a broom. Though I had found this initiation procedure ridiculous, when the new guy put a hand on my broom, to my horror I heard my voice booming “THAT’S MINE!”
My algebra classes used a book published by Holt (referred to as Holt Algebra). The team of authors include a math professor (Dr. Edward Berger) and a math reformer (Steven Leinwand). The book predates the Common Core and is fairly traditional, as evidenced by something the math department chair had said during the teacher workday I talked about earlier. Sally, the person from the District office had been telling us about the Common Core approach to teaching math—more open ended problems, more discussion, more working in groups, more problems that have multiple right answers. The math department chair brought up the point that it's hard to do all this because the books they use just don't have those types of problems in them. “Most of the problems can only be solved one way,” he lamented.
Nevertheless, the book does cater to some of the current groupthink trends in math education. When teaching the first unit for the first year Algebra 1 course, I wanted to focus on how to express certain English statements in algebraic symbols; for example, “4 less than a certain number” can be written as x - 4. While Holt Algebra does do this, it tends to focus more on the other way around—taking an algebraic expression such as 4/x and translating it into English. While most algebra books do this (as did mine from 50 years ago), the good ones tend to focus more on going from English to algebra. Holt Algebra spends more time going from algebra to English. In addition, it asks students to find two ways of expressing it, thus satisfying the “more than one way to solve a problem” motif that supposedly builds “deep understanding”.
“How are we supposed to find two ways to say this? What does this mean?” a girl named Elisa in my 6th period class asked me. She had told me on the first day that she was bad in math and requested to sit in front so she could see better and not be distracted.
“How would you say 4/x in words?” I asked. No answer. “What are you doing with the 4? Multiplying by x? Dividing by x?”
“Oh, dividing,” she said. “OK, so ‘4 divided by x’?”
“But what’s another way to say this?”
I told her to look at the page with the examples. In fact, for every student who asked, the dialogue was nearly the same, and I advised them to look on the page with the examples. (For those who are curious, the answer is “The quotient of 4 and x.”)
The problems in the book that asked for translation into algebraic expressions didn’t exactly start off with the easiest ones. In fact, Holt Algebra seemed to do this all the time—starting with complicated problems right away. One such problem that was stumping the students was: "William runs a mile in 12 minutes. Write an expression for the number of miles that William runs in m minutes."
The book offers the following hint: "How many groups of 12 are in m?” The hint seemed to confuse them more. It confused me. They needed a problem with a softer ramp-up, I decided. I told them to write an expression for the number of feet in m inches. While this also elicited blank stares, I then followed up by asking how many feet are in 24 inches. My student Elisa knew this at once. “How did you do it?”
“What do you mean?” she asked. This was going to be harder than I thought.
“I mean how did you solve it?” She thought a minute. “I just figured 12 times 2 is 24,” she said.
“Ah, so you divided 24 by 12,” I said. Asking students to explain something that seems obvious to them can present difficulties; I have no problem instructing them how to do that. (I realize however that her explanation in terms of multiplication could be viewed by some as superior to “I divided 24 by 12 because it shows her “deep understanding” of what division is.) I have not acculturated to that.
I then extended the questioning; try 36 inches, 48 inches. She repeated the pattern. What about if it’s m inches? Pause of uncertainty. “Uh, m/12?”
“Absolutely,” I said. For the next class, I devised my own worksheet with problems requiring translation from English to algebraic expressions. But despite my belief that I was doing the right thing, I still felt like I wasn’t quite legit. While I usually find it fairly easy to resist the acculturation process in math teaching it’s hard to escape it entirely. So I made sure they could also do the problems in the book that required the translation from algebra to English—in two different ways. I reasoned that such questions would be on the quiz and test that my teacher had devised. Then again, if anyone ever came in to my classroom to monitor me, they would see me teaching my students multiple ways of answering questions and think I was teaching them deeper understanding.
As it turned out, no one ever came in to observe me.
Friday, July 4, 2014
I. From the Everyday Math Student Math Journal, volume 2, final arithmetic problem set:
II. From Hamilton's Essentials of Arithmetic, final "Everyday Use of Numbers" problems, final speed test:
III. Extra Credit:
Compare and contrast the virtues calculator practice (integral to Everyday Math) vs. speed tests.
Wednesday, July 2, 2014
They're gone. Suddenly and completely. Just as it is sometimes with a suddenly shaved-off mustache, we didn't notice it right away. But something was different--calmer, quieter--and, looking back a few days after the change, we realized it dated to the day he turned 18.
J has been obsessed with the number two ever since we moved into a house with two staircases and had the rickety back staircase removed. The brief excitement of two staircases, with all the options for farcical, running-around-the-house mischief they afforded, was over. A fixation on which houses in our old Victorian neighborhood still had two staircases turned into repeated questioning about "how many staircases does x's house have?" (where x = anyone who lives in a house with two staircases), which, as we tried to minimize how much this question disrupted other conversations in progress, we generally answered, quietly, by signing the number two:
This "two" eventually turned into a floating signifier which J no longer solicited via staircase questions, but simply by requesting it outright: "Sign 'two'" or "Give me your twos." Or, as a work-around for the quotas we imposed, "Sign 'V'" (signed "V" being an ASL homonym for signed "two"). Or, when he eventually realized there was yet another homonym, "Sign 'peace.'" Seemingly morphing into J's inshaAlla, it became his routine sign-off on text and email messages: "I'm about to shower. Sign two." Except, of course, when he was pretending to be someone else.
For he long knew that "twos" made him different, and, in particularly that they could compromise his ability to get through a job interview and not get fired. Accordingly, he gradually managed to narrow the people whose twos he requested down to myself and his father. But we never expected them to go away entirely.
But gone they were, starting, as it turned out, on the day he turned 18. It was as if he'd made a quiet resolution some time ago about growing up--a resolution about which, for fear of causing a relapse, we didn't dare query him. But one month has passed, and the twos are still gone. So it now seems safe, if not to query J, at least to write about them here.
It's a milestone moment in two ways: both the end of a ten-year obsession, and the clearest indication of how much self-control he's gained in these final months of his childhood. There's even a bit of sadness: in a weird way, I'm missing those twos and the eccentricity they signified. More practically speaking, I'm missing what had been a handy negotiating tool--"Ok, then no twos for one week"--though I realize that seeking out more internal motivations is a good development for all of us.
Not that the obsessions are completely over. A new one appears to be coming out of the woodwork, but I'm guessing that he has a bit more company here:
"...if you pay me ten dollars!" It looks like we have new negotiating tool.
And then there's that other obsession: the one that has him forever starting conversations with guests at parties and staff members at restaurants and other establishments. The one that keeps him talking and socializing and scheming when there's nothing else obvious to talk or socialize or scheme about. Dating back as it does to when he was three months old and first able to express volition, it will, I'm quite confident, never go away:
Nor am I sure I want it to.