kitchen table math, the sequel: 1/13/08 - 1/20/08

Friday, January 18, 2008

help desk - "the minus sign"

Just spoke to C. who says he got the extra credit problem right on the test except he messed up the minus sign.

Messing up the minus sign is a chronic issue around these parts.

In the old days, C. would say things like, "I just forgot the minus sign!" Then I would say things like, "Well, the difference between plus-70 degrees and minus-70 degrees is the difference between life and death" and we'd go from there.

The other day I came across this post to Math Forum that made me think back fondly on those days:

I have wondered if anyone knows of a notation for negative numbers
> > that would help students avoid the crippling error of thinking that
> > they are just slightly flawed positive numbers. Have you not heard
> > a student say somethings like this," the solution of 2x = -6 is
> > 3 --oh, I mean negative 3.

Math Forum

Any more C. seems to take minus signs more seriously. But he's still losing track of them somewhere between Point A and Point B.

Any suggestions?

stupid in Brookline

In what may be the most innovative attempt to measure student progress in the state, Brookline officials recently unveiled a two-year effort to supplement the pencil-and-paper testing provided by the Massachusetts Comprehensive Assessment System with in-classroom activities that cover a broader range of topics and skills.

Junior high school science teachers have developed an assessment to measure skills in inquiry and data analysis, which are not measured by MCAS. Seventh-graders are presented with an activity involving a bouncing ball, while eighth-graders use a pendulum for their assessment, and how the students collect and analyze the results is graded according to guidelines that are shared by all of their teachers.

The resulting discussion among the teachers, according to Mark Goldner, a Heath School science teacher, explored the students' poor results in making sense of their data and how the teachers could deepen understanding of these skills.

In the past, Brookline officials have taken a strong stand against what they see as the narrowing focus by schools statewide to meet MCAS-tested standards, and the high-stakes requirement that students pass the MCAS to graduate from high school.


The new assessment system was constructed when teams of teachers, grouped by subject and grade, explored how to capture evidence of such learning.

Fischer-Mueller said Brookline officials read a lot of research on assessments and their role in closing the achievement gap, and look at what other schools are doing. "But we don't just transfer, we translate for what will work well in Brookline to get the results we want," she said.

note: They "read a lot of research."

Were any of them trained in assessment and psychometrics?


Did any of them go back to college and take a course or two in assessment and psychometrics?


Has the public been provided a bibliography of the "research" these educators read?


They read a lot of research, then spent hours of their time cooking up whole new tests to give their students on top of the state tests the kids already have to take. If you want your child to be tested and assessed all the livelong day, move to Brookline!

Brookline's fancy new tests will enable * Brookline teachers to send home report cards assessing the whole child, not just the child's knowledge of math, science, social studies, and ELA.

Here in Irvington we're way ahead of the game. Our Comments Bank [see here for sample Comment Bank including Positive and Negative Comments!] allows teachers to punch in a code that prints out such Comment-Bank gems as "finds subject matter difficult" or "inferential thinking needs improvement" on Quarterly Report Cards and Interim Reports. Lucky us. As a teacher told us this week, "Parents want to know these things."

Yes, indeed. Prefab essentializing comments about your child's innate cognitive abilities: so much more authentic than a simple B or C.

The public got its first extended look at the system during the School Committee's Jan. 3 meeting, where members praised the assessment presentations.

The public. Just getting its first extended look.

That rings a bell.

"This is such a culture change," chairwoman Judy Meyers said. "When I came to the School Committee eight years ago, there were pockets of excellence in individual schools, but no learning together and sharing what we knew. The whole approach has changed."


"We are mining MCAS data as well," Stone said. "But we put analysis and critical thinking in our learning expectations, and we must have a way to assess that. Paper-and-pencil tests just do not give us all we need."

More tests, please!

Because of state and national requirements, the paper-and-pencil tests aren't going away. But Brookline officials have long maintained that their existing curriculum is more than adequate, and students don't receive any special MCAS preparation. The new system won't demand significantly more time from students or teachers, officials said, but will provide a common way of measuring the performance of the district's teachers, students, and schools.

So....the new tests won't take any more time because a) we don't count the time we spent cooking up the tests, presenting the tests to the public, and flogging the tests in the education press and b) we're also going to ignore the time the kids spend taking the new tests and the teachers spend giving the new tests.

et voila

More tests in the same amount of time.

Board member and MCAS supporter Alan Morse questioned whether the science-inquiry assessment might interfere with students' ability to perform well on the MCAS science tests.

"No," said Sue Zobel, a Lincoln School science teacher. "A lot of us are coming out of mourning over MCAS and losing the inquiry strand because the test is a mile wide and an inch deep. We've been wanting to restore inquiry to science."

Superintendent Bill Lupini concurred.

"This doesn't just inform instruction, it's fun and motivational," he said of the new assessments. "In a lot of cases, it's using time a different way, rather than taking more time. These aren't assessments where you stop learning for a big test. You could argue that it is instruction: It's real-world learning."

In looking beyond MCAS, district seeks a sharper view
By Andreae Downs
Globe Correspondent / January 13, 2008

I have a suggestion.

Let's assess the fun and motivational part. Let's find out just how fun and motivational students find these fun and motivational non-tests. Then let's find out just how fun and motivational it is for parents to learn that their kids have hosed a whole new set of tests.

Suggestion number 2: I say we all send letters of support to board member Alan Morse.


* popular edu-term

Thursday, January 17, 2008

The Phonics Page


The Proficiency Illusion

I know that CJ wrote about it back in October, but it is worth revisiting:

Executive Summary of Fordham Institute's report, The Proficiency Illusion.

At the heart of the No Child Left Behind Act (NCLB) is the call for all students to be “proficient” in reading and mathematics by 2014. Yet the law expects each state to define proficiency as it sees fit and design its own tests. This study investigated three research questions related to this policy:

1. How consistent are various states’ expectations for proficiency in reading and mathematics? In other words, is it harder to pass some states’ tests than others?

2. Is there evidence that states’ expectations for proficiency have changed since NCLB’s enactment? If so, have they become more or less difficult to meet? In other words, is it getting easier or harder to pass state tests?

3. How closely are proficiency standards calibrated across grades? Are the standards for earlier grades equivalent in difficulty to those for later grades (taking into account obvious grade-linked differences in subject content and children’s development)? In other words, is a state’s bar for achievement set straight, sloping, or uneven?

This study used data from schools whose pupils participated both in state testing and in assessment by the Northwest Evaluation Association (NWEA) to estimate proficiency cut scores (the level students need to reach in order to pass the test for NCLB purposes) for assessments in twenty-six states. Here are the results:

• State tests vary greatly in their difficulty.

• Most state tests have not changed in difficulty in recent years.

• Improvements in passing rates on state tests can largely be explained by declines in the difficulty of those tests.

• Mathematics tests are consistently more difficult to pass than reading tests.

• Eighth-grade tests are consistently and dramatically more difficult to pass than those in earlier grades (even after taking into account obvious differences in subject-matter complexity and children’s academic development).

worth downloading and reading, especially if you live in a low-expectations state.

Here's the URL to download the PDF:

Sol Stern article online

School Choice Isn't Enough

I began writing about school choice in City Journal more than a decade ago. I believed then (as I still believe) that giving tuition vouchers to poor inner-city students stuck in lousy public schools was a civil rights imperative. Starting in the 1980s, major empirical studies by sociologist James Coleman and other scholars showed that urban Catholic schools were better than public schools at educating the poor, despite spending far less per student. Among the reasons for this superiority: most Catholic educators still believed in a coherent, content-based curriculum, and they enforced order in the classroom. It seemed immoral to keep disadvantaged kids locked up in dismal, future-darkening public schools when vouchers could send them to high-performing Catholic ones—especially when middle-class parents enjoyed education options galore for their children.

But like other reformers, I also believed that vouchers would force the public schools to improve or lose their student “customers.” Since competition worked in other areas, wouldn’t it lead to progress in education, too? Maybe Catholic schools’ success with voucher students would even encourage public schools to exchange the failed “progressive education” approaches used in most classrooms for the pedagogy that made the Catholic institutions so effective.

“Choice is a panacea,” argued education scholars John Chubb and Terry Moe in their influential 1990 book Politics, Markets and America’s Schools. For a time, I thought so, too.


During the 15 years since the first voucher program got under way in Milwaukee, university researchers have extensively scrutinized the dynamics of school choice and the effect of competition on public schools. The preponderance of studies have shown clear benefits, both academically and otherwise, for the voucher kids. It’s gratifying that the research confirms the moral and civil rights argument for vouchers.

But sadly—and this is a second development that reformers must face up to—the evidence is pretty meager that competition from vouchers is making public schools better.


Fifteen years into the most expansive school choice program tried in any urban school district in the country, Milwaukee’s public schools still suffer from low achievement and miserable graduation rates, with test scores flattening in recent years. Violence and disorder throughout the system seem as serious as ever. Most voucher students are still benefiting, true; but no “Milwaukee miracle,” no transformation of the public schools, has taken place.

incentivists versus instructionists

But does the school choice movement have a realistic Plan B for the millions of urban students who will remain stuck in terrible public schools?

According to Hoxby and Peterson, perhaps the two most respected school choice scholars in the country, no such plan is necessary. In their view, the best hope for education improvement continues to be a maximum degree of parental choice—vouchers if possible, but also charter schools and tuition tax credits—plus merit-pay schemes for teachers and accountability systems that distinguish productive from unproductive school principals.

That “incentivist” outlook remains dominant within school reform circles. But a challenge from what one could call “instructionists”—those who believe that curriculum change and good teaching are essential to improving schools—is growing, as a unique public debate sponsored by the Koret Task Force on K–12 Education revealed.


[I]n early 2007, members [of the Koret Task Force] did agree to hold a debate at the group’s home, the Hoover Institution at Stanford University: “Resolved: True School Reform Demands More Attention to Curriculum and Instruction than to Markets and Choice.” Hirsch and Ravitch argued the affirmative, Hoxby and Peterson the negative.


While the arguments about school choice and markets swirled during the past 15 years, both Ravitch and Hirsch wrote landmark books (Left Back and The Schools We Need and Why We Don’t Have Them, respectively) on how the nation’s education schools have built an “impregnable fortress” (Hirsch’s words) of wrong ideas and ineffective classroom practices that teachers then carry into America’s schools, almost guaranteeing failure, especially for poor minority children. Hirsch’s book didn’t just argue this; it proved it conclusively, to my mind, offering an extraordinary tour d’horizon of all the evidence about instructional methods that cognitive neuroscience had discovered.

If Hoxby and Peterson were right in asserting that markets were enough to fix our education woes, then the ed schools wouldn’t be the disasters that Hirsch, Ravitch, and others have exposed. Unlike the government-run K–12 schools, the country’s 1,500 ed schools represent an almost perfect system of choice, markets, and competition. Anyone interested in becoming a teacher is completely free to apply to any ed school that he or she wants. The ed schools, in turn, compete for students by offering competitive prices and—theoretically—attractive educational “products” (curricula and courses). Yet the schools are uniformly awful, the products the same dreary progressive claptrap. A few years ago, the National Council on Teacher Quality, a mainstream public education advocacy group, surveyed the nation’s ed schools and found that almost all elementary education classes disdained phonics and scientific reading. If the invisible hand is a surefire way to improve curriculum and instruction, as the incentivists insist, why does almost every teacher-in-training have to read the works of leftists Paolo Freire, Jonathan Kozol, and William Ayers—but usually nothing by, say, Hirsch or Ravitch?


Those who put their faith in the power of markets to improve schools must at least show how their theory can account for the stubborn persistence of the thoughtworld.

Instead, we increasingly find the theory of educational competition detaching itself from its original school choice moorings and taking a new form. Vouchers may have stalled, but it’s possible—or so many school reformers and education officials now assure us—to create the conditions for vigorous market competition within public school systems, with the same beneficent effects that were supposed to flow from a pure choice program.

Nowhere has this new philosophy of reform been more enthusiastically embraced than in the New York City school district under the control of Mayor Michael Bloomberg and schools chancellor Joel Klein. Gotham’s schools are surging ahead with a host of market incentives, including models derived from the business world.


the Bloomberg administration and its supporters are pushing markets and competition in the public schools far beyond where the evidence leads. Everything in the system now has a price. Principals can get cash bonuses of as much as $50,000 by raising their schools’ test scores; teachers in a few hundred schools now (and hundreds more later) can take home an extra $3,000 if the student scores in their schools improve; parents get money for showing up at parent-teacher conferences; their kids get money or—just what they need—cell phones for passing tests.


While confidently putting their seal of approval on this market system, the mayor and chancellor appear to be agnostic on what actually works in the classroom. They’ve shown no interest, for example, in two decades’ worth of scientific research sponsored by the National Institutes of Health that proves that teaching phonics and phonemic awareness is crucial to getting kids to read in the early grades. They have blithely retained a fuzzy math program, Everyday Math, despite a consensus of university math professors judging it inadequate. Indeed, Bloomberg and Klein have abjured all responsibility for curriculum and instruction and placed their bets entirely on choice, markets, and accountability.

in Massachussetts
Those in the school reform movement seeking a case of truly spectacular academic improvement should look to Massachusetts, where something close to an education miracle has occurred. In the past several years, Massachusetts has improved more than almost every other state on the NAEP tests. In 2007, it scored first in the nation in fourth- and eighth-grade math and reading. The state’s average scale scores on all four tests have also improved at far higher rates than most other states have seen over the past 15 years. The improvement had nothing to do with market incentives. Massachusetts has no vouchers, no tuition tax credits, very few charter schools, and no market incentives for principals and teachers. The state owes its amazing improvement in student performance to a few key former education leaders, including state education board chairman John Silber, assistant commissioner Sandra Stotsky, and board member (and Manhattan Institute fellow) Abigail Thernstrom. Starting a decade ago, these instructionists pushed the state’s board of education to mandate a rigorous curriculum for all grades, created demanding tests linked to the curriculum standards, and insisted that all high school graduates pass a comprehensive exit exam.


Now a professor of education reform at the University of Arkansas, Stotsky sums up: “The lesson from Massachusetts is that a strong content–based curriculum, together with upgraded certification regulations and teacher licensure tests that require teacher preparation programs to address that content, can be the best recipe for improving students’ academic achievement.”


The Hoxby/Peterson position is here.

I'm with Stern.

I think the terms used to describe this phenomenon are path dependence and relative autonomy.

Unfortunately, I'm not remotely confident that accountability has worked or will work, either, Massachusetts miracle aside, although I continue to support NCLB.

What I can't figure out is whether bad accountability, which is mostly what we have, is better than no accountability. If the state standards are fuzzy or trivial, is it better not to have standards at all?

Sol Stern in City Journal
pause for reflection

function machines


LA County

from Kings School?

from Utah State

from Math Forum

functions in algebra

Tuesday, January 15, 2008

Photosynthesizing pupils

I shouldn't be posting late at night when I should be asleep and can't think straight. But reading this piece about education on a leftist site again raises a nagging question in my mind about the influence of political belief on curriculum and instruction. I like to believe that there is no intrinsic connection between political beliefs and forms of instruction (pedagogy). Things are probably much more complicated than that. I can see an entanglement of curriculum and instruction in this piece that makes pedagogy susceptable to political orientation. At least this writer sees a political angle:

The proof and the power of these metaphors expressed through the progressive empowerment frame is in the programs and activities that come from them. In The Framing of No Child Left Behind (NCLB), I described some general activities that come from thinking in the empowerment frame:

It makes more sense to assess learning holistically, using projects and real-life activities and through descriptions of progress (intellectual "growth"), as much as possible. Further, assessment is integrated into each student's learning activities, rather than being done as an external process, by and for others. What the teacher does or says is not expected to be absorbed directly by the students. Rather, like the air, soil, and water that a plant converts into its green structure, students construct their knowledge from the resources and experiences provided to them by the teacher and student understandings will look and be different than exactly what the teacher taught. Thus, the teacher and students must continuously assess and communicate about lesson goals and student progress.

In conservative production frame, knowledge is thought of as discrete objects that are delivered by the teacher and absorbed directly by the student. This is why standardized tests make sense in the factory metaphors of the conservative production frame, but not in the gardening metaphors of the progressive empowerment frame. Measuring corn more often doesn't make it grow faster or taste sweeter.

So I guess if you are totally carried away by beautiful metaphors you can lose touch with reality. Measuring corn more often doesn't make it grow faster or taste sweeter but if you don't keep an eye on it you may end up with dried up, shriveled and stunted plants.

Saxon Math @ North Beach

For an unscripted kid talking about Saxon, you may want to see this one just for fun:


This is a copy of an email I sent out to my local math list shortly before the last algebra test:

I found a nice group of “linear function” word problems today (answers included - pdf file).

The kids in Math A are doing these problems. They aren’t classic Algebra 1 word problems, so if your child needs extra practice they’re not easy to come by.

As I understand it, algebra 1 used to be focused on setting up and solving linear equations. The classic Algebra 1 word problems were number problems, consecutive integer problems, distance problems (2 trains leave a station), coin & age problems, & mixture & work problems.

Today algebra 1 tends to be focused on the idea of linear functions. That’s why our kids did all those “function machine” problems back at the Main Street School. In a linear function problem you are given a pair of “solutions” and asked to write an equation to model the situation. My favorite linear function word problem thus far is the Celsius – Fahrenheit conversion:

The graph of an equation to convert degrees Celsius, x, to degrees Fahrenheit, y, has a y-intercept of 32°. Given that water boils at 212°F and at 100°C, write the conversion equation.

To write the equation you use the two coordinated pairs you already know:

When Celsius is 0 degrees, Fahrenheit is 32
When Celsius is 100 degrees, Fahrenheit is 212

These two points are coordinated pairs on the graph of the function:


You derive the slope of the linear equation from these two points; then you plug in one of the coordinated pairs and derive the y-intercept; et voila.

y = 9/5 x + 32


F = 9/5C + 32

I've never been able to remember how to convert from Celsius to Fahrenheit, and knowing how to use the two known conversion points to produce the formula is great.

I'm curious about people's reaction to this shift in content. One of our regulars is skeptical; he sees this as a move to turn algebra into statistics. Another sees this as another instance of pushing advanced topics lower down in the curriculum. Functions used to be introduced in calculus; now kids see "function machines" in 3rd grade. (And, of course, both motivations could be involved - both and more.)

Any thoughts?

Teaching to the Test

What is SO important that it makes flunking these questions OK? What part of these questions does not relate to understanding? Why would you need parent involvement or even homework to pass the NAEP test? Do schools have any way of determining if they screw up rather than the kids?

From NAEP 2007 Sample Math Questions and Results - 4th Grade

9. On the chart, circle all the numbers that have 4 as a factor.

1 2 3 4 5
6 7 8 9 10
11 12 13 14 15

Did you use the calculator on this question?

37 percent correct.

- - -

13. Mark's room is 12 feet wide and 15 feet long. Mark wants to cover the floor with carpet. How many square feet of carpet does he need?

Answer: _____________ square feet

The carpet costs $2.60 per square foot. How much will the carpet cost?

Answer: $ _____________

Did you use the calculator on this question?

24 percent correct.

- - -

14. There will be 58 people at a breakfast and each person will eat 2 eggs. There are 12 eggs in each carton. How many cartons of eggs will be needed for the breakfast?

A) 9
B) 10
C) 72
D) 116

Did you use the calculator on this question?

20 percent correct.

- - -

15. Ben bought 4 items at a bake sale and added their cost on his calculator. The total cost read 1.1 on the calculator.

What amount does Ben need to pay?

A) 11 cents
B) 1 dollar and 1 cent
C) 1 dollar and 10 cents
D) 11 dollars

Did you use the calculator on this question?

29 percent correct, in spite of the fact that they push calculators and understanding.

- - -

2. What number is 10 more than 5,237?

A) 5,238
B) 5,247
C) 5,337
D) 6,237

81 percent correct. Still about 20 percent got this wrong, and this is 4th grade.

- - -

This is not about constructivism, understanding, IQ, or SES. They complain about teaching to the test, but then what on earth do they want to teach that's more important? It should take them no more than 50 percent of class time to get kids to master this material - no homework required. Do schools think that these kids are stupid? At what point can you stop blaming external causes?

Teachers think that the problem with education is what walks into their rooms. They see kids who haven't mastered previous material and they can't imagine how they are going to fix that. The real problem is how did these kids get there in the first place. It used to be that social promotion was a last resort. Now, it's promoted as a normal part of (Everyday Math) spiral mastery. They are making the problem worse and getting nothing in return.

Monday, January 14, 2008

rightwinprof clears that one up

The first position in a sentence is topic position (this, by the way, is a pan-linguistic phenomenon), that is, the position of primary focus (focus is actually different from topic, but the distinction isn't relevant here). So "John went swimming" is about John, because John appears in topic position. We can, in English, shift something other than the subject to topic position when we want to make it, and not the subject, the topic of the sentence. This is called dislocation: "But as for Mary, John left her at the party." We also use cleft sentences to place something other than the subject in topic position, and make it the topic of the sentence. A cleft sentence looks like: "It was Mary John left at the party." Jefferson masterfully controls the topic throughout the document, and shifts it from section to section, beginning with the abstract, moving to King George III, and ending with the colonists. It's particularly effective in the long section of grievances, all of which begin with "He," meaning, of course, George III, even when the perpetrators of those wrongs were, in fact, other parties (ultimately, yes, Georgie was responsible).

Actually, to see what I was talking about wrt grammar and rhetoric, see Joseph Williams's Style: Lessons in Clarity and Grace (am I imagining it, or didn't you say you had ordered it?) He has an excellent section on using topicalization to create coherence.

I do have Joseph Williams' book (haven't read much thus far) -- it is fantastic.

My neighbor thinks so, too. I got it for her for Christmas.

Two Resources for Math Facts Mastery (1) Video Game, Timez Attack, (2) Re-introduction to Dave Marain's MathNotations Blog

Review from Math Notations

Here's the heart. of his review.

Having seen dozens of computer learning software products both online and on CD-ROM, I've reached the point where I can rapidly identify quality and Timez Attack is quality in every sense. I decided to email Ben Harrison, the president of the company, and share my thoughts about his product, the only item his company sells at this point. I explained that I would like to write an objective review of Timez Attack for MathNotations after I had an opportunity to try it myself and to observe a child using it. I asked him if he would be willing to give me access to the full version and he agreed. I haven't had time to thoroughly play around with this new version, but my son has been on it for about a week and he likes it. To derive the maximum benefit of this software, I feel I need to establish consistency for him to be on it for at least 15-30 minutes a day for about a month but that has not yet happened. I already see better retention but he is still only on a beginner level.

Here's the BigBrainz homepage:

Repetition is the key to mastery, so this does look really promising. The "starter version" is free.

is maintained by Dave Marain

Look for fully developed math investigations that are more than one inch deep, math challenges, Problems of the Day and standardized test practice. The emphasis will always be on developing conceptual understanding in mathematics. There will also be dialogue on issues in mathematics education with a focus on standards, assessment, and pedagogy primarily at the 7-12 level through AP Calculus.

Previous mentions of Dave Marain & MathNotations at KTM:

Interview with Lynn Arthur Steen one, two and follow up commentary: one two

Sunday, January 13, 2008

Ed Schools: Help or Hindrance?

“Evidence indicates that the traditional or teacher-centered approaches are more effective in helping students achieve academic goals than the learner-centered or “progressive” approaches.” -- George K. Cunningham

Wow, he actually said it.

George K. Cunningham's paper, Education Schools: Helping or Hindering Potential Teachers? is significant and if you're a parent, educator, or administrator it's worth your time. I promise.

If you're an education school student, please, please read it. Challenge the establishment, question authority, and dare to take the road less traveled. Come on, be a rebel and turn in an ed school research project about Project Follow Through, direct instruction, or precision teaching! Just think of it as retro-chic.

While Cunningham examines the University of North Carolina in his January 2008 paper for the Pope Center for Higher Education Policy, it is clear that education schools across the nation are entrenched in the same progressive/constructivist rhetoric he frowns upon. As a result, this rhetoric rules the day in most public school classrooms and our children are paying dearly for it.

“This advocacy of rhetoric as opposed to practical learning leads education students into realms far afield from normal education as most people understand it. It leaves precious little time to teach reading, writing, and arithmetic.”

I posted lots more about it here. Instructivist commented how enormously important Cunningham's paper is. I certainly agree.