Does anyone remember the math wars?

The New York Times is out with yet another entry on the failure of traditional teachers to teach math to Americans:

My hunch is that how we learn math in America has very little to do with best practices and a lot to do with how teachers remember learning math when they were children.

Can you spell Weltanschauung?

Free advice: Never trust a "hunch" when your hunch is identical in every respect with the worldview of a Most Emailed story in the New York Times.

If your hunch is identical in every respect with the worldview of a Most Emailed story in the New York Times, it's not a hunch. It's conventional wisdom and nobody needs to hear it again.

Contra Elizabeth Green, we do not have to accept that the traditional approach we take to teaching math — the one that can be mind-numbing, but also comfortingly familiar — does not work. It should be obvious to anyone who actually looked at our history that "the traditional approach to teaching math" worked perfectly well for many American students.

Here's Barry Garelick:

From the 1940′s to the mid 1960′s, at a time when math and other subjects were taught in the traditional manner, scores in all subjects on the Iowa Tests of Basic Skills increased steadily. From 1965 to the mid-70′s there was a dramatic decline, and then scores increased again until 1990 when they reached an all-time high. Scores stayed relatively stable in the 90′s.

TRADITIONAL MATH MEANS NEVER HAVING TO SAY YOU'RE SORRY (BARRY GARELICK)

The decline in SAT scores doesn't fit the narrative, either.

Do we know when state legislatures passed laws requiring teachers to have degrees in education?

I'm trying to track that down.

Speaking of the math wars, it's been a couple of decades now since the NCTM, an organization whose membership consists almost entirely of public school math teachers, accepted that the traditional approach we take to teaching math does not work.

Chris is turning 20. Neither he nor any of his peers learned their math facts at school.

dysteachia

In a well-publicized paper that addressed why some students were not learning to read, Reid Lyon (2001) concluded that children from disadvantaged backgrounds where early childhood education was not available failed to read because they did not receive effective instruction in the early grades. Many of these children then required special education services to make up for this early failure in reading instruction, which were by and large instruction in phonics as the means of decoding. Some of these students had no specific learning disability other than lack of access to effective instruction. These findings are significant because a similar dynamic is at play in math education: the effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.
Mathematics Education: Being Outwitted by Stupidity by Barry Garelick

The effective treatment for many students who would otherwise be labeled learning disabled is also the effective preventative measure.

and see Galen Alessi's classic: Diagnosis Diagnosed

math and race in Iowa

Among the possible explanations offered for the decline [in ITBS scores] are increased drug use in the mid-60′s, permissiveness, increase in divorces and single family homes, as well as the progressivist trends in education resulting in student-centered and needs-based courses. (See  Protecting Students from Learning for a more extensive discussion of this last item.) Another explanation offered is that the population of test takers starting around that time began to include more minority students, resulting in a dilution effect. That argument fails to explain, however, why the same pattern of declining test scores for the SATs exists for the ITBS and ITED test scores which were not limited only to college bound students. Also significant is the fact that the population of test takers in Iowa, Minnesota and Indiana remained primarily white which has been noted by Bishop (1989) and Murray (1992). Specifically, the U.S. Census of 1950 shows that the population in Iowa was 99.2 percent white, declining by 0.7 percentage points to 98.5 percent white by 1980. Similarly, the populations of Minnesota and Indiana were 99 and 95.5 percent white in 1950, dropping respectively to 98.2 and 92.8 by 1970. (Hobbes, 2002).
Barry Garelick: The Myth About Traditional Math Education
Education News

fractions in a 1940 arithmetic text

Terrific new article by Barry G: The Myth About Traditional Math Education.

The equal division of three cupcakes among four people, or the equal division of a 3 inch line into four parts, is an extension of the idea of division of a whole number by a lesser whole number, which students have already mastered. Students already know that if 12 cupcakes are equally divided among four people, then each person gets 12/4 cupcakes. This idea is extended by starting with the problem of 3 divided by 4, and expressing 3 as 12 fourths. The problem is now stated as 12 fourths divided among 3 people, so that each person receives 3 fourths. This idea is then applied to a line three inches long, so that fractions are ultimately related to a number line, and the final point made that fractions are a representation of division. This is a key concept and ultimately underscores an idea of representing a fractional part as a unit unto itself. That is, 3/4 of an inch can be thought of as a unit (i.e., there are four such units in a three inch line) which is a cornerstone idea when fractional division is studied later.

And here's Barry's chart showing the rise and fall in test scores:

practice in school

Hey everyone - I'm back from IL (didn't get to see Susan S - darn!) - and have just read a brilliant comment left by Lynne Dilligent on Joanne Jacobs' blog. Lynne's comment sums up a core frustration I've felt with the schools forever, re: the need for the school, not the parents, to be in charge of providing and overseeing the practice children need to learn what they're supposed to be learning.

Must go do "SAT work" with C. -- back in a little bit.

(Thanks to Barry for sending the link.)

getting used to it

Young man, in mathematics you don't understand things. You just get used to them.

Reply to Felix T. Smith who had said "I'm afraid I don't understand the method of characteristics."

—as quoted in The Dancing Wu Li Masters: An Overview of the New Physics (1984) by Gary Zukav footnote in page 208.

John von Neumann

I know people here have quoted this before (I think Vlorbik might have been the person who I first heard this from). But I just stumbled across it today and had to quote it again.

For what it's worth, this observation describes my own experience of learning math - quite a bit of it, at any rate.

no stacking at school

via StopTerc Nevada

daily quizzes beat concept maps

an oldie but goodie: the Gambill method

Revolutionizing Math at the School of the Future

(Cross-posted at Out in Left Field)

A front page article in Monday's Local News section of the Philadelphia Inquirer profiles a math class at Philadelphia's Microsoft-funded High School of the Future, whose teacher, Thomas Gaffey, placed second in Microsoft's U.S. Innovative Education Forum and was a semi-finalist in its Worldwide Innovative Education Forum. In Gaffey's ninth-grade algebra class there are:

No textbooks, no paper, no chalk, no desks, and no assigned seats.

Instead, students use laptops while sitting in rolling chairs at trapezoidal tables spaced out in hexagonal classrooms.

Just how newsworthy this sounds to you depends on whether you think chair mobility and table shape have a big influence on learning, on whether you've been following current trends in education over the last 50 years, and on how unusual you think it is for a teacher to "encourage his students to find answers to their own questions" and engage with them in exchanges like these:

"Is this an obtuse triangle?" one student asks.

"Well, what can you tell me about an obtuse triangle?" Gaffey replies.

"One of the angles has to be more than 90 degrees," the student answers.

"Are any of the angles here like that?"

"Yeah. Oh, I get it now!"

As the Inquirer explains:

This snippet of student-driven discussion is a glimpse of the style and approach that have earned Gaffey national and international recognition.

Student-driven? Who's asking most of the questions? But I'm splitting hairs here. What I should be asking is: Why does this kind of exchange warrant international recognition?

To fair, it wasn't this, specifically, that earned Gaffey his honors. Rather:

Les Foltos, one of the judges who reviewed Gaffey's work, was impressed by his emphasis on "actively engaging students in solving real-world problems." As Gaffey puts it, "If we want to teach math to learners, we should teach math how it is actually used. It doesn't matter how much you know. It matters what you can do."

Ah yes, "real world problems." Again, only if you've been out of touch with the last half century of educational reform, and with today's Reform Math in particular, will this strike you as revolutionary. Here is Gaffy's version of real world math:

In his classroom on a recent Tuesday, Gaffey's challenge to his "learners" - as students in the Parkside public school are called - was to estimate Earth's land area.

To solve the problem, the class first covered basic concepts about area and polygons - shapes with three or more straight sides.

Gaffey then asked, "If a shape has four sides, is it always a polygon?"

Learners who answered yes (the wrong answer) were asked to redefine what a polygon is, while those who answered no were asked to draw a four-sided shape that was not a polygon on the class "smart board."

Gaffey drew a shape with three straight sides and one curved side.

"Is this a polygon?" he asked.

"No," the class responded.
...

The class drew lines through each of the continents, chopping them up into complex polygons, then simple polygons.

The final phase was to derive formulas for the areas of the simple polygons, and add up the areas.

This sort of problem is not particularly new, as a quick survey through now-standard textbooks like Everyday Math and the Interactive Math Program makes clear. And it's been around long enough to have garnered some serious criticism--specifically in what Barry Garelick calls its "just in time" approach to teaching.

Among other things, "just in time" often means serious delay. For example, one would hope that students would already know the formulas for the areas of simple polygons, and how to derive them, well before they hit 9th grade.

But because so many students are so far behind where they should be, there is one thing in which I and Gaffey are in whole-hearted agreement. In Gaffey's words, as cited by the Inquirer:

"Math education, more than any other subject, is in need of drastic reform."

off-task

Chesterfield County middle schools this year are assessing all eighth-graders' knowledge of 21st-century skills, which include communication, collaboration, critical thinking, problem solving, innovation and use of technology.

"We obviously want our kids to have those 21st-century skills," said director of technology Lynda Gillespie. "The only way to make sure that you have those skills is through assessment."

Area schools assess students’ 21st-century skills
JUAN ANTONIO LIZAMA TIMES-DISPATCH STAFF WRITER
Published: May 24, 2010

Meanwhile, back on Planet Earth, life goes on:

Statistically speaking, however, having an expertise in statistics may help in getting a job: according to a survey conducted by the National Association of Colleges and Employers, graduates with math skills are more likely than their peers in other majors to find themselves promptly and gainfully employed.

The safest of all degrees to be acquiring this year is in accounting: forty-six per cent of graduates in that discipline have already been offered jobs. Business majors are similarly placed: forty-four per cent will have barely a moment to breathe before undergoing the transformation from student to suit. Engineers of all stripes—chemical, computer, electrical, mechanical, industrial, environmental—have also fared relatively well since the onset of the recession: they dominate a ranking, issued by Payscale.com, of the disciplines that produce the best-earning graduates. Particular congratulations are due to aerospace engineers, who top the list, with a starting salary of just under sixty thousand dollars—a figure that, if it is not exactly stratospheric, is twenty-five thousand dollars higher than the average starting salary of a graduate in that other science of the heavens, theology.

Economics majors aren’t doing badly, either: their starting salary averages about fifty thousand a year, rising to a mid-career median of a hundred and one thousand.

Learning by Degrees
by Rebecca Mead
The New Yorker
June 7, 2010

Ron Aharoni on calculations & calculators

Calculation isn't just figuring out the result of an exercise: it is figuring out the decimal representation of the result. Therefore, the ability to calculate is in fact tantamount to a profound understanding of the decimal system. This is one of the reasons why calculation is so important, and why it should not be replaced by a calculator.

Arithmetical operations can be calculated in many ways. The methods currently taught in school are the result of generations of thought, and much wisdom has been invested in them. Most are based on writing the exercises vertically, so that the ones digits are one above the other, the tends digits are one above the other and so forth.

Calculations are based on the knowledge of the addition and multiplication tables -- the sums and products of numbers smaller than 10. These must be memorized. The addition table should be well established in the first grade, and the multiplication table in the second or third grade. In addition, the children should be familiar with the rules that govern the operations, such as the distributive law and the rules of change.

The operation of division is the most difficult to calculate. On the other hand, the algorithm of division, called "long division," includes fundamental principles and therefore it should not be passed over.

Arithmetic for Parents: A Book for Grownups about Children's Mathematics
by Ron Aharoni
p. 95

Barry G on exercises vs. problems

By way of introduction, I am neither mathematician nor mathematics teacher, but I majored in math and have used it throughout my career, especially in the last 17 years as an analyst for the U.S. Environmental Protection Agency. My love of and facility with math is due to good teaching and good textbooks. The teachers I had in primary and secondary school provided explicit instruction and answered students’ questions; they also posed challenging problems that required us to apply what we had learned. The textbooks I used also contained explanations of the material with examples that showed every step of the problem solving process.

I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.

Optimistically believing that I could make a difference in at least a few students’ lives, I decided to teach math when I retire. I enrolled in education school about two years ago, and have one class and a 15-week student teaching requirement to go. Although I had a fairly good idea of what I was in for with respect to educational theories, I was still dismayed at what I found in my mathematics education courses.

In class after class, I have heard that when students discover material for themselves, they supposedly learn it more deeply than when it is taught directly. Similarly, I have heard that although direct instruction is effective in helping students learn and use algorithms, it is allegedly ineffective in helping students develop mathematical thinking. Throughout these courses, a general belief has prevailed that answering students’ questions and providing explicit instruction are “handing it to the student” and preventing them from “constructing their own knowledge”—to use the appropriate terminology. Overall, however, I have found that there is general confusion about what “discovery learning” actually means. I hope to make clear in this article what it means, and to identify effective and ineffective methods to foster learning through discovery.

To set this in context, it is important to understand an underlying belief espoused in my school of education: i.e., there is a difference between problem solving and exercises. This view holds that “exercises” are what students do when applying algorithms or routines they know and the term can apply even to word problems. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, we future teachers are told that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

As someone who learned math largely though mere exercises and who now creatively applies math at work, I have to question this thinking.
Discovery learning in math: Exercises versus problems
by Barry Garelick

Me, too.

What is new with the science on math disabilities?

Wednesday, December 02, 2009

Atypical numerical cognition, dyscalculia, math LD: Special issue of Cognitive Development

A special issue of the journal Cognitive Development spotlights state-of-the-art research in atypical development of numerical cognition, dyscalculia, and/or math learning disabilities.

Article titles and abstracts are available at Kevin McGrew's excellent IQ's Corner blog.



-----------
Joe Elliot on dyslexia:

"Contrary to claims of ‘miracle cures’, there is no sound, widely-accepted body of scientific work that has shown that there exists any particular teaching approach more appropriate for ‘dyslexic’ children than for other poor readers."

I am in agreement with Elliot.

I wonder if the same will be found to be true for dyscalculia and kids who struggle with math.

Math problems of the week: Systems of Equations in CPM vs. 1900's math

(Cross-posted at Out In Left Field)

1. The only systems of equations that students are required to solve algebraically in the CPM (College Preparatory Mathematics) Algebra Connections "Systems of Equations" chapter (published in 2006):

y = 1160 + 22x
y = 1900 - 15x

-----------

y = 6 + 1.5x
y = 2x

-----------

y = 2x -3
y = -x + 3

-----------

y = 2x -3
y = 4x + 1

-----------

y = 2x - 5
y = -4x - 2

-----------

y = -x + 8
y = x -2

-----------

y = -3x
y = -4x + 2

-----------

y = 2x - 3
y = 2x + 1

-----------

y = -4x -3
y = -4x + 1

-----------

2. A subset of the over one hundred systems of equations in the Wentworth's New School Algebra "Simple Systems of Equations" chapter (published in 1898):

5x + 2y = 39
2x - y = 3

-----------

x/3 + y/2 = 4/3
x/2 + y/3 = 7/6

-----------


x + y - 8 = 0
y + z - 28 = 0
y + z - 14 = 0

-----------

6x - 2y + 5z = 53
5x + 3y + 7 = 33
x + y + z = 5

-----------

2x + 3y + 1 = 31
x - y + 3z = 13
10y + 5x - 2z = 48

-----------

1/x + 2/y - 3/z = 1
5/x + 4/y + 6/z = 24
7/x - 8/y + 9/z = 14

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2/x - 3/y + 4/z = 2.9
5/x - 6/y - 7/x = -10.4
9/y + 10/z - 8/x = 14.9

3. Extra Credit:

(a) Discuss why CPM, but not New School Algebra, has to stipulate that the simultaneous equations be solved algebraically (rather than graphically or by "guess and check").

(b) Discuss the arithmetic and algebraic skills required by each problem set.

(c) Relate your answer in (b) to the final assignment in CPM's "Simultaneous Equations" chapter, the TEAM BRAINSTORM:

With your team, brainstorm a list for the following topics. Be as detailed as you can. How long can you make your list? Challenge yourselves. Be prepared to share you team's ideas with the class.

Topics: What have you studied in this chapter? What ideas and words were important in what you learned? Remember to be as detailed as you can.

pick one

Class size reduction or rapid formative assessment?: A comparison of cost-effectiveness

 Stuart S. Yeh, a, University of Minnesota, Educational Policy and Administration, 86 Pleasant Street, S.E., Minneapolis, MN 55455, United States Received 18 October 2007; revised 26 June 2008; accepted 25 September 2008. Available online 2 October 2008. 

Educational Research Review Volume 4, Issue 1, 2009, Pages 7-15

Abstract 

The cost-effectiveness of class size reduction (CSR) was compared with the cost-effectiveness of rapid formative assessment, a promising alternative for raising student achievement. Drawing upon existing meta-analyses of the effects of student–teacher ratio, evaluations of CSR in Tennessee, California, and Wisconsin, and RAND cost estimates, CSR was found to be 124 times less cost effective [emphasis added] than the implementation of systems that rapidly assess student progress in math and reading two to five times per week. Analysis of the results from California and Wisconsin suggest that the relative effectiveness of rapid formative assessment may be substantially underestimated. Further research regarding class size reduction is unlikely to be fruitful, and attention should be turned to rapid formative assessment and other more promising alternatives.

Middle-school Math Classes Are Key To Closing Racial Academic Achievement Gap

ScienceDaily (Apr. 22, 2009) — More challenging middle-school math classes and increased access to advanced courses in predominantly black urban high schools may be the key to closing the racial academic achievement gap, according to a University of Illinois study.


"Although we've poured a lot of money and resources into trying to reduce inequalities between black and white students, we've mainly focused on test scores and that hasn't been successful," said Christy Lleras, a U of I assistant professor of human and community development.

Why target middle-school math? Lleras said there's a feedback loop between math placement, student effort, and academic achievement.

"Over time, these three factors affect each other. Students who take more advanced math courses in middle school lengthen their lead over time, and the positive school-related behaviors developed in those advanced courses lead to even higher achievement.

"But the opposite is also true. Lower math placement in middle school significantly lowers a student's chances of getting into higher-level math courses in high school, which translates into fewer skills and behaviors and greater achievement gaps in high school," she said.

These gaps are largest in high-minority urban schools. "For kids in predominantly black urban schools, the biggest predictor of the math course they took in high school was the math course they took in eighth grade. For all other students, the biggest predictor was their prior achievement, not the course they took," she noted.

Lleras used data from the U.S. Department of Education's National Educational Longitudinal Study to follow the effects of math placement, school-related behaviors, and achievement in more than 6,500 public school students as they progressed from the eighth to the tenth grade.

Transcript data indicated the highest-level math course the student had taken at these levels. Math achievement was measured via tests given at the end of these school years. And engagement and effort were measured by teachers' evaluations of the student's attentiveness, disruptiveness, and homework habits.

Lleras believes that increased access to more advanced and rigorous math classes in high-minority urban schools can have a significant direct effect on all students' achievement and particularly that of African American students.

"Being in a classroom where the expectations are higher, the course work is more rigorous, and the climate is more academic has huge effects on student effort," she said.

Lleras worries that lower-performing schools will concentrate on teaching to the tests mandated by No Child Left Behind.

"Instead of focusing on test scores, we may be better able to affect educational trajectories by improving teacher quality and reducing class sizes, which helps to create school climates that foster both academic learning and student effort," she said.

Because racial achievement gaps were already significant by eighth grade, Lleras believes educators must begin to address gaps in achievement and opportunities to learn much earlier.

She argues that universal preschool and expansion of Head Start would go a long way toward reducing early racial inequalities because early-childhood programs tend to affect student-related attitudes and engagement more than achievement test scores.

"Children can't learn new material until they have the toolkit of skills and school-related behaviors to do so," she said.

"Then we have to make a sustained effort to keep these children learning over time. We need a persistent and additional effort to support urban minority students through tutoring programs and improved access to challenging material and high-quality teachers," she said.

"This study was a snapshot of three years in these kids' lives, and in just three years, they were falling farther and farther behind," she added.

The study was published in a recent issue of the American Educational Research Journal.

Why More Mathematicians Don't Oppose Reform Math: and why we desperately need them to

(Cross-posted at Out In Left Field).

Yesterday's NPR Weekend Edition Saturday featured an interview with Stanford University professor Keith Devlin on the importance of Algebra, and while I listened to it, it suddenly occured to me why more mathematicians don't oppose Reform Math.

Here's what I posted on the NPR website:

Keith Devlin suggests that, given calculators, students should focus less on accurate arithmetic calculations, and more on algebraic reasoning. But, as Devlin's fellow mathematicians (e.g., Howe, Klein, & Milgram) have argued, mastering the basic algorithms of arithmetic is essential preparation for algebra. And while the most mathematically inclined students--including Devlin himself--may be able to master these algorithms without much hands-on, numerical practice, the vast majority do need lots of practice, and striving for correct answers is an essential part of that practice.

Today's arithmetic, unfortunately, has been seriously watered down by the new "Reform Math". More mathematicians need to examine this curriculum and speak out against it; ironically, because they can get by without much arithmetic practice, and because so many of them found arithmetic boring, too few mathematicians have considered the potentially dire consequences that the latest trends in grade school math present to the rest of the population (and to the country as a whole).

When our most prominent, accomplished mathematicians, who themselves may well have gotten by without developing accurate arithmetic skills, discount the importance of teaching such skills to the general population, they do a terrible disservice to elementary school math education (and may themselves be horrified by the results, years later, when today's grade school students enter their classrooms).

Consider what one other NPR poster has taken away from the Devlin interview. As she writes in her post:

I want to thank Dr. Devlin for a great quote that I plan to post at the front of my classroom. "Mathematicians often make mistakes in elementary arithmetic because we have our minds on higher things." That will come in very handy!

Yikes!

Paul is back!

A while back we had a vertical meeting with math teachers from grades 4-8. The agenda was to familiarize ourselves with the standards for fractions outside of our own grade levels. The facilitators cut all the relevant standards into little strips, removing any identity as to the grade levels they came from. Our job was to put them back together again in the proper 3-8 sequence.

It couldn't be done except at a gross level. Each year, and within each year as well, there are nuances on the nuances on the nuances. It was a good exercise in cross polination. In hind sight it was also a dramatic demonstration of the ridiculous nature of those standards. If the teachers charged with delivering them can't see clear annual goals then how can the kids?

I have our cities math curriculum from 1958 (I was in middle school then) and it reads precisely like the A+ international standard in the link. Interestingly, it is riddled with demands for mastery at key milestones, something that is totally absent in our present day constructivist, spiral curriculum.

Funny how 50 years of 'research' has taught us how to do what we already forgot.

And here's Barry:

Also funny how the math of 50 years ago is said (by those pushing reform math) to have failed large numbers of students.

That is funny!

data driven loops and noise
data driven instruction redux

How does it all stack up?

A parent and I recently started up a Continental Math League team at our school, which uses Investigations math.

The response?  Enthusiasm from students and parents; skepticism from teachers.

Specifically, about "stacking." (Today's word for how we used to add, subtract, and multiply numbers by placing one number on top of the other.)

Kids love it. And not just the ones on our team. As as friend writes:

When I showed one of my sons how I had learned addition, i.e. the "stacking" method, he was very impressed. "Wow, that's so cool! That works great! I wonder if my math teacher knows about this?" was his innocent comment.

Yes, she does, and she doesn't like it. At least if she resembles the teacher who approached me after math practice yesterday and recounted the dismay she felt when she caught one of her students stacking numbers, thus abandoning the more "meaningful" and "faster" way he used to solve problems.

My co-coach and I tried to explain that the Continental Math League numbers are big enough, and random enough, that Reform Math's methods aren't faster and more meaningful, but inefficient and confusing. It's one thing to add 48 and 39 by reasoning that:

48 is 2 less than 50, and 39 is 1 less than 40, so add 40 and 50 and get 90 and then count backwards by 3 and get 87."

But take one of the problems we did at Continental Math League practice yesterday: 825 - 267. Restricting myself to the kinds of calculation that these second and third graders are able/expected to do in their heads, here's the most efficient non-stacking method I can come up with:

The closest friendly number to 825 is 800, and the closest friendly number to 267 is 250. 825 is 25 more than 800. 250 is 10 more than 260, and another 7 gets you 267. 10 plus 7 is 17. So 267 is 17 more than 250. So subtract 250 from 800. Well, 800 minus 200 is 600, minus 50 more is 550. Then subtract 17 from 25 by counting up from 17. Seventeen plus 3 more is 20 plus 5 more is 25. 3 plus 5 equals 8. Add 8 to 550* to get 558."

*By this point in the problem, how many people remember what they should be doing with this 8?

Anyone with a more efficient non-stacking method for subtracting 267 from 825 (no calculators allowed!) is invited to share it here.

(Cross posted at Out in Left Field).

Be afraid

I had lunch yesterday with an accountant who insisted on picking up the tab. When the waiter brought the check, the accountant pulled out his calculator.

To calculate the tip.