Wrong Questions About Spreadsheet Math

Spreadsheets are a ubiquitous and necessary tool these days. Students need to learn spreadsheet math.

"Our children still spend hundreds of hours perfecting their ability to add, subtract, multiply, and divide fractions. And the pinnacle of math for most of our K-12 students remains the ability to solve quadratic equations. When was the last time you used any of these skills? When did you last multiply two three-digit numbers together on paper, add two improper fractions with unlike denominators, or solve a quadratic equation?"

These are the questions people asked when it came to calculator use, and they are still the wrong questions. Spreadsheet math will not replace the ability to actually understand math any more than calculators did.

When the National Council of Teachers of Mathematics (NCTM) recommended calculators for even the youngest students, they rhapsodized about about how calculators would revolutionize math teaching, using the same sort of language that idealizes the potential of spreadsheet math.

"By teaching our children spreadsheet math we enable them to solve ...fascinating problems, problems without a single right answer, problems that can be explored, problems that get our children thinking "out of the box."

And that was exactly the wrong-headed pie-in-the-sky rationale for recommending calculators. It sounds great but does not work in practice. The problem with math instruction is not whether we should be using calculators or spreadsheets. The problem is the lack of skilled math teachers. The problem is the continued reliance on teaching tricks and shortcuts instead of math. Like calculators, spreadsheets have a similar tendency to replace thinking.

Beginning in 2001, I researched the calculator fallacy extensively culminating in a 78-page report in 2010. Briefly, I found that the research NCTM insisted supported the use of calculator in the early grades did not exist.

I agree that students need to learn spreadsheets, but not as a substitute for learning math. Since our elementary teachers lack an ability to teach math for understanding, abundant experience with mechanical processes, though far from ideal, is pretty much the only way kids learn to tell an unreasonable answer from a reasonable one, and even then they are not very good at it.

Just last week, a friend's eighth grade daughter (A+ in math per last progress report) was sure that if $27.50 could buy 10 lbs of hamburger, then $55.00 would buy over 150 lbs because "I followed all the steps correctly." When I told her that obviously she had not, she argued that even the calculator agreed with her, so I must be the wrong one. Just yesterday she insisted that -3 + ½ = -3½ (by analogy to 2 + ½ = 2½). In her mind, all you have to do is get rid of the plus sign and shove the fraction up against the whole number. When these kinds of misconceptions plague even good students, no wonder students who are not as “good” have math anxiety. Deep down, the anxiety is related to an unspoken and unspeakable suspicion that math makes no sense. They are right. When math is turned into a system of tricks and shortcuts, it makes no sense.

Surprise! Kids Value Rote Learning...

Surprise! Kids Value Rote Learning...

...just not when they are the one expected to memorize knowledge. Have you ever had a child ask you a question that require a memorized fact to answer? It happened to me recently. We were listening to a CD of classical music that had no printed table of contents. With nearly every piece, the (junior high) child asked, “Who wrote that?” Luckily, I can google the answer. She became exasperated with my lack of certainty and my need to look up so many of the composers. She asked impatiently, “Didn't you have to study music history when you were in school?”

Me: Yes, I did.

Her: Then how come you don't know who wrote all these songs?

Me: Do you like history class?

Her: NO, I hate it.

Me: Why?

Her: Because we have to memorize so many dates and other trivia.

Me: I guess you will start applying more gusto to your memorization.

Her: Why would I do that?

Me: Because clearly you think that memorizing facts is an important part of your education.

Her: Nooo. Whatever gave you that idea?

Me: Because you think it was an important part of my education.

Her: I never said that!

Me: But you clearly expect me to remember music composers off the top of my head better than I do. How would I have learned that information in the first place, except by memorizing it as facts? And you expect me to still remember it? Don't you think you should hold yourself to the same expectation?

Her: Well, of course.

Me: So I guess you won't groan anymore when teachers expect you to memorize stuff.

Her: Who said I minded memorizing stuff?

Me: Whatever.

Her: Hey, that's my line.

In Case of Nothing to Do, Break Glass...

...and then sweep up broken glass.

Americans have hopes and ideals for public education. As David Sirota explains,

Here in the industrialized world’s most economically unequal nation, public education is still held up as the great equalizer — if not of outcome, then of opportunity. Schools are expected to be machines that overcome poverty, low wages, urban decay and budget cuts while somehow singlehandedly leveling the playing field for the next generation. And if they don’t fully level the playing field, they are at least supposed to act as a counter-force against both racial and economic inequality.

The American ideal is that public education is supposed to be not only the engine of the American Dream, but also the primary mechanism for overcoming the social-economic obstacles of birth. Meanwhile, public educators consistently cite the poverty of their students as the number one reason public education fails to perform its promise. So, public education is supposed to give poor students the tools they need to overcome poverty, but public education cannot give them these tools because the students are poor. Teachers say education reform proposals that fail to address poverty are doomed to fail. We are trapped in a vicious circle. In such an environment, no wonder policy-makers, most of whom lack education experience, feel pressured to do something---anything. Education policies thus tend to be a perpetual cycle of creating and cleaning up messes.

And no wonder. Policy makers lack expertise themselves, so they turn to advice from those who seem to have proper credentials. Who do they ask? Education professors. Sounds reasonable, but guess what? Many education professors lack significant in-the-trenches experience in the very places teachers must implement policies handed down from on high. American society does not trust teachers. The main reason for the lack of trust is the double-mindedness of society. Sometimes we consider teachers to be professionals, and then undermine their professional judgment when we consider them hired laborers subject to dismissal for insubordination. We steadfastly refuse to put teachers at the head of the policy table even though, as Nancy Flanagan puts it, the teachers know where the carts are.

If we want to invest in a highly skilled teaching force, perhaps it's time to stop positioning teachers as drop-in observers who should be grateful for the chance to "represent" their peers in important decision-making bodies...Teachers should be at the head of the table, calling the meeting. The more professional responsibilities we take off teachers' plates, to standardize and homogenize, the more teachers' professional judgment is weakened.

Teachers should not be the target of reform, but the drivers. Right now, teachers are lucky if they can stand outside the door and listen at the keyhole.

At the start of the Obama presidency, when the Department of Education had lots of openings, the administration solicited applications. Many teachers applied. It turned that the administration was only interested in perpetuating more of the same, not ushering the change we all hoped for. Any PhD, especially a well-published PhD, whether they have actual significant experience or not, trumped highly effective, veteran teachers every time. Perhaps the number one disappointment educators have with the Obama administration is the refusal to listen to real teachers. They do not listen because they do not trust. They do not trust because our present policies prevent recruitment from among the most academically able students.

Teaching is not a top job choice, but a last resort. I routinely ask my education students why they want to be teachers. The answers are underwhelming. My top two favorite answers because they indicate the status of teachers in America: “I flunked out of hotel management,” and “It is either teaching or the Army.” In such an environment, it is difficult for the American public to accord teachers the respect and esteem they enjoy in other countries. Meanwhile, administrators break the glass and expect teachers to clean up the mess.

Smart People (You and Me) are Stupid

A recent provocative article from the New Yorker begins:

Here’s a simple arithmetic question: A bat and ball cost a dollar and ten cents. The bat costs a dollar more than the ball. How much does the ball cost? The vast majority of people respond quickly and confidently, insisting the ball costs ten cents. This answer is both obvious and wrong. (The correct answer is five cents for the ball and a dollar and five cents for the bat.)

Or try this one:

West and his colleagues began by giving four hundred and eighty-two undergraduates a questionnaire featuring a variety of classic bias problems. Here’s a example: In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake? Your first response is probably to take a shortcut, and to divide the final answer by half. That leads you to twenty-four days. But that’s wrong. The correct solution is forty-seven days.

Our first mistake is to assume the human beings are rational. We should start from the premise that human beings are lazy (Yes, I am talking about you, and I am talking about me).

When people face an uncertain situation, they don’t carefully evaluate the information or look up relevant statistics. Instead, their decisions depend on a long list of mental shortcuts, which often lead them to make foolish decisions. These shortcuts aren’t a faster way of doing the math; they’re a way of skipping the math altogether. Asked about the bat and the ball, we forget our arithmetic lessons and instead default to the answer that requires the least mental effort.

My son calls it the “slacker syndrome.” The first step to learning to make good decisions is to humbly acknowledge that deep down we are all slackers. The smarter we are, the more stupidly we decide, like the super-smart guy who thinks everyone else is an idiot, but HE can time the stock market.

...smarter people are more vulnerable to these thinking errors...intelligence seems to make things worse. The scientists gave the students four measures of “cognitive sophistication.” As they report in the paper, all four of the measures showed positive correlations, “indicating that more cognitively sophisticated participants showed larger bias blind spots.”....

Okay, now that I know I am prone to this weakness (maybe the next time a job interviewer asks me about my number one weakness, I should admit that I am stupid....hmmm...maybe not), I can use this new-found self awareness to avoid it in the future. Think again.

...“people who were aware of their own biases were not better able to overcome them.”...

Even worse, we all tend to think of ourselves more highly than we ought.

Perhaps our most dangerous bias is that we naturally assume that everyone else is more susceptible to thinking errors, a tendency known as the “bias blind spot.” This “meta-bias” is rooted in our ability to spot systematic mistakes in the decisions of others—we excel at noticing the flaws of friends—and inability to spot those same mistakes in ourselves.

Wow, that sounds like a modern version of old advise from Jesus.

"Why do you look at the speck of sawdust in your brother's eye and pay no attention to the plank in your own eye?” (Matthew 7:4)

Education and self-awareness do not work because the meta-cognition is not available.

The problem with this introspective approach is that the driving forces behind biases—the root causes of our irrationality—are largely unconscious, which means they remain invisible to self-analysis and impermeable to intelligence.

Therefore we are doomed to live in a Zen paradox.

The more we attempt to know ourselves, the less we actually understand.

Nevertheless, I cannot help but feel education and self-awareness of our biases must be better than ignorance. At least, we can learn to be more tolerant and forgiving of others.

Parents In Contempt

As long as schools hold the public (who pays their bills, by the way) and parents in utter contempt, I seriously doubt that they will be able to build the relational trust necessary to academic achievement. Even worse is the contempt demonstrated toward those parents and members of the public who are education colleagues who might actually have a message worth listening to. I am just going to tell the story. A friend's child brought home a math assignment on perimeter and area. One of the problems was unsolvable as presented. It looked something like this (I do not have the actual diagram since child already turned her homework in):

The problem is unsolvable because there is not enough information. There is no way to know whether the angles that “look” 90 degrees are in fact 90 degrees. There is also no way to know whether the vertical segment that “looks” like it bisects the base in fact does so. The child made these points in class, but the teacher shot her down. So thinking I was being helpful to a young teacher, I sent her an email. After some brief introductory remarks, I wrote,

... the last problem of a recent homework assignment on perimeter and area had insufficient data to solve the problem.  One of the principles of geometric diagrams is that we never go by appearance.  We solve using givens and proven facts.  Since that particular diagram gave no indicators of equivalent length, bisection, 90-degree angles, etc, no conclusions could be drawn regarding the length of the side opposite the one measuring 15 units, or any other non-given length, without making unsubstantiated assumptions.  We are miseducating children if we teach them, even indirectly, bad thinking habits.  One of the purposes of math instruction is the logical and critical thinking skills it cultivates. And if we are as worried about high-stakes tests as we say we are, we will teach students to think properly.  A favorite trick of bubble tests is to lead the students down a primrose path of faulty assumptions. If you would like discuss ways to help students learn to think, please let me know.  I spent a lifetime teaching math, first to junior high and high school, and later, college students.

The teacher sent a reply that seemed quite understanding, but showed that she somehow missed the point. She replied,

I completely understand what you are saying in the email. I also concur with what you said, however, I did verbally tell the students that angles that appear to be right are. ..However, if the student drew it out on the grid paper, then that student could find an area for the figure they drew. We are at an entry level with these problems, and so I looked at how the student drew the problem out, and determined if their area and perimeter matched that drawing.

Am I the only one who found the reply disturbing? So I wrote back,

Thank you for your reply. "I did verbally tell the students that angles that appear to be right are."  Angles that appear to be right are most decidedly not right just because of appearance.  I really think that if the student's math level is lower, then it is all the more critical to precisely teach thinking skills.  Drawing on grid paper does not really help, because it only pushes students harder to make unwarranted assumptions.

I closed by repeating my offer to help. Then I got this curt dismissal:

Hi again, Thank you for your response, it seems that on this particular problem I did not satisfy your criterion and I am sorry to have let you down. Have a nice day.

Okay, so she really does not want any help, I guess. And apparently she is happy to miseducate kids. I figured I would just move on, until....the school's guidance counselor contacted me, and let me know that she considers my communication with the teacher inappropriate and unprofessional. Seriously? She even challenged my right to have any conversation with the teacher by pretending that I was discussing a student, not a math problem. Now when I was teaching, the only time I passed communications on to administration was if it contained a personal insult of some kind, of course, a very rare occasion. Maybe it is unfair to extrapolate from one experience, but I assure you, schools routinely show contempt for parents, the public, and even the education-savvy members of the public. We must be careful that we as teachers refrain from thinking we get to define the terms of parental involvement. I know plenty of teachers who actually resent fully involved parents if those parents dare to challenge the teacher or the school. Then teachers vilify the parent as a "helicopter parent" in an ad hominem attempt to dismiss with contempt the parent's concerns. A lot of schools want to limit parent involvement to conferences and making cupcakes.

Unresolvable “Science” Debate?

Will the controversy between evolution and creationism ever end? Is it destined to swing forever on the pendulum of public opinion? The entire controversy is sustained on both sides by too much emotional investment in unexamined assumptions. The latest pretext for acrimony is a Tennessee bill intended to permit teachers “to help students understand, analyze, critique, and review in an objective manner the scientific strengths and scientific weaknesses of existing scientific theories covered in the course being taught.” In case that wasn't clear enough, the bill repeats its intention from the other way round. No teacher shall be prohibited from “helping students understand, analyze, critique, and review in an objective manner the scientific strengths and scientific weaknesses of existing scientific theories covered in the course being taught.”

I read the bill. It is only two pages of plainly-worded text. It mentions creationism not at all. It does refer to scientific theories, of which there are many, each with its own strengths and weaknesses. However, creationism is not one of those theories because it is not science.

You see, science is all about collecting only evidence that can be perceived with just the five senses. The sense may be amplified as when we use a telescope or other instrument. Science is concerned with explaining data collected only with the five senses. Other data is not considered.

An explanation that tries to account for extra-sensual data is, by definition, not a scientific theory. Nevertheless, due to public confusion and the desire of some that creationism be recognized as a scientific theory, it will be in science class that students ask their questions. Teachers need to be prepared to answer them while respecting deeply held religious beliefs.

The bitter acrimony is really unnecessary. It is easy and reasonable for students to accept that science attempts to explain only sense-based data. Most of the problem stems from a widespread misunderstanding of what science is.

As far as evolution goes, it suffers from historical bar-lowering, as it has weaknesses that do not adequately account for the scientific facts. Even within my lifetime, scientists have weakened the definition so much as to create a near tautology: evolution is change over time. Many science texts state it just like that. Others pretty it up a little, “evolution can be precisely defined as any change in the frequency of alleles within a gene pool from one generation to the next.” However, such a definition in non-controversial. Organisms do change over time. The biblical Jacob realized it thousands of years ago when he made a deal to work for Laban, receiving only the spotted sheep as his wage. Laban promptly removed all the spotted sheep from the herd. Nevertheless, by careful breeding, Jacob was able to create a herd of mostly spotted sheep from a herd of un-spotted sheep.

Years ago the National Association of Biology Teachers (NABT) used a fairly stringent definition: "The diversity of life on earth is the outcome of evolution: an unsupervised, impersonal, unpredictable and natural process of temporal descent with genetic modification that is affected by natural selection, chance, historical contingencies and changing environments." Unwritten, but understood and unquestioned, was the additional idea that mechanisms of descent were robust enough to account for the change from say, sponge to zebra.

In fact, this unstated implication is the root of the controversy. Pro-evolutionists (as distinguished from scientists) believe the implication; Anti-evolutionists (again, as distinguished from scientists) do not. For many, the implication goes directly to deeply-held belief systems. Later, the NABT deleted the words “unsupervised” and “impersonal.” Today, there is no definition on the website at all. One of the reasons that the definition of evolution has gotten weaker and weaker is that the data, especially as regards speciation, is inconclusive, and fails to support the more robust definition. There are lots of instances where it is not at all clear whether two organisms are members of different species. A high-quality university level biology book addresses the speciation continuum and other issues, but it can be a tough read.

As inconceivable as it may be to some, it is possible to discuss the weaknesses of evolutionary theory without smuggling in creationism. Only ideologues would consider the mere mention of evolution's weaknesses as an attack upon evolution. For critical thinkers, it is the grist of intelligence-making. As F. Scott Fitzgerald said, “The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function.”

How Rigor Empowers Academic Achievement

Maybe we do need another word besides “rigor”, but “challenging” and “rich” are weak alternatives.

Rigor in my teaching practice means conscientious excellence. For example, rigor requires students to differentiate solution from answer. Suppose the question is, “What are the best dimensions for a particular garden?” The student will algebraically calculate two perfectly legitimate solutions. One solution will show two negative numbers; the other will show two positive numbers. Students must choose the solution which answers the question. Since the question is about a garden, that would be the solution with the two positive numbers,because a garden cannot have negative measurements. A different question might require the negative solutions to be the answer.

Or perhaps the question is, How many cars do we need for the field trip. The solution might be 7.2, and it can be the correct solution, but the wrong answer. The correct answer is, "We need 8 cars." Mindless rounding also yields a correct solution,but a wrong answer. I require answers written in complete sentences that also include the unit. I would mark all three of these so-called answers wrong: "x=8", "8," and "We need 8."

I also require students to keep units attached to numbers when they calculate. So the area of a room is not 9X12, with the ft^2 attached later. When students show their work, I want to see 9 ft x 12 ft = 9 x 12 x ft x ft = 108 ft^2. Please do not dismiss my simple example as trivial. This sort of training, call it rigor if you like, pays off big when students must do chemistry or physics calculations with lots of units running around. My physics students learned that if the resulting unit is not what they expected, they probably also made a more serious mathematical error somewhere. That self check is lost when units are divorced from numbers and remarried at the end of the calculation.

In the earliest grades, rigor may imply making sure students understand the role of an equal sign, and knowing that the horizontal line separating a column of numbers from the result of a calculation is not a substitute equal sign. Many adults graduate from high without a proper appreciation of an equal sign.

Every field has similar examples of the value of conscientious excellence. Most people prefer the two syllables of “rigor” over the seven syllables of “conscientious excellence.” Just because three of the definitions seem negative and harsh does not invalidate the value of the fourth definition. Rigor, properly used, is not a blockade to academic achievement or educational accessibility, but its open door.