Math education: Arguing over false choices

By Dr. Joseph Ganem, professor of physics, Loyola University

"The astronomer may speak to you of his understanding of space, but he cannot give you his understanding.The musician may sing to you of the rhythm which is in all space, but he cannot give you the ear which arrests the rhythm nor the voice that echoes it. And he who is versed in the science of numbers can tell of the regions of weight and measure, but he cannot conduct you thither. For the vision of one man lends not its wings to another man."----Khalil Gibran, "On Teaching," The Prophet

Imagine a football coach who does not spend practices drilling his team and running plays. Instead, players watch videos of football games, analyze and diagram the actions, discuss the reasons that some plays work and others don't, and plan strategies for upcoming games. His reason for this approach is that drill work is tedious, repetitive, and exhausting. Players will enjoy practice much more if they can study the underlying strategies and concepts of football, have engaging discussions, and learn to think like a professional football player.

We would call such a coach delusional, not because of what he is doing, but because of what he is not doing. Obviously everything he is doing needs to be done, but his team will not stand a chance on an actual football field without putting in hours of tedious, repetitive, and exhausting drill work.

For an activity that has a kinesthetic component it is immediately obvious that learning it will only be possible through repetitive drill work. No one would entertain the notion that they could learn to play tennis by watching the Wimbledon tournament on television, learn to play piano by attending a concert at Carnegie Hall, or learn to dance by going to a performance of the New York City ballet. But, if the activity lacks a kinesthetic component somehow, what should still be obvious no longer is.

Consider the debates on math education that have run on for decades. Should students be taught standard algorithms for operations such as multiplication and division and focus on getting correct answers, or should students be taught conceptual thinking and focus on discovering mathematical knowledge on their own? Educators have argued both sides of  this issue, but in reality it is a false choice.

Without a conceptual understanding of math the subject is of little use. Applying math to real-world problems and knowing if the results of a mathematical analysis make sense requires an understanding of the concepts. But, it is not possible to have a conceptual understanding without with the extensive practice, memorization, and drill work needed to achieve computational fluency.

I tell my students that expertise in any subject, math or otherwise, has three components - facts, skills, and understanding. Each of these components is learned in a different way. Facts are static and must be memorized. Skills are actions that must be practiced in order to become proficient. Understanding evolves and comes only through experience and reflection.

This way in which I think about learning is different than the widely influential Bloom's taxonomy. Bloom saw learning as a hierarchical process, while I see it as an iterative process. Bloom saw separate learning domains - cognitive, affective, and psychomotor - that each had their own hierarchy, while I see the iterative learning process as being much the same in each of the different learning domains.

In Bloom's taxonomy, first published in 1956, the hierarchy in the cognitive domain from the bottom up is: knowledge, comprehension, application, analysis, synthesis, and evaluation. In this model of learning, comprehension (or understanding in updated terminology) is necessary before students can actually do something with their new knowledge. Hence many educational reform movements in the decades following the taxonomy have emphasized "conceptual" learning over practice. However, I disagree with the idea that a conceptual understanding is necessary before higher order activities, such as application, analysis, and synthesis can take place, because understanding is an ongoing process.

Chess as Example

For example, consider learning chess. It is an activity without a kinesthetic component hence it would fall under Bloom's cognitive domain of learning. But no one would believe that the game could be mastered without practice, or that novice players could discover the principles of strong play on their own.

To learn chess an aspiring player must memorize the names and movements of the pieces, and the object and rules of the game. These are what I refer to as facts. But the acquisition of skill in playing the game requires a program of study and practice. In order to improve, players must read texts on chess tactics and strategies and attempt to implement those ideas by playing actual games. There is no substitute for practice, but at the same time players must learn additional facts (acquire more knowledge).

However, an understanding of chess evolves in time. A novice, a skilled player, and a grandmaster can all look at the same chess position. The novice will see individual pieces. The skilled player will see groups of pieces. The grandmaster will see the entire position.

But if the grandmaster articulated his understanding of the entire position to the novice, the narrative would be of limited use. The novice would not have the knowledge base and the skills necessary to make sense of most of what a grandmaster would say about a given chess position. But that does not mean that the novice is incapable of applying, analyzing and synthesizing chess ideas. Those ideas might be relatively crude, and obvious to the grandmaster, but the process is necessary to reach a high level of understanding. It is for these reasons that I view learning as an iterative process.

Expertise

Experts are experts because they do think about their subject of expertise differently than novices. But those thought processes cannot be transferred directly to a student, they must develop through study and practice, and there is no shortcut to that development. This should be especially obvious in a subject such as math but apparently it is not.

Many years ago, before calculators and optical scanners had been invented, I made a purchase at a bakery counter tended to by a young woman who had to pencil in prices on the bags of pastries being sold. I asked for 5 donuts priced at 26 cents each. She placed them in a paper bag and on the outside of the bag she computed 26 x 5 using the standard algorithm for multiplication that I, and countless other students, had learned in grade school. She of course was very proficient at multiplication problems using this method, because throughout the day, everyday, a steady stream of customers patronized the bakery counter.

Before she could write out the problem, I said to her: "It's $1.30." She completed the problem, writing all the steps on the bag, and the result was $1.30. Startled by my seeming clairvoyance, she looked at me for an explanation. She knew of no other way to multiply but the standard algorithm, and that process required time and writing. How could I multiply the numbers instantly in my head and arrive at the correct answer?

I said to her: "If the donuts were 20 cents each how much would 5 cost?"
She replied:" A dollar."
I said: "And what is 5 times 6?"
She understood immediately what I had done, but only because she was already proficient at multiplication. If I tried to teach my methods for doing mental math to people not already proficient in the use of standard algorithms, my explanations would lead to confusion rather than enlightenment.

Real learning is iterative, not hierarchical, and it doesn't matter whether the subject is, to use Bloom's terminology, in the cognitive, affective, or psychomotor, learning domains. However, the desire of educators to systematize learning often leads to rigid ideologies riddled with false choices. The argument over whether math instruction should focus on concepts or computation is in many ways analogous to the argument on whether reading instruction should focus on phonics or whole language. Fluent readers use and understand both approaches.

Likewise, learning math is an iterative process that cycles between concepts and computation. Experts in math are proficient in both because it is impossible to master one without the other.

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Joseph Ganem, Ph.D., is a professor of physics at Loyola University Maryland, and author of the award-winning book on personal finance: The Two Headed Quarter: How to See Through Deceptive Numbers and Save Money on Everything You Buy. It shows how numbers fool consumers when they make financial decisions. For more information on this award-winning book, visit TheTwoHeadedQuarter.com. His article is reprinted here with permission of Dr. Ganem. This article was previously published on The Daily Riff.


Note from Laurie Rogers: If you would like to submit a guest column on public education, please write to me at wlroge@comcast.net . Please limit columns to about 1,000 words, give or take a few. Columns might be edited for length, content or grammar. You may remain anonymous to the public, however I must know who you are. All decisions on guest columns are the sole right and responsibility of Laurie Rogers.

Leadership seems filled with predators and sheep. Where are the sheepdogs?

 By Laurie H. Rogers

"Truth is isolating."
- according to a community member who would know

"It's easy to win when you cheat."
- according to a child who cares about fairness

A good friend of mine, whose perspective I value, said there are three kinds of people in the world: Sheep, sheepdogs and predators. It’s an immediately clarifying way to view oneself and the world. I argued that there are more kinds of people than that. There are:

• sheep pretending to be sheepdogs,
• sheep pretending to be predators,
• predators pretending to be sheep,
• predators pretending to be sheepdogs.

(True sheepdogs don’t pretend to be other than what they are, unless operating undercover. Some need a significant emotional event before finding their inner sheepdog. Some need a break from sheepdoggery, thus making them appear to be sheep. But once awoken to a threat, I think all true sheepdogs begin sheepdogging. Look at what happened after 9-11. And for a powerful look at some real sheepdogs, see the 2012 movie "An Act of Valor.")

My friend shook his head. He accepted my premise, but he said leopards don’t change their spots, thus mixing his metaphors yet nevertheless making his point. The pretense doesn’t change the nature of the beast, he said. A predator pretending to be a sheep still is a predator. A sheep pretending to be a sheepdog still is a sheep. The question is: How much damage does the pretense do, beyond the damage done by the inherent nature of the beast?

I thought about this analogy, while pondering the public-education mess. And make no mistake about it: It’s a mess.

• The mission of sufficient academics is not being accomplished;
• Taxpayers pay through the nose while continually hearing how stingy they are;
• Legislators attempt to undermine the people’s right to make decisions for their children, while also attempting to raise taxes and place more burdens on schools;
• The federal government (via the U.S. Department of Education) is staging what many see as an illegal coup;
• Vendors hover like vultures, waiting to pick at the taxpayer carcass;
• Children struggle, fail, dropout or test into remedial classes that about half cannot pass;
• School districts claim that student data have gone up and budgets have gone down, when the truth for most of us is that student knowledge has gone down and budgets have gone up;
• (Allegedly grass-roots) citizen groups beat voters over the head demanding that we “vote yes” – allegedly for the kids;
• Unions have excessive power with insufficient accountability;
• Media sycophantically suck up to districts while blaming dissenters.

It’s bleak out there, and I wonder, “Where are the sheepdogs?” Everyone in education claims to look out for the people, yet the mess keeps getting worse and more expensive. At one time, we had leaders who fought for the right thing, stood up vocally against the wrong thing, and worked on the people’s behalf rather than for their own interests. Where the heck are they?

Many sheepdogs work in military installations around the country and world. I’ve been lucky to have known several. Quiet sheepdogs in the general population make a living and raise their family. But in leadership, and especially in education leadership, outspoken sheepdogs are hard to find.

No one said being a sheepdog is easy or popular. Standing up to a mob isn’t for sissies. The only people who love sheepdogs are other sheepdogs and a very few sheep. (The predators certainly don’t love them.)

Sheepdogs need courage – not an absence of fear, but a determination to persevere in the face of fear. They must find courage when they don’t feel it. In addition, sheepdogs need knowledge. They have to be able to see the threat in order to fight it. Knowledge is power, which is why the predators work so hard to keep knowledge from them.

Although I have multiple character flaws (just ask my teenager), I really do care – about right and wrong, about the vulnerable, about serving my honor, and about doing my best. We’re all responsible for making wrong things right, even when the going gets tough or we just don't feel like it. But I have run up against a mob, and I believe my values aren’t widely shared there.

From 2007-2012, I came to see that many state and district administrators and superintendents, governors, government appointees, edu-wonks, board directors, union leaders, members of the media, businesspeople who feed off of the public trough, and a disturbing number of elected officials show little sign of knowing or caring what other people’s children are learning -- or about what students know when (or if) they graduate. They don't appear to see their role as providing absolute truth to the people. Instead, their priorities, it seems, are money, ego, power and allies; getting along with the powerful; proving they’re right; pushing their political/social agenda; and squishing out the dissenters.

Some would go so far as to rip the Public Records Act right out of the people’s hands.

• See Spokane Public Schools’ 2012 Legislative Priorities, Item 3B, which was to push for a law that would allow school districts to charge the public for the cost of providing public records. 
• Spokane Sen. Lisa Brown’s bill SB 6576 would require all school districts to charge the public for the cost of providing public records.
• SB 6351 would allow public agencies to limit responses to public records requests. The bill leads with a discussion of inmates, but the new language is written generally, so as to encompass everyone. It allows public agencies (the government) to threaten citizens (the people) with legal action over public records requests, to file for injunctions, and to reject future requests from repeat requesters.
• SB 5062 would place the onus on requesters to know what's missing from public records, and to somehow itemize missing records before taking legal action. It would allow all agencies to take 30 extra days to produce missing records if someone notices their absence, thus avoiding penalties for a willful failure to provide them initially.
• SB 6345 would give eight appointed people the power to redo state government. Meetings would not be subject to the Public Records Act, or to the Open Meetings Act. Decisions could be made in executive sessions that exclude the public. Proposals may not be amended in committee, and passed bills would be final. (That's a little kingdom, right there.)

SB 6576 and SB 6351 would effectively eliminate the Public Records Act -- a people's initiative -- for 99% of the citizens. SB 5062 would take the teeth out of the Act. SB 6345 would exempt eight citizens from open-government laws. These bills don’t reflect concern for the public’s will, needs or best interests. They don’t reflect concern for truth, transparency or full disclosure. And why would a school district work for a bill that would remove the public's ability to know what it's doing?

In 2011, I asked a board director who remained silent in a public meeting about math, “Why didn’t you tell the people the truth?” His answer: “It wouldn’t be good for me.” I asked a principal who remained silent why he didn’t speak up. “I just came to watch.” I asked a teacher who remained silent why she didn’t speak up. “I was scared.” Later, I asked a legislative aide why his boss didn't help me with my efforts. “He’s staying out of it,” was the answer.

K-12 education isn’t supposed to be about the adults. (Naturally, the adults don't appear to see it that way.) The schools’ mission is to impart sufficient academics. If they don’t do that, they have failed. Clearly, children aren’t being adequately prepared, yet we keep hearing that things are improving and they just need a tighter grip on our wallet. Do we even have any sheepdogs in district leadership?

Some teachers appear to be sheepdogs, although being a teacher and a sheepdog in Spokane certainly is fraught with peril. When a local teacher put on an excellent candidate forum last election season, she faced harsh union/district/media pushback, as if she had done something wrong. More to the point, district employees again got the message: Do NOT speak up. At all. (And they don't.)

But those who care about the children must find a way to speak up … or risk being complicit.

Meanwhile, parents have been called "sheep" by various so-called leaders. The 1% in charge gives parents a warped view of the school district and its outcomes, then blames them and calls them sheep. Parents and teachers don’t realize how their views have been shaped by hidden agendas, little real accountability or transparency, pots of money, barrels of ink, and/or few apparent scruples. They don’t know that most high-school graduates and college hopefuls leave the K-12 system with few usable skills in math or grammar. And they don't know that many in leadership and the media chose to not leave their kids to founder in a failing public system.

It’s time we reconsidered who’s in the 1%. We’ve been well-trained to think of the 1% as Wall Street “fat cats” and execs in large companies who care more about profits than people. But the 1% isn’t just about money. It’s also about political influence, opportunities and social position.

The 1% includes district superintendents, board directors and administrators who seem more interested in ego, money, power and pet education theories than in the children. The 1% includes union leadership – accountable to almost no one – which uses its clout to heavily influence elections and ballot propositions. The 1% includes influential, allegedly “grass-roots” groups such as the League of Education Voters, Stand for Children, and Citizens for Spokane Schools (CFSS).

CFSS spread its well-heeled influence around Spokane with daily ads in the newspaper (“vote yes”), huge billboards (“vote yes”), signs everywhere (“vote yes,”) and an embarrassing amount of media assistance (“vote yes”). How much money was that, anyway? A more accurate name for CFSS might be: “Citizens – Using Pots of Money from Somebody to Get More of Your Tax Dollars.”

The 1% includes a huge number of legislators who seem unaware of the real problems in public education. “I know a lot about it,” a representative assured me. Yet, they push tax increases, useless mandates and counterproductive programs on all of us. How many have actually examined district budgets, claims, curricula, outcomes, election activity, threats or expenditures?

The 1% encompasses most of Congress, along with certain people in the White House.

The 1% includes Bill Gates, who carries a perplexing amount of influence over public education, despite the fact that he and other corporatists aren’t accountable to the public for this influence in any real way. When you have pockets as deep as his, everyone listens, whether or not a) you know what you’re talking about, b) you’re effective, or c) it’s appropriate for you to interfere. Math advocates can’t get a sliver of respect anywhere, whereas Gates can get it while still in his jammies.

The 1% includes the U.S. Department of Education, the NGA, CCSSO, Achieve, the WEA, NEA, AFT, Bill & Melinda Gates Foundation, Broad Foundation, Rockefeller Foundation, Texas Instruments, Pearson Education, OSPI, Educational Service Districts, most of the media, and many other people and groups who work with the districts, make money off the districts, hope to make money off the districts, or just prefer to hang out with the 1%.

I wonder: How many sheepdogs are in that 1%? Sheepdogs wouldn’t stand by while children are harmed by loopy edu-fads. Sheepdogs wouldn’t allow taxpayers to be ripped off, voters to be obstructed, and parents to be deceived (then blamed). Sheepdogs wouldn’t dream of saving their own children and then staying silent as other people’s children are betrayed.

True sheepdogs know the job is tough, there are barriers, others will obstruct – perhaps even be dangerous –and that there are consequences to being a sheepdog, but they accept that the job is the job, what’s right is right, and it’s their job to protect the flock. Education sheepdogs are knowledgeable and experienced, with solid research and data. The predators and sheep have little more than, “We want it this way.” Yet, despite our solid arguments, public education doesn’t change – other than to cost us more as it continues to deteriorate.

And so I wonder: How many of the “sheep” and how many of the “sheepdogs” in government and leadership actually are the predators, just engaging in pretense? Or, perhaps, and this is generous, living in denial.


P.S. For a powerful look at some real sheepdogs, see the 2012 movie "An Act of Valor," in which key roles are played by active-duty SEALS.
Honest, unflinching and disturbingly realistic, "An Act of Valor" is a rare show of respect from the entertainment industry for the nation's military. I tip my hat to all of those involved.

Please note: The information in this post is copyrighted.
The proper citation is: Rogers, L. (February 2012). “Leadership seems filled with predators and sheep. Where are the sheepdogs?" Retrieved (date) from the Betrayed Web site: http://betrayed-whyeducationisfailing.blogspot.com/

This article was published Feb. 19, 2012 on Education Views at: http://educationviews.org/2012/02/19/leadership-seems-filled-with-predators-and-sheep-where-are-the-sheepdogs/