Zero is a Real Number

Zero is a real number. Could such a headline possibly be click bait? If so, it is pretty lame. Of course everyone knows zero belongs to the set of real numbers. The problem is the word “real.” A sentence such as “zero is a real number” immediately puts people into mathematics mode wherein they consider the word “real” in only its mathematical sense. Sometimes people recall set theory theory and the curious case of a set containing only one member, zero, as opposed to an empty set with no members. The problem here is that set theory leads people to objectify zero. They think of zero as an object rather than a number.

Zero is a real number. When the truth of this statement dawns, the world changes forever. If you are thinking, “Well, of course zero is a real number. What a stupid waste of time to write about it,” you may be one of those people for whom the realization of this truth in all its depth and beauty has not yet occurred.

My student teacher this year was one of those people in September. 27 years old and she never knew zero was a real number. She thought she knew it, but she betrayed herself when she began teaching first graders to answer the question “how many?.” Although she never explicitly said so, she gave her charges to understand that the minimum answer to the question was “one.” I surprised her by reminded her that “zero” is a legitimate answer to the question, “how many?” She did not quite believe me. “Think,” I said, “Of a time when you may have looked for eggs in the refrigerator and found there were zero eggs.” Her eyes widened. “Oh...yeah!” she said, “I hadn’t really thought about it.” I reminded her that when she set up her counting situations, to let zero often be the answer. Children come to school already preconditioned to disregard zero. Their parents and preschool teachers have given them 6 years of experience ignoring zero. One of the first math tasks at school is to undo that misconception.

Zero is a real number. Tax season provides a perfect example. Consider two taxpayers. One person may complete a tax return and find that his tax liability is zero. Therefore when he pays his taxes, he pays zero dollars. Another person does not even complete the form. One person paid no taxes. However, the other one did pay his taxes, and he paid zero dollars. “Zero” and “nothing” are not the same thing. Set theory was supposed to make this distinction clear, but too often we go into math mode and miss the point.

Zero is a real number. I will never forget the day in November when this realization struck my student teacher. She was in the middle of teaching first grade math when she looked at me sitting in the back of the room and said incredulously, “Zero is a real number,” as if it were her own discovery and not something I had said again and again for more than two months.

Is Division Repeated Addition?

One of Stanford mathematician Keith Devlin's pet peeves is the common “division is repeated addition” meme . He despises it so much he has something like a mantra, “Repeat after me. Division is NOT repeated addition.” Naturally, math teachers give him a lot of pushback because division is indeed repeated addition (except when it's not).

It seems there are actually two topics in play here. 1) Multiplication as repeated addition and 2) the skill of elementary math teachers. I completely understand his frustration with prospect of undoing the poor math instruction college students typically receive during their elementary school years. I also experience the same frustration as a secondary and college level instructor.

Multiplication as repeated addition is not a definition of multiplication, even though many elementary math teachers erroneously think the definition of multiplication is precisely repeated addition. Repeated addition is merely another name for the group model of multiplication. There are other models, such as the array model, the area model and the number line model, to name the ones most commonly presented to elementary students. Devlin rightly maintains that it is inaccurate to say that multiplication is repeated addition, period. As a misleading misstatement, it is right up there with “you cannot subtract a bigger number from a smaller number.”

However, as properly taught (a giant qualifier, I know), the group model is merely the first element in a teaching sequence which eventually progresses to the area model, then to the use of the area model to multiply fractions, and beyond. For example, you can definitely model a positive whole number times a negative rational number on the number line where it very much looks like repeated addition of the given negative number. Turn that number line vertically, and it makes even more sense to students because it reminds them of another number line they are very familiar with, the thermometer.

Devlin writes, “Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum.” Exactly, and this is precisely the way a good teacher presents the group model. Children sometimes ask questions like, “Instead of saying 8 + 8 + 8 + 8 + 8, and then saying the answer, can't we just say “8, five times” and then say the answer?” Of course we can, and that is what we do when we say 5 x 8 = 40. The group model is meant to express this particular connection between addition and multiplication. The group model is not meant to be a definition of multiplication. Nevertheless, I agree there are too many elementary math teachers who fail to make the distinction, or properly progress through the models.

An umbrella idea I like involves the word “of” as an English language expression of multiplication. We can say “5 groups of 3,” or “5 groups of -3,” or “1/2 of 3,” or “1/3 of 4/5,” or “16/100 of 40” or “75% of 200,” etc and neatly cover most examples of multiplication that children are likely to encounter before junior high. Devlin prefers scaling as the dominant meme and argues that children should readily understand scaling because examples of scaling surround them. The problem is most eight-year-olds have difficulty comprehending scaling as a model and effect of multiplication. Even though they can readily see that a scale model is a perfect replica of the original, they do not understand how it is possible that doubling the dimensions of a garden (to take a simple example) results in a garden four times larger. Most of the scaling children see is usually on maps where the scale is for them an unimaginably large (or small, depending on viewpoint) number.

Teachers are better off working their way up to the scalar model of multiplication. I have found this is best done by reminding younger students early and often with the idea that we have not yet exhausted the possible models and applications of multiplication. I have found it useful to show some examples of these applications, and say something like, “Later you will learn how you can use multiplication to produce an exact scale model, or use multiplication to produce a real-life-sized object from a scale model.”

Actually, most students get their first solid grip on scaling when they work with similar figures (typically triangles) during high school geometry. Personally, I have found success with older elementary students by giving them basic practice in scaling on the coordinate plane or increasing recipe yield and other types of problems. Students also enjoy the products of their work whether it be art or good eats. The number line model is also a good introduction to scaling because you are scaling only one dimension, as opposed to the two and three dimensions involved in scaling area and volume, respectively.

Devlin also laments the constant push to make math “real.”

No wonder children arrive at college not only having little or no genuine understanding of elementary arithmetic, they have long ago formed the view that math has nothing to do with the world they live in...many people feel a need to make things concrete. But mathematics is abstract. That is where it gets its strength.

His comments seem contradictory, but they are not. One of the most enjoyable aspects of teaching math is showing students the leap from concrete to abstract. For example, I love showing students a cube and showing them the edge e, the face e x e, then showing them cube e x e x e. I usually let the idea hang, and love when someone asks if I can show e x e x e x e on the cube. No, I answer, I have nothing to show them that. And therein lies the power of math. Math can help us express ideas we can understand, but for which we have no physical representation. I am thrilled when someone asks if e x e x e x e can be time. I ask what would e x e x e x e then express. The children are really products of this century, and one of them is likely to answer, “the coordinates of a time-traveling spaceship.” So much fun. One time, a child said, “Maybe someday we will have a real meaning for more es.”

To the extent that teachers present repeated addition as a property, which in some applications, connects the operations of addition and multiplication, no harm done. However, considering repeated addition as THE definition of multiplication is a serious problem, and Devlin is right to be concerned about it. As Denise Gaskins pointed out, “...if (a particular) model doesn’t work universally, then (the model) certainly cannot be used to define the operation.”

Devlin asks an interesting, and in today's pedagogical environment, nearly taboo question:

The "learn the technique first and understand later" approach is very definitely the only way to learn chess, and millions of children around the world manage that each year, so we know it is a viable approach. Why not accept that math has to be learned the same way?

I would say that technique over understanding has been the preferred approach for centuries, but by all accounts, many adults have never made it to the “understand later” stage. In my experience teaching concept concurrently with technique works the best. The problem I am seeing is that these days too many elementary teachers attempt to teach concepts they barely understand, and then give short shrift to technique because the kids have calculators for that. The result is legions of kids who not only have faulty understanding of the concept, but also lack the ability to perform the technique quickly and accurately.

Additional Discussion:

https://www.quora.com/Why-is-it-incorrect-to-define-multiplication-as-repeated-addition

https://denisegaskins.com/2008/07/01/if-it-aint-repeated-addition/

http://scienceblogs.com/goodmath/2008/07/25/teaching-multiplication-is-it/

https://numberwarrior.wordpress.com/2009/05/22/the-multiplication-is-not-repeated-addition-research/

http://www.quickanddirtytips.com/education/math/is-multiplication-repeated-addition

http://billkerr2.blogspot.com/2009/01/multiplication-is-not-repeated-addition.html

http://rationalmathed.blogspot.com/2008/07/devlin-on-multiplication-or-what-is.html

http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html

http://homeschoolmath.blogspot.com/2008/07/isnt-multiplication-repeated-addition.html

http://www.textsavvyblog.net/2008/07/devlins-right-angle-part-i.html

http://mathforum.org/kb/thread.jspa?threadID=2045768

http://www.qedcat.com/archive_cleaned/114.html

Nobody Understands Place Value

Parents often ask me to help their children with math homework. My reply is always the same. I am not interested in “helping” with homework. I am very interested in addressing the gaps and misconceptions that give children difficulty with their homework in the first place. Chief among these is a pervasive lack of understanding about place value.

Children's understanding is generally limited to identifying the place value of a given digit or inserting a digit in a given place. I do not blame the children. Most curriculum asks them to do nothing else. Replacing the terms “borrowing” and “carrying” with terms like “exchanging” or “regrouping” represented a tremendous improvement in math education. Even though many elementary math teachers these days play trading games, and the kids appear to know what they are doing, every junior high or high school math teacher has observed that they do not profoundly understand place value, so crucial to understanding the quadratic equation, bases other than base ten, and other topics.

Therefore, I usually start my homework help by first playing simple trading games with the student. The problems I use for the games are ones I know students can calculate correctly, such as 48 + 17. Maybe they can even do it in their heads. No matter.

The first thing I do is dispense with the usual place value names. I use “loose ones” and “packages of ten.” Loose ones is a better term than simply ones. The ones are ones precisely because they are not in a group. They are loose. Children often do not realize that the loose ones' place is fundamentally different from all the other places. Thus they often have trouble with the ones' place in other bases. For example, teachers tell students that when they are working in, say, base five, the largest possible digit in the ones' place is a 4 because “there is a rule that the one's place can be no larger than one less than the base.” Although it is true that the ones' place can be no larger than one less than the base, it is not because of a rule. The reason is much more fundamental than a mere rule. Understanding the ones' place as “loose ones” is key to discovering that fundamental principle.

After we play the trading game for a little while, I have the student do a simple addition problem. When students calculate a problem like 18 + 25, they put a 3 in the ones' place and a 1 above the 1 of 18. Then they add 1 + 1 + 2 and write a 4 in the tens' place, resulting in the answer of 43. Then I ask, “How did you get that answer?”

They usually reply, “I put a 3 in the ones' place and a 1 above the 1 of 18. Then I add 1 + 1 + 2 and write a 4 in the tens' place, so my answer is 43.”

That's fine. Of course they answer in a mechanical, non-mathematical way. They have heard teachers repeatedly explain addition problems to them in much the same way. Then I ask, “Yes, but why did you do that?”

Students invariably reply, “Because that is how the teacher told me to do it.”

Then I ask, “Yes, but why did the teacher tell you to do it that way? Why does it work?” Now they are stymied.

So I show them how to “prove” (not really prove, more like demonstrate) the answer using a picture (similar to this one, but simpler).

I show them how to draw the picture and talk their way through it. “See, you have 15 loose ones. That is enough to make a package of ten. So you gather up a package a ten and put it with all the other packages of ten. You still have 5 loose ones left. Because 5 is not enough to make a package, you leave them loose and show them in the loose ones' column. You add up the packages of ten and put that total in the packages of ten column.”I have them illustrate several problems by drawing the picture.

Very often students realize for the first time that place value is all about making groups of ten. Subtraction is all about breaking groups of ten into loose ones and dumping them with the other loose ones. Every place except the loose ones is a group of ten something. Teachers tell students that each succeeding place is larger by a magnitude of ten, but somehow children fail to grasp the significance of this fact. The reason the standard addition algorithm works is because you are gathering up groups of ten at every place. Likewise, the reason the standard subtraction algorithm works is because you are breaking a group of ten at every place.

Instead of the usual place value mat, I like to use a mat that labels the places a little differently. I start with the loose ones, then packages of 10 loose ones, then cartons of 10 packages, then boxes of 10 cartons, then cases of 10 boxes, then pallets of 10 cases, and so on. I usually stop at cargo ship with 10 shipping containers. Kids love it. Even second graders can easily calculate a multi-digit addition or subtraction problem. In fact, after kids master place value as groups of 10, they often ask me to set them problems with any number of digits. I usually refrain from a problem with more than 13 or 14 digits because even though the kids find the problem easy, they also find it tedious and time-consuming. But hey, tedious and time-consuming is a whole sight better than hard when it comes to doing homework.

Rote Does A Lot of People A Lot of Good

Recently I read a forum comment somewhere to the effect that “rote does not do anyone much good.” Ironically, this comment was part of a comment whose main idea was that educators should question buzzwords and dubious tenets of pop-education. In spite of the current popularity of this sentiment, rote actually does a lot of people a lot of good. I agree that sometimes educators have abused rote. I remember a fourth grade science book from around thirty years ago that included a whole unit on state birds and flowers as if such information had even the remotest relevance to science. Naturally, since most schools consider the textbook to be THE curriculum, many teachers felt compelled to “teach” children the state birds and flowers. Inquiry or constructed learning will not work for this kind of arbitrary knowledge. Rote is really the only way. State birds and flowers should never have been part of the science book in the first place.

Today we have Common Core. Just as in the past, of course publishers are scrambling to roll out new textbooks that reflect Common Core. Thus yet again, the textbook will become the default curriculum. Yet, even within Common Core and even within a commitment to teaching concepts over rote, there are still a number of topics for which rote is still the best method. The order of the alphabet, sight words, broad overviews of history, and arithmetic facts are just a few. People forget the intense amount of repetition and memorization involved learning a first language, let alone subsequent languages. Churchgoers know memorizing the books of the Bible greatly aids in finding the preacher's text. If you want to pass your first driver's license test, it would behoove you to memorize the rules of the road. Rote memorization of poems provides an avenue of future pleasure. If you live in a character-based literary system such as Chinese, you will need to memorize several thousand characters to simply be a literate person.

Another related canard holds that the purpose of education is not knowledge itself, but the ability to find knowledge. However, students who lack a substantial reserve of memorized knowledge have a great of difficulty even figuring out what search terms to use. Personally, as much as I hated memorization when I was young, I have come to appreciate the easy access to information, internet or no internet.

Finally, there is an important reason to refuse an ideological stance against rote learning. Sometimes it is all a student has left. Math is a subject area well-suited to discovery methods. All mathematical procedures are based on the real and predictable behavior of numbers. Very little math knowledge is actually arbitrary. The best math teachers who consistently use the best discovery methods to help students acquire mathematical concepts still sometimes encounter students who simply cannot get it. If they cannot acquire the concept, and we also also deny them rote learning, we leave them with nothing. Although rote should never be the first resort in a non-arbitrary domain such as mathematics, rote still remains the best last resort to ensure that all students acquire the basic skills they need for their adult lives. As educators, we need be at the forefront of confronting ideological statements wherever found

Nothing Wrong with Rote

A popular view among educators is that rote learning is bad, bad, bad. Point out how well Asians do in international tests compared to Americans, and defenders will likely counter that maybe so, but Asian education depends on that bad rote learning, but we Americans, no matter how poorly we compare, are superior because we emphasize concepts and creativity.

To be honest, as a math teacher, for a long time I believed that rote learning, while undeniably effective, was merely second rate. Kids must need lots of memorization of mathematical recipes and homework practice to pound poorly understood mathematical procedures into their brains. As a math teacher, I believed that if skilled math teachers developed a strong conceptual foundation within children, the logic and elegance inherent in math would minimize the need for tedious, time-consuming homework.

As a junior high and high school math teacher, my ideas about foundation building were only theoretical. It was easy to look at my students, the products of elementary school instruction and conclude that elementary math teachers were doing a terrible job of building foundations. After all, there is plenty of documentation for the inadequate math teaching skills of elementary teachers. I could be charitable. I did not blame elementary teachers too harshly, because they themselves did not acquire mathematical foundations when they were elementary students. They cannot teach what they do not know.

However, until I came to China to work with elementary students, I had never had a chance to test my hypothesis that all students really need is a great foundation. Actually, for most students my hypothesis worked. They could demonstrate a deep and thorough understanding of the concepts I taught. I often assigned homework of only five to ten problems, and after this little bit of practice, they could reliably get the right answers.

However, there were a few students for whom concepts were not enough. One day they would demonstrate terrific understanding. The next day we had to start almost from scratch. I tried everything, every approach I could think of, including a lot more practice. What worked when nothing else did was lots of practice---yes, tedious, time-consuming practice designed to activate rote memory.

I finally concluded that if a child can successfully utilize the concept to solve problems, that's great. However, if the only way they can master the procedure is to repeatedly execute it until they reliably get right answers (my standard was 80% or more), then so be it. Better that then sending them on their way with nothing.

Sadly, as comparative studies show, too many American children lack essential conceptual foundations because the documented lack of teaching skill means teachers fail to actually effectively teach the concepts. Students also fail to do adequate procedural practice because of the American educational aversion to “boring” homework. So I say let's teach concepts and teach them well, AND let the students practice until they can get the right answers. Some students may need more practice than others. So be it.

Interestingly, recent research on children's and teen brains explains why children have such great memories, and how practice, even in an environment of complete concept understanding, is necessary to build brain pathways.

the whole process of learning and memory is thought to be a process of building stronger connections between your brain cells. Your brain cells create new networks when you learn new tasks and new skills and new memories. And where brain cells connect are called synapses. And the synapse actually gets strengthened the more you use it. And especially if you use it in a patterned way, like with practice, it gets even stronger, such that after the practice, you don't need much effort to remember something.

When we dismiss rote learning, we forfeit a valuable tool for building neural pathways in the brain.

See related posts:

Patient vs Impatient Problem Solving

Common Cart---Cart Before Horse

I Love Manipulatives...But

Cultural Sacred Cows of American Education

MacDuff: The New Math

Slaying the Calculus Dragon

No doubt about it. Many students consider calculus scary, right up there with monsters under the bed. Calculus is the Minotaur or St George's dragon of math at school. Sadly, schools have done little to undermine its almost mythological reputation, what with “derivatives” and “integrals” and those frightening numberless equations recognizable by the initial elongated “∫.” There is like, what? 100 equations, that, according to most teachers, need to be memorized.

It is a pity, because calculus is really the Wizard of Oz, terrifying to behold, but quite tame behind the curtain. Did you know that most people do calculus in their heads all the time? In fact, because the numbers associated with calculus are ever-changing, moment by moment, doing calculus with numbers is a bit pointless. A mother filling the bathtub very often does not want to stand around watching the water. She knows that the water is coming out of the faucet at a certain rate. She knows the bathtub is filling at a certain rate. Every moment the volume of water is changing. Yet, she reliably comes back to check the tub before it overflows.

The high school quarterback and his wide receiver communicate an even more difficult calculus on the field, seemingly by telepathy. The quarterback never aims the ball at the place where the receiver is standing. That mental math is too easy, more like algebra or even arithmetic. No, he aims the ball toward the place he hopes the receiver will be. In his head, he calculates the trajectory of the ball, the amount of force necessary (oh my gosh, not physics, too!), the speed of the receiver, and every other factor. And most of the time he gets it right, and the pass is completed. The fun thing about calculus is that the numbers are ever-changing. It is like hitting a moving target, whereas algebraic numbers thoughtfully stand still.

At its core, calculus is nothing but slope. Remember humble slope, change in y over change in x. Slope is an expression of rate, such as, change in miles over change in time, most commonly called “miles per hour” or mph. The graph is a straight line, so you can pick any two points to find the slope. But what if the graph is curved? If you were to magnify each point and extend its line, every point has a different slope. That is because straight lines illustrate rates like speed (velocity), while curves illustrate rates like acceleration (getting faster and faster each moment), just like the football getting slower and slower until it reaches the top of its path, comes to dead stop (but only for a moment) and then gets faster and faster.

A graph has three basic pieces of information, the x data set, the y data set, and slope. “Derivatives” are used to find the slope of a curve at any point when the x and y data are known. When you know one of the data sets and the slope, you can use “integrals” to find the other data set. Techniques like finding the area under the curve are used to get as close as possible to the exact answer. First you divide the area under the curve into rectangles all having the same “x” length. Then you add up all the area. Obviously, some of the rectangles are a little too small for the curve and some are too big, so your answer is only an approximation. If you shorten the “x” length to make narrower rectangles, your approximation will be closer. Integration allows you to find an exact answer instead of an approximation, however close the approximation may be. But there are “limits” (you have no doubt heard of limits). Your “x” length may be very small indeed, but it can never be zero, because then the sum of the rectangles would illogically be zero.

If you can find a skilled dragon slayer, that is, a teacher who can demystify it, studying calculus can be great fun.

What do Employers Really Want from College Grads?

The summer 2015 issue of the Phi Kappa Phi Forum featured an article summarizing a recent survey of 300 employers entitled “The Value of Student Agency.” It was an interesting article in light of the fact that so many people are lobbying for universities to be job training centers. On line forums are replete with comments like, “No one forced students to accept substantial loans to finance their educations, all too often in fields for which there were no jobs following graduation.” Or, “That person making your latte probably has a masters degree in something no employer finds useful.” Or, “There is a lot of slop that colleges pass off as "education."”

A typical university mission state reads as follows:

ABCU aims to graduate lifelong learners with the courage to challenge boundaries, ask questions and ignite knowledge with creativity. ABCU students take charge of their own intellectual and artistic development and integrate an active, independent, critical and reflective perspective into their lives as a whole.

I suppose to many ears, that sounds like “slop.” So the question is what do employers want from College graduates.” The survey sought to answer that question. According to the results, employers want that “slop.” 90+% employers say they want employees who think critically, communicate clearly, solve complex problems, promote innovation and that these qualities are more important than any specific undergraduate degree. In other words, employers say they want employees who went to college to get educated, not to get job training.

Traditionally, college has been the place to get an education, and employers provided the job training. If college is for education, then, for example, whatever your major, one thing all students should learn at college is how to formulate and defend ideas without resorting to the logical fallacies and ad hominem so prevalent in online forums. However, if college is for job training, then choosing a marketable major/job is the important thing. Given that 65% of incoming freshman need remediation in English and/or math, apparently college is truly the new high school. Right now society needs to decide what it truly wants of college, education or job training.

The second question is whether the employers interview in the survey answered honestly, or told the interver what they thought he wanted to hear. Employees who actually have the desirable qualities say they often find themselves actively discouraged from displaying just those qualities. Thinking critically and communicating clearly implies a contradiction with being a team player. And “being a team player” is usually code for going along to get along.

The biggest problem I see with college students is their lack of seriousness. t would be great if students took their K-12 education way more seriously and qualified themselves for some of the millions of dollars in scholarship money that goes unawarded every year. Instead, what we have are students who think K-12 means "doing time," and then choose their college based on its party-school ranking. Think not? Ask any high school guidance counselor.

Data-Driven Fallacy

“Data- driven” sounds like a great way to make decisions. It even sounds scientific. What could possibly go wrong?

When data drives decisions, stakeholders, especially stake holders under pressure, will find a way to influence the data in their favor. Therefore Well Fargo employees, pressured to meet sales quotas, open dummy accounts. HRBlock tax preparers, pressured to demonstrate the success of the Second Look program, begin regular tax returns as Second Look returns and convert them midway to a standard tax return without the client’s knowledge, and teachers change answer sheets for standardized tests.

When statistics become the most important consideration, employees will create the statistics they need. It is only human nature. Performance evaluations MUST be holistic, not only to prevent fraud, but also, and most importantly, to form a comprehensive evaluation of the employee including factors not easily quantified, yet crucial to student success. Noth8ing is a panacea, not even and maybe not especially “data driven” decisions.

Preconceived Bias Always Trumps Critical Thinking

In the last article, I discussed the strange phenomenon that whenever critical thinking and preconceived bias go head to head, dollars to donuts, preconceived bias will win. A big part of the problem is too often our self worth and identity is tied up with our opinions. If we could learn to separate our opinions from ourselves as persons, perhaps we could make progress discussing and solving the pressing issues that surround us. In this article, I will examine an extremely common current example of preconceived bias trumping critical thinking. In so doing, I expect UI am sure to offend anyone who overly identifies themselves with their opinions. Finally, I will present one professors explanation of why bias is so powerful and his suggestion for overcoming bias.

Yahoo! Forums, (unsurprisingly), is a great repository of examples of confirmation bias. “Bob” says illegal aliens go to our schools and use all other taxpayer funded infrastructure. They use fraudulent ID to get tax refunds they're not entitled to. They use our emergency rooms and never pay a dime. “Ken” says if the government went after illegal aliens using SSNs fraudulently they could eliminate SSN tax fraud. “Rick” says someone files a tax return for 30 years at the same address and then some illegal files using the number a thousand miles away. “Whatever” says a job that used to pay 40K is now paying 25K and illegals are doing it without paying income tax “Patrick” says a drive through “illegals town” shows you exactly who is filing fake returns. “Tickle” says illegal aliens can file taxes claiming only $10,000 of income and get $24,000 in tax refunds. “SuperBaby” says the government released 35,000 illegals who committed crimes (rapes, murders) on the street.

Looking over the threads these comments appeared in, I was struck by the lack of counter response. Very few people challenged these statements; however, there were a few and here is a sampling:

Bob, you are dead wrong. They do not use fraudulent IDs to get tax refunds they are not entitled to. They apply for, and after much scrutiny by the IRS, may or may not receive a number, called an ITIN, solely for paying taxes. Because they do not have SSNs, they do not qualify for tax credits including EIC. Usually their withholding is too low, so most of them end up paying balance due. They also pay into Social Security and Medicare. Their payments are passed through to Social Security beneficiaries with valid SSNs. Therefore, they help fund your SS benefits. Is there some fraud by a small minority? Yes. However, the vast majority of refund fraud is perpetuated by citizens with valid SSNs. ... “They go to our schools and use all other taxpayer funded infrastructure.” True, and they pay taxes, just like you. They pay income tax, sales tax, and even property tax (which funds schools) as a component of their rent. Now with drivers licenses, they also get to pay gas tax. ... “They use our emergency rooms” sometimes and pay for them. Generally they utilize neighborhood clinics for medical care. Do a few go to the emergency room and never pay? Sure, but again the vast majority of people who do that are actually US citizens. ... Apart from the misdemeanor of being in the country illegally, they are generally law-abiding taxpayers who keep their heads down. They are not "criminals" in the sense you mean. The criminals most of us have to worry about are actually white-collar citizens. It amazes me how people persist in believing untruths in the face of facts. If you oppose illegal immigrants, you need to find some actual valid reasons. ... The vast majority of tax fraud is committed by citizens, not illegal aliens. Sorry. ... An illegal cannot use a SSN to file a tax return. The IRS computers WILL reject those instantly as name and number will not match. So you do not have to worry about that part. ... Illegal immigrants do pay income tax. they also pay the social security and medicare tax, but will never draw benefits because they do not have a valid SSN. The invalid one on their W-2 was put there by the employer. … No Patrick, illegals are not the ones filing fake returns. They do not have access to the personal info they would need to pull it off. However, they do pay taxes using and IRS-ssued tax account number called an ITIN because they do not have SSNs. They generally pay a significant balance due because their employers think they are doing them a favor by withholding nearly nothing. … If illegal aliens "say" on their tax return that they made $10,000, either there will be a w-2 or a Schedule C. If there is a schedule C, there will be a balance due, not a refund, because of the required Schedule SE. … Even if all 35,000 who were released committed violent crimes, that would be only 0.3% of all illegal immigrants, the remainder of whom who, apart from their entrance into the country, are otherwise law-abiding. They keep their heads down to avoid unwanted government attention. ...Illegal aliens do NOT submit phony SSNs for their children. The tax return would be rejected for name-number mismatch. If the children have an SSN, then no birth certificate is necessary. By the way, it is citizens with proper SSNs that commit nearly all the EIC fraud, not the illegal aliens.

I have no intention of debating illegal immigration. The point is not to defend any particular opinion, but to examine the logic. Faulty logic does not necessarily mean an opinion is wrong. However, valid logic naturally lends better support to an opinion.

I just made an allusion to the possibility that an opinion could be wrong, thus implying that an opinion can also be right. One of the most unfortunate principles of the fake critical thinking lessons in our schools is the idea that there is no such thing as a right or wrong opinion. The principle is true as far as it goes. The thing is some opinions have higher quality than others. The main determinate is the quality of logical support for the opinion.

Let us put aside for a moment the hot partisan arguments over the issue of illegal immigration, and examine the original comment and the responses using the tools of logic. Surely the first prerequisite of logic is to determine the facts of the matter. Intriguingly, a perusal of the thread these comments appeared in show that apparently that no one fact-checked the responses. For some strange reason, these challenges also reliably garnered a collection of thumbs-down, even though a bit of research supports the factual basis of each challenge. Preconceived notions and confirmation bias certainly at work.

Alan Jay Levinovitz explains that throwing facts at preconceived biases will not work because these biases “... are based on really powerful narratives, stories about how we construct our identities..You have to deconstruct the narrative (first).

Why Critical Thinking Lessons Do Not Work

Daniel Kahneman studies thinking. Although the interview* is discussing bias, not critical thinking, the implication is inescapable.

Daniel Kahneman: So ...students were asked to evaluate whether an argument is logically consistent – that is, whether the conclusion follows logically from the premises. The argument runs as follows: ‘All roses are flowers. Some flowers fade quickly. Therefore some roses fade quickly.’ And people are asked ‘Is this a valid argument or not?’

Quick. Ask yourself. Is this a valid argument? Don't peek at the answer. Have you decided? OK, if you said no, it is not a valid argument, why did you decide so? If you said yes, it is a valid argument, why did you decide so? If you said yes, you agree with the majority of the students. They said it was a valid argument because they have observed with their own eyes that the conclusion is true. Some roses certainly do fade quickly. Do you agree with the students' reasoning?

Daniel Kahneman: It is not a valid argument. But a very large majority of students believe it is because what comes to their mind automatically is that the conclusion is true, and that comes to mind first. And from there they naturally move from the conclusion being true to the argument being valid. And people are not really aware that this is how they did it: they just feel the argument is valid, and this is what they say.

I have bolded the important words. People do not really think; they feel. Then they draw their conclusions on the basis of feeling. Perhaps you agree with the interviewer who suggests that direct teaching of logic will solve this problem.

Nigel Warburton: Now in that example I know that the confusion between truth and falsehood of premises and the validity of the structure of an argument that’s the kind of thing which you can teach undergraduates in a philosophy class to recognise, and they get better at avoiding the basic fallacious style of reasoning. Is that true of the kinds of biases that you’ve analysed?

It is quite reasonable to expect that with a few lessons, we can teach people to at least pay attention to the question. The question asked if the conclusion follows from the premise. That means start with the premise, NOT start with the conclusion.

Daniel Kahneman: Well, actually I don’t think that that’s true even of this bias.

I read that and thought, well, why not. It seems pretty obvious that if students learn how to evaluate an argument in terms of logic, they will certainly be able to apply that valuable skill in their daily life. After all, the whole point of education, and especially critical thinking skills is to apply the lessons in daily life. Students expect education to be thus applicable. Otherwise they would not continually ask, “When are we ever going to use (fill in the blank)?”

Daniel Kahneman: The thinking of people does not increase radically by being taught the logic course at the university level. What I had in mind when I produced that example is that we find reasons for our political conclusions or political beliefs, and we find those reasons compelling, because we hold the beliefs. It works the opposite of the way that it should work, and that is very similar to believing that an argument is valid because we believe that the conclusion is true. This is true in politics, it is true in religion, and it is true in many other domains where we think that we have reasons but in fact we first have the belief and then we accept the reasons.

So according to Kahneman, we so cherish our preconceived biases that no amount of logic, facts, or reality will dislodge them. And in fact, this stubbornness is exactly what we perceive everywhere in our society, within our political parties, in online forums, and on our neighbor’s porch over lemonade. However, even though critical thinking lessons do not work, I say we need to not only continue to teach critical thinking skills, and do so in an even higher quality way. Better to give students access to the tools and hope some students will actually use them, than to deny the tools to all students.

*If the pdf link to the interview does not work for you, try this non-pdf link.

How Should Students Show Their Math Work?

In this post, I am pinging off Maria Miller of Math Mammoth. I recommend Math Mammoth for its concept-based lesson development and worksheets.

Many students resist showing their work. They feel they are demonstrating their smartness by not showing their work, as in “See, Ma. No work.” However, when you ask these students how they got the answer, they cannot remember what they did. Sometimes they say they used a calculator. OK, I say, but what numbers did you put into the calculator? They cannot tell me. I explain that since we cannot record thoughts the way we can record voices, students need to make a record of their thoughts when they solve a problem. Dispensing with the work is not actually smart at all.

Now, here is where we see the real difference between strong students and weak students. Strong students respond to my words, and start showing work ever after. Weak students respond (eventually) only to action. I make them do their homework again, and I mark right answers wrong if there is no work.

As Maria says:

The purpose of writing down the work allows someone else to follow the person's thought processes. This is of course important for students to learn no matter what their future occupation: they need to be able to explain to others how they solve a problem, whether a math problem or a problem in some other field of life!

As strong as Chinese math teachers tend to be, they do not encourage students to show their work. Chinese teachers expect “clean” papers, with only answers. Chinese teachers check whether answers are right or wrong. They are completely unconcerned with why the student got a wrong answer, or if the answer is coincidentally right for the wrong reason. Retraining my students has been quite a challenge. Today they appreciate the need to show work, and they work hard to demonstrate that their work flows in a logical manner. Today, they show off their work instead of showing off the lack of work.

Even though Chinese teachers do not want to see work in the final product, they actually have high standards for the format of work. They train students from first grade in this format, and one reason they do not care to see the work in, say, fifth grade is they trust the student followed the format to get the answer the teacher does see, a dubious assumption at best.

Maria says she would ask primary student to verbally explain how they got an answer. Chinese teachers expect students to translate verbal (or written) math problem to mathematical expressions. Students learn to write “number sentences” from the very beginning. Perhaps there is a picture of a tree branch with three birds and two more birds landing. The child translates this picture in the number sentence “3 + 2 = “, and then writes “5 birds”.

I modify this approach a little. I expect children to write “3 birds + 2 birds = The idea of ignoring the units and then plugging them back in at the end leads to all kinds of confusion in later grades. Leaving the units out of the work is a major reason students persistently forget to square the unit when finding area. The math sentence should be 4 cm x 5 cm =

When students first begin studying area and perimeter, I make them write intermediate steps. In the case of area the intermediate step is: (4 x 5) x (cm x cm) = 20 cm2. In the case of perimeter, the intermediate steps might be (2 x 3) cm + (2 x 5) cm = 2(3 +5) cm = (2 x 8) cm = 16 cm. There are many types of problems where keeping track of the unit is vital. An early example is division, especially division with remainders. Often the unit for the quotient is different from the unit for the remainder. Knowing the difference is the key to understanding the solution.

I also require the box. The box makes the number sentence a complete sentence. Later, we will replace the box with a variable, and later still the variable may appear somewhere besides the end. Take this problem for example: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away? I expect children to translate this sentence to math as written, without doing any preliminary math in their heads. Thus “5 birds - = 2 birds”.

Most teachers have the children write this math sentence as 5 birds - 2 birds = 3 birds. Doing so requires the students to do some math in their head first. The purpose of the number sentence is to accurately translate the problem to math terms. The number sentence must follow the story. The number sentence for a multi-part story should incorporate all parts into one number sentence. When problems become more difficult, the ability to translate the story to math as written becomes essential. The crucial part of solving a math problem is the number sentence. When the number sentence is correct, absent any silly mistakes in the work, the solution will most certainly be correct.

Finally, I require the students to answer the question with a complete sentence. The purpose of answering the question is to help student differentiate the solution from the answer. For example the solution to the question, how many cars do we need for the field trip might correctly be 5.2 cars, but the answer is 6 cars.

Summary

The work for a word problem needs to have three parts.

1.  A translation of the word problem into a complete math expression that includes the units and follows the story.

2. The arithmetic which tracks the units all the way through to the solution and may include intermediate steps for as long as necessary for mastery.

3. The complete answer to the question.

Sample

Math Expression: 10 x [$10.50 – (2/5 x $10.50)] = n

Work: 10 x {$10.50 – [($10.50 ÷ 5) x 2]} = n

10 x [$10.50 – ($2.10 x 2)] = n

10 x ($10.50 – $4.20) = n

10 x $6.30 = $63.00

Answer: Annie's total bill is $63.00 or Annie paid a total of $63.00 for the shirts.

Well-trained fifth graders have no trouble displaying their work as in the sample. This vertical work format, started in first grade, gives the students excellent preparation for mathematics involved in algebra, chemistry, physics and calculus. In fact, starting in third grade, I often have students format their work in two vertical columns, the second column for the math property used, as in this simple sample:

Problem: There are 5 birds in the tree. After a certain number fly away, there are 2 birds left. How many birds flew away?

Number Sentence: 5 birds – n = 2 birds

Work:

---

Arithmetic	Property
5 birds – n = 2 birds	given
+ n              + n	both sides rule
5 birds + 0 = 2 birds + n	additive inverse (opposites rule)
5 birds = 2 birds + n	additive identity
-2 birds = -2 birds + n	both sides rule
3 birds = 0 birds + n	math fact/additive inverse
3 birds = n	additive identity

Answer: Three birds flew away.

Class Policies for High School

These are the class policies I use for junior high and high school math and science classes. They are quite brief, but effective because students perceive right away that I say what I mean and mean what I say. Being straight-forward and authentic is probably the number one key to classroom management. Educators will debate the validity of grades forever, but as long as colleges pay attention to GPAs, teachers will have to figure out a way to determine grades. The following system has worked well for me.

Duties of Responsible Students:

1. Responsible students come to class on time, with their homework and materials laid out on their desk, pencils sharpened, and ready to begin before the scheduled start of class.

2. Responsible students do everything in their power to make it as easy as possible for their classmates to concentrate and achieve.

3. Responsible students turn in work that is neat, complete and on time.

Components of Grade:

1. Classwork 40% of grade (includes quality of work completed in class and responsible behaviors during class. Giving your work your professional best effort will raise this grade. This grade starts at 100% for all students.

2. Homework 30% of grade. The grade is the number of completed assignments out of possible assignments. Unacceptable assignments will receive an “R” which means “redo within one week.” Otherwise, the grade for that assignment becomes 0.

3. Tests 20% of grade. Test are graded as a straight percentage.

4. Quizzes 10% of grade. These are generally pop quizzes. 51% or better on a pop quiz earns a P for pass. 50% or below is a “no pass.” Announced quizzes are graded as a straight percentage.

Formatting Your Work:

1. All work must be done on standard 3-ring notebook paper, or specified graph paper. Do not fold your work.

2. Pencil is acceptable for certain work done in class and for math. Products like lab reports and essays must be written in cursive using blue or black ink only. You may write your work on a word processor, however printer malfunction is not an acceptable excuse for failing to submit the assignment on time.

3. Remember to use your English skills. Even when the work is not for English class, you are still expected to indent paragraphs, maintain margins, proofread and rewrite your work as necessary to submit your best work.

4. All papers must have a proper heading as previously instructed.

How to Evaluate a Math Textbook

Regardless of Common Core, everybody knows that practically speaking, the textbook IS the curriculum. Therefore, it behooves textbook adoption committees to choose carefully. First, ignore the beautiful graphics. The beauty may truly be only skin deep. Reject books that teach tricks, procedures and shortcuts. Choose books that teach the profound understanding of fundamental mathematics. You do not have to read the entire book. Look especially for how the book handles the following topics:

Place value---Place value is arguably the most essential foundation stone of all future math understanding. Yet most textbooks provide only a rudimentary presentation of place value. Students are expected to do no more than name the place of a given digit or write a certain digit in a given place. The understanding of place value actually begins with counting. Make sure children name what they are counting and start with zero, “0 frogs, 1 frog, 2 frogs, 3 frogs...there are 7 frogs altogether.” Remember, place value depends on fully knowing the name of what is being counted, and not simply as part of a memorized pattern. 203 means you have counted 2 hundreds, 0 tens, 3 ones. 203 can also mean you have counted 20 tens, 3 ones. Which version is more useful depends on the context of the real life math. An early emphasis on place value helps students with later concepts such as fractions (203 thousandths), volume and area (203 cubes vs 203 squares), or the difference between like and unlike terms (3a + 2b). There are many more math concepts that depend first on naming what is being counted and understanding the significance of the name to place value.

The equal sign—An equal sign means everything around the equal sign is equal to everything else. Therefore an expression like 2 + 3 = 5 x 4 = 20 is not allowed because 2 + 3 does not equal 5 x 4. However the separator bar within the vertical format is allowed, because the separator bar does not mean equal; it is a separator bar.

Long Division---Although the idea that division is nothing but repeated subtraction is a bit oversimplified, the long division algorithm exactly depends on repeated subtraction because when you multiply within the algorithm, you are multiplying negative numbers. That is why you subtract the result of the multiplication. Look for a text that presents long division as more than memorizing the steps of the algorithm.

Multiplication and Division of Fractions---½ x 2/3 means one-half of two thirds. This example highlights the value of word problems. Word problems put math where it belongs and from where it arises, that is, math is the solving of real life problems. All math problems have a story. A page of naked problems has simply lost the stories. Suppose I have a ribbon 60 cm long. 2/3 of the ribbon is 40 cm, and half of that is 20 cm. 20 cm is 1/3 of 60 cm. Through examples like this, students can see that ½ x 2/3 = 2/6 = 1/3.

Division works the same way. Say I need to measure ¾ cup sugar and all I have is a 1/8-cup measuring cup. How many times do I need to fill my measuring cup to get ¾ cup sugar? ¾ cup divided by 1/8 cup therefore equals 6 times. (Notice again usefulness of knowing what you are counting. In this example, the answer is counting “times,” not “cups”). Texts should require kids to solve math problems by drawing pictures. When the student can reliably use a diagram to solve a problem, they are ready for the algorithm. Only at the end of the learning process should we teach the shortcuts. Math first, then shortcuts. Pictures are also the first step to proofs.

Absolute Value---Make sure absolute value is presented as distance from zero, NOT as simply a negative number turning into a positive number. A football analogy may help. If the quarterback is sacked, the ball may be 5 yards from the scrimmage line, but from the quarterback's point of view, it is still a negative 5.

Canceling---I loathe this word. Students are not “canceling.” They are simplifying a fraction. Simplifying a fraction means finding “1.” It does NOT mean crossing off numbers. Canceling leads students to lose track of the difference between “0” and “1.”

Multiplying and Dividing Decimals---Multiplying and dividing decimals has nothing to do with moving decimal points. It has everything to do with multiplying or dividing by powers of ten. 12 x 1.4 means 12 times 14 tenths. 14 tenths means 14 divided by ten, so 12 x 1.4 means [(12 times 14) divided by 10], which means 168 divided by 10, which equals 16.8. Students can tell where the decimal point goes, not by counting decimals places but by realizing the answer must be a number close to the product of the whole numbers. 12 x 1 = 12, so the answer must be close to 12. 1.68 is too small. 168 is too big. Therefore the answer is 16.8.

It is easy to confuse students by changing the problem slightly to 12 x 1.40. They will likely say they need to count 2 decimal places so the answer is 1.68. Giving them a new rule about ignoring zeroes does NOT build math understanding. Shortcuts are just that: shortcuts---and should be taught only when the student knows the actual road, not to replace the actual road.

Ignore the glitzy graphics and choose textbooks that handle all these topics well.

Exactly Those Contrary Ideas

Twenty years ago the late comedian Bill Hicks felt obliged to defend the blasphemous content of his stand-up routine. In so doing he said something remarkably insightful.

‘Freedom of speech’ means you support the right of people to say exactly those ideas which you do not agree with.

Hicks is right. However, it is too bad he did not actually mean what he said.

The Founding Fathers promoted Freedom of Speech because they did not want to lose their heads merely for disagreeing with the king. They wanted to be able to say exactly those ideas the king would not like. Therefore, the established the right of people to say exactly those ideas other people, especially people in power, do not agree with.

So far, so good.

The problem is that today, many people toss off the phrase “Freedom of Speech” as if it is a constitutional defense of any expression. Freedom of Speech protects ideas, especially ideas that might threaten the interests of the powerful in exploiting the weak. It was never intended to let people say (or draw, or film) anything they want.

If you cannot express your idea in a non-”blasphemous” way, perhaps your idea is not worth expressing at all.

Over time, people have gradually lost the ability and the social censure to restrain themselves. There are kids in school who seem unable to speak an obscenity-free sentence. Voltaire said, “The man of taste will read only what is good; but the statesman will permit both bad and good.” Our society seems less and less capable of producing children (and finally adults) of taste. As Paul of Tarsus wisely advised, “...fill your minds with those things that are good and that deserve praise: things that are true, noble, right, pure, lovely, and honorable.” A mind full of treasure has no room for trash.

Increased Reporting DOES NOT Increase Achievement

So here I am, in the land of tiger moms, where supposedly parents are highly involved in their child's education, even riding that pendulum to the other extreme. I don't see it. What I see is a lot of complaining, but little to no positive action at home.

In my school, some parents complained that if they do not know about the homework, they cannot insure its completion. So the principal decided to send a picture or description of the homework assignments to the parents' cellphones. It made no difference. Kids who regularly completed homework before the messages continued to do do. Kids who did not do their homework still do not do their homework. In fact, the parents of five of my twenty-one students admitted to doing the homework for their children. People always like to recommend more communication. Sounds good theoretically, but "communication" is not the cure-all everyone supposes. However, as the principal said, one benefit is the parents stopped calling.

Of course, I expected the students to make a note of the assignments everyday. It is not the teacher's job to tell the parents what the homework is. Parents should check their child's assignment book. If their child is not writing down the homework as instructed, parents should deal with the noncompliance at home. Schools need to stop giving already busy teachers more useless duties. Parents need to emphasize that knowing and doing the homework is the student's responsibility.

A trend over the past thirty years has been to hold the children less and less responsible for their schoolwork and put that burden on the teacher. Years ago, as long as the child was behaving not too badly, parents heard virtually nothing from the school except for the four quarterly report cards. Parents of high achieving children were fine. Parents of low achievers began complaining. They said they could do nothing at home to mitigate a failing grade if they do not know before report cards come out that their child is failing, In response to these complaints, schools started issuing mid-quarter “progress report”. It made no difference to final report cards. High achievers continued to achieve highly; low achievers continued to fail. The only discernible outcome was that teachers had double the reporting work.

Eventually, even mid-quarter reports were deemed too few and some schools began mandating weekly progress reports. It still made no difference. Schools began requiring students to purchase expensive “planners” on the dubious assumption that students were not writing down their homework assignments because they had no little notebook to record the assignments. This assumption is beyond silly, and of course, made no difference. Responsible students have always written down their assignments, long before planner became the soup du jour. Then schools began requiring teachers to post the homework online. The only apparent effect is to create more busy work for the teacher, and stop parental complaints.

Some teachers take matters into their own hands and require students to do the homework during lunch. This tactic is at least partially effective because it generally ensures the homework gets done. I do not like using the lunch hour because children need to run around and play before settling down to an afternoon of work. We invite behavior problems when we deny them this energy outlet. Furthermore, research shows that exercise increases thinking ability and concentration. I prefer to keep kids after school. I have found it to be more effective at promoting self-responsibility.

There is one major caveat: the assigned homework needs to be worth doing.