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.