Fisking a Developmental Math Petition

A favored author of my, Larry Correia, will engage in taking an article or blog post and systematically discussing its failings. Using his format as an example, I am going to address a recent article in the Ogden Standard Examiner about a student who started a petition against Weber State's developmental math program. The original article can be found here. The original article below is in regular text while my comments are in bold/italics.

Student petition blasts WSU math program

OGDEN -- It’s not unusual to see students break down into tears because of Weber State University’s Developmental Mathematics Program, according to Lauralee Stephens Kohl.


It's not uncommon to see student break down because of math anywhere. Math is widely considered a difficult and involved topic. If crying students was the only reason to change a program then the Interior Design program at USU should be change. Or nursing programs all over the country. Or Culinary Arts, medical school, law school, anatomy courses, chemistry courses, computer science courses. The list is nearly endless.


The program was already difficult for students who are struggling with math, and changes to the program only add obstacles to graduation, she said. That’s why she’s circulating a petition to stop the changes.

“It’s like an abyss,” Kohl, of Brigham City, said of the program. “Once you get in it, you feel like you just can’t get out.”

Kathryn Van Wagoner, director of the Developmental Math Program, says many of the concerns listed in the petition are based on misinformation or misunderstandings.

“I have received some emails from students about some information they were given, and it’s actually not correct information,” she said. “We are transitioning to this new plan, so we've been refining and verifying our internal communications prior to making a formal announcement of the new plan, and student concerns have brought to my attention some information that has to be corrected on the website.”


So you are saying that the math program shouldn't change? They shouldn't try to incorporate new technologies, teaching methods, or more accurate assessment tools? If people didn't like change in mathematics we would still be using lines and circles. We wouldn't be past Roman numerals. That would make everything better. Consider this. A recent article in the Chronicle of Higher Education shows that the STEM fields (Science, Technology, Engineering, and Math) primarily rely on lecture based teachingf much to the determent of their students. Lecturing has been shown to have limitations particularly "that lecturing is not suited for higher levels of learning: comprehension, application, analysis, synthesis, and evaluation." Those are the very levels that math is designed to stimulate. Why would you want to stop a math program from making changes to their program in an attempt to improve instruction and student experience?


Jeff Henry, president of WSU’s Student Senate, sat down with Van Wagoner to discuss the rumors about the program changes.

“I wanted to get an understanding of what exactly is happening,” he said. “From what she’s explained ... the new changes are going to be beneficial to students.”

So, what Kathy said was right, many of the concerns are due to misinformation and misunderstanding. Once Jeff Henry got his understanding he saw the changes as beneficial. So maybe it's not the program.


Developmental math is, essentially, pre-college math for students whose skills are not quite up to college standards. According to WSU’s website, 75 percent of public four-year universities across the nation offer developmental classes.

The first complaint Kohl lists in her petition is the practice of placing holds on students who have not completed developmental math classes within a certain period of time. Until those classes are completed, no other classes can be taken.

Van Wagoner says that practice has been gone since January. The idea was to motivate students to complete the classes, instead of causing bigger problems for themselves through procrastination. It didn’t work.


"It didn't work" So they changed it. There were many people who celebrated that change. I know of sparkling cider and treats being shared in some offices. Students were not the only ones that didn't like that. Again, they wanted to help students and when they saw it didn't work they got rid of it. Also, if your first concern on your petition was resolved six months ago I would say that you're off to a bad start. Have you done your homework about the issue before asking for signatures?


At the heart of Kohl’s petition is the use of Accuplacer exams to determine in which class to place students, based on their math skills.

“When I started ... you could take unlimited AccuPlacer placement tests,” Kohl said.

It cost students $10 per test, but many were willing to take them several times in an attempt to score high enough to get out of taking certain classes. Now the number of times a test can be taken has been limited, she said.

“We made that change quite some time ago,” said Van Wagoner, “because students were repeatedly testing and not making any improvement in their placement, and basically throwing their money away.”

Now students have two attempts at the placement test, which is enough to allow a retake, said Van Wagoner. More should not be necessary, because the point of a placement test is to find out what math do you know, so you are put into the right class.

Henry agrees that taking the test multiple times isn’t helpful.

“If I’m taking it 15 times, I’m really guessing — I don’t know what I’m doing,” he said, adding that the current policy of allowing the test just twice a year is fair and reasonable.

Can you argue with their reasoning? "were repeatedly testing and not making any improvement...basically throwing their money away." Kathy basically is saying "We don't want your money if it's not helping you." But this change limited the effectiveness of Acer Placer - and that's what a lot of student object to.

Acer Placer is a private company that "offer[s] personalized instruction in a class of no more than 8 students. Our instructors guide you through the materials covered on the Accuplacer math placement test." They claim, "students from any background in math can test our of Math 1050." I cannot condemn them nor can I recommend them. I don't really want to do either. This is because I've seen them work for some students and not work for other students. Students take an eight week course which involves taking the Accuplacer test multiple times, often more than once a week. When WSU limited the number of times a student could take the Accuplacer students enrolled in Acer Placer courses claimed it was because we were trying to shut down Acer Placer - that WSU felt threatened by Acer Placer.
It wasn't, WSU doesn't. It's because of the reasons that Kathy gave above. It is also worth considering that students that take the same test multiple times will learn how the test reacts. So, are those students learning how to succeed in a math class, are they demonstrating that they know the basis of algebra well enough to succeed in a class or are they just learning how to take a test? I will never understand a statement I've heard so often. "I'm not good at math, so I'm going to test out." If you aren't good at something what makes you think you can test out? If students want to spend money on a private company which wants $1000 dollars that is their right. If Acer Placer is teaching students mathematical concepts and principles then it doesn't matter what test they take - math is math whether its Accuplacer, ACT, Math Mastery, or SAT. Math is the same in English, French, Russian, and Chinese. Do what you want to study for the test. Just don't expect us to endorse wasting your time taking the same test twelve times with minimal results.


Starting Aug. 25., the Accuplacer test won’t even be used for math placement. It’s being replaced by a weighted rubric of high school GPA and ACT math scores.

“We’ve eliminated the need for students to take a placement test,” Van Wagoner said. “We did an internal review of past student success, and found ACT scores and GPA were a good indicator of student success.”

Look! You don't even have to take the test. You don't have to spend your money if you don't want to.


Students who don’t have an ACT score, or who want to improve their placement, will now be given the “Math Mastery” exam.

If you want to test out of classes go ahead! We're not trying to stop you.


Kohl objects to the “Math Mastery” exam, saying it’s based on the school’s 0950 and 0990 math courses.

“The math program has a 60 percent fail rate at Weber State, so you’re not teaching the essential skills needed,” she said, adding that the curriculum should be fixed before it’s the basis of testing.


Of course! The curriculum is at fault! The topics that they cover in Math 0950 and 0990, which are almost identical to other developmental math courses offered around the state. The subject of mathematics, one of the most standardized topics across the state is at fault for the 60% fail rate. Only 40% of people in college can do fractions, real numbers, decimals, exponents, and ratios. The curriculum of the developmental math classes was taught 500 years ago and last week in millions of elementary schools across the country and the world. Of course it's the curriculum that's at fault. Blame the professors. Blame the technology, Blame astrology. Blame any variable you care to, but to say that the basic curriculum of the developmental math classes of Weber State is not suitable material to gauge a student's ability to succeed in math is laughable. If you can't do fractions, it's not the fractions' fault. It might be yours, it might be your sub-par teacher's fault, it might the fault of dyscalculia , but it's not the fractions.


She says she spent three times the required hours working on math assignments, and never improved until she took math classes from a private tutoring business.


Students in developmental math courses that use the Hub meet one day a week (50 minutes) and are required to meet 100 minutes to simulate the time they would normally spend in class if they were in a regular lecture course. Research has shown that for every hour in class, students can expect to spend two hours outside of class on homework and studying. This is widely acknowledged. Let's say she spend 300 minutes a week in the Hub, that is three times the required amount for the class. Based on our ratio of hours in class per hours out of class she is short another 100 minutes a week. A common misconception is that those 100 minutes is the minimum to succeed in class. It is not. It is the same simply attending class. Homework is extra. She made a great start and just needs to keep going.

Van Wagoner says changes have been made. In addition to online classes and a TERM (Technology Enhanced Redesign of Mathematics) self-paced course, which students do on their own with help available from tutors, the program is offering developmental math classes in flipped form. Flipped classes ask students to study lessons at home, and then meet in a classroom, with a teacher, four days a week while working through assignments.

The program is also starting a new “Pathway to Contemporary Mathematics” (Math 0810) class, for students who aren’t going into science, technology, engineering or math.

“It’s not necessarily less rigorous, it’s just more relevant to the student,” said Van Wagoner, explaining that it has less emphasis on algebra.


Again, the developmental math department is expanding their offerings in an attempt to meet the needs of more students. And they are petitioning to stop that? You want math instruction to stay the same and not look for new and improved methods of teaching and reaching out to students?


Kohl’s petition also says the “Math Mastery” exam can only be taken once a year, and will be more difficult to pass because it’s fill in the blanks instead of multiple choice.

“It’s tricky. If you put in 0.5 instead of 1/2, it’s wrong,” she said.

That is true, said Van Wagoner, but each question will tell you the format in which the answer should be written.


If you can't follow instructions.....


Joanna Bushell, of Ogden, signed Kohl’s petition. She is one of several students who has been taking private math classes to learn skills, and then taking AccuPlacer tests multiple times at an applied technology college.

“It’s not a trick — you have to know math to pass,” she said. “There’s no way to just fumble your way through AccuPlacer and get an accidental passing grade.”

Taking the test multiple times is for practice.


Good! If they teach you want you need to know and you can still practice what's the problem?


“Once you feel confident you can pass it consistently, you go to Weber State and hope you can pass,” she said.


Hope you pass? If you know the stuff and you pass it consistently, why wouldn't you be able to do it at Weber State? Math is math (as shown above).


Carin Mann, of Layton, is using AccuPlacer to test out of math completely.

“I’m not even signed up for math at Weber State University,” she said. “I heard horrible things about it, and that nobody can pass it.”

And a person who has not even tried something is a good source for information about the situation. This is also a good place to mention that many student that don't take developmental math courses at Weber, take developmental math classes is at one of two Applied Technology Colleges near by. Weber does not discourage students from taking classes there. What is interesting is that they use a self paced modular layout for instruction. There is no teacher and no tutor support. We use a similar system: modular and with minimal class time, so students can work as fast or slow as they want within the usual semester time line. So students that attend the ATCs are taking classes using the same format that they would here but they don't have a designated professor or free tutoring services. I've learned that some students after taking classes at the ATCs returned to Weber because we can provide additional support unavailable at the ATC. We're okay with students taking classes somewhere else and transferring them in. I just find it funny that they avoid us because they hear that "Weber math is hard!" just to go somewhere that is just like our program with less help.


Bushell says eliminating AccuPlacer tests forces students into developmental math classes at WSU.

“They’re gradually narrowing it down so your only option is to take two years, or four semesters, of their math classes,” she said.


Because we only want to to take our courses. That's why we accept transfer work. That's why we allow students to complete their math requirement how ever they want. That's why we still allow students to test out of math. That's right, not just test out of developmental math, but students can still completely test out of math. The only thing that has changed is that they need to demonstrate their competence in fewer attempts.


She’s afraid the real reason for doing away with AccuPlacer is monetary, because students would have to pay for WSU classes instead of private tutoring.

“I would imagine it’s a great money maker for the university,” she said.


Because the $1000 dollars they pay for to Acer Placer is such a loss to us. The average class size is 3 credits. Based on current tuition rates if they only took that one class the cost would be $920. Yes, we might feel that, but Weber's job is not to make money. It's what we need to do to help enhance our programs and offerings, but nobody who works with students is being told by administrators, "If students don't register for developmental Math classes then you're out of a job." But very few students take just one class because it's not as economical as going full time (12 credits), or even just going part time. If a student who traditionally attends 12 semesters says, "I'm only going to do nine semesters and take my math somewhere else" the difference between the tuition and fee's cost is just under $500. If a student did that for three semesters we would lose $1500, less than what one additional student would bring in by taking six credits. While the student in question would save $500. Unless you are going to try to test out of at least two classes there is no real cost benefit to Acer Placer. Not to mention, if the student continues to take 12 credits we essentially don't lose anything. I can understand their reasoning to save money. The argument that we will suffer monetarily simply because of a small percentage of our students not taking upwards of 10 credits over the course of their 60 to 120 credit university career is silly. Not to mention tuition and fees are used to pay for services and resources to support students - most of which are not used by the students. Yes, we are money grubbers and developmental math is our method of staying in the black.


The new policies apply to all students, no matter when they started at WSU, according to Kohl.

Because the “Math Mastery” test takes over on Aug. 25, students who were counting on testing out through AccuPlacer are in a pinch.

“They should have to give more notice,” said Bushell. “I just need to test out of math to get my degree. ... I have approximately 30 days to do it.”


Interesting if you can test out of three classes over the course of 8 weeks, why would you have problems simply completing once class over 16 weeks? How much notice would you require? 8 weeks? The change was made at the beginning of Summer semester which was the first week of May. It's been at least 11 weeks since then not to mention another four weeks until the deadline goes into effect. If you had wanted to do the Acer Placer course, you could have done it twice in the time give. Thirty days is four weeks, what were you doing the other four weeks?


Van Wagoner says every change that’s been made has been with the goal of having students take fewer math classes, and get to graduation.

“We want them to be successful in their math,” she said.

She knows the changes make for a complex situation, and encourages students to meet with advisers on a regular basis.


Ever since we heard about this change I have asked every student how they intend to complete their Math requirement. Those that said that they were doing Acer Placer, or even that they "intended to test out" that there were changes and that they needed to talk to the Developmental Math Department before paying for anything. Most of the things that students experience difficulty with could have been either prevented, fixed, or avoided by talking to an advisor. Yes, we try to contact students about policy changes, but we are limited by the information students give us. I've been involved in initiatives to contact students and many do not have current or correct contact information. Others still do not return correspondence which may indicate that they don't receive it, read it, or care about it. If students don't try to stay in contact with us there is no way to verify that we can reach them.


“If there is any student who has any kinds of concerns about math, I would be glad to meet with them,” she said.

Contact reporter Becky Wright at 801-625-4274 or bwright@standard.net. Follow her on Twitter at @ReporterBWright.


Overall, I thought the reporter did a decent job showing both sides. I am continually amazed at the lengths people will go to find a way around math. It seems that if some people put the effort they dedicate to complaining, circumnavigating, and trying to work the system to just doing what is asked of them in the system they would be done. Yes, math is hard. Yes, it is abstract, Yes, you will rarely, if ever, see it the same way that it is shown in the text book. Yes, it takes time you would rather use to do other things. So does a lot of other things and yet we do them. Taxes, insurance, immigration, legal proceedings, raising children, dating, recovering from illness. Yet people do those things too. Yes, those examples are all individual and different. People still do it. Math is required for a college degree and so you do it if you want the degree. In your work life there may be times you have to do something undesirable for your boss. Think of this the same way. If you have difficulty ask for help. If that person can help you, ask someone else - just like anything else that's hard. There are legitimate disabilities that prevent people from succeeding in math classes - get help for them, we will accommodate you as needed. I am more than willing to acknowledge the difficulties these students have experienced in math. I am not willing to consider this petition as being the slightest bit useful.

Update: As it turns out this article was useful to the Developmental math Department at Weber State. Kathy VanWagoner said that she wanted to do a press release when they made changes but the PR department wouldn't do it. Then this article came out and she was required to do a press release.

Mathematical Beauty and Creativity

http://vimeo.com/77330591 - Beauty of Mathematics

http://www.firstpost.com/topic/organization/pixar-pixar-and-math-disney-pixars-brave-wonder-moss-video-EnaA9ZRPXiE-35299-1.html - Wonder moss

http://fineartamerica.com/featured/4-flow-of-life-flow-of-pi-cristian-ilies-vasile.htmll - Flow of life / flow of pi

I got thinking about this topic when a friend on Facebook liked this quote. I fully agree with this quote in all aspects and I want to draw attention to point II. Music is mathematical - pitch, frequency, rhythm and meter - really all of music can be described by mathematics.  Destin from Smarter Every Day did a great video of a beatboxer (Flula) in slow motion In it he shows how overtones produce a unique sound signature. It also goes over the different mathematical components that make up sound. I'm not saying that music can be replaced by mathematics, only described by it. We can do the same with art. In fact, in the special features of the Disney-Pixar move Brave they describe how math is used to create the amazing world shown in the film. Really, mathematics is a language that can be used to describe just about everything within our world. Not just the sciences but also the arts. I am not saying that it can be used to predict things. Nassim Nicholas Taleb would tell us that anytime we have a human element involved, predictions go out the window. But mathematics can be used to describe it in its present state.

Many think of mathematics as cold and unfeeling. Statistics and probability are guidelines not rules. The mathematical analysis of a painting will never look the same as the painting itself. Algorithms can't put their heart and soul into producing a piece of music like a composer might. To call someone "calculating" is usually considered, if not an outright insult, a derogatory remark used for villains and along the lines of "heartless." I think that this view of mathematics is one of the big reasons that many see the arts and mathematics as mutually exclusive. It is not uncommon for me to hear, "I'm one of those English people / art people / music people and so can't do math." I think my previous examples demonstrate that mathematics is not separate from the arts, but rather is something that can help us understand it even more.

So rather than continue my reasons why people don't like math, I want to demonstrate some of the beauty of mathematics. It is my hope that the following examples may be appreciated regardless of mathematical ability. Just like art or music you don't have to know how to create it in order to enjoy it.

First, fractals.

[picture citation]
Simply Googling "fractals" and looking at the images will give you lots of cool pictures, both from nature and artificial. Fractals are produced you have a design replicate itself so that ever successive generation or iteration looks like the initial one. The picture above shows how every new iteration branches the same way that the first do. If you were to zoom in (or out on a fractal you would find yourself looking at the same image, they are infinite in their presentation. One of the most famous fractals is the Dragon Curve. Numberphile did a great video on the Dragon curve and how it works.

The second example I want to look at is an artist by the name of Cristian Ilies Vasile. Numberphile introduced me to his art in one of their videos - Pi is Beautiful. Looking like a web of colored lines Cristian plotted the digits 0-9 in a circle and then starting with 3 he drew a line to 1 then to 4, back to 1 and continued along the digits of pi. He's done the same with e and the golden ratio. Like the Dr. James Grime points out, this is math for art's sake. There is no real mathematical use for these pictures, but they are quite striking. Check out the link on Cristian's name above. I would love to pick up one of his pictures myself and am hoping to in the near future.

The Unknown Expectation of College

This is a topic that I used to teach in a full day lecture and have presented a profession conference. I'm a big fan of cognitive theory and the things we can gain from it. While this isn't one of the "student development" theories it is a very real one when it comes to college. It also explains why algebra is required for college degrees. You might have seen it before in regards to note taking, asking questions, but I think that a basic understanding of this concept teaches students what their professors expect from them.

This pyramid was developed by Benjamin Bloom back in the 1950s and is referred to as Bloom's Taxonomy of Learning. The basic idea of the taxonomy is that not all learning is equal. As we learn new things we progress through the different levels of leaning. We cannot move into higher levels of learning without first achieving the levels below them. The levels are broken down as follows:

Knowledge (basic learning) - being able to repeat information as it was given to you. Defining terms, listing components, repeating concepts you have heard.

Understanding (simple comprehension) - the ability to explain in your own words. Taking the information out of context and replaying it to someone else.

Application (initiation of action) - using the information to accomplish a task or other such action. seeing how the information works in the context of the material.

Analysis (Deep understanding) - breaking the information down into its respective components and seeing how the parts interact. Also, seeing how the information fits into the greater context and outside of the initial context. Identifying relationships between the information and itself and other topics and facts.

Synthesis - (deep application, creation) - using the information to create new ideas, conclusions and applications. Using the component pieces to produce new ends and means.

Evaluation (defending your conclusions and actions) - having the ability to defend your conclusions and actions to others using clear and developed arguments and evidences.

The best way to break down the levels is with the use of action verbs - verbs that require action or demonstrate clear objectives. These verbs are often found in the assignments, tests, and research proposals that professors give. Being able to associate the verb with the level helps students understand what kind of depth or quality the professor is looking for. There are many lists of action verbs, but I like this one from Clemson University. It's got a nice layout and a good list of verbs.

Now, the expectation that I explain to college students. In high school the basic standard of teaching and learning is that students are expected to demonstrate that they know and understand the material as it was presented to them. Teachers and students do not progress much beyond the first to layers of the pyramid. However, when student arrive at college, the professors will often times help them achieve those same levels of a topic, but then will expect work on the higher levels. They try to give the tools to help students move from one to the next, but they are not content if students just linger in the levels of knowledge and understanding. This is often why new freshman will complain that their tests are not fair. "The test was about stuff that we didn't cover in class." or "I don't remember seeing this in class." These responses are often the case because the student leaned the material on a minimal level but the professor is testing their higher level learning.

What does this have to do with algebra? If the goal of college is to get people to think on a higher level then algebra is the natural gateway. Example:

Knowledge level - Define addition: calculating the total of two or more amounts (citation)

Understanding level - Explain addition: what you get when you combine two numbers

Application level - 2+2=? answer: 4

Analysis level - 2+?=4 answer: 2

Basic algebra, even in such a simple calculation, elevates a persons thinking - requiring them to analyze, break down, and find the relationships in the equation. When my old math teach said, "This is to make you think" the desire was not just a cognitive process. Bloom shows us that not all thinking is the same. This is to get you to think deeper then you might normally. This is to stretch your brain so that it doesn't return to its original, limited state, but to enlarge your capabilities. Like I said last week "This is to make you think" is the most accurate response to the question "why do I need to study algebra." i also think that it is the most valid and the best. This is why colleges set a standard on the minimum level of mathematics students must complete. So they can be sure that students are going to have an opportunity to think on those levels that can actually empower students to be proactive and not just reactive.

When will I ever use this?

I've commented on Mathematics in the past but some recent discussion are moving me towards it again. Last time I discussed reasons people may despise math. This time, I would like to look at why math is considered a critical skill college - or at least critical enough to be required for general education. The most well known course for the Quantitative Literacy requirement is MATH 1050 - College Algebra, but it may be one of up to five course options. The office I work in has been tasked with enforcing the math completion policy for our university, so I hear plenty of students complete about, berate, and in so many other way despise the policy, math, math teaching, and everything else related. As the messengers of this policy our staff has be metaphorically shot at on a regular basis. This was the catalyst of the previous post, but I would like to provide an answer to those students who may ask, "When will I ever use this?" It is a fair question and deserves an answer.

My high school math teacher, when confronted with this question used to say, "You won't [use this]. This is to make you think." I feel that this is the most accurate answer to the proffered question. There is a lot in mathematics that is either specialized for specific calculations, or they are more abstract concepts that are the foundation for those specific calculations. With that said, that instructor taught my trigonometry class and as well as other classes required for calculus. Algebra, the basis for all other advanced mathematics, is used more often. I would like to split the usage into two areas: unconscious calculations and intentional calculations.

Unconscious calculations are the ones that you do without even thinking about it. Every time you get in a car and drive you brain is processing velocity, acceleration, position, distance, and time equations faster than your reflexes can even respond. The same is true of any physical activity or sport - to catch a ball is an exercise in those same factors so you know where to move your hand. The world you live in can be described in numbers, vectors, shapes, solids, and velocities. Without your brain's ability to do math people would never had killed enough food to evolve past subsistence level of living. All of this is algebra. This is not to say that if you have difficulty in solving algebra problems in your math book that you are a bad driver. The math that our brain does without thinking about it is intuitive. But if we consider those calculations that we do intuitively - then the answer to our questions is, "Every moment of every day."

Intentional calculations are the ones we actively choose to do or the ones that our math classes require us to do. Even thought these are a active choice there are certain calculations we do all the time without thinking about it. Anytime you look at a clock to see how much time you have, you are doing algebra.

current time + (how much time) = target time. 

Any time you use money, particularly cash, you use algebra. This is very true when figuring out a tip, a discount, or other promotions.

available funds - purchase price = new available funds

original price * discount percentage = new price 

Factor that into the original equation:
available funds - (original price * discount percentage) = new available funds

These two examples may be obvious to most people. Those same people may come back and say that use calculations do not require exponents or fractions. That is true, for the most part - some purchases may be very complex. Even if such calculations are not very complex by studying algebra you are practicing skills that you use every day. The more practice you have the better you get at them, the better you get the more consistently accurate you are and the fast you are. Until such equations, no matter how complex may be.

There are other areas where intentional calculations come into play. Many hobbies that people do use math, including: wood workings (or other building hobbies - measuring, scale, etc.), cooking (one of the few places that people intentionally calculate fractions), gambling (probability), and many other games of all kinds (angles in strategy games, keeping and maximizing score, fields of view, the list goes on.) Again, these may not require exponents or logs, etc, but those develop complexity and give practice. You cannot be good at something unless you are stretched.

The last place that I want to mention that people use algebra every day is in problem solving. I don't think it's a coincidence that the individual calculations that we are required to solve in math classes are referred to as math problems and that the process of resolving difficulties as problem solving. I don't know which came first, perhaps it's an example of  the chicken vs. the egg, but regardless math problems and problem solving go hand in hand. Just like we can describe the vast majority of our physical world in terms of mathematics many of our social, emotional, and mental situations can also be labeled in terms of mathematics.

If X happens, then Y will occur. But if I don't want Y to occur, then I must do Z.

Anytime you have a situation with any number of possible outcomes the ability to reason is key. That reasoning is often termed logic and is the basic language of mathematics. The ability that you gain in learning how to arrange, manipulate, and see the relationships in  numbers translates to other kinds of problems as well. So even though you may never use numbers in your future equations, algebra is again proven to be helpful in everyday interactions and situations.

Perhaps next week I'll go into the biggest reason that math is taught in college in what I call the unknown expectation of college. But until then, I think this is sufficient.

The modern Dr. Jekyll and Mr. Hyde

Working in education gives me a lot of opportunities to meet, talk with, and learn more about students. Over the last couple of years I have seen a real trend that is disturbing. One of my colleagues compared it to The Strange Case of Dr. Jekyll and Mr. Hyde. Like the odious potion in the book it can literally transform students from gentle, inspiring persons into hate filled, aggressive, belligerent alter-egos. It is considered the ultimate four letter word of academia and is the most infamous curriculum ever imposed upon students. It is math.

I have heard math referred to as "the bane of [a student's] existence," "impossible," "useless," "a waste of time," and "[a student's] worst subject." When talking about math or asking why they have to take it many students are rude, angry, upset, and demanding. I know of no other topic in academia that has such a negative stigma as math. I've seen cases when students would rather wait an hour to have an opportunity to complain to someone, try to work around the system, or rant to a person then perform a simple task (such as watching a 30 minute online presentation) in relation to their math requirement. I have seen people turn from Jekyll to Hyde because of math. I want to know why!

This is a conversation I've had with one of my coworkers several times. She helps keep me grounded because we have had different experiences, have opposing views, and have different perceptions on math and math education. With that said we do agree on a variety of things relating to math (how it might be improved, difficulties in the system, etc) but we do approach it from two very different starting positions. It was she that described her own experiences as Jekyll and Hyde and found herself puzzled by the fact. I won't try to recreate our conversations here, but I do want to express some of the musings that resulted from them. The overarching question that I want to know is: Why do people hate math so much?

Is it because math is difficult? Math is an abstract concept used to describe our real world. You don't see the number three in nature, the same way you don't see the letter A going for a walk. Just as letters are used to provide a visual component to our spoken language, math is used as a medium to express real world patterns and systems. There is plenty of debate on if numbers actually exist, or if math is humans attempting to explain the universe or is it a natural part of the universe we discover, but the bottom line is that it may not be a natural way to think for many people. It is not often intuitive to think in terms of x and y. And because it is difficult to see how the quadratic equation relates to anything you deal with in daily life, many people question it's practicality. Without a clear objective use of it people lose interest, don't remember it well, and generally struggle with learning it. Yet, many people do hard things all the time and continue doing them despite them being hard. Athletes train in rough conditions, computer techs debug code to get their programs working, musicians practice long hours, artist and writers struggle through blocks of creativity. People run marathons, hike mountains, fight wars, win noble prizes, discover stars, cure plagues, go to law school. Our society holds people who overcome difficulty in high honor. Couches tell players to push past pain. Therapists tell clients to work through their problems. And heroes of all kinds tell children to reach for the stars. Yet if the difficulty is math, people are prone to give up, give in, and accept the idea that "I can't do it." So, I think we can discount it being difficult as a reason why it is so hated. Most things are difficult and we get through just fine.

Maybe it is the"impractical" nature of mathematics? As I've described above, it is not a natural way for many people to think and because you will rarely ever see a situation that calls for a logarithmic function there is little use in remembering it, even for the test you have on Friday. However, a couple of years ago a law maker in Utah argued that a liberal arts degrees was a "degree to nowhere." And thinking back on your own education when have you ever used those facts you learned about the Civil War in high school or college history? When have you ever listed the romantic era composers or painters? Since when has world geography, to quote Fred son of Fanny sister to Ebenezer, "put a scrap of money in [your] pocket?" There are many other "impractical" topics out there. In fact, if you talk to enough people you will find that every topic is impractical in one way or another, yet people have studied them for years many times without complaining, and many times while complaining, but rarely with the vim and vigor of the utter detestation that people have for math. Now I've mentioned how it is the facts that are often impractical and I stand by it, but I firmly state that the skills students learn in the classes that I disparaged above are very important. Yet, math contains both skills and facts that are relevant to everyday life. From telling time to figuring out a tip you use small math facts and skills everyday. So, not only are people studying "impractical" topics all the time, but math is more practical than many others. As Adam Savage would say, I think this one is busted.

I've heard lots of students say that they've had negative experiences with math. Generally a good health (or rather unhealthy) dose of negativity will jade most people. This is one that I can't necessarily disagree with, but I can claim that it is not also the case, nor do I believe it can be the only cause. My own experience with math in junior high is not a positive one. I failed it in seventh grade and was required by my mother to lug a math text home and new an set of problems every day before I could do anything else. An entire summer of doing pre and beginning algebra before I could enjoy the nice weather outside or books and games inside. Also, for both my seventh and eight years my calculators broke early in the school year. Any advanced computations I had to work on on paper with pencil. I went through learning my basic algebra without technical assistance. Geometry  was the same until we got to sine/cosine/tangent. I didn't have a list of ratio tables, or I'm sure I would have had to do that by hand too. Despite this, I never hated math. I hated homework, but I had to do that for all of my classes. I didn't begin to really enjoy math and become an advocate for it until college, during which ironically I never had a math class, but my negative experiences in seventh and eight grades did not cause me to despise math. Now I fully acknowledge that my experience and me as a person is far different from everyone else. Again, I do feel that a sever or series of negative experiences this may cause people to hate math. But what is it can causes many students to have those negative experiences? As part of the Jekyll and Hyde transformation many students are not hesitant to complain about math instructors. Out of all of the instructors I'm told about, math teachers rank the highest in the number and extend of "bad teachers." Not that they are the only source of negative experiences, but they appear to be a big one. Let's look at them for a second.

I think it is very clear that math teachers have an effect on whether students enjoy math or hate it. Students that enjoy math often refer to a teacher that supported and helped them through difficult parts or "made math fun." Students that turn into Mr. Hyde at the mention of math often will complain about the quality of math teaching or refer to a particular instructor as being useless, unhelpful, bad, etc. I have to point out that many students are unwilling or do not want to claim responsibility for their grades which leads to statements like "the teacher failed me" or "he teacher was bad," but I don't want to dismiss teachers and instructors that are not good teachers. While there are students that do not understand that they are required to participate in the learning process there are also teachers that do not provide the support necessary to give students a fighting chance of learning the material. My wife, Angel, who coordinates math tutors at a branch campus of the university we work for felt that the lack of support from instructors is the biggest part. A supportive instructor can help students through difficult times, find practicality and creative means within math, and tends to provide more positive experiences than negative. Angel's own experience was negative and then positive. In sixth grade she was told that she would never succeed in math and to choose something that wouldn't require it. This simply hardened her resolve to prove that teacher wrong. In junior high she had an instructor that supported her in math and helped her learn it. Since then she has completed multivariable calculus, linear algebra, and differential equations which is the math base for a degree in mechanical engineering. She loves math, even though she's always had difficulty with it. Instructors have an undeniable effect on their students.

And this leads to what I ultimately believe is the reason that math is such a hated topic. We hate it because we have been taught to hate it. One of the criticisms I've heard of an education program I once worked for was that students who dislike math went into elementary education. Those students became teachers who then went on to teach math. Because they did not like it themselves they instilled that dislike into their students. Math was a chore for them, so it because a chore for the children. etc. Also, think about the perception that people who are good at math have. They are seen as nerd, geeks, or out of touch. There are very few positive perceptions of mathematicians. Anti-social, awkward, shy, naive, unpopular, etc. Who would want to be good at math when there are very few positive role models. Even the discussion of a student having excessive negative experiences in math is a classic example of psychological conditioning. To boot you have stereotypes associated with math - "women and minority populations are bad at math," "math is hard," "math is useless," etc. Ironically, the government is clamoring for more nerds. Industry is calling for more engineers, computer programmers, scientist,  mathematicians. From what I can see, society is both begging for and discouraging people who like math. Generally it's not the same people doing both, but both messages are getting across.

I do know of groups that are doing cool things with math and science. Numberphile, ViHart, Smarter Every Day, Veritasium, Sixty Symbols, Periodic Videos, Minute Physics, Purple Math, Khan Academy and many other discuss math and science (often with the math) and show some really cool things. I'm glad these folks are out that there are resources available to help students learn. I've watched some of their videos with my children (ages 5 and 3). I think the best way to end the hate (and thus increase the "nerds") is to teach that math is not bad. Yes, it can be hard but it can also be interesting, practical, and even (heaven forbid) fun. Math is a critical skill that develops our creative and critical thinking abilities and helps us make sense of abstract and concrete ideas. Stop the negative self talk. Stop the hate. Stop the blogger from harping on this again. 

Multiple relationships: looking at the numbers

I did this exercise a couple of months ago, but I thought I would discuss it here in light of another math moment. Above is a standard 10 X 10 multiplication table. Most of us were required to memorize this at one point or another in grade school. The line on the diagonal shows the squares of each number. You probably noticed in grade school that the table is identical when mirrored across that line, i.e. 3*4 is the same as 4*3.

But now look at the group of boxes. If we look at the number in the middle we can refer to the numbers around it by location: so the red number is North, Green - East, Purple - South, Blue - West. What I want to point out is the relationship between these for numbers. So when I say NW that means 24 and 21.

Now that I've set this up, take a look at the NW and SE numbers. The difference between them is the same, i.e. 3. Now look at NE and SW. Again, the difference is the same, i.e. 11. It doesn't matter what number you pick, the differences between NE:SW and NW:SE will be the same. Perhaps I missed the boat on this back in grade school, but I think that is kind of cool. Also consider this, if you shift the center number down and over (say to 40), the relationship stays the same for the NW:SE numbers, and if you moved it down and over the other way (to 24) the NE:SW relationship is the same.

Now what happens if you just shift the number down (to 32). The difference for NE:SW becomes increases to 12 and NW:SE increases to 4. But if you shift it right (to 35) the NW:SE difference drops to 2 while the NE:SW difference stays at 12. How does this work, you ask?

LIKE THIS!

This was the original excel sheet I set up when I looked into this. This is the way you read it.

1 - Locate a blue line (e.g. in the upper left corner), look at the numbers that are on either side of that line (2 and 6). The difference between those numbers is equal to the axis number the line passes through (3) plus one.

2 - For the red lines, it is the axis number minus one. The easiest example is the longest that runs through the squares. all of the numbers mirrored across it are the same.

So, go back to our original example - 28. The blue line (the NE:SW difference) crosses the axis at 10, giving us a difference between numbers of 11. the red (NW:SE) line crosses at 4 resulting in a difference of 3.

Any movement along any single line will result in the same relationship between the numbers. Now, what happens if you don't know where the line will cross the axis? For instances, we can see where the blue line would cross on the number 336 and we will pretend we don't see the red line either. The key is to find the factors with the least difference. In this case it's 16 and 21. To figure out what the blue (NE:SW) difference will be take the absolute value of the difference of the two factors (i.e. 16-21 = 5) for the red, you take the sum of the factors (i.w. 16+21=37).

Now this is probably way more than you wanted to know about the multiplication table (or what I do in my free time), but there is an even simpler way to do this. It's called finding the slope! Finding the slope of a line has been done for centuries. And I just discovered it!

Not really. the multiplication table is basically one quadrant of the coordinate plane. When viewed like this (with the factors on the top and left) we are looking at the lower right quadrant (+X, -Y). If you remember from those grafting equations classes the slope is rise (the difference in Y) over run (the difference of X). Technically all of these lines have a slope of 1 because they are straight and at a 45 degree angle, but this is how it works.

As we treat this as a coordinate plane all of the factors at the top (the X axis) will be positive while the numbers at the left (the Y axis) will be negative. So we take our original box:



The differences (or slopes) between our coordinates are:
S and E =  -4 (rise) plus +7 (run) = +3
S and W = -4 (rise) plus -7 (run) = -11
N and W = 4 (rise) plus -7 (run) = -3
N and E = 4 (rise) plus +7 (run) = +11

Since we only care about the distance (not the direction) we can eventually just use the absolute values which give us 3 and 11.

There you have it. I have proven that slope exists within the multiplication table. What does this mean for the world of mathematics? Probably very little. But It was a fun adventure to figure this out. As I work in education, I have been asked by many students, "When will I use math?" My 10th pre-calculus teacher, Mrs. Mead, use to answer many of those questions with, "You won't. This is to get you to think."

As this post has gone on long enough, I will simply close by agreeing with Mead. The math you see in a textbook is not the math you are doing in your head when you look at a clock, try to figure out a sale price, or even the advanced equations of speed and distance when you drive. However, the basic nature of algebra is such that it forces your brain to think on higher levels and puzzle out solutions. That is the nature of math. And that is why it is needed.

Muse on that.