Week 5: What do YOU think should be the kind of math competence we require students to learn? And, is it important for students to see the relevance of math to jobs and everyday life?

I don't think I was entirely clear.  I meant to say, if the math class requirements were done away with and it was only required that you take four years of math, any class, would classes that are currently viewed as slacker classes but that are useful in life become more popular?  Do you all think that less people would feel pressured to take calculus and be more likely to take stats, or econ, or something of the sort?

Nora, Given Michael's point about people seeing discrete mathematics courses as a joke, I don't know if choosing any 4 math courses would have helped you avoid the calculus predicament you're in? Something more seems necessary... Which brings me to my next question (which I'd love for you to weigh in on).

Michael, do you think implementation would be as hard as the job of getting people to widen their definition of what can be considered achievement in math? I think you've gotten at a really big aspect of what it would take to gain acceptance. What are barriers to acceptance and what do you suggest for overcoming them?

In answer to Michael's question there are several branches of discrete mathematics which are both deep and useful.  Two that come to mind are game theory and graph theory.  These are subjects which can be studied at various levels and can certainly be introduced in high school.  They differ from calculus in that they don't require an enormous amount of algebraic machinery.  Both of these branches have applications to decision making, such as the kind of behavioral economics that Michael suggests.  They simply provide the ability to explore these areas in more depth.  

I also think that it is a good thing when different subjects overlap.  Certainly every college physics course teaches two weeks of calculus, so that students know what velocity and acceleration mean in order to talk about gravity. Any serious biology course today has to do some combinatorics if it is to explain anything about genetics.  In fact, I believe that the increased importance and popularity of the life and environmental sciences will eventually make my case. 

It seems to me that nearly everyone agrees that math is an important part of life but there is debate on what should be required of students in terms of math.  I am in calculus class not by my own choosing but by that of my parents.  I have been on the math fast track, which led up to calculus this year, for years.  I have had a hard time not only understanding the subject but also in wrapping my head around how I will ever use it. I do not plan on going into the math field, so calculus will most likely never benefit me, besides in applying to colleges.  While I don't think that this was the right course for me to take, I definitely believe that I should be taking some math course, just maybe one that will be useful for me later on.

Do you think that there should continue to be required math courses to take to graduate high school or would just taking four years of math suffice?  What if you just had to pick any four math classes to take during high school?  Do you think that that would be helpful in spiking interest in math?

The main issue logistically here seems to be implentation. Dr. Garfunkel, how would you suggest different branches of math be added into the high school curricula? Earlier on Marie mentioned the topic of real-world applications such as checking books, calculating interest, maintaing a budget, ect. As it happens, at my school anyways, these topics are covered--albeit in a very different setting. Where I'm attending high school all enrolled students must take a semester's credit of economics. The standard year, as recomended by our guidance advisors, to take the course is Junior year. One has two choices. One may take the 'advanced' option, which focuses on preperation for the AP Macro Econ test, or the 'regular' option, which ignores theory altogether in favor of personal economics like the formentioned skills brought up by Marie.

I have no doubt there are highly challenging branches of math outside of calculus, but as it stands the alternatives sometimes already exist (although I would be interested to hear what specificaly has not, but should, be introduced in highschool), but are considred "joke" classes. Because no student in the class is relying on a good education in the subject matter for AP, SAT, or ACT tests, the teacher (in my experience) is often under much less pressure to teach it rigorously, or worse, in some cases at pressure not to teach it rigorously.

Actually, Kim has raised a mathematical point as well, i.e. what do you mean by best.  The theory of optimization is almost never mentioned in high school math, but it is crucial in the world of work and in daily life. How do I make the most money? How do I incur the least costs? What's the fastest way to do something? where's the best place to put the fire station? These ideas from discrete mathematics are simply not addressed.

I want to be clear here. I think that learning mathematics and doing mathematics is a good thing.  It might even be reasonable for colleges to look at achievement in math as a predictor of future success. But what math? The mathematics we teach in schools today is mathematics (for the most part) that we've already taught machines to do.  Adults are teaching children to do perform computations that they themselves never perform - see long division. There is almost no recognition of technology whether it be computational or algebraic.  Machines can solve equations and perform routine algebraic manipulations.

Think about other subjects. when you take a 10th grade biology course or a college course called biology 101, you learn something about the different parts of the subject and what a biologist does. The same for chemistry and physics. In mathematics you get a run up to calculus.  while calculus is an important branch opf mathematics, it is just that - a branch.  If after high school or after a year or two of college mathematics you were to ask a student what mathematics is about, you would likely get an answer that was partial at best and more likely to be total nonsense.

Students need breadth as well as depth. 

Michael and Annika -- Welcome to Students Speak Out. I am glad you are part of the discussion and contributing so thoughtfully.

Michael -- I, too, hear three commentators in defense of continuous mathematics. But I also hear two questioning it. And I hear 1 saying that better connections between math and life/work could be made in any case. You make a good point, Michael, that perhaps we're all caught up in the game. It can be hard to see clearly while playing...

So, here's a question: To what extent does the current definition of math achievement influence the defense of continuous mathematics?

Marie wrote about the state of Colorado requiring some kinds of math, but not being open to other kinds. I also read Sol's 2nd paragraph below. Their points are that getting into college requires all to meet certain continuous math standards. But there is much evidence that when school is said and done there are many people who are in well-paying careers, and even doing well in colleges or other post-secondary work, who are using other kinds of math. Perhaps poorly because they weren't educated in this kind of math.

IMPORTANTLY: This is not to say we ought to abandon math. This is also not to suggest that we abandon continous math across the board (there are some who will use it and need it). This is to say that all math is not continuous math, and to question whether we can see that as truth.

Semeo made a point that students' views on "what is math learning" comes from what they've been told and surrounded with. I see Semeo's point, and extend it to adults. Among policymakers and education leaders there are a lot of people who think we must solve achievement questions by accepting as given what we currently do. Make tweaks within those givens... We don't have to address our problems that way. Part of why we're here and working on these challenging questions is to examine where we might, as a nation, need to stretch ourselves in the discussion about "what is student achievement".

Some related questions:

1. Do you think there are well-paying jobs that require math, but not necessarily continuous math? Brett has put down some points here that I sense contibute to how people think about this question, which leads me to my next question.

2. What is the balance between "access" and actual experiences of life? The nation's discussion about student achievement is focused a lot on "access". Sol touched on this in his question. Every student should have the opportunity to get into the "best" colleges. Every student should have the opportunity to get into the "best"; most well-paying careers. But, from here is where we start to make leaps. The leaps come at many levels. Can you identify any leaps? 

Consider some of these ideas:

--What is "best"?

--How much are plumbers (who use a ton of math) making now that supply is low given the continuous mathematics focus?

--If you were a teen parent (of your own child, of your siblings), would you want access to college admission? Would you want options in math-learning? Why would you want those? If you want both, in what balance? What if you were a teen from a long line of plumbers who stood to inherit a profitable business? Does our prevailing math learning structure assume "one path for all"? Or do we accept there might be other pathways?

--How much is the "access" discussion related to our "adolescence" discussion from a couple of weeks ago?

--Is access to college and post-college entrance really a reason for our nation to ask ALL students to score well in continuous mathematics? Or should math learning for jobs people will be in matter more?

--I suspect we all hear the stats in Brett's post more than we hear the stats in Sol's post. The question that emerges after I read Brett's post is who else uses math and how else. Arguably, many students had access to continuous math and the options Brett outlined. They didn't choose them. At the same time, they're now in college with many choices. Have we now, in the name of access, lost the opportunity to teach them other math skills they might use more in their jobs?

As I said, our job here is to consider new ideas and push out of the box in terms of what could be. That said, ALL answers are respected here. If our current way of doing things still reigns supreme in your mind, by all means say so.

I have no doubt that some very bright mathematics majors do well on entrance exams in those respective fields. It is not so surprising to me, that discipline definitely has the rigor to help prepare its majors for further schooling. Yet, to say continuous mathematics is making better lawyers/doctors seems to me to be stretching the truth. According to tables on national medical school acceptance, a whole host of unrelated fields had statistically high acceptance rates. Of Mathematics majors who applied to a medical school, only 40% were admitted, compared to 50% of Philosophy majors, 50.5% of English majors, 53% History majors, and 53.5 % anthropology majors. The point here is not to begin recommending prospective medical students to read more Dickens or brush up on Leibniz hoping to maximize their chances, but to recognize the only thing these statistics show decisively is that medical schools like diversity among their students—which should hopefully not be news to anyone. The truth of the matter could have easily been guessed, students of all majors can be accepted if they are bright enough, but ultimately, for every one accepted applicant majoring in a non-science field there are three accepted Biology majors.

http://www.knox.edu/statistics.xml

On a side note, I’d like to take a liberal quote from the paper linked by the guest speaker. I think it very well matches my own opinion and would be interested to hear others’ agreement or protest:

“I mentioned that the colleges must act first, if we are to have change at the school level. Colleges can give lip service day and night to any set of ideas, but admissions criteria and college placement tests rule! Students and parents are less interested in what colleges say than what they do. Any alternative path into mathematics must be valued and endorsed at the college level if we are to expect any change at the secondary level. And parents must be brought along so that the change is seen to make sense and to benefit their children.”

If a person does partake on the continuous mathematics pathway, what benefits are out there for someone with a major in mathematics? Well, it is statistically one of the best majors for getting high scores on either the LSAT or GMAT (graduate school entrance exams). Math majors taking entrance exams for both law school and medical school do better than their pre-law and pre-med student counterparts (just to provide more specifics on that fun fact).   When compared to English majors, math majors make 37.7% more on their starting salaries immediately after college. On that note, math related jobs fall in the top five spots in job satisfaction ratings.

Sources-

http://www.math.duke.edu/major/whyMajor.html

http://www.math.uga.edu/~curr/WhyMath.html

http://www.ams.org/profession/data/emp-survey

That being said, is our continuous mathematics pathway making better lawyers, doctors, business men/woman, etc.?

Are students who continue on this continuous pathway through college just statistical outliers?

Are these sources just bias?

Other thoughts?

In reading the thoughts of this week's guest and his respective student commentators, I see several points seem to be in consensus among near everyone here. The ‘real-world’ applicable skills taught by of high school level mathematics are dubious at best and more often, laughable, such as in the case of the cynical student’s mock thankfulness described in Annika’s post. Nevertheless, among the student commentators, one upheld the current requirement of Algebra II as minimum graduation requirement, one wanted to see the minimum be moved to pre-calc (a class, as a side note, shadier in intent than any other in the high school math curriculum; my school, in the desire to teach a host of techniques necessary in some Calculus courses—trigonometry, vectors, mathematical induction, etc.—gave up the title pre-calculus entirely in favor of the vaguer “Math-Analyses”), and one postulated a four-year requirement of math courses, regardless of the level that the students reach at graduation. This staunch defense of what I believe falls under Dr. Garfunkel’s definition of continuous mathematics, despite recognition that much of the math taught in a high school is irrelevant to the average person’s life, is a clear sign of what is wrong with the American education system.

Let me be clear: I am a hypocrite. I am in an advanced track in my school’s math program. I am not interested in pursuing a career that benefits from higher math, yet I plan on taking all four years of math myself—likely culminating with an AP test or two. I wish this was not the case. I do not blame those who recommend the previously mentioned requirements. They are on to something—that certain math courses, even when not actually used in the career of choice, are nearly essential to success. Like Dr. Garfunkel said, “[students] take the exam to impress the admissions officers at the colleges of their choice, not because they ever intend to do anything with the mathematics they mastered.” It is my belief that math, rather than be taught for regularly applicable skills or recommended as a necessity for those who wish to specialize in a study or career that requires it, has become a mere benchmark; just another way for colleges and universities to help determine which students to admit to their gates.

There needs to be change. The system, as it stands, is being clogged by the bureaucratic insistence by school administration that every student achieve a certain level of achievement in fields that, by practical applications anyway, are irrelevant to most. When I was taking an Honors Geometry course, my teacher was taken away mid-year and our class split up among other class periods. Why? All because so many students, poorly taught and inspired by earlier courses, failed Algebra I and had to be moved into a newly created class, requiring a two-hour class block, to prepare them for Algebra with another year of Pre-algebra. My teacher had the lovely assignment of teaching it. I am blessed to go to a good school with high academic standards, but all the same my learning is disrupted by these sorts of shenanigans. I can not help but fear for those less lucky than I am and how their schools handle similar problems.