A few years ago I was teaching the majesty of the cosine function to some students in a suburban Raleigh high school. The lesson was part of the widely dreaded trigonometry unit in a course titled “Advanced Functions and Modeling”—a fancy name for what was formerly referred to as Algebra III. Essentially, this course was for college-bound students who had completed Algebra II but did not wish to pursue Pre-Calculus yet. Most of the students in the class knew that they would not become future mathematicians or nuclear engineers, but they were, nonetheless, headed for some type of higher education, either at a four-year college or a community college.
Standing at the front of the classroom that day, armed with overhead projector sheets and dry erase markers, I mustered up all of the enthusiasm, passion, and interest one could possibly have for the cosine function and uncovered its mysteriousness to the set of decidedly unfanscinated students staring up at me. I demonstrated how the cosine function is constructed from the unit circle, explained why it oscillates back and forth between a maximum and minimum, and, yes, even showed how sound waves from guitars can be modeled with these springy curves!
Me: So as you can see, the periodicity of this function, if not manipulated, is 2π, just like the basic sine function.
As I scanned the classroom I noticed that a girl in the very back corner had her hand raised, waving it in the air energetically, desperately wishing to be called on. This is every teacher’s dream, for it usually provides proof that at least one person is interacting with the knowledge you’re presenting.
Me: Yes, [Student]?
Student: What is the point of this? I don’t get why we have to learn the cosine function.
Uh oh. Not the question I was hoping for. Why did this girl need to learn the cosine function? When would she encounter it or use it in the course of her life outside of this classroom?
Me: Well, what do you want to be when you’re all grown up?
Student: A beautician.
Again, not the easily pitched answer I wanted. I went on to describe how the business cycle oscillated between recession and expansion, much like a cosine function, and how she would have to follow this trend as a small business owner. The young woman was unimpressed.
Me (giving a vague, cop-out answer): On the whole, mathematics is a useful subject because it teaches you how to think logically, problem solve, and justify the things you do. It also stretches your mind as you deal with abstract complexities and forces you to be concerned with small, but important, details. Now, let’s take a look at example three…
I suspect that many teachers often encounter this same type of question from students, regardless of grade level or subject. And it’s a natural philosophical question for humans to raise: Why do we do the things we do? Why, with all of the infinite knowledge out there, are high school juniors learning about cosine functions? If students are required by the state to daily attend school for thirteen years, they deserve to know why they’re there and why the state decided that they should learn trigonometry.
This question of why students learn what they do is one that education standards-setters and curriculum experts are confronted with constantly. Although I don’t want to wade into the messy political questions surrounding common/national standards, or the ones set by states, I do want to suggest that the justification for each educational standard should be clearly articulated to educators so that they understand why they’re teaching certain pieces of knowledge and skills. When students can fully comprehend why they’re learning about a scientific principle or a literary device, they have much more incentive to want to master it and apply it.
So, when teachers, parents, and students see a cluster of standards like the one below from the Common Core State Standards for Mathematics, it’s important for them to grasp why this cluster has been chosen and how students benefit from learning it. Are they learning it merely because it’s tradition? Will 10% of the Algebra III class eventually use this in their professional lives? Are students, in reality, learning to generally solve problems and confront complexities?
Prove and apply trigonometric identities
8. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to
calculate trigonometric ratios.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and
tangent and use them to solve problems.
Articulating the justifications for individual education standards is key for successful teacher instruction and student learning. And standing in front of a room full of honest high school students may be the best way to practice defending them.
Click Image To Enlarge
5/11/2012 | U.S. News & World Report highlights ideas from "Moving On Up: How Tuition Tax Breaks Increasingly Favor the Upper-Middle Class." 
5/11/2012 | In Diverse, Steve Burd comments on the Obama administration's plans to address student debt. 
5/11/2012 | Kevin Carey talks to Salon about the government's role in rising student debt.
{ 20 comments }
Hey buddie, I’m just wandering through the web to find few data and encountered your page. I am engraved by the content that you own on this website. It manifests how well you follow this issue. Bookmarked this blog, will return for further. You, my friend, ROCK and ROll!!!
Watch 300 Online Free
Tri-gonometry is most obvious when dealing with large-scale triangle surveying in which you have a combination of any 3 measures of angle or side length. Sine, cosine, tangent and/or cotangent tables will give you the value for another particular angle or side. If you know two angles and the side measure you can determine lengths of the other sides. If you know two side lengths and the angle between them, you can determine the other side and the other angles. Trig is most obvious and useful in design, architecture, navigation, etc.
For all HS students, statistics is the most useful and interesting math application that will be used daily for a lifetime. Trig requires many examples in many fields in order to connect to concrete thinkers (me).
In my lifetime, I use statistics in both my work and newspaper browsing.
Trig has never been necessary for any of my work, including local surveying where simple level and angle determinations made trig both unnecessary, more complicated, and would produce more margin of error through calculation.
Trig is for engineers, surveyors, and math majors. For the rest of us,
please teach us the many uses of statistics.
Can all of you distinguish between a Petrachan and Phakespearian sonnet??
Name the general who won the most Revolutionary War battles?
But cosines?????
Hello, neighbor (a Cary resident here). I think if you wait for “them” (whoever they are) to take care of your needs, you may be forever disappointed, because “they” just don’t care enough and don’t have the vision. I am referring to this: “the justification for each educational standard should be clearly articulated to educators.” I suggest taking matters into your own hands – but especially inviting the kids to take the matters into their own hands.
I bet most beauticians never use trig, and in your situation, I would start the conversation from fully acknowledging this. Then I would invite the whole class to dream, on behalf of that girl. After all, what beauticians use now is very different from what they used 100 years ago, and the progress accelerates. It’s up to these kids to change their fields in some meaningful ways. It’s up to you to help them develop the power to do so.
NASA has a project for imagining a future colony on Mars. How about planning for a beautician THERE? What are POSSIBILITIES for making the field better, using trig and other advanced math? What are some COOL beauticians doing even now?
As an example, I present to you the blog of a modern matchmaker. Your students may like some of his graphs: http://blog.okcupid.com/
obviously you weren’t teaching at enloe
This is a question I always dread as a math teacher, “Why do we have to know this?”. In some cases I could easily respond, “You might not have to know this, but Billy might need it when he goes to college.” Now that would be considered extremely rude for me as the teacher to insult a student with the assumption that they themselves will never achieve a higher education than high school (some probably won’t even accomplish that task). I really agree with Paul that we can not limit students opportunities based on their lack of motivation as a hormonal teenager. I also think that there is a lot of good reason for Forrest to introduce this topic and request a why. If the people writing the curriculum can not justify why it should be taught, then why teach it? I feel that we are given an impossible number of items to teach to our students. It then becomes the task of districts or individual teachers to determine what is and is not important. I can honestly say that I have no clue what colleges or universities are looking for from high school graduates, so I would prefer if a panel of experts could determine a reasonable amount of material for me to deliver as well as provide good reasons for why each topic is important.
Matt & Paul,
The way I see it, states require all children living in their bounds to attend school for thirteen years. During that thirteen years, there is (quite obviously) a finite amount of time for learning and a finite number of resources, including teachers, textbooks, and technology. Yet there is an infinite amount of knowledge that we could choose to present to students.
An educational standard is a statement that a particular topic should be known by all students when they exit a particular level of schooling. It represents a conscious choice by a group of adults (or a society) of which knowledge should be mastered by all.
The purpose of this post is to encourage more people to ask why certain standards are selected and to increase awareness of the reality that there are limitless possibilities for what could be taught in school. Arguing that particular topics are taught because it’s tradition or it “just is” is poor reasoning that must be drastically improved. When you make the case for learning cosines, you must weigh it against the cases that could be made for learning other topics (e.g., how to invest money or debating what makes an ideal society).
Young people are very frequently the freest thinkers and the best philosophers because they constantly ask “why?” Their questions, which have not been as limited by the arbitrary constraints of the past, are important to consider in education policy.
Designing a curriculum that only includes skills that will be needed by the majority of students is a horrible diservice. Most students won’t need chemistry, physics, or higher math. However, is it really up to someone’s 10th grade guidence counseler to decide that they should have more limited options to persue careers that use those skills at a later point in life, or to restrict a child from trying the material and learning they do have a love for it?
My view is that a college bound curriculum should have a core set of skills, including trig or algebra III, that allows students to have a wide choice of majors in college. Electives can be used to explore that particular interests in more depth.
Non-college bound students are another topic, but once again, do we want to pick classes that restrict a student from some more selective colleges just because they don’t think thats what they want to do when they are 15?
As an experiment I asked my teenage daughter if she knew what a cosine was. She’s had one algebra class and no trig and had no idea. I said it described periodic things and she said “it sounds stupid.” Ok, that’s the baseline. She’s seen olde tyme phonograph records and likes music so I explained how vibrating things make sound and sound breaks down into a bunch of different vibrations. Each of those is a cosine. So the needle is following a bunch of cosines. That’s analog music recording. Today we use digital recording, and we actually take the vibrations, break them down into cosines and save them as data on a disk, then we play them back by decoding those cosine functions back into the vibrations. She thought that was cool. Maybe something like that strategy might help for some students.
Some people get through their whole lives never using anything from math other than basic arithmetic and a little geometry. Others can find ways to apply math to things the former group would never think of. Math is a set of tools that help you solve problems better that you could often solve without it. It isn’t essential for most people, it is just a great tool for those clever enough to master it and incorporate it into their thinking.
It’s certainly nice to be able to show why something is helpful and show a practical application of relevance to a student, but not everything is like that for every student and they have to also be able to tolerate a certain amount of material that they can’t directly apply. Mathematical functions let you put numbers on things so you can solve problems. Cosines describes things that go back and forth or up and down or around and around repeatedly. Examples abound in music, electrical power, electronics, electromagnetics, toys, games, puzzles, etc., etc.. If a student can’t even appreciate music, electricity, toys, games, or anything electronic, then they have to treat it as a puzzle to solve. If none of that grabs them, they aren’t going to have much fun with math, and that’s a shame. Their thinking tools will be more limited.
Why do we need a justification? What’s wrong with learning something because it’s good to learn? If a kid is in trig, then he is going to learn all the trig functions. I know that here in NC all kids are required to take 4 maths, and beginning next year, they must have Alg 2 to graduate. That in itself is the problem. If we are going to require 4 maths, then provide 4 meaningful maths with useful applications (tech math, business math, etc.) instead of forcing every kid into the 4 year university track.
BTW, I’m not sure how I feel about common standards period, much less whether they need justification. My opinion is to let orgs like NCTM determine math standards, NCTE determine reading/writing standards, NSTA determine science standards, etc.
Two examples to address the critical thinking aspect.
Many chemicals are deadly in large amounts but not in trace amounts should they be treated the same? NO, things like heavy metals build up in the system so a LINEAR model is appropriate on the other hand saccharine and other organic chemicals have a threshold below which the body washes it out PERIODICALLY so one could consume over time large amounts without ill effect. This must be true since every mouthful of even vegetables contains trace amounts of 1000’s of carcinogens from eons of evolution.
In a more artistic vein, there is a wonderful Itunes U. video by MIT Prof. W. Lewin on rainbows e.g. why the color sequence ROYGBIV, why a 2nd bow, which direction rainbows are in the sky etc. His explanation relies fundamentally on cosines and sines and angles of reflected light in water drops.
So, government resources were dedicated to teaching trig to so-and-so because they might want to become engineers someday or they might want to be a journalist and understand threshold theory or they might simply want to learn why a rainbow is as it is. I dream that H.S. finance or philosophy approached this which is why I mentioned accounting, Spanish and chemistry.
It now occurs to me that I should emphasize that I do not wish to start a curriculum war nor undermine the value of cosine functions. I have no strong opinions about cosines, really.
There seem to be only a few justifiable reasons why people choose to, or are required to, learn about any topic:
a. They find the topic interesting and studying it increases their happiness.
b. Learning the topic assists them in mastering other topics.
c. The topic is useful in their professions or for projects they wish to undertake, like trigonometry for the engineer or basic geometry for Joanne’s canopy.
d. Studying the topic refines mental reasoning, clarifies moral understanding, or widens opportunities, similar to ABCDE’s argument.
The trouble for those who set educational standards or design curricula is that, as children grow older, stipulating uniform expectations becomes more difficult. Most of us can easily defend teaching all kindergartners the English alphabet, but why were government resources devoted to teaching trigonometry to Princess Mom’s kids or to Joanne? Wouldn’t Joanne have benefited more from studying finance or philosophy in high school? After all, there is almost an infinite amount of knowledge that we could present to students.
I suppose the conclusion I’ve reached is that each uniform educational standard must be well justified to the public, including students. If some standards cannot be defended, students should be given more flexibility to learn knowledge related to the professions they wish to pursue or to their individual interests.
When students (and teachers) do not understand why they’re learning about a topic, their motivation is deflated and boredom and apathy are likely to set in.
Didn’t Bardolf answer this question? There is periodic phenomena everywhere that can be modeled with Sine and Cosine functions. Why does every little bit of knowledge have to be useful? Isn’t it enough that it models sound and light and waves of all kinds? If we only taought useful stuff we would only have vocational schools.
I hated trig. When I asked my teacher why we needed to learn sines, cosines and such, he said we’d use it in college if we majored in math or science. I assured him I had no interest in a math, science or engineering major. I was taking the class so I could apply to a “good” college. But why did colleges want humanities types to learn trig? He had no answer.
I might add that he gave me a B for the second semester when my calculations showed that I’d earned a C. I was so surprised, given how I’d harassed him, that I asked if he was sure it was the right grade. He said it was. I think he gave me extra credit for asking questions.
I scored a 699 on SAT math, got into Stanford and majored in English and Creative Writing. I’ve used my arithmetic skills quite a bit during my career in journalism. I once used geometry to design a canopy for my daughter’s bed and figure out how much material to buy. Sines, cosines, logarithms . . . No.
Firstly there is a contradiction if the student is going to be a beautician then she need not be in a class designed for someone going into college. There are an endless number of things she could be learning with the same vague justification e.g. accounting, Spanish, chemistry, …
Second, perhaps an astrology lesson which is heavily dependent on cosines would be in order since it holds just as much validity as the business cycle hokum and is more appealing.
The best explanation for cosines is in periodic phenomena which occur everywhere. Musical waves, compression algorithms on her IPOD, signals and waves for her cell phone, light itself. Students get trapped into a linear mental framework for problem solving when many things are periodic or worse exponential.
BTW one might be able to get students into college w/o trig, but they won’t become engineers at a Big Ten University.
As a HS English teacher, I often find myself in the same tough spot as you. Students always ask “why?” The problem that I have is that I, myself, don’t even know why it’s important to learn some of the things that I am teaching. I resort to vague, general answers about the importance of knowing certain concepts for college and life beyond.
To some degree, education involves learning things without knowing why. We may not consciously see the value of learning, for example, about cosine functions, but subconsciously we are probably synthesizing and applying all the knowledge we have learned (and retained) throughout our educational career in our everyday lives.
I am in Engineering Services. I do quality testing on asphalt, concrete, aggregates, soils, etc. I do not have an advanced degree and I am not a professional engineer. I need algebra on a daily baisis, Algebra II at least 3 times a week and Algebra III, Trig and some calc probably on a monthly basis. You really do have to use these formulas if you work with things like density, volumes, psi, viscosities and a host of other things. We can hire brilliant people with advanced degrees that if you give them the results of the testing, literally have a page of numbers in front of them. The comprehension of advanced mathematics helps you interpret the results, investigate why something failed, why something is working, etc. It may not be for everyone but yeah, some of us geeks really do use them.
You do realize that you never answered your own (and your student’s) question. I’ve managed to get all grown up–and into three Big 10 universities–without any math beyond high school Algebra II. So why do my children need to learn about cosine functions?
Going along with this, the standards writers should have the courage to not include a standard if they can’t come up with a justification.
Comments on this entry are closed.