Should calculus be taught in high school?

[mXSCNT]

My personal opinion on math education in the US is that our problems stem from the anti-intellectual culture that many youth get drawn into. The culture glorifies soldiers, musicians, actors, athletes, anything but scientists, who are derided as stuffy and useless. There isn't much emphasis on a work ethic, either. It's all about quick gratification. The result is, most students don't value math much, and if they do value it they are less inclined to work at it. The best students, who both value achievement and are willing to work, are ostracized as geeks. With that kind of peer pressure who would want to be smart?

 

[j93]

My personal opinion on math education in the US is that our problems stem from the anti-intellectual culture that many youth get drawn into. The culture glorifies soldiers, musicians, actors, athletes, anything but scientists, who are derided as stuffy and useless. There isn't much emphasis on a work ethic, either. It's all about quick gratification. The result is, most students don't value math much, and if they do value it they are less inclined to work at it. The best students, who both value achievement and are willing to work, are ostracized as geeks. With that kind of peer pressure who would want to be smart?

This still doesn't account for the fact that in the US kids spend 7 years learning how to add and subtract due to the curriculum.

 

[thrill3rnit3]

I believe that if reform is to be done to the curriculum it should start with the bottom (preschool - elementary education), working its way to the top (high school curriculum).

 

[Astronuc]

Math is not emphasized enough at those levels. For heaven's sake kids don't fully understand how to add/subtract "unlike" fractions until the 6th grade...

I learned that in 4th grade in the US. But I've found US schools uneven. Some are great and many are poor. I probably had the best teachers in the schools I attended, but that's because I got shuffled into Major Works (MW) or Honors courses.

 

[thrill3rnit3]

I learned that in 4th grade in the US. But I've found US schools uneven. Some are great and many are poor. I probably had the best teachers in the schools I attended, but that's because I got shuffled into Major Works (MW) or Honors courses.

It's supposed to be "taught" at that stage. But because of the lack of emphasis by the teachers, and thus the lack of interest by the students (I'm talking about the middle tier-lower tier students), they don't fully understand the concept until middle school.

Which is pretty pathetic IMO.

 

[PhysicalAnomaly]

Maybe you could adopt the asian method and just make the students do more and hope it works. XD

The australian maths syllabus is a year behind malaysian and singaporean syllabi and their students are no more competent at what they learn either. The students in the asian countries do more questions a day and by the time they graduate from high school, they are expected to have done thousands of calculus questions. There's also the massive peer and parent pressure. They go for tuition classes and spend a lot of time just doing problems. We also learn so many different methods of doing things that it's quite shocking to find that the australian students only know a single method.

I personally don't think much of mindlessly doing hundreds of questions. But if it works, it works.

 

[thrill3rnit3]

Maybe you could adopt the asian method and just make the students do more and hope it works. XD

The australian maths syllabus is a year behind malaysian and singaporean syllabi and their students are no more competent at what they learn either. The students in the asian countries do more questions a day and by the time they graduate from high school, they are expected to have done thousands of calculus questions. There's also the massive peer and parent pressure. They go for tuition classes and spend a lot of time just doing problems. We also learn so many different methods of doing things that it's quite shocking to find that the australian students only know a single method.

I personally don't think much of mindlessly doing hundreds of questions. But if it works, it works.

Well I don't think doing a lot more problems would solve the issue either. What I'm talking about the way it is being taught to the students. Here in the U.S. the "plug and chug" method is the prevalent method in use by most of the students AND teachers alike.

So when the kids are given a problem a little bit different from the sample exercises, they are lost and have no clue where to even begin.

 

[Andy Resnick]

I've said this many times before, and will say it again-

Looking over this thread, there is not one single comment (excepting mine) that admits the role of *parents* in their child's education. Part of the problem with the US public education system is that many parents have completely ceded their role in the education of their children to the schools. As long as parents consider their children's mathematical education (or any other part- history, composition, etc) not worth discussing over dinner, and parents make no effort to show their children that the material they learn in school has value outside of the classroom, no amount of time and effort spent in the classroom will compensate.

Teachers have an incredibly difficult job and get paid very little money. Is it any wonder that high-caliber educators are not created and nurtured? The US curriculum is now results-based: school funding hinges on how well the students perform on idiotic standardized tests. Is it any wonder that increasing amounts of classroom time are spent teaching to the test rather than providing an educational environment?

To you folks who claim to be so concerned about how poorly students are being educated, I challenge you to do something about it- offer to teach a 'science day' in an elementary school classroom. Volunteer for "Teach for America". Stop whining about how the larger public doesn't give a rat's a** for the subjects you hold so dear. Engage the public and get their attention.

 

[Feldoh]

Eh, maybe my experience is unique but I took AP Calculus, got a 5 and passed out of Calc I and II at college. I've gone on and passed Calc 3, and differential equations easily with a's. Next semester I'll be taking real analysis, and I've been going over the book over the summer and although it's difficult doesn't really seem over-the-top. On top of that the only way I make money is tutoring students in math (Calc I-III).

Basically my point is is that I've done just fine without ever having to retake the first few intro calculus courses in college, so to be honest I really don't think it's a solution that really makes sense.

On top of that people I know that have taken the AP test and opted to skip on college credit now find that they (two of my close friends) dislike math just because they've had the same old information for two year in a row, which starts to get stale.

 

[symbolipoint]

Andy Resnick:
Some people, even I, share your opinion that parents need to care and encourage. On the other hand, some parents mishandle this, destroying the childrens' motivations for Math and are unable to give or find sensible help. A few children are lucky that their Math instruction in their school may actually be good; better than just "plug & chug" Algebra.

Feloh:
Some students NEED to study material or courses more than once. They also need opportunities to use the Mathematics which they study. Part of this is just having good variety of Algebra and Calculus exercises with derivations and analytical thinking; and some of this is having science lab exercises or real-life work situations which can be understood or managed with Mathematical topics.

 

[Astronuc]

I've said this many times before, and will say it again-

Looking over this thread, there is not one single comment (excepting mine) that admits the role of *parents* in their child's education. Part of the problem with the US public education system is that many parents have completely ceded their role in the education of their children to the schools. As long as parents consider their children's mathematical education (or any other part- history, composition, etc) not worth discussing over dinner, and parents make no effort to show their children that the material they learn in school has value outside of the classroom, no amount of time and effort spent in the classroom will compensate.

Teachers have an incredibly difficult job and get paid very little money. Is it any wonder that high-caliber educators are not created and nurtured? The US curriculum is now results-based: school funding hinges on how well the students perform on idiotic standardized tests. Is it any wonder that increasing amounts of classroom time are spent teaching to the test rather than providing an educational environment?

To you folks who claim to be so concerned about how poorly students are being educated, I challenge you to do something about it- offer to teach a 'science day' in an elementary school classroom. Volunteer for "Teach for America". Stop whining about how the larger public doesn't give a rat's a** for the subjects you hold so dear. Engage the public and get their attention.

Yep - parents' lack of involvement in their childrens' education is a big problem - has been for 3+ decades since I left high school, and years I went to primary and secondary school.

After the students leave the school, teachers cannot make the students do homework or study. That is when the parents need to enforce the discpline and ensure their children do their homework and class assignments. That should the be the priority - not watching TV, or playing video games, or running the streets, or playing sports or some other extracurricular activity in place of studying. But this is OT.


Analysis and calculus should be taught as early as possible, and certainly by 12th grade, but that requires the pre-requisites be taught in earlier classes. One difficulty is disparity in the ability of students and also in the capability of teachers - not only from state to state or from school to school, but even within schools.

By the time I was in 11th grade, I was well ahead of my parents ability, so I pretty much took responsibility for my studies and academic program. I was one the fortunate students who got the best teachers in the school who were also the heads of the mathematics and science departments, and I had the best academic counselor who was well aware of university programs around the country, and each year directed students to NSF and university summer programs in the academic subjects of interest. Many of us in the honors/major works/AP programs did a summer program between 11th and 12th grade. One of my classmates went to MIT for a summer program in math and science, and he attended MIT out of high school. I did a summer program at Colorado School of Mines in EE and NucE. One of the kids from that same summer as CSM is now a professor of astrophysics at Caltech.


The high school I attended in 11th and 12th grade was on the trimester system and one typically took 4 courses per trimester period. I took 5 courses in order to add an extra course. I had taken geometry and trigonometry in 10th grade, so 11th grade was a second year of algebra, with some linear algebra and more trigonometry. The 12th math program consisted of analytical geometry (one trimester) and two trimesters of differential and integral calculus. Only about 30 students out of more than 700 did that math program. All the rest did up through analytical geometry, if that, in their senior year.

 

[buffordboy23]

I definitely agree that parents are a large part of the problem, but we shouldn't place all the blame concerning the poor state of our education system on them. If we allow ourselves to believe this folly, then the only solution toward a better education system is to change the mindset of a nation…highly unlikely. Educating our youth does not require us to educate their parents. Therefore, we should focus on educating qualified and competent mathematics/science teachers, modifying instruction to actually connect with students lives, and modify the K-12 curriculum to the student’s educational goals. This “educational” population is smaller than the parent population and values education more as well, so reform in this area should be more realizable.

First, let’s look at the number of qualified math/science teachers. I don’t have recent statistics in front of me, but I bet that many teachers out there teaching these subjects are not qualified to teach them but do so with a temporary certificate or something similar. So students of these teachers get the shaft.

Now, the remaining teachers are qualified to teach these subjects. What exactly does it mean to be a qualified teacher? Usually, it means that the teacher has a degree in the area they are teaching and passed a general and subject-specific certification exam. Do you really think that graduating students with teaching degrees really know their subjects? From what I have seen, the methodologies often employed in college instruction only require memorization to pass a multiple-choice test, so there is no real understanding to be had unless the student takes their own initiative. What about the general and subject-specific certification exams? It’s a net to catch the dumbest of the dummies and keep them from actually entering the classroom. Don’t believe me? I knew an elementary education major that had her degree for two years and still couldn’t pass the mathematics portion of the general exam, and so she couldn’t teach in the classroom. So while a student may have a “qualified” teacher, they are actually getting the shaft.

This is a difficult problem to overcome, since our educational system is cyclical through time and is affected by numerous factors.

In the future, I plan to discuss some simple solutions to this dilemma and discuss the other two points when I get the time. This post may appear somewhat to wander from the OP (teach calculus in school?), but I assure you that I am working my way there.

 

[buffordboy23]

The simple solutions to the first point of my last post are to modify college instruction to ensure real understanding of the subject, to learn appropriate and creative methodologies to transmit this knowledge effectively to the student, and to provide incentives to aspiring teachers entering these fields. All of these are done in some college-level settings, yet it hasn’t become widespread. Even if it were widespread, we would have to wait years to see real results.

Now, let’s suppose that your one of the few to get such training in college or that you acquired it on your own through initiative and hard work. Would you actually employ these skills in the classroom? With so many cookie-cutter lesson plans available on the internet and resources offered by textbook manufacturers, the demanding work load that our current teachers face makes it so tempting to sacrifice the time needed to employ the skills learned in college in favor of these time-saving crutches. This leads to my second point, modifying instruction to actually connect with students lives.

In regard to mathematics, the typical and most simplistic form of instruction is rote memorization. While I do agree with this at the elementary level, since this is the foundation of all advanced mathematical subjects, like algebra and so on, I disagree with this method of instruction during math education in grades 7-12, yet this method still persists.

The connection of elementary mathematics education is easy to connect to students lives (they see examples everyday), but advanced subjects are more of a challenge to convey. It requires a large sacrifice of time on the part of the teacher to develop such lessons, since the resources readily available don’t usually have the necessary focus—check the research studies done on mathematics textbooks and their associated resources and you will find that they are rated poorly in most instances. Furthermore, real-life scenarios/problems for math subjects offered in grades 7-12 require more critical and creative thinking on the part of the student…something they are not used to and is a skill in and of itself. Research shows that, in general, students value the learning of a subject if it appears useful or important to them, so we must not neglect this fact and target it in our instruction.

Here’s a simple example to show why knowledge of trigonometry is important to the student. In the future, the student will likely buy a house. They may eventually decide that they want to cut down a tree that resides on their property, and that they want to do this task themselves to save money, yet the layout of their property and the general appearance of the height of the tree makes this appear like a risky endeavor. If the tree has only one cut at its base, will it fall on the house, or will two cuts and the extra work be necessary? Using their critical thinking skills and knowledge of trigonometry, the former student realizes that they can accurately measure the baseline from some position to the tree and the angle from this position to the top of the tree and compute the height of the tree to good accuracy. Thus, the question is answered and learning trigonometry has proved useful to the student. There are many more examples that can convey the value of advanced mathematics, but it requires competency on the part of the teacher to show this to students, and unfortunately, this does not generally happen in our classrooms.

The last point, modifying the K-12 curriculum (mostly 7-12), is connected to the discussion of the second point. What mathematical knowledge is really necessary for students who choose vocational studies vs. college prep studies in the sciences or liberal arts? Usually, students in the vocational studies don’t take calculus, while students in college prep do and for many of them it will never be of any use except for a well-rounded educational background. Instead of requiring calculus for these particular students, it should be offered as an elective vs. another class that explores familiar mathematical subjects and their connections to real-life scenarios in order to build problem solving skills. This aspect of mathematics education should be specifically tailored to the student's chosen path of study and should provide the student with freedom of choice rather than required restraint. So, if a student is planning to pursue the sciences then calculus should definitely be taught in high school.

 

[Astronuc]

I took introductory mathematics and science courses through a local university program during junior high and high school. I had to go out a buy my own analytical geometry and calculus textbooks, and I think that was during the summer before I started 10th grade. My dad took me the main technical bookstore in the city, so I could browse the aisles for math and science books. I was able to learn bits and pieces, but I had no formal direction from a mentor.

buffordboy23 raises several good points, which are all aimed at improvements in primary and secondary education, which in theory would lead to having schools in which calculus is taught in high school to those students are prepared to learn it.

Teachers need support and appropriate training.

Parental involvement is essential.

 

[mgb_phys]

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html

 

[thrill3rnit3]

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.

 

[buffordboy23]

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.

There are only two reasons why an individual would choose to learn something. They are practicality and pleasure. Neither is significantly evident in our mathematics curriculum but they should be.

Practice with real-world scenarios develops experience and foresight, which then enables one to solve problems when actually confronted by them. Here's an example a former colleague of mine gave to her middle school students. The student got to choose any car that they wanted to have in the future and that they think they could someday afford. Many students picked really expensive vehicles. They calculated their monthly loan payments as part of the project. Most of them crapped themselves when they saw the final figures, and some noted that their parents salary wasn't even sufficient. Yet we still see people placing themselves in bad financial situations due to lack of critical reasoning or just plain temptation. If you are fortunate to have tools, but have no experience or knowledge of using them, then they are useless.

 

[physics girl phd]

There are only two reasons why an individual would choose to learn something. They are practicality and pleasure. Neither is significantly evident in our mathematics curriculum but they should be.

Practice with real-world scenarios develops experience and foresight, which then enables one to solve problems when actually confronted by them. Here's an example...

And here's another:

Our rising fifth grader has been picking at dinner then sneaking peanut butter in the middle of the night (after having some for breakfast and lunch)... so to motivate him to do otherwise, I prepared a worksheet for him where he estimated his daily consumption of peanut butter (8 servings in his case) and he then would look at the nutritional value of peanut butter on the jar and see how much of various dietary needs were being neglected, met or exceeded by his daily consumption. After seeing he gets 144% of an adult's daily fat needs, and none of certain nutrients, we're hoping that now reconsiders his decisions on things.

Unfortunately he didn't get to the part where he looked at the few other things he eats and how they might fill the gaps... mostly what little he does eat at dinner, a packet of raisins, a snack bag of crackers, and a bottle of apple juice (and boy did he protest when I insisted he get a 100% no sugar added juice when we were at the store!). However, I noticed this morning that his preferred bread contains 5% of saturated fat per slice and no vitamin A or C (two of the biggies that were missing from the PB). Ouch!

While yeah, it was practice with math... but he was genuinely interested (probably was hoping things would turn out better for him and his peanut butter diet would be justified). And then for us it had the desired result -- at least he ate his regular dinner and didn't sneak peanut butter last night!

 

[Andy Resnick]

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html

I have not seen this before- thanks for posting it!

 

[Astronuc]

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.

Some students need tangible examples, while others have no problem with abstractions like n-tuples.

 

[Astronuc]

Probably everyone here is familiar with this article (Lockhart's lament), but it is a good read
http://www.maa.org/devlin/devlin_03_08.html

I too have not heard of Lockhart.

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

How do the educators tap into the imagination of students' minds?

 

[buffordboy23]

And here's another:

Our rising fifth grader has been picking at dinner then sneaking peanut butter in the middle of the night (after having some for breakfast and lunch)... so to motivate him to do otherwise, I prepared a worksheet for him where he estimated his daily consumption of peanut butter (8 servings in his case) and he then would look at the nutritional value of peanut butter on the jar and see how much of various dietary needs were being neglected, met or exceeded by his daily consumption.

Your right. This is just mild form of coercion though. In my post, I was specifically referring to the freedom of choice concerning the individual.

The educational standards require that coercion be the epitome of our education system and coercion usually doesn't work by itself. It also needs an offering of practicality (how will this benefit me now or in the future?) or pleasure (will learning this be fun? will I have large freedom with my approach?). It appears that your son saw the practicality of your proposed lesson and complied. I can't confidently comment on the pleasure though.

 

[buffordboy23]

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

I agree. Also, Lockhart suggests that math can be perceived as an art form, and this is true.

How do the educators tap into the imagination of students' minds?

Initially, it probably requires carefully planned use of scaffolding. For example, it's easy to provide scenarios that can lead elementary-level students to the concept of infinity without initially telling them what infinity is. How does the student then make sense of the results of this scenario and analogous ones? They can create a definition that characterizes it. Another example that borrows from Lockhart is to let students choose the geometric figure inscribed in a square and to determine how much area of the square that the figure consumes--the scaffolding is in place, but with a sense of individual freedom for exploration in this case.

By the teacher modeling this type of behavior and by seeking original and slight variations on the problems from the students, we expect that they eventually will ask their own questions. This is analogous to how a scientist operates. By studying the questions asked by other scientists (mainly the teacher and sometimes the students) and learning about their results, they (mainly the students) learn to ask new questions that are relevant to the current body of knowledge that exists.

This is usually absent from our education system.

 

[Sankaku]

Why are we so concerned about connecting math to "everyday lives"?? I think the teachers try to hard to make that connection and application, in which they are losing the theoretical side of their lectures.

I think there is an important distinction here. Kids need to be INSPIRED to want to do math. One way to do it is to make it something that they see as useful. When I was in high-school, half the boys wanted to know mechanics because they liked cars. A means to an end.

Although the "useful" angle is sometimes good, I think the "Lockhart's Lament" article points out that you get more interest if something is "beautiful" rather than just "useful."

You can read "beautiful" as "cool" if it fits modern semantics better. For me, I have a rebellious love of things that are beautiful but of no (obvious) practical importance whatsoever. Whatever the motivation is, you can't expect 13 year old kids to be interested in math just because some grownup thinks that "the theoretical side" is important. The "theoretical side" has to have some kind of relevance, no?

As far as teaching Calc in high-school. Yes, if kids are ready for it and enthusiastic about it. Otherwise, No. The question seems to be devolving into "should we force smart kids to study Calc in high-school?" I don't like this. Why are we in such an awful blinding rush?

 

[SonyAlmeida]

Taught but not forcibly so, which is... the status quo, pretty much.

 

[Andy Resnick]

I too have not heard of Lockhart.

I like Lockhart's idea "that there is a playground in their [students] minds and that that is where mathematics happens." That's brilliant.

How do the educators tap into the imagination of students' minds?

That's a good question, indeed! Unfortunately, I have to agree with Lockhart's conclusion- in order to treat Math like the other Arts, one must give up standardized tests. Personally, I don't think that is ever going to happen as long as I am alive- it's too hard to argue against "minimum competency requirements", because there are some perfectly valid reasons for having minimum standards.

Perhaps this means moving toward a more European (i.e. German) model, with separate 'vocational' tracks established early on- but OTOH that is exactly the system that drives away talent (to the US, currently).

 

[thrill3rnit3]

I'm not definitely not against teaching applications of the mathematics, but it seems like teacher nowadays are too focused on the application that the theory behind it is lost.

Same reason why calculus books nowadays are considered "watered down", for example they are relying too much on the calculator which I think is counterproductive.

 

[buffordboy23]

I think there is an important distinction here. Kids need to be INSPIRED to want to do math. One way to do it is to make it something that they see as useful. When I was in high-school, half the boys wanted to know mechanics because they liked cars. A means to an end.

Exactly. There is practicality here. This is exactly why I believe that parental involvement is not the most important factor in regard to the education of our youth. Whether the parents are neglectful or not to their child's education, their sons/daughters know how to use many technologies, such as ipods, cell-phones, computers, etc., better than their parents and most of us older adults. Why? Because one major reason is that it keeps them socially connected to their peers...social Darwinism in effect. They see value in learning how to use the technology and take initiative to teach themselves.

 

[thrill3rnit3]

Simplicio and Salviati's conversation about teaching the "practicality" of math is definitely what I was talking about.

 

[RedX]

I didn't read everything so apologies if my opinion is repeated.

Maybe you can have a compromise: only teach differential calculus in high school, and spend the other half of the year strengthening algebra. Integrals you can learn in college.

A lot of calculus is, in my opinion, memorization, until you take an analysis course. But teaching students d/dx(x^n)=nx^(n-1) is not a waste because they'll have to learn it eventually, and eventually everyone has that down by rote. It's the age old question - do you learn by rote first, and then teach why? Or do you teach why first, and then the rote? High school can lay down the rote, and college can provide the aha that's why. Maybe reform should be on the college side. Each major has a definition of what amount of math is acceptable, each college has applicants of differing abilities, so maybe the colleges should cater to the students and not the high schools to the college?

The AP calculus test should weed out students with inadequate understanding. In theory at least. But I don't trust them. Perhaps it's better to ask for AP-test reforms than refining what's taught in high schools.