How can we make math less scary?
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 KestrelLowing
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How can we make math less scary?
So something I've been thinking about recently, spurred on by other threads here, is that math is typically seen at the most difficult subject there is. All anyone needs to do is look around at popular culture to see that. It's popular to say that 'I'm not good at math'. How can we change this?
Math is a difficult subject, but no more difficult than any other. However, it typically has been able to stay away from grade inflation due to the pretty much right/wrong nature of most math subjects. Is this the major reason math is seen as so difficult?
One thought I've had is that from the very beginning, math is not always taught by the best people to teach math. Elementary school teacher are notorious for not being that great in math (although this is obviously not always the case). This causes math classes that are very formulaic and not terribly insightful as the teachers themselves learned math in a formulaic manner without much meaning behind the method. It could be that many teachers during those years when people decide what their favorite subject is are not particularly fond of math themselves and pass that onto their students.
So, does anyone else have any ideas how we can make math less scary without lessening the content learned? I really do think this is important as the world needs more people more comfortable with math.
Math is a difficult subject, but no more difficult than any other. However, it typically has been able to stay away from grade inflation due to the pretty much right/wrong nature of most math subjects. Is this the major reason math is seen as so difficult?
One thought I've had is that from the very beginning, math is not always taught by the best people to teach math. Elementary school teacher are notorious for not being that great in math (although this is obviously not always the case). This causes math classes that are very formulaic and not terribly insightful as the teachers themselves learned math in a formulaic manner without much meaning behind the method. It could be that many teachers during those years when people decide what their favorite subject is are not particularly fond of math themselves and pass that onto their students.
So, does anyone else have any ideas how we can make math less scary without lessening the content learned? I really do think this is important as the world needs more people more comfortable with math.
Re: How can we make math less scary?
To put my credentials on the line, I am a volunteer math tutor in an adult vocational program, mostly dealing with materials in grades 69 (fractions through beginning algebra and practical geometry). I am also studying to get my M.S.Ed in Adolescent Education, but the following opinions are primarily formed by my lay perspectives.
I think you have some interesting hypotheses. Frankly, I suspect that every subject has some good and bad teachers in primary school. But math is unusual (and perhaps unique) in that every years' curriculum is built on the previous years'. For instance, I sense that it only takes one good reading teacher to make up for a string of marginal reading teachers, or if you forgot nearly all of high school chemistry it won't have a large impact on your physics class next year. But if you didn't learn multiplication properly, you're going to start in a hole next year when it's time to learn fractions, and takes an extraordinary effort to not start algebra even deeper a year or two later. Under these circumstances, it's not hard to predict exactly what we see in the real world: students who struggle and have selfesteem issues and antipathy to the subject, teachers who can't reach half their students without abandoning the other half, and adults who forget everything they ever learned and are grateful for the ignorance.
There are a lot of interesting things that we can tinker with going ahead  multimedia learning, computer assisted instruction, selfpaced remedial labs, handson experimentation, more of an appeal to the realworld utility of the concepts  that can make math education more accessible to the students. But if I were put in charge, the thing I would do immediately is to assess all students at the BEGINNING of the term and place them in the math class that was appropriate for their knowledge level instead of naively assuming that it's the course after the course that they passed last year. That way, if nothing else, you can hold the teachers and students accountable for their performance in advancing their knowledge over the course of the term.
I think you have some interesting hypotheses. Frankly, I suspect that every subject has some good and bad teachers in primary school. But math is unusual (and perhaps unique) in that every years' curriculum is built on the previous years'. For instance, I sense that it only takes one good reading teacher to make up for a string of marginal reading teachers, or if you forgot nearly all of high school chemistry it won't have a large impact on your physics class next year. But if you didn't learn multiplication properly, you're going to start in a hole next year when it's time to learn fractions, and takes an extraordinary effort to not start algebra even deeper a year or two later. Under these circumstances, it's not hard to predict exactly what we see in the real world: students who struggle and have selfesteem issues and antipathy to the subject, teachers who can't reach half their students without abandoning the other half, and adults who forget everything they ever learned and are grateful for the ignorance.
There are a lot of interesting things that we can tinker with going ahead  multimedia learning, computer assisted instruction, selfpaced remedial labs, handson experimentation, more of an appeal to the realworld utility of the concepts  that can make math education more accessible to the students. But if I were put in charge, the thing I would do immediately is to assess all students at the BEGINNING of the term and place them in the math class that was appropriate for their knowledge level instead of naively assuming that it's the course after the course that they passed last year. That way, if nothing else, you can hold the teachers and students accountable for their performance in advancing their knowledge over the course of the term.
 doogly
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Re: How can we make math less scary?
Yeah, math teachers are a hard commodity to come by. There is actually very little expectation that they would be any good!
Let's say you're at your parent teacher conference, as a parent. Meetin the teachers. You ask the English teacher, "What was the last book you read? you know, for pleasure." If they said "Nothing," trouble. Havin a chat with the principal about who is being allowed near your precious angels, ruining their educations. If you ask the math teacher what the last math problem they did for pleasure was, you're the weirdo. What's that all about?
Math is widely considered to be a chore. It is practical. Ugh. Gross. Math is joyful. You cut music education from the schools, and the kids still form bands. They rock out. They play instruments. Better than shit on the radio even, from time to time! I say the answer is to cut math and science from schools. They are sucking at it anyway. At least this way there's a chance the kiddies band together where their parents can't see them and make some robots or some shit. Maybe fiddle with circles.
Perhaps that's not quite the answer. But I think if you offer math teachers a little more salary and freedom in the class room, you can attract more and better mathematicians. People who can impart some sense of the joy, can share it and for starters, experience it themselves!
Let's say you're at your parent teacher conference, as a parent. Meetin the teachers. You ask the English teacher, "What was the last book you read? you know, for pleasure." If they said "Nothing," trouble. Havin a chat with the principal about who is being allowed near your precious angels, ruining their educations. If you ask the math teacher what the last math problem they did for pleasure was, you're the weirdo. What's that all about?
Math is widely considered to be a chore. It is practical. Ugh. Gross. Math is joyful. You cut music education from the schools, and the kids still form bands. They rock out. They play instruments. Better than shit on the radio even, from time to time! I say the answer is to cut math and science from schools. They are sucking at it anyway. At least this way there's a chance the kiddies band together where their parents can't see them and make some robots or some shit. Maybe fiddle with circles.
Perhaps that's not quite the answer. But I think if you offer math teachers a little more salary and freedom in the class room, you can attract more and better mathematicians. People who can impart some sense of the joy, can share it and for starters, experience it themselves!
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 Proginoskes
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Re: How can we make math less scary?
Read "A Mathematician's Lament" by Paul Lockhart, at http://www.maa.org/devlin/LockhartsLament.pdf .
A paragraph from the paper, to whet your appetite:
A paragraph from the paper, to whet your appetite:
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or
physics (mathematicians conceived of black holes long before astronomers actually found any),
and allows more freedom of expression than poetry, art, or music (which depend heavily on
properties of the physical universe). Mathematics is the purest of the arts, as well as the most
misunderstood.

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Re: How can we make math less scary?
Remove the abstract hard Math as early as possible. Kids should be able to take it as an option but after basic algebra there's no reason to force them to take a separate Math class. If they happen to need some Math which they haven't covered before outside of a Math class then teach it in the context of that subject.
From what i've seen most complaints are not about it being hard but more about it being useless.
From what i've seen most complaints are not about it being hard but more about it being useless.
Re: How can we make math less scary?
KestrelLowing wrote:Math is a difficult subject, but no more difficult than any other.
Would disagree with this somewhat. Humans are not exactly made for calculation, but we are quite capable with language. Of course, this is not an excuse for the difficulty of math as much as an indictment about what math has become, or rather not become. Further, math tends to be an abstraction, and students are not ready for abstraction for several years, and students are not proficient with abstraction for several years after. Imagine learning to read without seeing text.
KestrelLowing wrote:However, it typically has been able to stay away from grade inflation due to the pretty much right/wrong nature of most math subjects. Is this the major reason math is seen as so difficult?
This is part for the first reason above. Math education tends to be calculation driven which gives this right/wrong questions; either you choose the correct formula and calculate correctly to get the right answer, or you are wrong.
KestrelLowing wrote:One thought I've had is that from the very beginning, math is not always taught by the best people to teach math. Elementary school teacher are notorious for not being that great in math (although this is obviously not always the case). This causes math classes that are very formulaic and not terribly insightful as the teachers themselves learned math in a formulaic manner without much meaning behind the method. It could be that many teachers during those years when people decide what their favorite subject is are not particularly fond of math themselves and pass that onto their students.
What you say is correct. Teachers do not enjoy math. Teachers were taught math by people who did not enjoy math, and as people who do not enjoy math teach a new generation to not enjoy math. The problem is simultaneously teachers do not want to expend the effort and do not understand what interesting math curriculum is.
[quote="KestrelLowing"}So, does anyone else have any ideas how we can make math less scary without lessening the content learned? I really do think this is important as the world needs more people more comfortable with math.[/quote]
Calculation and abstraction needs to be dismissed. We have ubiquitous computers and a century of developmental study. What students need to understand is when we use a formula or process and how to gather the numbers, and this should be done by gathering numbers from the world. This is not to dismiss the usefulness of mental calculation for estimation purposes, essential to using computers properly, but something tells me professionals steeped in mathematics are not busy calculating by hand as we make students do, and those professionals have something real to work with besides.
 doogly
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Re: How can we make math less scary?
Ah, the solution to problems with math is to do less math. Wonderful, wonderful.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: How can we make math less scary?
doogly wrote:Ah, the solution to problems with math is to do less math. Wonderful, wonderful.
Welcome to the math wars. It is obvious that the solution is less standardized testing, more effective communications about problemsolving strategies, more standardized testing, and a handsoff approach to let students discover the harmony of math themselves. Oh, and of course we need to give all math teachers large raises and then fire them.
(Having reread what I just wrote, I'm strangely enough broadly in favor of five of those six planks. Which makes discussing reform so much harder, because very few people are just flatout wrong.)
 Bakemaster
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Re: How can we make math less scary?
doogly wrote:I think if you offer math teachers a little more salary and freedom in the class room, you can attract more and better mathematicians. People who can impart some sense of the joy, can share it and for starters, experience it themselves!
I'm skeptical about the extent to which the above statement is more applicable to math than to any other K12 subject area.
As far as experiencing and imparting joy, it doesn't matter how excited or animated you are about math unless one of the things you're sharing with your students is the ability to perform highly on quantitative assessments. Not because standardized testing is superfantastiche for students, but because it's going to be a reality for both students and teachers in the foreseeable future.* The student who's struggling to perform in a manner compatible with the manner of his or her assessment is going to be a hundred times more difficult to attract, entertain or excite with antics about the joy of math.
*Barring a paradigm shift in education, for which, as we should all be painfully aware, there really is no budget in these hard times. Or ever.
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 doogly
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Re: How can we make math less scary?
I don't know what makes math rather different. I do know you're much more likely to find an English teacher who reads for pleasure at present than you are to find a math teacher who maths for pleasure, but those two suggestions are likely to be attractive to people in any field. I can't imagine someone who wouldn't enjoy money and freedom.
Standardized testing exists, for better or worse, but I still wouldn't teach to it. It can just happen. Consider English again. Instead of reading a vocab book and doing spelling and grammar drills, you read books. Well written books. At least, I hope so. Please let that be the case in other people's educations...
Standardized testing exists, for better or worse, but I still wouldn't teach to it. It can just happen. Consider English again. Instead of reading a vocab book and doing spelling and grammar drills, you read books. Well written books. At least, I hope so. Please let that be the case in other people's educations...
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 Magnanimous
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Re: How can we make math less scary?
markop2003 wrote:From what i've seen most complaints are not about it being hard but more about it being useless.
I also work as a volunteer math tutor, and I definitely get this a lot. Some students don't bother to learn the math not because they can't, but because they don't see it as applicable to the real world. (This starts at approximately algebra, I think.)
So along with the OP, another question is "How do we show people that math is a useful field?" I... can't really think of a good way to answer this to someone who doesn't already know a lot of math.
 Bakemaster
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Re: How can we make math less scary?
doogly wrote:Standardized testing exists, for better or worse, but I still wouldn't teach to it. It can just happen.
That attitude absolutely reeks of privilege. What about environments where you can expect fewer than half of students to graduate high school? Do you really think standardized tests can "just happen" for students who can't easily assume they'll pass?
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 doogly
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Re: How can we make math less scary?
I don't mean just happen as in nobody cares, I mean that the most effective way to teach math and english is not by test prep, it is by doing interesting problems and reading interesting books.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
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Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: How can we make math less scary?
doogly wrote:Ah, the solution to problems with math is to do less math. Wonderful, wonderful.
Ideally, we want to provide students opportunities to apply the academics within school because the end should be students apply the academics within life. We want students to use the writing process to create meaningful works, read authentic literature and integrate the experience and information, perform scientific experiments and record and share data, and examine primary source documents to draw conclusions. Using the writing process is more writing than meaningless prompts. Reading authentic literature is more reading than leveled readers. Performing scientific experiments is more science than taking notes from a textbook or lecture. Gathering numbers and working with how to use them to create an answer is more math than calculating endless equations.
Bakemaster wrote:As far as experiencing and imparting joy, it doesn't matter how excited or animated you are about math unless one of the things you're sharing with your students is the ability to perform highly on quantitative assessments.
Wonderful thing is authentic experiences with the material helps students retain and recollect the material as well as recognizing the material in other forms. Teaching to the test is less effective on such assessments; the reason this is done is the simplicity of teaching to the test rather than efficacy.
Magnanimous wrote:(This starts at approximately algebra, I think.)
Before this students create this concept where math has no real use. Around the time students work with specific branches of math we can begin to see students who normally work and achieve show boredom and disinterest in work. This is my experience, at least, with working with several classes across multiple grades.
Bakemaster wrote:That attitude absolutely reeks of privilege.
He is not wrong in general. Some test strategies are useful, but modeling the curriculum around the assessment with the district or state or whatever provided summative assessments and worksheets and materials in general is ineffective in several ways for development and less effective than designing proper experiences.
 KestrelLowing
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Re: How can we make math less scary?
I guess I never really thought that people loose interest at algebra  that's when I finally started being interested. That's when it actually seemed realworld to me. Of course, I'm currently an engineering student, so that's obviously a huge bias here.
Actually, I think abstraction is absolutely the most useful portion of math  to take that abstraction, know how to use it, and realize that things can be used in more than one case. Humans are awesome in that regard. For example, when working with dogs, you have to 'proof' the behavior. Say you're teaching a dog to sit. If you teach that inside in your living room and nowhere else, chances are they won't sit anywhere else. You have to teach them in each place. While dogs will pick up on it quicker, they still need to be taught how to do it in multiple environments.
Humans are pretty awesome that they have a much greater capability of 'learning to sit' in one place and then automatically applying it to other regions. However, this obviously is encouraged by practice, which mainly takes place in math classes.
However, I do completely agree that things that will take numbers from the 'real world' is needed. I remember seeing video of a math teacher who did this and I thought it was very effective  you know those rate problems where you've got a tank and it's draining at a certain rate and you're filling it at a certain rate? He actually did that with real stuff. The numbers obviously weren't perfect, but I think it was very effective and really helped give a complete understanding of that problem.
Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
Vangor wrote:
Calculation and abstraction needs to be dismissed. We have ubiquitous computers and a century of developmental study. What students need to understand is when we use a formula or process and how to gather the numbers, and this should be done by gathering numbers from the world. This is not to dismiss the usefulness of mental calculation for estimation purposes, essential to using computers properly, but something tells me professionals steeped in mathematics are not busy calculating by hand as we make students do, and those professionals have something real to work with besides.
Actually, I think abstraction is absolutely the most useful portion of math  to take that abstraction, know how to use it, and realize that things can be used in more than one case. Humans are awesome in that regard. For example, when working with dogs, you have to 'proof' the behavior. Say you're teaching a dog to sit. If you teach that inside in your living room and nowhere else, chances are they won't sit anywhere else. You have to teach them in each place. While dogs will pick up on it quicker, they still need to be taught how to do it in multiple environments.
Humans are pretty awesome that they have a much greater capability of 'learning to sit' in one place and then automatically applying it to other regions. However, this obviously is encouraged by practice, which mainly takes place in math classes.
However, I do completely agree that things that will take numbers from the 'real world' is needed. I remember seeing video of a math teacher who did this and I thought it was very effective  you know those rate problems where you've got a tank and it's draining at a certain rate and you're filling it at a certain rate? He actually did that with real stuff. The numbers obviously weren't perfect, but I think it was very effective and really helped give a complete understanding of that problem.
Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
Re: How can we make math less scary?
KestrelLowing wrote:Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
You remember correctly.
I like this concept, but I'd like to know more about the comprehensive lesson plan. I'd like to think that eventually, once the students came to realize the critical independent variables for a problem, that they would then solve at least some traditionallyphrased word problems. I don't think his students would be weak at demonstrating their mastery of the skills they developed on one of those boring ol' multiple choice standardized tests  indeed, the point is that his excellent pedagogical skills would make that test a cakewalk.
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Re: How can we make math less scary?
Vangor wrote:Bakemaster wrote:That attitude absolutely reeks of privilege.
He is not wrong in general. Some test strategies are useful, but modeling the curriculum around the assessment with the district or state or whatever provided summative assessments and worksheets and materials in general is ineffective in several ways for development and less effective than designing proper experiences.
He's not wrong in theory. In reality, the freedom to teach material in the best way for the deepest understanding is a privilege afforded only when teaching in certain environments. Your students have to have a certain level of preparation for the material, which is contingent on the quality of every teaching experience they have had before entering your classroom, which is contingent not only on the quality of their past teachers but also on resources, funding, curriculum, home environment, and so forth every year they've been in school.
As teachers encounter more and greater limitations on the time and resources available to them and on the aptitude or preparation of their students, teaching to the test becomes more and more attractive. What it's not is a good way to learn math; what it is is the fastest route to a student being able to demonstrate proficiency on the next test. It hurts the student's education in the long run, but the system is not designed in such a way that it effectively holds teachers accountable for their students' performance a few years down the road. It is designed to pressure teachers to produce quantifiable results for a group of students within the span of a single year or term. And it only takes a few weak links in the chain to undermine students' preparation to the point that their next instructor is going to have to cut a few more corners and gloss over a few more topics to continue to meet his or her standards for performance.
The value of being able demonstrate proficiency on the next test is further exaggerated when passing the test is a requirement for high school graduation (e.g. MCAS). If you have the luxury and privilege of teaching kids in honors and AP courses, who are expected to pass with ease, you can poohpooh test prep and turn up your nose at the thought of teaching to the test. Not so when you're working in an environment where your students are expected to fail. In such an environment, having the diploma is what's truly important to the student, not a rigorous, intuitive understanding of mathematical concepts. Having the diploma is what comes first because without it, no amount of excitement or authenticity is likely to break the cycle of poverty that prevents these children from realizing the successes and opportunities of the "good" students.
In all of this, I'm not trying to argue against making math exciting and authentic and favoring conceptual learning over rote learning. I'm simply suggesting that the benefits of this approach are disproportionately realized by the students with the least need, which is to say, those students who have yet to fall particularly far behind. There are few exciting, conceptual math topics to cover until after basic arithmetic, by which point the damage has already been done for the students with the most need.
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 KestrelLowing
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Re: How can we make math less scary?
Bakemaster wrote:He's not wrong in theory. In reality, the freedom to teach material in the best way for the deepest understanding is a privilege afforded only when teaching in certain environments. Your students have to have a certain level of preparation for the material, which is contingent on the quality of every teaching experience they have had before entering your classroom, which is contingent not only on the quality of their past teachers but also on resources, funding, curriculum, home environment, and so forth every year they've been in school.
As teachers encounter more and greater limitations on the time and resources available to them and on the aptitude or preparation of their students, teaching to the test becomes more and more attractive. What it's not is a good way to learn math; what it is is the fastest route to a student being able to demonstrate proficiency on the next test. It hurts the student's education in the long run, but the system is not designed in such a way that it effectively holds teachers accountable for their students' performance a few years down the road. It is designed to pressure teachers to produce quantifiable results for a group of students within the span of a single year or term. And it only takes a few weak links in the chain to undermine students' preparation to the point that their next instructor is going to have to cut a few more corners and gloss over a few more topics to continue to meet his or her standards for performance.
The value of being able demonstrate proficiency on the next test is further exaggerated when passing the test is a requirement for high school graduation (e.g. MCAS). If you have the luxury and privilege of teaching kids in honors and AP courses, who are expected to pass with ease, you can poohpooh test prep and turn up your nose at the thought of teaching to the test. Not so when you're working in an environment where your students are expected to fail. In such an environment, having the diploma is what's truly important to the student, not a rigorous, intuitive understanding of mathematical concepts. Having the diploma is what comes first because without it, no amount of excitement or authenticity is likely to break the cycle of poverty that prevents these children from realizing the successes and opportunities of the "good" students.
In all of this, I'm not trying to argue against making math exciting and authentic and favoring conceptual learning over rote learning. I'm simply suggesting that the benefits of this approach are disproportionately realized by the students with the least need, which is to say, those students who have yet to fall particularly far behind. There are few exciting, conceptual math topics to cover until after basic arithmetic, by which point the damage has already been done for the students with the most need.
I actually think that a good portion of these issues could be alleviated if instead of staying with your grade for all subjects, you simply went to the class you needed to go to. I realize that especially in underprivileged areas, this could create some significant issues initially due to social pressure, but I think that overall, this could allow students to truly understand the material instead of just understanding how to pass the test.
And you make a very interesting point about the 'damage already being done'. I don't think this just applies to those in difficult academic environments. I'm starting to believe that this is the main reason that students are very annoyed with math in general  being unprepared because the real understanding isn't there.
The reason I started this thread was because I'm seriously considering trying to help elementary schools with math and science. I do not have the temperament or patience to be a fulltime elementary school teacher, but I really do enjoy teaching about math and science and I really think the most effective place for any type of math and science enrichment is in elementary. I've been a FIRST Lego League mentor for a few years, but I'd like to try and reach out to more students in the actual curriculum  whether this is with helping teachers truly understand the implications of the math and science they're teaching, or coming in and doing more project type things with the students directly. So, while nothing is really in motion yet, if anyone has any ideas, I'd be really grateful.
Re: How can we make math less scary?
KestrelLowing wrote:Actually, I think abstraction is absolutely the most useful portion of math  to take that abstraction, know how to use it, and realize that things can be used in more than one case. Humans are awesome in that regard. For example, when working with dogs, you have to 'proof' the behavior. Say you're teaching a dog to sit. If you teach that inside in your living room and nowhere else, chances are they won't sit anywhere else. You have to teach them in each place. While dogs will pick up on it quicker, they still need to be taught how to do it in multiple environments.
This is not the abstraction I am discussing but the abstraction of the teaching materials. For instance, teaching measures of central tendency is frequently done with data sets which are superficially real such as height of nonexistent students in a classroom. The information lacks concrete visualization and is irrelevant to the student performing the work. Instead, the class could measure themselves. Maybe our student does not have a good approximation for 4' 6" or 1.37m, but they know how tall the student who we got the measurement from is. As well, this data helps visualization of outliers, and we can discuss measures of central tendency and contrasting usefulness in our data set.
Abstraction to generalization happens from this concrete activity because you are still teaching the same concepts, and I would argue this ability to apply the concepts to other scenarios occurs more rapidly with concrete activity because you have simply taken data from the world rather than needing prepared sets which are made to function.
KestrelLowing wrote:Ah, here's the video, it's a ted talk. It's really good if I remember correctly.
Big fan of Dan and his methods and of TEDtalks in general (which includes a talk on computers for calculation in math education: Conrad Wolfram). Recently used his water cooler filling activity with several groups of students by getting an actual water cooler and hose and having them ask the essential question and gather all of the data. One of the most important moments in the activity came when students were measuring diameter and height of the outside of the cooler to calculate volume. All of the objects are conveniently wireframed with volumeless walls in abstractions.
Bakemaster wrote:In reality
Not to disagree with the points you are making, but I do not see this as much of a reason to teach to the test. Teachers do, often, by choice, while many do by requirement, but those who do not teach to the test will do students a far greater service.
Bakemaster wrote:what it is is the fastest route to a student being able to demonstrate proficiency on the next test.
The lessons often require as much time in class as any others. What is lengthened is teacher preparation time to create an authentic experience. In the watercooler lesson above I taught calculating circumference, volume of cylinders, calculating rate of flow, and reinforced several skills of measurement of length and time, multiplication, division, and changing fractions to decimals. As well, those students understand how to annex zeros to avoid remainders. This took me an hour and a half to complete, or two days worth of math for more than two days worth of concepts. The measures of central tendency referenced above was done in about the same time, and this is about a week of material.
Teachers lock themselves into this notion of teaching to the test being the expedient route, but this is not true. Students acquire concepts faster, retain them longer, and have deeper understanding, all within the same period of time. Part of the advantage of my method is the material is developmentally appropriate and interesting, which helps to reduce when lessons do become bulky, because they can be.
Bakemaster wrote:I'm simply suggesting that the benefits of this approach are disproportionately realized by the students with the least need, which is to say, those students who have yet to fall particularly far behind.
This is completely inaccurate. Such methods activate multiple modalities and create meaningful experiences and emotion. Those help with the retention and recollection of all students as well as serving to maintain interest. The thing with students with the least needs in the stereotypical school is those students are able to succeed in environments of academic abstraction. The thing with students with the most needs in the stereotypical school is those students are not able to. Hence, we minimize the academic abstraction.
Bakemaster wrote:There are few exciting, conceptual math topics to cover until after basic arithmetic, by which point the damage has already been done for the students with the most need.
You are viewing the point through a narrow window of what is exciting in mathematics to you, I feel. Addition is not exciting. Snapblocks are according to a group of primary ESE students. Addition with snapblocks is therefore exciting. Using the snapblocks gives them kinesthetic and visual information while I provide auditory information, and the students are active and engaged and happy.
Interesting point which intersects with previous comment about students with most needs is support strategies for ESOL, ESE, Title 1, and such students tends to be helpful for all students.
 doogly
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Re: How can we make math less scary?
Or, the tl;dr version:
Bakemaster: "For underprivileged, underpepared students, you may have to go with the more efficient techniques, even if they are more superficial."
Alia: "They is not actually more efficient. This particular cake is a lie."
Bakemaster: "For underprivileged, underpepared students, you may have to go with the more efficient techniques, even if they are more superficial."
Alia: "They is not actually more efficient. This particular cake is a lie."
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
 Bakemaster
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Re: How can we make math less scary?
I don't think I've made any claims about efficiency. (Who is Alia?) I can cut up a fresh pineapple in ten seconds with a dull knife, but it's not going to be done well or efficiently. The problem isn't that I'm using a bad technique, it's that there is no good pineapplecutting technique that can be performed in ten seconds with a dull knife.
Maybe part of the reason I feel the way I do is that I've never had the experience of being able to direct the course of study of a student over an extended period. As a tutor, I've always played second fiddle to the lesson plan dictated by the instructor. For one group of students studying conceptual physics in particular, I was able to spend a good amount of time doing things in what I felt was the "right" way, but of course the entire nature of the course was conceptual. When I showed the group e.g. how to make a cube out of a flat piece of paper as part of a demonstration about relationships between surface area and volume, that was a successful activity because their understanding was exactly what was being tested. The way they were being assessed had good synergy with the best way for them to learn. But for many of the students with whom I worked on algebra and trig, it often seemed that an interesting presentation only inhibited their ability to solve the problems they were assigned. Even when they clearly understood the concept, they would require so much more time and guidance in order to translate that understanding into something that allowed them to solve a problem that I started to feel like I was doing more harm than good. At least one of the instructors agreed, and we discussed how I might modify my approach to be more effective. When I went back to hammering things like FOIL and SOHCAHTOA—tools for learning by rote, pretty useless conceptually—I was much more successful with these students, insofar as they would make fewer mistakes as we went through problems, feel more confident about their ability to achieve, and retain more from meeting to meeting. I didn't get them excited about math but I did get them excited about the fact that the math was becoming approachable when it hadn't been previously, and that was reflected in their performance.
So, if I had been their instructor instead of their tutor, what would have been different? This is not a rhetorical question. Instead of having an hour or two of oneonone time with a student each week, I would spend three to five hours a week teaching several dozen students at once. My intuition tells me that this is a much more difficult task. If this is a group of students who have never been on an honors track, most of whom are struggling both inschool and out, what's the best favor their teacher can do them? How many deficiencies in prerequisite material can any teacher possibly repair in the course of a term while also covering a full term's worth of new material, such that the students not only can pass the standardized test that determines whether they graduate, but also have a firm conceptual grasp and an appreciation for the role that math can play in their daily lives? Much less become excited about a subject that has plagued them and everyone they know for years in school. I like Freedom Writers as much as the next guy, but where does Hollywood end and real life begin? (Stand and Deliver might be the better reference but I have to admit I haven't seen it.)
EDIT: This all comes across as very cynical and for that I am sorry. I don't want to be cynical. What I would prefer to be is skeptical but welcoming of the evidence that will show me what is possible. I keep thinking about this one guy, I don't want to talk about him much because he's such an extreme example; a mid50s excon with practically no powers of retention, studying basic algebra at a community college. He had so many conceptual deficits that I would try to repair, and as we worked together he would understand, and in the next meeting it would be gone. What he needs is extreme and very different from what students in general need, but I can't stop thinking about it.
Maybe part of the reason I feel the way I do is that I've never had the experience of being able to direct the course of study of a student over an extended period. As a tutor, I've always played second fiddle to the lesson plan dictated by the instructor. For one group of students studying conceptual physics in particular, I was able to spend a good amount of time doing things in what I felt was the "right" way, but of course the entire nature of the course was conceptual. When I showed the group e.g. how to make a cube out of a flat piece of paper as part of a demonstration about relationships between surface area and volume, that was a successful activity because their understanding was exactly what was being tested. The way they were being assessed had good synergy with the best way for them to learn. But for many of the students with whom I worked on algebra and trig, it often seemed that an interesting presentation only inhibited their ability to solve the problems they were assigned. Even when they clearly understood the concept, they would require so much more time and guidance in order to translate that understanding into something that allowed them to solve a problem that I started to feel like I was doing more harm than good. At least one of the instructors agreed, and we discussed how I might modify my approach to be more effective. When I went back to hammering things like FOIL and SOHCAHTOA—tools for learning by rote, pretty useless conceptually—I was much more successful with these students, insofar as they would make fewer mistakes as we went through problems, feel more confident about their ability to achieve, and retain more from meeting to meeting. I didn't get them excited about math but I did get them excited about the fact that the math was becoming approachable when it hadn't been previously, and that was reflected in their performance.
So, if I had been their instructor instead of their tutor, what would have been different? This is not a rhetorical question. Instead of having an hour or two of oneonone time with a student each week, I would spend three to five hours a week teaching several dozen students at once. My intuition tells me that this is a much more difficult task. If this is a group of students who have never been on an honors track, most of whom are struggling both inschool and out, what's the best favor their teacher can do them? How many deficiencies in prerequisite material can any teacher possibly repair in the course of a term while also covering a full term's worth of new material, such that the students not only can pass the standardized test that determines whether they graduate, but also have a firm conceptual grasp and an appreciation for the role that math can play in their daily lives? Much less become excited about a subject that has plagued them and everyone they know for years in school. I like Freedom Writers as much as the next guy, but where does Hollywood end and real life begin? (Stand and Deliver might be the better reference but I have to admit I haven't seen it.)
EDIT: This all comes across as very cynical and for that I am sorry. I don't want to be cynical. What I would prefer to be is skeptical but welcoming of the evidence that will show me what is possible. I keep thinking about this one guy, I don't want to talk about him much because he's such an extreme example; a mid50s excon with practically no powers of retention, studying basic algebra at a community college. He had so many conceptual deficits that I would try to repair, and as we worked together he would understand, and in the next meeting it would be gone. What he needs is extreme and very different from what students in general need, but I can't stop thinking about it.
c_{0} = 2.13085531 × 10^{14} smoots per fortnight
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 doogly
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Re: How can we make math less scary?
Bakemaster wrote:(Who is Alia?)
Latin for others. Yay for Latin.
But yeah, tutoring is a whole different ball game. You gotta play by the teacher's style, cause assessment happens once a week. There's just no time to play in there. I got two sections of calc, we have a common section on the final, mostly straightforward computations, that I'm thinkin the kiddies should rock. Certainly do worse on for having a somewhat more conceptual sort of time in my class. (the differences may not be profound but I do what I can.)
I noticed the same thing going from a TA to an instructor. I mean, there was some success having a different style from the prof, people who hated lectures could get some refuge with the TA, but for the most part the students want to see the styles match.
Prereqs is a big problem for me right now. I got freshmen. Lots of issues with preparation, and it is what it is, I can't even yell at other people in the department for passing them when they shouldn't. I could yell at admissions, but they pay tuition so that wouldn't go over well. So, I try to cope. I think it's actually not so awful with calc, because calc unifies a lot of topics. All that triangle trig gets extra context with waves and d/dx sin = cos and new relationships and so on. And what the hell are rules for exp and log doing in precalc, who cares about e^x unless you are taking calc? In principle they should know that ln(x^2) = 2lnx before showing up but I don't mind spending new time on logs here. Now logs have a purpose though. It works out.
If they can't do (a+b)^2 = a^2 + b^2 + that pesky 2ab in calc 1 though, shit be serious, we might be in the wrong place. Classes exist for you to take that maybe you should be in. As a sort of cruel joke we expect you not to notice, that class is called "College Math."
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: How can we make math less scary?
If a third grader hasn't studied the way lasagna takes longer to cook than spaghetti, he or she is vulnerable to miseducation to the effect that "size is relative, the genius Einstein said so."
Re: How can we make math less scary?
Bakemaster wrote:For one group of students studying conceptual physics in particular, I was able to spend a good amount of time doing things in what I felt was the "right" way, but of course the entire nature of the course was conceptual.
Older students are generally able to work with the abstract with little difficulty. Further, older students generally work with material impractical or unsafe to experience directly. The more authentic and more modalities engaged the better, but there is no detrimental effect to teaching via the abstract for students fully able to grasp the abstract. However, students have generally been lost to math before this development occurs, and this is the period I am primarily discussing.
Bakemaster wrote:But for many of the students with whom I worked on algebra and trig, it often seemed that an interesting presentation only inhibited their ability to solve the problems they were assigned.
Should differ between "interesting" for the sake of generating interest and "interesting" by nature of developmentally appropriate and active. Teachers, sadly, incorporate a whole mess of fluff in order to generate interest in lessons which remain developmentally inappropriate and passive. This is the basis of the discussion, for me at least; math is scary due to being developmentally inappropriate and passive for the learner.
Bakemaster wrote:So, if I had been their instructor instead of their tutor, what would have been different? This is not a rhetorical question. Instead of having an hour or two of oneonone time with a student each week, I would spend three to five hours a week teaching several dozen students at once.
Oh yeah. Managing whole group instruction is an entirely different beast. Simply having individualized attention provides gains for students, not to mention the fluid differentiation and pacing. Whole group, you have to maintain pace and provide learning which is conducive to the learning of the broadest section of students, and differentiating is impractical.
Bakemaster wrote:If this is a group of students who have never been on an honors track, most of whom are struggling both inschool and out, what's the best favor their teacher can do them? How many deficiencies in prerequisite material can any teacher possibly repair in the course of a term while also covering a full term's worth of new material
Every aspect of education needs change, to me. How we progress students is major. This teacher should not responsible for taking care of deficits in previous material; review, yes, but not the instruction itself. But, the current reality is what it is, and you do what you can. Covering the material and making standard progress is not accidental or simple in itself, and several years of deficits in instruction and intervention creates further problems.
 doogly
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Re: How can we make math less scary?
Vangor wrote:Should differ between "interesting" for the sake of generating interest and "interesting" by nature of developmentally appropriate and active. Teachers, sadly, incorporate a whole mess of fluff in order to generate interest in lessons which remain developmentally inappropriate and passive. This is the basis of the discussion, for me at least; math is scary due to being developmentally inappropriate and passive for the learner.
Ah yes, word problems where every piece of information is pure window dressing / distraction! My favoritest ever.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: How can we make math less scary?
People are scared of math as it is some kind of "science fiction". Please don't be! Everything you will need to learn you can accomplished with practice. Persistent people always better pass in life that smart ones. So keep in your mind that you do not have to be smart to learn math, just you might have easier initial understanding of math, doesn't mean that you will not learn it. So just be persistent, that is key in life.
Re: How can we make math less scary?
I think the most useful thing to making math less scary to students is to change the cultural perception of math. I remember when I was a kid, while I was still learning all the basic functions, many Saturday morning cartoon shows (and the like) would use algebra as the goto school boogeyman. A lot of adults followed suit, treating algebra as a rather difficult subject. After learning algebra, it wasn't bad at all. Math is perceived by a majority of society as hard, very hard even. If you can make it so people think of it as just like any other subject with respect to difficulty, students will follow and stop seeing it as a scary subject.
Of course, that's a bit of a chicken / egg solution, so possibly not that useful to the discussion.
Going back to something mentioned at the beginning of the thread, another big thing would be making sure math teachers enjoy and are good at math my best math teachers in k12 included one who was really really into math, and another who studied nuclear physics in college, and both would frequently divulge into offtopic diversions to show us interesting problems or solutions. The worst stuck with a by the books method. If the person teaching the subject doesn't like it, or isn't good at it, they won't do a good job teaching it.
Of course, that's a bit of a chicken / egg solution, so possibly not that useful to the discussion.
Going back to something mentioned at the beginning of the thread, another big thing would be making sure math teachers enjoy and are good at math my best math teachers in k12 included one who was really really into math, and another who studied nuclear physics in college, and both would frequently divulge into offtopic diversions to show us interesting problems or solutions. The worst stuck with a by the books method. If the person teaching the subject doesn't like it, or isn't good at it, they won't do a good job teaching it.
Re: How can we make math less scary?
Here's a radical solution: don't teach math. Teach Physics. You need all the way up through easy Calc problems in an entry level college Physics course. So, replace all higherlevel math (algebra, trig, etc.) with a real subject that uses that math. If physics doesn't cover everything, teach Statics. The concepts are easy, and perfectly suitable to highschool.
By mixing the math with problemsolving, you will not only make it more interesting, but more useful as well, at least to anyone entering one of the STEM fields. So far as I can tell, that is most of what higherlevel math is prep for anyhow; you don't need trig to become a lawyer.
By mixing the math with problemsolving, you will not only make it more interesting, but more useful as well, at least to anyone entering one of the STEM fields. So far as I can tell, that is most of what higherlevel math is prep for anyhow; you don't need trig to become a lawyer.
 doogly
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Re: How can we make math less scary?
You just called trig higher level math. I lolled.
But anyway, yes, that seems to make sense. But what kind of math class isn't full of examples from physics, biology, economics, and of course the joys of pure math? There ought to be mutual reinforcement, not subjugation of one of the fields.
This is essentially what I did in high school though, what you are proposing. I took AP Physics that was calc based, and took no math at all senior year. Coreqs be damned, my teacher was happy to sign off on exceptions. AP Calc is just stretching a month's worth of ideas into a year's worth of homework. Terrible. So I am certainly sympathetic to the proposal. Just make sure to give physics doublelong periods if that's what the deal is.
But anyway, yes, that seems to make sense. But what kind of math class isn't full of examples from physics, biology, economics, and of course the joys of pure math? There ought to be mutual reinforcement, not subjugation of one of the fields.
This is essentially what I did in high school though, what you are proposing. I took AP Physics that was calc based, and took no math at all senior year. Coreqs be damned, my teacher was happy to sign off on exceptions. AP Calc is just stretching a month's worth of ideas into a year's worth of homework. Terrible. So I am certainly sympathetic to the proposal. Just make sure to give physics doublelong periods if that's what the deal is.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.
Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?
Re: How can we make math less scary?
doogly wrote:You just called trig higher level math. I lolled.
But anyway, yes, that seems to make sense. But what kind of math class isn't full of examples from physics, biology, economics, and of course the joys of pure math? There ought to be mutual reinforcement, not subjugation of one of the fields.
Well, trig is kind of higher level. Certainly, lowerorder creatures seem to have difficulty with it. (Seriously, ever try teaching your pet slime mold sohcahtoh? Tough row to hoe, that is...)
The difference between math with a side of science, and science with a side of math, is the difference between a simple problem drowning in irrelevant details, and a minimally constructed problem with an elegant solution. One way I hate you, one way I love you.
I felt the same way in undergrad too; I learned more calculus in Fluids, in less time, than I ever did in calculus.
 Zamfir
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Re: How can we make math less scary?
doogly wrote:You just called trig higher level math. I lolled.
Then again, the far majority of people will never use trigonometry in their lives outside of high school tests. Even people who got a good grip on it in high school will mostly ignore it for the next years, and when they ever enocunter a situation where it could come in handy, they have forgotten too much of the details. So perhaps it's "lower maths", but there's not really a higherthantrig maths for most people.
It's very tempting to see high school maths as an introduction, a preparation for the real stuff later on. Typical high school curricula are filled with that stuff: trigonometry, calculus, logarithms, they're taught because they are breadandbutter for hardscience people. But not the obvious things to teach as last maths that someone will study.
And as far as I can tell, people catch on to that pretty quickly and accurately. If they have to study intromathformathypeople, it makes perfect sense to see the subject as pointless.
There might be a similar risk in using physics as guidance for maths. There's a lot of physics that's important for people to know about, but being able to calculate stuff is far less important. If you turn physics into a calculation class, it becomes less interesting for people who want to have a grip on how the world around them works conceptually, in favour of people who are preparing for a more indepth schooling after high school.
Re: How can we make math less scary?
So, I may or may not be stealing this directly from Lockhart; that essay inspired me to read a lot (first saw about two years ago) and I've since had a lot of authors and ideas muddled in my mind. It seems to me that there is a subject called "Mathematics" which truly ought to be an elective beginning around high school, and a subject called "Computation" which might be tested, and/or applied in other subjects. Mathematics should be one of those, but so should the more applied sciences.
Excessive? No. Don't tell me this is ridiculous when English was split into two classes called "Literature Studies" and "Language Arts"  'reading' and 'writing' respectively, for those not used to the edubabble. I'm not against splitting English. In fact, I feel this split was, if not totally reasonable, at least entirely justifiable. And it seems that Math has two components also, and the fact that one is practical and the other is theoretical doesn't help. Now, we could argue the same for any of the sciences, but it seems to me that in (say) Physics, there is a great deal of communication between the theorists and the experimentalists. But in math, there is no 'real world' to keep the theorists in check; the only way to test the theory is against more established theory. This is a facet of the sciences that I am particularly ignorant about, so if I am totally wrong about this I'd like to hear why.
What wouldn't be in Computation that is currently in math classes? Introductory Algebra should probably be outsourced to Mathematics because it would allow more time to get students comfortable with the concept of a variable, the distinction between equations and expressions, etc., etc., meanwhile more time could be spent with making sure that they get fractions, dammit (also solving equations, and later systems).
There are some people who would argue that Computation is unnecessary in the age of computers. I'm not discounting this viewpoint, but it's not a claim with which I have ever been completely convinced. Without resorting to hyperbole ("How will I solve 4+3 without pulling out a calculator every time if I never learn it?!") I think that I probably would never have learned mental multiplication if I didn't have to, and there are times when that would have been completely obnoxious.
In any case, I think that this change would be a rather easy one to sell to the parents ("Your kids will be learning MORE math!"), and the bureaucrats ("Twice as much math means SUPAR test scores!"). Historically, though, saying you're going to "eliminate drillandkill"  regardless of how flowery you dress up that statement and how much data you have to back it up  has tended to create a rather polarizing atmosphere, to say the least.
So that's what we want of Computation; what do we want in Mathematics? Proofs, of course! And not Geometrystyle proofs (Ah, hello there, Lockhart!); maybe later we can talk about that level of rigor, but not as a first introduction. For now, let's focus on games, let's talk about puzzles, let's learn some number tricks, then figure out why they work. We can teach proof techniques, maybe some logic. If I had (a lot) more time, I might want to think about curriculum, because it would obviously have to be done very carefully, to be developmentally appropriate  even if it's an elective, it still has to be effective.
** It occurs to me that it might be less arrogant to call it "Logic" instead of "Mathematics" but I definitely don't know enough of the history or psychology of the situation to say much.
In response to some other people: Zamfir's post is actually what made me think that my "Mathematics", or at least highschool Mathematics, should be an elective. I remember my high school chemistry course, and it definitely had the feel of "People who like Chemistry should know this stuff and the rest of you… well, yeah." I happen to be blessed with sufficient smarts/work ethic to have made it through the class relatively unscarred, but a lot of the material left me bored for that reason.
Byrel: The idea is interesting  I feel that might be a bit too much pressure on the science classes, but like I said, I'm rather ignorant on science pedagogy, so I'll refrain from commenting more than a positive grunt. I do know it would be hard to sell to the parents for the obvious reasons, but I feel that with a bit of effort the textbooks might come around.
ars111: I feel that you should probably read Ghostbear's post right below yours. You have the right idea in terms of thinking positive, but you can't think positive in the face of years of difficulties combined with the cultural situation as it is.
doogly: Dan Meyer has been on a rant about the terrible use of media to present math problems since basically his blog started.
Vangor: I'm not sure what the politics are for not doing away with NCLB, because I feel that we have enough of the populace enraged about education (whether or not the standardized tests are their concern) that it wouldn't be complete political suicide to try to repeal it. However, I agree that our current testing structure doesn't do what it's designed to do, and what it does do doesn't seem to be particularly useful. On the other hand, the everpresent question of what will replace it; that's a much harder problem to tackle.
Excessive? No. Don't tell me this is ridiculous when English was split into two classes called "Literature Studies" and "Language Arts"  'reading' and 'writing' respectively, for those not used to the edubabble. I'm not against splitting English. In fact, I feel this split was, if not totally reasonable, at least entirely justifiable. And it seems that Math has two components also, and the fact that one is practical and the other is theoretical doesn't help. Now, we could argue the same for any of the sciences, but it seems to me that in (say) Physics, there is a great deal of communication between the theorists and the experimentalists. But in math, there is no 'real world' to keep the theorists in check; the only way to test the theory is against more established theory. This is a facet of the sciences that I am particularly ignorant about, so if I am totally wrong about this I'd like to hear why.
What wouldn't be in Computation that is currently in math classes? Introductory Algebra should probably be outsourced to Mathematics because it would allow more time to get students comfortable with the concept of a variable, the distinction between equations and expressions, etc., etc., meanwhile more time could be spent with making sure that they get fractions, dammit (also solving equations, and later systems).
There are some people who would argue that Computation is unnecessary in the age of computers. I'm not discounting this viewpoint, but it's not a claim with which I have ever been completely convinced. Without resorting to hyperbole ("How will I solve 4+3 without pulling out a calculator every time if I never learn it?!") I think that I probably would never have learned mental multiplication if I didn't have to, and there are times when that would have been completely obnoxious.
In any case, I think that this change would be a rather easy one to sell to the parents ("Your kids will be learning MORE math!"), and the bureaucrats ("Twice as much math means SUPAR test scores!"). Historically, though, saying you're going to "eliminate drillandkill"  regardless of how flowery you dress up that statement and how much data you have to back it up  has tended to create a rather polarizing atmosphere, to say the least.
So that's what we want of Computation; what do we want in Mathematics? Proofs, of course! And not Geometrystyle proofs (Ah, hello there, Lockhart!); maybe later we can talk about that level of rigor, but not as a first introduction. For now, let's focus on games, let's talk about puzzles, let's learn some number tricks, then figure out why they work. We can teach proof techniques, maybe some logic. If I had (a lot) more time, I might want to think about curriculum, because it would obviously have to be done very carefully, to be developmentally appropriate  even if it's an elective, it still has to be effective.
** It occurs to me that it might be less arrogant to call it "Logic" instead of "Mathematics" but I definitely don't know enough of the history or psychology of the situation to say much.
In response to some other people: Zamfir's post is actually what made me think that my "Mathematics", or at least highschool Mathematics, should be an elective. I remember my high school chemistry course, and it definitely had the feel of "People who like Chemistry should know this stuff and the rest of you… well, yeah." I happen to be blessed with sufficient smarts/work ethic to have made it through the class relatively unscarred, but a lot of the material left me bored for that reason.
Byrel: The idea is interesting  I feel that might be a bit too much pressure on the science classes, but like I said, I'm rather ignorant on science pedagogy, so I'll refrain from commenting more than a positive grunt. I do know it would be hard to sell to the parents for the obvious reasons, but I feel that with a bit of effort the textbooks might come around.
ars111: I feel that you should probably read Ghostbear's post right below yours. You have the right idea in terms of thinking positive, but you can't think positive in the face of years of difficulties combined with the cultural situation as it is.
doogly: Dan Meyer has been on a rant about the terrible use of media to present math problems since basically his blog started.
Vangor: I'm not sure what the politics are for not doing away with NCLB, because I feel that we have enough of the populace enraged about education (whether or not the standardized tests are their concern) that it wouldn't be complete political suicide to try to repeal it. However, I agree that our current testing structure doesn't do what it's designed to do, and what it does do doesn't seem to be particularly useful. On the other hand, the everpresent question of what will replace it; that's a much harder problem to tackle.
Approximately 100% of my forum contribution is in Nomic threads! In fact, if you're reading this signature, you probably knew that because you're reading a Nomic thread! But did you know that I've participated in both Nomic 16.0 AND Nomic 15.0? Woah!
Re: How can we make math less scary?
Zamfir wrote:doogly wrote:You just called trig higher level math. I lolled.
Then again, the far majority of people will never use trigonometry in their lives outside of high school tests. Even people who got a good grip on it in high school will mostly ignore it for the next years, and when they ever enocunter a situation where it could come in handy, they have forgotten too much of the details. So perhaps it's "lower maths", but there's not really a higherthantrig maths for most people.
It's very tempting to see high school maths as an introduction, a preparation for the real stuff later on. Typical high school curricula are filled with that stuff: trigonometry, calculus, logarithms, they're taught because they are breadandbutter for hardscience people. But not the obvious things to teach as last maths that someone will study.
And as far as I can tell, people catch on to that pretty quickly and accurately. If they have to study intromathformathypeople, it makes perfect sense to see the subject as pointless.
There might be a similar risk in using physics as guidance for maths. There's a lot of physics that's important for people to know about, but being able to calculate stuff is far less important. If you turn physics into a calculation class, it becomes less interesting for people who want to have a grip on how the world around them works conceptually, in favour of people who are preparing for a more indepth schooling after high school.
You can say that about any subject though. When am I ever going to need to use this second language I was forced to take for 4 years? Many students won't ever be in a job or situation where that skill is required. Same thing for English, if a student goes into a trade there aren't likely to ever need to know how to analyze poetry. Same thing for history, yes its good to know about large historical figures, but do you need to know about family life in Medieval Europe or about the Reformation in great detail? If high school subjects are so useless why not just stop in 8th grade?
Trig and Calc can have a few uses in the everyday world you just need to know how to look for them, teachers should provide more examples and base their classes of real world situations if student interest is a problem.
However, the far more important part about math and any hard science is that it teaches students to think. Math teaches you quantitative and analytical problem solving skills which are extremely valuable. History and English teach you critical thinking skills. A second language opens up another culture to you. These classes have value outside of the direct knowledge of the subject one learns.
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 Zamfir
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Re: How can we make math less scary?
There are lots of topics that can be used to teach people to think. Within mathematics and outside of it. The question is, which of those many options do you choose for the curriculum.
Trigonometry or calculus are hardly particularly good things to help people learn to think. At the basic level they are mostly mechanical recipes to solve particular kinds of problems, without much need to understand the background. Type the "cos" button in this situation and the "sin" button in that situation, put the superscript in front of the x then lower the superscript by 1, freaking doubleangle formulas. They just happen to be recipes for problems that are extremely common in the physical sciences and in engineering, so for some people they are important tricks to master.
If you want people to think, some of the mainstays of high school math just aren't very good examples. They tell a story of math that comes down "a bunch of formulas, buttons on your calculator, figure out which ones to use in this situation". Realworld example cases mostly end up with thinlydisguised engineering problems, and that's no accident. Learning a fairly mechanical approach to solve certain probems can be inspiring and rewarding, but only if you wanted to solve those problems.
I really can't comment on your secondlanguage example, for obvious reasons. But my wife is a history teacher in training, and one thing that stands out is that historians do discuss the contents of school curricula a lot, with each other and with people outside of the field. And the curriculum changes all the time, in response to changing opinions. While math curricula are very static, and basically determined by the entry requirements of hardscience departments. It's treated far more as preparation for more advanced subjects than as a goal in itself.
Trigonometry or calculus are hardly particularly good things to help people learn to think. At the basic level they are mostly mechanical recipes to solve particular kinds of problems, without much need to understand the background. Type the "cos" button in this situation and the "sin" button in that situation, put the superscript in front of the x then lower the superscript by 1, freaking doubleangle formulas. They just happen to be recipes for problems that are extremely common in the physical sciences and in engineering, so for some people they are important tricks to master.
If you want people to think, some of the mainstays of high school math just aren't very good examples. They tell a story of math that comes down "a bunch of formulas, buttons on your calculator, figure out which ones to use in this situation". Realworld example cases mostly end up with thinlydisguised engineering problems, and that's no accident. Learning a fairly mechanical approach to solve certain probems can be inspiring and rewarding, but only if you wanted to solve those problems.
I really can't comment on your secondlanguage example, for obvious reasons. But my wife is a history teacher in training, and one thing that stands out is that historians do discuss the contents of school curricula a lot, with each other and with people outside of the field. And the curriculum changes all the time, in response to changing opinions. While math curricula are very static, and basically determined by the entry requirements of hardscience departments. It's treated far more as preparation for more advanced subjects than as a goal in itself.
Re: How can we make math less scary?
Turiski wrote:So that's what we want of Computation; what do we want in Mathematics? Proofs, of course! And not Geometrystyle proofs (Ah, hello there, Lockhart!); maybe later we can talk about that level of rigor, but not as a first introduction. For now, let's focus on games, let's talk about puzzles, let's learn some number tricks, then figure out why they work. We can teach proof techniques, maybe some logic. If I had (a lot) more time, I might want to think about curriculum, because it would obviously have to be done very carefully, to be developmentally appropriate  even if it's an elective, it still has to be effective.
** It occurs to me that it might be less arrogant to call it "Logic" instead of "Mathematics" but I definitely don't know enough of the history or psychology of the situation to say much.
You know, I hadn't really thought about the pure mathematics when I suggested totally eliminating it. You are right; there is a place for number theory, axiomatic geometry, game theory, etc. Perhaps some such elective should be available. But this math is completely distinct from what will be useful in most technical backgrounds. The real reason most people learn Algebra, Trig, Geometry (aside from being forced to), is not to then reason about the results. It is to use them. This math is the core of practical science and engineering.
So, I guess I agree that a Math (or Logic) course should be maintained, but I have to disagree at where to draw the line. Introductory Algebra? Algebra is at the core of any mathcentric field. Algebra, with trig identities and calculus stacked on top. If you want to separate useful math from pure math, then the useful math has to include algebra and trig at a minimum.
On the other hand, this is still teaching people math that won't be necessary for their occupation. (How many lawyers need algebra? Burgerflippers? Truck drivers?) Maybe we should split math into two parts, Arithmetic (fractions, etc.), Logic (pure math, number theory, etc.) and absorb the classes that are direct STE prep into the science classes (particularly Physics).
And, as a couple folks have pointed out, increase the time spent in physics to compensate for the additional material.
On the other, other hand, maybe teaching people math they won't need will be helpful. As big boss pointed out, a lot of high school is not designed to prepare us specifically for our careers. My dad went so far as to say the goal of highschool was a trained mind; not really any of the knowledge in it. Certainly most college students I knew that had trouble in mathbased classes didn't struggle with the math; they struggled with problem solving skills, when applying it to math. They could recite all the formulas, properties, integrals. (As Zamfir noted, d(x^n)/dx is trivial.) But they still got Fs. Why? Because they couldn't solve the problems. They didn't understand when to apply those rules, functions, etc. Like the old joke says, you earn one dollar for hitting things with hammers, and 100 dollars for knowing where to hit them. This is a skill that isn't taught in history, theoretical math, english, etc. So far as I know, it is only taught in hard science classes and math. (Don't get me wrong, those other classes do teach skills. Just not this skill.) So I wouldn't be opposed to forcing people who will never use this particular field in their career to take Math/Mathematical Physics. It may very well improve the efficiency of their mind in the long run.
Re: How can we make math less scary?
KestrelLowing wrote:I guess I never really thought that people loose interest at algebra  that's when I finally started being interested. That's when it actually seemed realworld to me. Of course, I'm currently an engineering student, so that's obviously a huge bias here.
Well, arithmetic is pretty obviously applicable to the real world. Remember the basic word problems: if I have two apples and you give me three, I now have five apples; if I have six apples and give you two, I now have four; if two people each have two apples, there are four apples total; if there are eight slices of pizza and four people, each person gets two slices.
Now bring in fractions/decimals. Well, okay, if two people are sharing a five dollar cost, each pays 2$50. There's a money analogy for adding and subtracting negative numbers, too: if I have 50$ in my bank account and write a 75$ cheque, I'm 25$ overdrawn. If I then deposit 30$, I have 5$ in my account. People deal with money almost every day, so it's easy enough to understand.
Then come the formulas. I need to buy some notebooks for two dollars each and erasers for three dollars each  how much will that cost? Again, easy, realworld problems.
Now we're in algebra. Some things make sense (I spend 5$ a month to get 100 texts and each text after that costs 5¢) but others... not so much. Most people have no reason to find the intersection of three parabolas, for example. Once at this stage, for most people, it's just mathturbation. It may be fun, but it doesn't get anything done.
Re: How can we make math less scary?
This is a huge thorn in my side! I do tutoring at a community college, and we get all sorts of people. A lot of them have had terrible experiences with math previously, and didn't understand it in high school or even grade school. Some of them haven't touched it in years and are now coming back to school. They are often frightened, stressed, and most have convinced themselves (or been convinced by others!) that they are bad at math. The idea of people doing math for fun is something totally foreign to them. They don't care about it because they have no reason to care; it's merely a gatekeeper for their real goals. I tend to agree with them. The algebracalculus sequences seem mostly designed for "just in case you want to do engineering" and students are rushed through it with very little real justification or motivation otherwise. I'm coming from a perspective of attempting to repair damage that has already been done; it's probably much easier to avoid some of the damaging behaviors in the first place.
I strongly suspect that any standardized curriculum for math won't cut it. Some people want or need more real world applications, but if there's one thing I've noticed, it's that different people approach mathematical ideas with different intuition and they also get stuck in different areas. What's obvious to one person is not obvious to another, so there will always be some people who are alienated. I've personally had more luck with analogies and applications to music theory and art than I've ever had with physics or engineering ideas. It's much easier to get a student to enjoy math by relating it to abstract ideas that they already love.
The scariest thing about math that I've seen? The culture. The ability to have correct and incorrect solutions seems to somehow make all sorts of really irritating behavior perfectly acceptable. Students come to fear being wrong, and won't make an attempt or talk about their ideas for fear of being shot down. If students aren't willing to jump in and try a few things to see what happens, they're going to be doing math only by formula and will only be able to deal with problems that they have a template for. A computer can do that; you may as well develop ability to follow instructions by assembling modular furniture in my opinion. Math decreases rapidly in utility for most people with this approach after arithmetic, not to mention it quickly becomes much more difficult without intuition and understanding to build on. There's a lot of advice handed out about practicing more math problems, but not general problem solving tactics.
The knowitall attitude is more common than it should be, and sometimes it almost seems as though it's encouraged. There is a great deal of ego that some people have wrapped up in their mathematical abilities, and there is a lot of reluctance to admitting to being wrong or even unsure. However, there's no problem with pointing out when other people are wrong. You see this in the "advanced students" very frequently, and unfortunately also in teachers and tutors. Sometimes they are indeed very good at the subject, but they can still be very damaging to others trying to learn and often are quick to label others as stupid. The ones who know relatively little mathematics and have this attitude are perhaps the most dangerous. I certainly dislike being around these sorts of people for extended periods of time, and I most definitely do not aspire to be one. Thankfully, not all math people are like this, but we have some bad eggs that make themselves known!
There is also a reluctance to acknowledge the intuition of others; most everyone I encounter has some "best way" of looking at a concept and cannot fathom teaching or explaining it any other way. Granted, it is much more difficult to work on the fly with other people's thought processes, but a total lack of willingness to even try can be very demoralizing for the student. People can come to feel stupid when they can't see things the "obvious" way that some "expert" does.
The very best math teachers and tutors I've seen are very flexible, and will entertain all sorts of student questions. I encounter a surprising number of students who aren't doing particularly well when it comes to grades, but have wonderful ideas and questions about creating equivalent definitions and methods in algebra and calculus. These people will ask lots of questions about material that gets glossed over in their classes, and very frequently they will get shot down with a "that's wrong" or at best a "that's too advanced and you can't understand it."
The most powerful technique I've found is "I don't know, let's find out!" It's great for students to see processes of discovery and even better if they're involved themselves. Get them asking why questions, and ask them why questions if they aren't already asking themselves. If they're wrong, lead them to figure out why they're wrong and encourage them to keep generating and evaluating their own ideas. If they're right, do the same thing. This is something that's difficult to do in the time constraints of a classroom, but not impossible. As a mentor or tutor you have much more flexibility to do this.
Above all, it would be very nice to see some better attitudes and a more supportive culture. Lockhart's dream classroom would fall on its face if students weren't encouraged to try and fail.
I strongly suspect that any standardized curriculum for math won't cut it. Some people want or need more real world applications, but if there's one thing I've noticed, it's that different people approach mathematical ideas with different intuition and they also get stuck in different areas. What's obvious to one person is not obvious to another, so there will always be some people who are alienated. I've personally had more luck with analogies and applications to music theory and art than I've ever had with physics or engineering ideas. It's much easier to get a student to enjoy math by relating it to abstract ideas that they already love.
The scariest thing about math that I've seen? The culture. The ability to have correct and incorrect solutions seems to somehow make all sorts of really irritating behavior perfectly acceptable. Students come to fear being wrong, and won't make an attempt or talk about their ideas for fear of being shot down. If students aren't willing to jump in and try a few things to see what happens, they're going to be doing math only by formula and will only be able to deal with problems that they have a template for. A computer can do that; you may as well develop ability to follow instructions by assembling modular furniture in my opinion. Math decreases rapidly in utility for most people with this approach after arithmetic, not to mention it quickly becomes much more difficult without intuition and understanding to build on. There's a lot of advice handed out about practicing more math problems, but not general problem solving tactics.
The knowitall attitude is more common than it should be, and sometimes it almost seems as though it's encouraged. There is a great deal of ego that some people have wrapped up in their mathematical abilities, and there is a lot of reluctance to admitting to being wrong or even unsure. However, there's no problem with pointing out when other people are wrong. You see this in the "advanced students" very frequently, and unfortunately also in teachers and tutors. Sometimes they are indeed very good at the subject, but they can still be very damaging to others trying to learn and often are quick to label others as stupid. The ones who know relatively little mathematics and have this attitude are perhaps the most dangerous. I certainly dislike being around these sorts of people for extended periods of time, and I most definitely do not aspire to be one. Thankfully, not all math people are like this, but we have some bad eggs that make themselves known!
There is also a reluctance to acknowledge the intuition of others; most everyone I encounter has some "best way" of looking at a concept and cannot fathom teaching or explaining it any other way. Granted, it is much more difficult to work on the fly with other people's thought processes, but a total lack of willingness to even try can be very demoralizing for the student. People can come to feel stupid when they can't see things the "obvious" way that some "expert" does.
The very best math teachers and tutors I've seen are very flexible, and will entertain all sorts of student questions. I encounter a surprising number of students who aren't doing particularly well when it comes to grades, but have wonderful ideas and questions about creating equivalent definitions and methods in algebra and calculus. These people will ask lots of questions about material that gets glossed over in their classes, and very frequently they will get shot down with a "that's wrong" or at best a "that's too advanced and you can't understand it."
The most powerful technique I've found is "I don't know, let's find out!" It's great for students to see processes of discovery and even better if they're involved themselves. Get them asking why questions, and ask them why questions if they aren't already asking themselves. If they're wrong, lead them to figure out why they're wrong and encourage them to keep generating and evaluating their own ideas. If they're right, do the same thing. This is something that's difficult to do in the time constraints of a classroom, but not impossible. As a mentor or tutor you have much more flexibility to do this.
Above all, it would be very nice to see some better attitudes and a more supportive culture. Lockhart's dream classroom would fall on its face if students weren't encouraged to try and fail.
Re: How can we make math less scary?
maxh wrote:Now we're in algebra. Some things make sense (I spend 5$ a month to get 100 texts and each text after that costs 5¢) but others... not so much.
Yeah, the thing is that the vast bulk of introductory algebra is dealing with linear equations in a single unknown, and those word problems can just as easily be solved by arithmetic reasoning. (Just for lulz, I decided to practice making a word problem inspired by what I took away from Dan Meyer's TED talk, plus the way my GED students would rather solve it to show off my thinking here.)
Spoiler:
That's a good illustration of why I'm not as much on Team Lockhart as I could be. It's not just that I think that being asked to independently recreate the accumulated work of generations of gifted mathematicians is a high risk activity that threatens the selfesteem of average students. But if a student waited until being asked to solve quadratic equations or systems of equations in multiple unknowns to deduce that a more systematic abstract approach was called for, they'd be hardpressed to realize what sort of approach would be necessary.
Re: How can we make math less scary?
( This post is a work in progress  from the little I've read of the rest of this thread, I really want to keep going )
@ Byrel: You bring up some good & thoughtprovoking points. Without putting the effort into them that they deserve, here are my first reactions:
We agree on this, I think.
Yeah, I definitely understand this sentiment; and it was something I thought about when I said "elective at the high school level." This definitely didn't come through well in my last post, but I think it should be required through middle school (how early it would be started is a whole mess of things I don't really want to think about) for the precise reason of algebra  so important and yet such a radically different level of abstraction than was required in any math class beforehand.
This is a good point, and I'm not sure we disagree on this either. Personally, I'm okay with saying "Man, Biology. Wasn't very good at that," and moving on, which I suspect is how many people would treat this Mathematics/Logic class. But at least when they decide to leave Math and not look back, they at least know what they're walking away from. In the current system, "what is math" is a question whose answer at best muddled, and at worst nonexistent, in most classrooms. So lawyers could walk away from trigonometry, burgerflippers and truckdrivers could walk away from algebra, etc.
(I feel like I'm missing an important nuance but the specifics are not coming right now)
I don't know enough of the typical American culture to know whether your dad's view is particularly radical. It's certainly one that a lot of academics agree with… but math always seems to be the exception. However, it's not really  if high schools taught mathbound students the same critical thinking skills it teaches literaryanalysisbound students, then I'm not sure a college would mind much that they don't know trigonometry (after all, it's not a conceptually difficult subject by itself), conics (besides circles, they won't need them for a while), probability (which is often an elective), or statistics (which almost necessitates a conceptual understanding of the integral).
I believe this is the single biggest problem facing math education. All the things about student engagement and higher test scores, etc. etc  they are secondary to this. Perhaps this comes from a very oldfashioned view that high school should be a "liberal arts" education, but I feel the goal of fixing this memorizationconceptualization gap should drive the direction of public education policy in whatever capacity it can.
@ Byrel: You bring up some good & thoughtprovoking points. Without putting the effort into them that they deserve, here are my first reactions:
Byrel wrote:You know, I hadn't really thought about the pure mathematics when I suggested totally eliminating it. You are right; there is a place for number theory, axiomatic geometry, game theory, etc. Perhaps some such elective should be available. But this math is completely distinct from what will be useful in most technical backgrounds. The real reason most people learn Algebra, Trig, Geometry (aside from being forced to), is not to then reason about the results. It is to use them. This math is the core of practical science and engineering.
We agree on this, I think.
Byrel wrote:So, I guess I agree that a Math (or Logic) course should be maintained, but I have to disagree at where to draw the line. Introductory Algebra? Algebra is at the core of any mathcentric field. Algebra, with trig identities and calculus stacked on top. If you want to separate useful math from pure math, then the useful math has to include algebra and trig at a minimum.
Yeah, I definitely understand this sentiment; and it was something I thought about when I said "elective at the high school level." This definitely didn't come through well in my last post, but I think it should be required through middle school (how early it would be started is a whole mess of things I don't really want to think about) for the precise reason of algebra  so important and yet such a radically different level of abstraction than was required in any math class beforehand.
Byrel wrote:On the other hand, this is still teaching people math that won't be necessary for their occupation. (How many lawyers need algebra? Burgerflippers? Truck drivers?) Maybe we should split math into two parts, Arithmetic (fractions, etc.), Logic (pure math, number theory, etc.) and absorb the classes that are direct STE prep into the science classes (particularly Physics).
This is a good point, and I'm not sure we disagree on this either. Personally, I'm okay with saying "Man, Biology. Wasn't very good at that," and moving on, which I suspect is how many people would treat this Mathematics/Logic class. But at least when they decide to leave Math and not look back, they at least know what they're walking away from. In the current system, "what is math" is a question whose answer at best muddled, and at worst nonexistent, in most classrooms. So lawyers could walk away from trigonometry, burgerflippers and truckdrivers could walk away from algebra, etc.
(I feel like I'm missing an important nuance but the specifics are not coming right now)
Byrel wrote:On the other, other hand, maybe teaching people math they won't need will be helpful. As big boss pointed out, a lot of high school is not designed to prepare us specifically for our careers. My dad went so far as to say the goal of highschool was a trained mind; not really any of the knowledge in it.
I don't know enough of the typical American culture to know whether your dad's view is particularly radical. It's certainly one that a lot of academics agree with… but math always seems to be the exception. However, it's not really  if high schools taught mathbound students the same critical thinking skills it teaches literaryanalysisbound students, then I'm not sure a college would mind much that they don't know trigonometry (after all, it's not a conceptually difficult subject by itself), conics (besides circles, they won't need them for a while), probability (which is often an elective), or statistics (which almost necessitates a conceptual understanding of the integral).
Byrel wrote:Certainly most college students I knew that had trouble in mathbased classes didn't struggle with the math; they struggled with problem solving skills, when applying it to math. They could recite all the formulas, properties, integrals. (As Zamfir noted, d(x^n)/dx is trivial.) But they still got Fs. Why? Because they couldn't solve the problems.
I believe this is the single biggest problem facing math education. All the things about student engagement and higher test scores, etc. etc  they are secondary to this. Perhaps this comes from a very oldfashioned view that high school should be a "liberal arts" education, but I feel the goal of fixing this memorizationconceptualization gap should drive the direction of public education policy in whatever capacity it can.
Approximately 100% of my forum contribution is in Nomic threads! In fact, if you're reading this signature, you probably knew that because you're reading a Nomic thread! But did you know that I've participated in both Nomic 16.0 AND Nomic 15.0? Woah!
Re: How can we make math less scary?
Attitude is everything. When I try to help my fellow students with math they typically bring the wrong attitude. They assume that math isn't supposed to make sense and that I'm just going to tell them another stupid formula or method they can apply to this weird sequence of numbers and letters to get another random sequence of numbers and letters. I think that if a different attitude is fostered from the beginning, this will be different and students will feel more open to math. Unfortunately, I don't know how to do this.
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