In College, Working Hard to Learn High School Material
nytimes.com
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I've had AP students (high school seniors!) write things like this:
7x - 4 = 10, therefore 3x = 10.
It took a moment for it to sink in that 7x - 4 is not the same as 3x.
Students don't grasp the basic concepts, and nobody takes the time to catch them up, impeding their ability to learn until they catch up and develop a better understanding. Unfortunately the second step often never happens.
This is why I'm excited about methods like the Khan Academy's exercise system, where you're encouraged to stay on a concept until you truly understand it. I'm interested to see if they can pull it off successfully, and if it'll be put to good use. Education certainly needs all the help it can get.
I wonder what a physics major and hobby programmer like myself can do to help.
The whole thing is open source: http://github.com/khan/khan-exercises
We've had people who claim to have never been professional programmers come in and create impressive exercises that thousands of students are using daily to fill gaps in algebra, trig, calculus, etc.
Here's a couple examples of the sort of thing I mean. Along with another physicist, we've made a complete special relativity simulator (still rather beta, though -- UI needs improvement, and the demos don't have much explanation):
http://schroedingers-hat.github.com/jsphys/relativity/relati...
There's some premade demos, and the goal is to make it embeddable so it can be used along with a video or text to illustrate a point. I've also done a bit of an interactive tutorial on wave interference, where you can play with waves to see the result:
http://schroedingers-hat.github.com/jsphys/waves/wave_summat...
Eventually I'd like to finish off these simulations and use them to teach, as part of a complete curriculum. There's been some past efforts (with Java applets, for instance), but nothing integrated into a curriculum. Someday...
A) taking small steps every day to make our exercises more and more interactive (see @jpulgarin's link below)
B) thinking about longer-term projects, simulations, whatever you want to call 'em.
We welcome anybody to step in and blow the doors off our current exercise framework with a full-fledged simulation of some sort.
A large part of the issue is that the classrooms aren't differentiated by ability so we have students who can't really do or understand basic math but we're trying to teach them algebra and trigonometry. Some of them may pull through but do they really understand the subject matter when they don't understand the foundation from which it builds upon?
One of my education instructors told me once that it doesn't matter if a student can calculate 20x15 as long as they understand that we're looking for a number that is 20 times larger than 15. The problem is they really don't understand and they must rely on a calculator for even basic math. It really is disappointing and I'm not sure I want to teach anymore.
I feel like the system is broken and just because someone, somewhere said that when we educate students like this they can perform just as well or better than if we actually made sure they were in a class that suits their abilities we've latched onto it as a method of reform.
(I taught developmental math for 2 years at a local community college. My classes were difficult and I expected a lot of my students but the instructors that ended up with my students later on were all grateful of the foundation that I made sure my students had.)
As other replies you have received have mentioned, Khan Academy is working hard at improving its exercises in that regard. An even better current example, a service that costs money (except for UNLIMITED short-duration free trials) is ALEKS
which my wife and I are using to learn accounting and all four of our children have used or are using to learn mathematics and chemistry. ALEKS is a very useful tool for beginning college students who have gaps in their knowledge, which is most college students. See
http://www.aleks.com/about_aleks/Science_Behind_ALEKS.pdf
for a detailed explanation of the "knowledge spaces" theory behind ALEKS.
Thanks for asking the useful follow-up question, which I upvoted to encourage that kind of behavior here in what is meant to be an informative online community.
My experience as an actual, PERSONAL user of ALEKS only began yesterday, when my wife (the registered user) started taking the accounting course, with me second-chairing as we discuss the exercises together. But all of my children have used ALEKS
for a while, so I will discuss its strengths in relation to Khan Academy
and in relation to the Art of Problem Solving (AoPS),
https://www.artofproblemsolving.com/
another Web resource that all HN participants ought to know about. The strength of ALEKS is its thorough research base on curriculum topic interconnections based on the "knowledge spaces" theory
http://www.aleks.com/about_aleks/Science_Behind_ALEKS.pdf
that gives the ALEKS program its name. The online assessment that begins each ALEKS course identifies exactly what each learner already knows (in the scope of the syllabus of an ALEKS course) and suggests next topics to cover based on just what will broaden the learner's knowledge base enough to link to new unlearned topics that the learner can learn after first learning the current unlearned topics. ALEKS uses adaptive, periodic assessments to keep a learner in his or her "zone of proximal development," to use a term from another learning theorist, and is relentless in checking for gaps in knowledge or previous topics that the learner has forgotten. For my oldest son, who got past the top level of the ALEKS curriculum years ago, the ALEKS courses were perfect for spot review of a course (for example, high school algebra 2) just before a semester final in the distinct, accelerated mathematics course he was in at the time.
http://www.mathcep.umn.edu/umtymp/
For my younger children, ALEKS is less repetitive than the courses at the local high school that my second son tried out for a while (we are mostly homeschoolers) and very accommodating of the radical acceleration (precalculus topics at sixth grade age) that my third son is now pursuing in math studies to support his interest in science. The advantages ALEKS has over Khan Academy include more open-ended online problems, very careful research on interrelationships of topics based on knowledge spaces theory, thorough and complete syllabuses for each course based on international best practice, and smooth transition from one level of the ALEKS program to the next. ALEKS has a moderate list price (with discounts for longer enrollment terms or multiple enrollments from the same family, and UNLIMITED free trials). The cumulative progress information available to learners who pay the moderate price for ALEKS is quite useful and increasingly familiar to schools.
Advantages that Khan Academy has are low list price (free), a syllabus that goes up to higher-level topics than ALEKS (multivariable calculus and linear algebra, with the sky the limit for future courses), and the engaging online videos. We are still light users of Khan Academy in our household, but precisely because several HN participants are improving the Khan Academy exercise engines, I will encourage my children to use Khan Academy more and more over the next few years, and I could use it myself to learn math beyond precalculus.
The Art of Problem Solving (AoPS) is an online community I have participated in for years. (My screen name originated there.) The free program Alcumus there
https://www.artofproblemsolving.com/index.php?mode=interacti...
illustrates some ways that other online programs could serve up problems rather than mere exercises. The online community there supports some extremely interesting online classes that get into MUCH harder problems than those served up by Khan Academy.
Does this answer your question? Thanks for asking for more details.
Do you guys think there is a market for printed textbooks?
I hope to earn a living from selling print-on-demand copies at lulu dot com. When people ask me "what is your startup" and I tell them that it is a math and physics textbook they laugh at me, but I think it is actually a pretty good business idea.
Am-I on brain crack?
Oh I forgot to say -- the book is really kicks ass and targeted to low-attention-span youth. I do a coherent rendition of high school math in 90 pages, calculus in 60 pages, mechanics in 50 pages, etc. The page format is half-of-letter. email me if you want to see.
I hope you can be successful. Learning by video is so incredibly ineffective, I really don't get why people like it so much. Even coders learn Ruby by video???
I am doing the Stanford AI course and ML Course and it is OK to be kept on track with the weekly scheduled exercises. But I wish I could just read up the stuff (which I occasionally do, but it is not always easy to find the same information in written form). For example, the linear algebra section on Wikipedia seemed to be not that much help.
I once thought about buying a math text, as I noticed my Math skills were atrophying. Nothing seemed excellent, in terms of being aimed directly at adults wanting to learn math for no particular purpose.
But khan academy is excellent for that purpose.
Your book could sell, but you'd need to be clear in who it's for and how they'll benefit. Book marketing is a whole other Challenge, even if your book is good and does meet an unfilled need. An meeting a need isn't easy.
As the other commenter said, there are math books to be had by the hundred. There are free math exercises online. Most people don't care, and are busy.
Start by testing/marketing your book with students. Have them write reviews. Find out what they seem to like about it, an try to work that into your message if it's a need you hadn't considered. Good luck.
I am really trying to understand, because my brain seems to be wired differently than the modern generation. I find it harder to learn from videos than from books (although most books are too verbose, too).
Learning is an active process no matter which medium is doing the delivery (book, video, in person, etc.) You need to work problems, do examples, prove things, and review to really learn something.
Actually come to think of it, that might be really interesting data the Khan Academy is sitting on: how often people try to rewatch a part of the video. Could be an indicator if they are really learning, or pseudo learning. Or maybe it is only me who sometimes needs to rewatch parts.
Because in-a-de book me speak like Jiamaican. Tell a dem youth 'bout dem formula man. No trouble with no exam, once you overstand them proof from first principle.
> And is it for adults or youth?
I am targeting the adult market. There is some swearing and adult subject matter [not necessarily //adult//, but still inappropriate for high school students]
Thanks you --- and everybody else who commented --- for all your advice!
Do you think a student could pass the AP Calculus BC exam, using only your 60 pages of Calculus as a reference? Would a student be able to go through 1st grade to 12th grade using nothing but your book? If not, why would I buy your book?
I still want to see "The Simple Math of Everything" http://lesswrong.com/lw/l7/the_simple_math_of_everything/
The basic math sections in the book are quite fast. The intent was not so much to teach/drill/annoy the reader with the high school stuff, but to have a self-contained book, instead of saying "complete the square which you should know from high school".
> you think a student could pass the Calculus using only your 60 pages of Calculus as a reference?
Honestly, yes. But my belief is unimportant here -- I have to test things out with real students and real exams.
> 1st grade to 12th grade
A student no, but an adult who doesn't remember that stuff yes.
You could've picked a fresh subject, like a niche of bioinformatics, or cutting edge solar technologies, but you picked one of the most generic topics with books that don't really age. Basic math doesn't really change, you could pick up a math book from 100 years ago and use it just fine.
I was hoping to give away (part of) the content for free (http access) to attract people. Then try to push visitors to order the printed version like: http://betterexplained.com/articles/math-betterexplained-ebo...
> Textbooks is a cutthroat business, > there are thousands of textbooks on math
I know. That is why I don't want to enter the "course textbook" market -- I am not versed in the steaks and bitches selling tactics ;)
I am trying to position myself as an add-on book, kind of like the Schaum's outlines or the "for dummies" series, but without treating my readers as retards.
> math book from 100 years ago and use it just fine
Correct, and in fact the older the book you pick, the better it will be: more concise. It seems the current bloat of Stewart (300+ pages) is due to the publisher's desire to sell a more expensive product.
I know that in the 1800s few schools taught the calculus we now call "calculus." Back then that term was used for many fields of math, including logic ("the propositional calculus"). Nowadays it's almost only applied to differential and integral calculus. The rigorous foundations of calculus were only laid starting in the mid-1800s, so I seriously doubt that any "backwoods" student would have been doing anything remotely like ε/δ proofs then; that being the basis for our modern understanding of calculus. It might also have been that "1st year calculus" is what we now call "pre-calculus."
That's not saying it's impossible. Here's an example book on calculus from De Morgan in the early 1800s: http://books.google.com/books?id=zA4TAQAAMAAJ&ots=iIzsbl... . This is from a book series meant for adults/non-researchers interested in learning more about the latest science. You can see it has a more geometric approach. I find fascinating the footnote remarking the lack of consensus on how to use the integral sign. The comment on page 52 is telling:
There is then in this method neither the rigor of geometry, nor that approach to truth, which, in the method of Leibnitz, may be carried to any extent we please, short of absolute correctness. We would therefore recommend to the student not to regard any proposition derived from this method as true on that account; for falsehoods, as well as truths, may be derived from it.
But let's suppose his father did study differential and integral calculus. That would make him the exception, and not indicative of the average. For evidence: http://pballew.net/mathbooks describes math education in the US and mentions that in 1871 for the University of Nebraska in Lincoln:
"It was early accepted that mathematics, in particular calculus, would not be a required study for all students. In the first University Catalog it is noted that "Surgery and Calculus are not required of students in the Classical course." Exemptions from other mathematics were soon included."
I could apply the same questions to what "geology" means. Here's a textbook from the latter part of the 1800s. It goes into detail about all sorts of different rocks, where they are found how they are use commercially, and so on. It describes land forms. But what of geological processes? Nothing - because we hadn't figured it out yet! It says that folded strata probably come from cooling of the earth's "heated nucleus." In the section "Drift or Boulder Period" it notes what we now say is evidence of the Ice Age, but which then was a curious unknown "produced by Icebergs, Glaciers, or both." They didn't know if it was ice covering the surface, or if the land submerged and icebergs came.
They certainly didn't know about continental drift.
So the geology of the 1800s would have been quite different from the "earth science" I learned in school.
PCC had a large high-school completion program which they tried to steer many entering students towards, along with all the corresponding remedial classes; these served the dual functions of providing an alternative to traditional high school and remedial education for people who supposedly graduated high school. PCC took the very sensible step of providing a placement test when you first enrolled to find out what level of classes you should start with; you could place anywhere from "need 3 years of remedial classes" to "skip the first year or so of english and math".
In my case, I lucked out and got a good advisor, who confirmed what I'd found from the handbooks: rather than spend 3 years earning a high-school diploma (graduating at the same time I would have received one in standard K-12), I could spend 2 years earning an A.S. and enter PSU as a junior. I chose the latter. :)
In Florida, the most important thing to teachers (and the school boards) is students passing the FCAT (state wide tests). The more in your district who pass the more funds you protect. It's no longer about teaching kids a good education. It's a ridiculous reward system to protect your job.
Most public schools have cut out "unnecessary" courses like; music, band, art, history (except American history - only teach what is needed to pass. You looking for an in-depth discussion of the civil war? Not gonna happen), advanced math (hell even basic math in some schools), the sciences, drama, english lit, etc, etc.
It's shameful down here. And I think it's a major factor in our "class divide".
You want your kids to get a good education, you send them to private school. If you can afford it.
The issue today is that the federal government in the US has made it a priority that every kid go to college, including the ones in the lower half of the bell curve. That's a recipe for diluting the value of a college degree, which historically only the top quartile of students would receive.
Do you really think that the system worked before statewide tests?
The tests aren't perfect, but "the kids can pass the test" is a huge improvement over before.
If you've got a better way, great, but if it doesn't have an objective way to verify performance, we already know that it will work out very badly.
Indeed it don't.
I think that one of the big missing things in math education is the application. I can remember all of the stuff that I learned when using a compass and a straight edge, but when I came out of private school, and went to a public school, and started learning out of books, nothing stuck.
We could do worse than to just send all high-school students to a community college, or programs structured like them.
Compared to universities, you're probably right. Most big universities reward faculty and grad students based on research, not on teaching. So faculty and grad students optimize what they're rewarded for, on average. (I made this point in an essay I wrote for students called "How Universities Work:" http://jseliger.com/2010/09/26/how-universities-work-or-what... ). If you want to get better teachers or instructors, the universities have to start rewarding staff for that. If they don't, they get what we have now: universities optimized mostly for research with an incidental sideline in teaching.
It doesn't help, too, that a lot of students aren't in universities for learning—they're in to party or hang out or burn time or whatever. But that's another subject: http://jseliger.com/2011/10/07/student-choice-employment-ski... .
I agree there is a problem here, but am afraid it is not as simple as "durr, their high school is too easy." There are kids in those schools who will _need_ that diploma; of course, making things easier will devalue the diploma in the long run, so making things easy is no solution, but you can't entirely fault them for focusing hard on getting the students to pass exams. It's more like a symptom of larger system we live in, where having that diploma is simply far, far better than not having it. (whether it carries any real meaning or not)
Thank you "No Child Left Behind"?