Afraid of Your Child's Math Textbook? You Should Be.
open.salon.com
open.salon.com
Just use the Singapore Math series for K-6 and be done with it. Sound pedagogy, known to work, has been debugged, no frivolous cutting edge theories from M.Eds who have never taught math successfully.
[Citation needed]. The methods my kids currently use, seem more effective that the ones I used at school. I am in my late 40s.
edit: I'm based in the UK, however.
2. The average state of math knowledge of contemporary american high school graduates, coupled with statistics on needed remedial education for college entrants with high school diplomas.
Teaching math is easy. I can teach any kid math. It's not a problem at all because we know how to do it. Many countries do it successfully. Observing what schools are doing, the textbooks, the approaches, and the level of math competency of elementary school teachers completely explains what is going wrong. It's not a mystery. Also explains why kids homeschooled by parents, even parents who were high school drop outs, end up with considerably better math skills than most public school graduates, and even many private school graduates.
Typical high school graduates don't understand fractions, percentages or decimals. They can't do division. They don't understand algebra at all. We know this is true because of college placement tests.
This is absurd. 100% of high school graduates should have passed differential and integral calculus. That should be the minimal acceptable standard. That it is not, and that graduates fail at even moderate arithmetic skills, shows the complete failure of the system.
Downvote all you want while shouting "the dedicated teachers are not the problem, the students and their parents are!" That won't change a thing. Meanwhile, those who utilize proven approaches succeed and those who make excuses and follow fads fail.
I do some SAT tutoring, and I find that students are helpless at basic arithmetic if they can't use a calculator.
My suspicion is that calculators prevent students from developing mental math skills, and that this hurts conceptual understanding across the board.
But I can only see the results of the system, so I don't know what actually produced the types of students I'm seeing. I may be wrong about calculators - from what you say, it could simply be that the system in general is bad.
Even before that, mechanical skills such as polynomial factorization or algebraic reorganization ("fiddly symbol manipulation") were actually, in my opinion, aided by having a calculator which could perform them for me. By releasing the burden of actually performing the computations, I was focused on larger patterns which made the computation meaningful and possible.
There are two camps here, of course. As an engineer or mathematician, doing mental arithmetic is a dead skill---all the effort comes from modeling and understanding the structure of math. However, "practical" math such as computing tips or adding up grocery bills---and this was the kind of math application stressed in my public school education---probably benefits much more from rapid mental sums.
So, ironically, I'd support the heavy involvement of calculators (even ones as powerful as Mathematica itself) in education if it's to be aimed to produce engineers, scientists, or mathematicians. Otherwise, they're probably damaging.
In many cases, the end results appear identical. However, I'd rather the upcoming engineers, scientists and mathematicians be banned from using calculators for anything more than basic arithmetic until they prove they do understand the concepts.
We don't pine for the days when people had to hand-calculate logarithms, because we've come up with better ways of using that tool. CASes and graphing calculators are the same thing.
Factoring quadratics is mostly the same every time, while factoring in general is not, and graphing definitely is not.
Graphing, in particular, is a crucial math skill, because it forces you to understand what is actually going on with each function. As a rule, students who have no idea what a graph is going to look like before they press "graph" also don't understand many important underlying fundamentals.
There is no reason middle/high school students should be using graphing calculators on a regular basis. Scientific calculators for things like logs are fine, but beyond that the calculator replaces actual learning.
As a physicist, I use math environments like Matlab and Mathematica to do calculations all the time. Some are simulations that are too massive to even contemplate without a fairly powerful computer. Sometimes I just use them to speed up things I could do by hand. In the latter case, I usually do some samples by hand just to check if my program is working as expected. Later, when using the program, I always try to have some estimate of what the results should look like, even if that intuition of what is correct comes from a very different way of thinking. This is something a lot of students I've taught simply do not do, and it really bites them in the ass a lot!
For example, there's one standard junior lab experiment where students work with a very weak sample of a radioactive isotope. It's just barely strong enough to register a small amount of clicks if you practically stuff the sample inside of a Geiger tube. As an exercise, we have students calculate the equivalent dose of radiation they receive over the course of the lab. If the calculation is done correctly it should come out to being barely above background exposure and nowhere near as much as an intercontinental flight or diagnostic X-Ray. However, without fail, every year at least some students will calculate an exposure that is fatal, either by incorrectly converting units or just simple "calculator error". "Calculator error" is what we call it when a student writes out their calculation and everything is correct except the final answer. They got all the input numbers and units right, but they just failed to punch it into their calculator correctly.
Many students probably make this mistake but catch it right away. Others do not catch their mistake and how they deal with it in their lab reports varies quite a bit. Some simply don't notice it. They clearly had no idea what to expect from the calculation and didn't bother to compare their result to what was in the lab materials. Other students are at least smart enough to note that they think their answer is too high, although they don't know why. Occasionally, a student will actually write that they are concerned that the experiment was unsafe, perhaps because a horrible mistake was made and they were given the wrong isotope samples. These last students are easy to laugh at, but at least they are thinking, unlike the first group of students who write down a dose that would have left them dead without any further comment.
That first group of students who trusts their calculation blindly (or simply don't care) is much larger than the other groups, and that's my greatest beef with our education system. It turns children into calculators. i.e. Devices that can mechanically turn input into output without understanding the process they are executing or having any intuition about what the answer should look like. Grade school texts rarely teach students to critically examine their results. They present students with a way to do a calculation, ask them to execute it like a mechanical device, and then knowing whether they're right or wrong boils down to checking the back of the book to see if the book's answer matches their own. Grade school students can get perfect marks without having any understanding at all, provided they practice the methods enough.
Understanding the method and having the ability to recognize what results should look like are skills that should be at the very core of math curriculum's. Learning to be a calculator is boring. Math is infinitely more interesting when you know the why and not just the mechanical how.
I end up retraining students to do mental math and only use calculators to confirm a calculation already done mentally or on paper.
I believe this helps them spot possibilities, and understand what the less than straightforward questions are asking them to do.
I wonder how my college students today have already estimated the total of their purchases when they arrive at the cashier. Such thinking doesn't seem to be part of the culture anymore.
I've become an old fogey faster than I expected to.
The goal is definitely to avoid turning people into calculators. I found that for myself and my peer group, having a calculator was the best panacea to doing that. It meant that we'd use those extra free moments wondering why things worked and what it meant instead of "carrying the ones".
What would have been much more useful for me would have been a stronger grounding in statistics.
It is also really really far from what is being accomplished on average. High schools have seniors who can't do division and don't understand operations on fractions. Calculus is something anyone can understand, but it is seen by teachers as some super advanced mysterious stuff for rocket scientists while they struggle along to teach subtraction to 18 year olds.
Imagine a physical education fitness curriculum whose goal after 13 years of training was for 18 year olds to be able to turn over by themselves and begin to crawl. That's what we are doing, being satisfied with goals for graduates that shouldn't be challenging even for 6 year olds. It shouldn't be tolerated at all.
Schools that are graduating students with no math skills have had 13 years of failed instruction with these students. Not just one teacher along the way. All of them. The system should be burned to the ground and started over. That will never happen though, it's too corrupt and incompetent. Myself I've given up hope reform of the system is possible.
In any case, I believe the OP was asking why it's important that every student takes calculus? I'm very curious about that as well. Why again is that so important?
Here as well - please don't just down vote but please explain why you are if you are going to.
I know a lot of people who struggled in classes like Algebra 2 and Precalculus (which is kind of an abomination in retrospect) who finally learned to appreciate the power of mathematics by going through Calculus. Not everyone will walk away with that perspective, but I feel like society can only benefit from a greater respect for math.
But yes, statistics is very important too, for similar reasons.
Probability, graph theory, statistics, number theory, game theory, groups, sets, logic , these are topics which can be interesting and basics of them can be taught before college. Similarly, linear algebra is not that scary.
I completed Calculus BC test in my Junior year in HS and went to nearby University for Calculus 3(forget what it was called exactly), got a bad professor with worse English than mine and burned out of math for a long time. I was under the mistaken impression that Calculus was end of all math.
Secondly -- You say "Teaching math is easy." What does that mean? Just from personal experience, 10 years or so ago I took Calc 1 and 2 in high school, and calc 2 again in college, and did well in all, and know how to integrate and differentiate and use various formulas due to a recognition of certain inculcated patterns, however would I say I am skilled at actually using calculus as any sort of real life thinking/problem solving tool? Absolutely not and not even close. Is this a success? Absolutely not. One of the most important things was completely overlooked by the "traditional" method I was taught in.
I'd argue that a similar pass rate for introductory statistics is vitally important for our continued viability as a society. Calculus is, frankly, less important than knowing when a politician is lying to you, or whether that test result means you need to draw up a living will now.
Education differs significantly between countries, international comparison is hard.
Your argument is the antithesis of the reason for hackernews!
In many schools today, math is taught as a "holistic" subject, where algebra, geometry, trigonometry are intermixed and taught as a "integrated" whole. In theory, this would let students appreciate that there are many ways to approach a problem and that the same theorem can be proven via many methods. In practice, all it ensures is that students have an insufficient background in three subjects rather than thorough knowledge in one.
Bruce Lee said, "I do not fear the man that has practiced 10,000 kicks once. I fear the man that has practiced one kick 10,000 times." What our schools are doing is teaching different "kicks" (mathematical skills) but without following it up with the necessary practice required for students to achieve mastery of those skills. What we end up with are kids that know of mathematical theorems and mathematical results without knowing the results themselves.
Maybe the issue isn't with the "frivolous" theories but rather with the treatment of them. As in depend on empirical evidence through studies prior to applying them to a wide student base. The first +-10000 light bulbs weren't released to the public. Why are new educational methods?
"Among the first users was Mark Twain, who fiddled around with it before putting it aside. Yes, Twain did become the first person to submit a novel in typed form to the publisher, but that wasn't until much later ("Life on the Mississippi,"1883) , and he didn't type it himself... it was a typed copy of his handwritten manuscript."[1]
That manuscript was produced at least 8 years after the first Remington model appeared. Although Clemens was clearly a technophile (and pal of Tesla), he considered the first models to be curiosities rather than immediate replacements for pen or pencil and paper.
Calculus Made Easy http://www.gutenberg.org/ebooks/33283
An oldie but a goodie.
Edit: This prefigures the For Dummies books by what, 80 years?
The books were useless. It was nearly impossible to find concepts in it, and when you did the description was often relatively short, relying on an example that was often poorly constructed.
In the end, if the book didn't jog my memory I would have to Google the concept to find a better page. The book might as well have been nothing but problem sets.
I never met any of her teachers. I got the impression that at least one of them wasn't qualified to teach math, but that was 2nd hand information through the lens of my sister, so I can't be sure how accurate it was.
(People always deride others in debates and arguments for "throwing money at the problem", but they never stop to explain why it is a bad thing. Now I know.)
I suppose that what this shows is that simply "throwing money at the problem" is not a good thing. You have to throw money at people who are genuinely driven to solve the problem.
http://minnesota.publicradio.org/display/web/2011/11/06/onli...
appears to be a working link, from a website better about keeping links alive, to the same AP news story.
Maybe some children would have worked out to create their own key or to write in the book, but this boy just stopped. An activity that he's good[2] at, that he enjoys, is being slowly poisoned by poor quality materials.
It's not just math books either. Where's the Feynman quote about the horse evolution?
[1] Proper diagnosis from a set of real doctors. [2] Got a scholarship to a decent school partly based on his math ability.
First, I think he meant instilling math phobia.
And second, this "golden ration == attractiveness" myth is un-scientific and an urban legend perpetuated by Dan Brown and similar authors.
I was a software engineer at an educational software startup, and we were continually "course correcting" to meet these mandates (and yes, they did seem arbitrary and "change for change sake").
In my sophomore year of college, I discovered there were massive price disparities between old and new textbooks. Thermodynamics, 1'st edition (~1970) - $10, Thermodynamics, 7'th edition (~2000) - $120.
I used the old editions, and it didn't hurt me much (I lost a few points due to renumbering of homework problems). All they did was renumber the chapters and problems, no new material was added, most of the time the text was identical.
In this space, wholesale churn is rampant. Math series only have a several year shelf life, before old-and-broken is replaced with new-hotness. My wife and mother and sisters are all elementary school teachers, and they haven't ever stayed with a math series for more than 5 years. In our district, they are required to change them every 7.
K-12 level mathematics, compared to college level curricula, is much more "how" to teach than "what" to teach, and much more subjective. There were knock-down drag-out fights about the use of math manipulatives (which is the use of physical elements to show arithmetic operations).
But why is the author of the submission complaining about textbook publishers rather than state legislatures and school administrators?
1) Student test scores get worse or a new state test is implemented and shows that students are doing poorly.
2) The school board is pushed to fix the problem. They're not math experts.
3) The textbook publishing industry pushes their experts to advocate their latest series of textbooks and teaching methods.
4) The school board then changes the district to a new curriculum with the new textbook.
This pattern plays out all over the country. And repeats over time.
Yes, that's a good tip for students to be aware of.
Here is another. In college I discovered that I could buy used textbooks in the college bookstore if I got there early enough, and I could sell the same textbook for $1 under the bookstore price by showing up at the class the next semester and standing outside the door at the end of class. Using this method I spent on average less than $10 to use the latest versions of required college textbooks. (There were many I kept though.)
"In creating the Department of Education, Congress specified that:
No provision of a program administered by the Secretary or by any other officer of the Department shall be construed to authorize the Secretary or any such officer to exercise any direction, supervision, or control over the curriculum, program of instruction, administration, or personnel of any educational institution, school, or school system, over any accrediting agency or association, or over the selection or content of library resources, textbooks, or other instructional materials by any educational institution or school system, except to the extent authorized by law. (Section 103[b], Public Law 96-88)"
So even if we wanted to have national standards for math textbooks, the Dept. of Education is prohibited by law from doing so.
http://www.education.gov.uk/schools/teachingandlearning/curr...
The alternative is to specify the National Curriculum and then second guess the changes each time the government changes.
School maths is a socially determined thing. It has little to do with mathematics in University departments or the ways in which mathematical reasoning is used in other areas of life. Definitions vary.
PS the new textbooks by pearson, stanley thornes, longman &c are relatively error free.
Also, http://ck12.org for open textbooks.
Granted this isn't actually being accomplished. The reason why new textbooks are needed is independent of why new textbooks are being bought. However, I feel that your argument "Subject X hasn't changed, so we shouldn't textbooks on X" is off target. Why are there new web frameworks keep coming out? Because this time, we're going to do it right.
I agree with the comment by HN participant droithomme above that it would be perfectly possible for school systems in the United States simply to adopt the Primary Mathematics textbook series from Singapore (which already exists in two editions adapted to the United States in general and adapted to California curriculum standards in particular)
http://www.singaporemath.com/Primary_Mathematics_US_Ed_s/39....
http://www.singaporemath.com/Primary_Mathematics_Stds_Ed_s/1...
and thereby do much better than most United States schools do with any of the other existing textbook series.
Based on international textbook content analyses and other evidence, gathered and digested with the help of professional mathematicians aware of international K-12 teaching practice, the great majority of states in the United States have voluntarily adopted the Common Core State Standards in Mathematics
http://www.corestandards.org/the-standards/mathematics
and the next iteration of textbooks in the United States will CLAIM (with truthfulness of that claim yet to be verified) to follow that set of standards. I like what I have seen of those standards. I have yet to see any new textbook series that fully embody the best practice in mathematics teaching in other countries. The best elementary mathematics textbooks I have seen are in the Chinese language, published either in Taiwan or in China. Among English-language textbooks, the Singapore Primary Mathematics Series has been plainly superior for at least the last twenty years.
I can't imagine there's new branches of maths that we need to teach school kids about. (and same goes for a lot of other subjects)
I can more readily imagine new developments since my childhood in physics than in mathematics that would have to be reflected in elementary textbooks, to be sure. But textbooks would have to be updated in any country that does not yet have the best available textbooks, even if the subject of mathematics has not changed. There has been a growth in understanding during my lifetime about how to teach mathematics to elementary-age pupils. Content analysis of different textbooks used in differing countries has helped develop that growth in pedagogical understanding.
Moreover, at least two aspects of mathematics as encountered by most members of the public have changed in my lifetime:
1) today anyone in a developed country can buy an inexpensive calculating machine that is more generally useful than a dedicated office business machine of a generation ago, and
2) discrete mathematics plays a much bigger role in business and in industry than it did when the march to calculus was the main path in mathematics education for technical careers. Both of those changes would have to be reflected somehow in curriculum design for K-12 mathematics instruction.
That sounds an awful lot like reality. Not that I think it's a great way to teach maths (though maybe it is a great way to teach problem solving) but an educated adult should have little trouble spotting the missing information or contradictions and either guessing, pointing out, or assuming them away.
I realize that it wasn't intentional on the part of the authors (neither are real problems), and that it's not the point of the article but merely a hook.
I see your point, I see the article's point and you're both right -- a maths textbook probably shouldn't look like that. It's a mistake, an oversight, a bug, and, as someone noted in the comments, it would be great if books were being improved instead of written anew every few years. It's why I can run the most current Linux kernel and actually have a better experience than with a stable one -- it was improved, not rewritten.
What I didn't like was this idea that a parent could be scared because of some imperfection in the way a homework problem is framed, pandering to this light version of maths anxiety many people seem to have that actually prevents them from learning maths. If the problem is poorly stated, reframe it, point out the problem, guess or find the missing information, assume whatever needs to be assumed to get some sort of workable solution (which you can then improve upon :) There is no reason to panic, be afraid, or even really bothered by that.
Again, I know, it was just a hook but it bothered me enough to post these two comments.
Open-ended questions and problems are indeed awesome. Moreover, they are an essential part of a sound education in mathematics, even at the K-12 (primary and secondary schooling) level of learning. But open-ended questions used for teaching purposes should be carefully written for sound teaching points, and teachers using them should have sufficient background in mathematics to guide student approaches to grappling with them. One of my favorite authors on mathematics education reform (Professor Hung-hsi Wu of UC Berkeley) began writing on that issue in 1994 with his article, "The Role of Open-ended Problems in Mathematics Education,"
http://math.berkeley.edu/~wu/open-role.pdf
and he followed up on that article with a wonderful article in the fall 1999 issue of American Educator, "Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education."
http://www.aft.org/pdfs/americaneducator/fall1999/wu.pdf
Since then, Professor Wu has written many more useful articles on mathematics education, including guides for parents, teachers, school administrators, and teacher educators on how to apply the new Common Core State Standards in mathematics better to improve mathematics education in the United States.
http://www.corestandards.org/the-standards/mathematics
A good example of a beguiling textbook by a world-famous mathematician with lots of open-ended problems is Algebra, by the late Israel M. Gelfand and Alexander Shen.
http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773
Some of the problems in this book are HARD, but they are generally well posed problems of actual research interest to mathematicians, that just happen to be accessible to pupils just beginning to learn algebra.
AFTER EDIT: answering the question kindly posted below, one example I had in mind is that Gelfand asks students to figure out how many different ways there are to group terms in an expression with parentheses as the number of terms increases. This essentially asks the students to discover the Catalan number sequence.
Could you share a few examples? I've looked through the books from Gelfand's correspondence course (which are indeed excellent) but don't remember any problems that fit your description. Some of them would certainly be challenging for young children--I'm more interested in the second half of your statement.
I'll volunteer one potential example. There was a sequence of problems that dealt with the solvability by radicals of palindromic polynomials. That certainly motivates some ideas of Galois theory in no small way, but it's very basic and of no interest to research mathematicians.
Addendum: Now that I have looked it up in the book, I see it was a single problem, Problem 270, not a sequence of problems. That sequence of problems was from a mathematics competition for young children.
Math literacy is generally poor, so we want schools to do as good as they can.
Not all children have "educated" adults around to help them with their homework, and we especially want these children to have good education to help lift them from poverty.
Someone is paying money for those text books, so that person at least has a reasonable expectation of suitability for purpose.
And why do we accept[1] educational methods without insisting on good quality research to back them up? Why is it acceptable to churn out a book that not only has small errors but which causes confusion and delay in children?
[1] If we do accept them. I'm not a teacher, so maybe there's lots of research and it's all rigorous and great.
But now what? Who should we be talking to who can affect change?
I've thought about working on a free site to help parents/students locate the best free online math resource that can help them with a specific question, worksheet, homework assignment, etc. As a tutor, I've frequently heard from parents that they know these types of things exist on the internet, but that they have no clue how to find the right Khan video that will help with tonight's homework.
If anyone is interested in this sort of thing, please get in touch.
To your point, if he is unqualified to write exercises for a textbook, he would also be unqualified to create these for his own class. If he is underpaid...that is largely irrelevant- if he were really underpaid, he wouldn't have taken the job.
The article was of the anti-corporate bias perspective, which is why it was on Salon.com, that this is because of the "profit motive" of the publishers. However, that perspective is also nonsensical. The examples do not change that much from edition to edition- they are seldom created from whole cloth. Even in his tortured example of complaining about the golden ratio problem was one of textbook translation- pointing out that, as an author, he cared about things other than the effectiveness of the text.
It is true that many textbooks are of low quality...but considering the bias of the writer, the whole article is a bit self-serving, while also being hypocritical.
All points about the incompetence of textbook buyers stand though. They choose for the worst of reasons, with effectiveness almost never even on the table as a concept.
Similarly the author gets royalties on all copies, but the salesperson only gets commission on the books they sell. A thousand books sold by four sales staff is 1000 books for royalty purposes, but only 250 books on average for commission purposes.
There is some slight irony in assuming something is wrong with the number just because they don't appear superficially linearly correct, when criticising the quality of maths books...
Apparently, that's not the case anymore for educational book writers. The article states:
Today, royalties are a thing of the past for most writers and work-for-hire is the norm.
...and during the article this new state of affairs is mentioned quite a few times, and loathed.
Textbook selection committees are usually staffed with people who do not understand the subject, and when domain experts are asked to donate their time to evaluate materials, their feedback is ignored as the only purpose of having them aboard was to apply to the dog and pony show evaluation proceedings a sheen of legitimacy.
Textbook publishers spend millions each year wining and dining members of committees, providing them with gifts and trips and sometimes even setting them up with speaking fees that are a masked form of bribery and kickbacks. It's a well known problem and has been going on for decades. It's very similar to the way the pharmaceutical rep industry works.
I strongly recommend everyone read "Judging Books by Their Covers" by Richard Feynman for his experience in the process, which remains the standard modus operandi to this day:
http://www.textbookleague.org/103feyn.htm
As the editor's postscript to that article summarizes:
"As a rule, however, state agencies don't want legitimate evaluations of the textbooks that publishers submit for adoption, because the agencies are allied with the publishers. The adoption proceedings staged by these agencies are not designed to help school districts, to protect students, or to serve the interests of taxpayers. Rather, they are designed to serve the interests of the publishers, to generate approvals and certifications for the publishers' books, and to help the publishers sell those books to local schools."
These dynamics are part of why I strongly oppose calls for national standardization, the streamlining and centralization of would lead to a much larger and more difficult to combat corruption as greasy palm edu-bureaucrats rake in the dough for selling out the country.
Of course, things like tablet-based (eInk or LCD) reading and khan academy are the future of learning.
How you design texts can make a difference (google "conceptual change texts" for example), but how much do textbooks really influence the quality of instruction? If a teacher is just blindly following a textbook (and not doing other things like supporting students, using interactive software, etc.), then there are bigger problems.
Ironically, too, students would learn more from a textbook with mistakes in it if they were made aware of that fact and encouraged to find them. I know I learned the most in geometry where I had a teacher who couldn't even draw a circle and made mistakes every single day.