Knowing and teaching elementary mathematics

What Hacker News says

1. jacobolusNov 2016
   Elementary mathematics took thousands of years to develop, and understanding basic arithmetic in a deep and fluent way is a highly advanced skill (really, a constellation of dozens of inter-related but distinct skills), which most students and even most teachers never fully develop, but which is incredibly helpful for becoming a fluent mathematician, scientist, or engineer.

   Beyond basic understanding, teaching elementary mathematics effectively is a difficult job which can take decades to fully master. There are a large number of potential conceptual and procedural errors which various students can make, some of which are subtle and hard to correct. Someone who can perform elementary mathematics is not necessarily able to properly teach it.

   As for world-class expertise, that really depends on what our goals are. There are undoubtedly world-class experts in elementary mathematics, who have spent decades doing extensive research in the history and practice, and have thought very deeply about the nature and meaning of numbers and mathematical relations. My comment was about the “ideal” case, if you want the student to learn as fast and effectively as possible.

   But you can also get by okay with just a typical trained professional teacher (ideally a specialist in mathematics education).

   I recommend Liping Ma’s book Knowing and Teaching Elementary Mathematics for some basic analysis.

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2. tokenadultMay 2015
   I lived in China for a few years and the education system there is pretty much all rote-memorization.

   你好. I used to live in Taiwan, and I have a big bunch of mathematics textbooks from China, as well as some recent math students in my local mathematics classes who grew up in China. Mathematics instruction in China is not at all about memorization, in the sense used in this article. A good book on the topic of mathematics instruction in the United States and China is Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma.[1] Her field study, with sponsorship by the United States federal government, made clear that elementary mathematics instruction in China is much richer and deeper and more conceptual than in the United States.

   But the idea of the article is true: When you just memorize, it makes it difficult to apply the principals that you are learning to other situations/problems.

   Yes. To memorize without underlying understanding is no more useful than not memorizing at all. Developing deep understanding of a topic usually brings along with it sufficient memorization of fundamental facts that the problem-solver will be able to solve some problems on the spot, and know when to look things up if needed.

   [1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

   http://www.aft.org/sites/default/files/periodicals/amed1.pdf

   http://www.ams.org/notices/199908/rev-howe.pdf

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3. tokenadultDec 2014
   Another Hacker News participant recently kindly shared in a comment a 2007 report by Educational Testing Service (ETS) titled "Teacher Quality in a Changing Policy Landscape: Improvements in the Teaching Pool".[1] The report showed that even though changes in teacher license requirements are boosting selection of higher-SAT-score rather than lower-SAT-score aspiring teachers, elementary teachers are drawn from college students who scored significantly lower than most college graduates on the SAT math section (Figure 21 in the report). The mean score of a recent prospective elementary school teacher in the United States when that teacher took the SAT for college entrance was barely above 500 on the math section.

   Alas, in general, American schools are underperforming in teaching the most advantaged members of American society,[2] especially in mathematics,[3] while not doing well by the least advantaged young people in American society either.[4] The current efforts in preschools reported in the interesting article submitted here should be followed up by further reform of elementary school mathematics instruction. The book Knowing and Teaching Elementary Mathematics by Liping Ma[5] is especially informative about what to do to help all learners in the United States learn mathematics better in early schooling.

   [1] http://www.ets.org/Media/Education_Topics/pdf/TQ_full_report...

   [2] http://educationnext.org/when-the-best-is-mediocre/

   http://educationnext.org/young-gifted-and-neglected/

   [3] http://educationnext.org/teaching-math-to-the-talented/

   [4] http://www.oecd.org/pisa/pisaproducts/pisa2009/48165173.pdf

   http://www.ams.org/notices/200502/fea-kenschaft.pdf

   [5] https://www.engageny.org/sites/default/files/resource/attach...

   http://www.ams.org/notices/199908/rev-howe.pdf

   http://www.nytimes.com/2013/12/18/opinion/q-a-with-liping-ma...

   http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

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4. tokenadultMay 2014
   students do not have to study for 14 hours a day in their prime of their youth to acquire a stable career

   Students don't have to do that anywhere. And they don't do that anywhere. I have lived in east Asia (and my three older children all have) and I challenge your assumption that pupils there study for fourteen hours a day (they most certainly do not) or that everything about schools there is a "pressure cooker environment." There are trade-offs involved in living anywhere, rather than living somewhere else, but you really owe it to yourself to find out more about how effective primary teaching is done before concluding that it shouldn't be tried in the United States. As they say in Chinese-speaking countries, "百聞不如一見" (hearing about something a hundred times isn't worth seeing it once). I offer for your reading pleasure a reading suggestion, full of food for thought, namely Liping Ma's book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States.[1] Reviews of this book appear below the book link.

   Anyway, we can get other reality checks on this issue by seeing what people who have studied the United States school curriculum say about secondary schools here, for example "What’s Holding Back American Teenagers? Our high schools are a disaster."[2] The last federal sample survey on the issue resulted in news reports such as, "School is too easy, students report,"[3] suggesting that a lot of learners here don't appreciate dumbed-down school lessons that waste their time without providing a return in new knowledge and skills.

   [1] http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

   http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

   http://www.ams.org/notices/199908/rev-howe.pdf

   http://www.jstor.org/discover/10.2307/2695436?uid=2&uid=4&si...

   [2] http://www.slate.com/articles/life/education/2014/02/high_sc...

   [3] http://usatoday30.usatoday.com/news/education/story/2012-07-...

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5. tokenadultApr 2014
   You and onlookers may enjoy reading the article, "Basic Skills versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education"[1] (1999) by Hung-hsi Wu, for a discussion of what leads to deep understanding of mathematics. Another classic article on the topic is Richard Askey's review[2] of Liping Ma's book Knowing and Teaching Elementary Mathematics, a very good read indeed.

   [1] http://math.berkeley.edu/~wu/wu1999.pdf

   [2] http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

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6. tokenadultFeb 2014
   Thanks for the interesting comments. Taking the top-level comments in order of posting, I read

   This article is lacking examples. What exact case makes the standard curriculum bad? What are the good examples of the alternatives?

   The article mentions, "more importantly, the gifted, interested student should be exposed to mathematics outside the core curriculum, because the standard curriculum is not designed for the top students."

   The whole site that the article comes from serves as an example of mathematics teaching that goes deeper and connects topics together better than the standard curriculum in United States schools. Other authors have written on the same topic. Professor John Stillwell writes, in the preface to his book Numbers and Geometry (New York: Springer-Verlag, 1998):

   "What should every aspiring mathematician know? The answer for most of the 20th century has been: calculus. . . . Mathematics today is . . . much more than calculus; and the calculus now taught is, sadly, much less than it used to be. Little by little, calculus has been deprived of the algebra, geometry, and logic it needs to sustain it, until many institutions have had to put it on high-tech life-support systems. A subject struggling to survive is hardly a good introduction to the vigor of real mathematics.

   ". . . . In the current situation, we need to revive not only calculus, but also algebra, geometry, and the whole idea that mathematics is a rigorous, cumulative discipline in which each mathematician stands on the shoulders of giants.

   "The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."

   Personally, I hated calculus. It turned me off of math for a long time -- there was too much emphasis on memorizing heuristics for solving problems, like integral tricks and trigonometric identities. Worse, calculus was used as the canonical example of "college-level math", so it seemed that further math courses would just be about memorizing more and more problem solving tricks.

   This second comment to be posted expresses what many students miss out on if their secondary school curriculum rushes to get to calculus as early as possible without also being designed to help them understand mathematics as well as possible. That's what the submitted article is about.

   If I were to make a recommendation between following the curriculum, and pursuing extra-curriculars, I'd say: do both.

   Yes, the both-and approach is helpful. That's what the article says when it says "Developing a broader understanding of mathematics and problem solving forms a foundation upon which knowledge of advanced mathematical and scientific concepts can be built. Curricular classes do not prepare students for the leap from the usual ‘one step and done’ problems to multi-step, multi-discipline problems they will face later on. That transition is smoothed by exposing students to complex problems in simpler areas of study, such as basic number theory or geometry, rather than giving them their first taste of complicated arguments when they’re learning a more advanced subject like group theory or the calculus of complex variables."

   Lockhart's Lament [0] comes to mind.

   [0]: http://www.maa.org/sites/default/files/pdf/devlin/LockhartsL...

   Lockhart's Lament is indeed also a response to an era (different from the era I grew up in) when many high school students are rushed into a calculus class before reaching a profound understanding of fundamental mathematics.

   "you’re in ninth grade and you’ve already taken nearly all the math classes your school offers" I thought most high schools taught calculus. Both my and my wife's did. Why is this 15 year old going to a local community college or university for that?

   The author is indeed writing for a particular audience (which, as you correctly point out, is growing in size) of young people who have blazed through the United States mathematics courses that are now typical at ages once thought unimaginable. My late dad took his calculus course in the late 1940s as a second-year college student. I had just seven high school classmates in the mid-1970s who took calculus in high school at all. Most students in my generation who took calculus at all took it as a first-year university course. My oldest son began a formal course in calculus at eighth-grade age, through an accelerated local program that was founded in the 1980s. My second son is taking AP calculus BC as high school junior (eleventh grader). People are rushing into calculus much more rapidly than ever before in the United States, but often lack "profound understanding of fundamental mathematics (PUFM)" before starting the calculus course. A link that furthered my process of pondering how students might learn mathematics better was Richard Askey's review of the book Knowing and Teaching Elementary Mathematics by Liping Ma.

   http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

   Another review of that excellent book by mathematician Roger Howe

   http://www.ams.org/notices/199908/rev-howe.pdf

   is also food for thought. In some countries, elementary mathematics is not considered "easy" mathematics, but rather fundamental mathematics, which must be understood in full context to build a foundation for later mathematical study.

   However, my personal experience was the exact opposite. Calculus classes, along with the accompanying physics-with-calculus classes, were what transformed my concept of math from a game you play with symbols into a powerful way to describe the way the world works.

   There are definitely a lot of students who enjoy a calculus course for that experience. That seems to be a form of enjoyment that especially comes to students who have had time to learn about other topics on the way to learning calculus. Russian mathematical instruction tries harder than instruction in the United States to bring in examples from physical science at all ages, so that the mathematics that explains physics is taught to students who have a decent background in physics.

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