Category: What is mathematics

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What is good mathematics?

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Terence Tao

It may not be for the majority of learners but it remains an important goal of math education in the basic level to hone future mathematicians who in turn are expected to produce good mathematics.  This post presents Terence Tao‘s personal thoughts on what good quality mathematics could mean. Terence was a child prodigy. When he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Terence Tao currently holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. He was one of the recipients of the 2006 Fields Medal.

What I like about Terence’ descriptions of good quality mathematics is that it is possible at K-12 levels for students to actually produce good mathematics within the universe of their knowledge given the right motivation and challenge, the appropriate activity, enough opportunity, etc. I think students have the right to participate in the production of mathematical knowledge. Activities such as problem solving, mathematical investigations, and  modelling are all about training students to “produce” good mathematics.

Good mathematics could refer to any of the following:
  1. Good mathematical problem-solving (e.g. a major breakthrough on an important mathematical problem);
  2. Good mathematical technique (e.g. a masterful use of existing methods, or the development of new tools);
  3. Good mathematical theory (e.g. a conceptual framework or choice of notation which systematically unifies and generalises an existing body of results);
  4. Good mathematical insight (e.g. a major conceptual simplification, or the realisation of a unifying principle, heuristic, analogy, or theme);
  5. Good mathematical discovery (e.g. the revelation of an unexpected and intriguing new mathematical phenomenon, connection, or counterexample);
  6. Good mathematical application (e.g. to important problems in physics, engineering, computer science, statistics, etc., or from one field of mathematics to another);
  7. Good mathematical exposition (e.g. a detailed and informative survey on a timely mathematical topic, or a clear and well-motivated argument);
  8. Good mathematical pedagogy (e.g. a lecture or writing style which enables others to learn and do mathematics more effectively, or contributions to math- ematical education);
  9. Good mathematical vision (e.g. a long-range and fruitful program or set of conjectures);
  10. Good mathematical taste (e.g. a research goal which is inherently interesting and impacts important topics, themes, or questions);
  11. Good mathematical public relations (e.g. an effective showcasing of a mathematical achievement to non-mathematicians, or from one field of mathematics to another);
  12. Good meta-mathematics (e.g. advances in the foundations, philosophy, history, scholarship, or practice of mathematics);
  13. Rigorous mathematics (with all details correctly and carefully given in full);
  14. Beautiful mathematics (e.g. the amazing identities of Ramanujan; results which are easy (and pretty) to state but not to prove);
  15. Elegant mathematics (e.g. Paul Erdos’ concept of “proofs from the Book”; achieving a difficult result with a minimum of effort);
  16. Creative mathematics (e.g. a radically new and original technique, viewpoint, or species of result);
  17. Useful mathematics (e.g. a lemma or method which will be used repeatedly in future work on the subject);
  18. Strong mathematics (e.g. a sharp result that matches the known counterexamples, or a result which deduces an unexpectedly strong conclusion from a seemingly weak hypothesis);
  19. Deep mathematics (e.g. a result which is manifestly non-trivial, for instance by capturing a subtle phenomenon beyond the reach of more elementary tools); Intuitive mathematics (e.g. an argument which is natural and easily visualisable);
  20. Definitive mathematics (e.g. a classification of all objects of a certain type; the final word on a mathematical topic);

You can find the link to the complete paper in Terence Tao’s WordPress blog. You may also want to read Terence’s books.

Posted in What is mathematics

Bedrock principles of math and what it means to understand math

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In one of my LinkedIn group, someone started a discussion with this question What  is the bedrock principle of mathematics? May I share some of the answers.

1. “The bedrock principle of mathematics is the axiomatic system. The realization that there are propositions that must be taken for granted in order to have something to build upon.”

2. “Speaking about foundations, in my opinion the bedrock should be enlarged at least as follows:

  • discerning that two things are different;
  • identifying two things which share same property;
  • discovering relations among properties.”

3. “I would say the bedrock principles of mathematics are:

  • The ability to differentiate two things
  • The ability to rank two things (as to most value, shortest route, least danger, etc.)
  • The ability to expand the above to more than two things”
Bedrock principles of any discipline of course can’t tell us how one can know if he or she understands a piece of that discipline. So I asked How can one tell if he/she understands a piece of mathematics?
According to Peter Alfeld you understand a piece of mathematics if you can do all of the following:
  • Explain mathematical concepts and facts in terms of simpler concepts and facts.
  • Easily make logical connections between different facts and concepts.
  • Recognize the connection when you encounter something new (inside or outside of mathematics) that’s close to the mathematics you understand.
  • Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

You may also want to read my post To understand is to Make Connection.

The challenge of course will still be this question: What does all these imply about teaching mathematics?

Posted in What is mathematics

The fun in learning mathematics is in the challenge

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Just like the games we play, the fun in learning mathematics is in the challenge. In the post Math is not easy, I argued that we love a sport because of the challenge it presents, the opportunities it gives us to make prediction, analyze, strategize, make our stand and defend it, etc and not because it is easy to play! Learning math is like playing our favorite sport. I shared that post in Math, Math Education, and Culture community in LinkedIn and I got interesting comments and insights.

  1. Andrea Levy • If a game is too easy, it no longer is fun! That is why kids move on to more and more challenging games. Math is fun when it is challenging, but not overwhelming. Chess is a wonderful game because the rules are simple, but the game is more challenging when you play with people who are at, or within a certain range, of your own abilities. If you play with someone who is too far below you in understanding strategy then the game is boring. If you play with someone too far advanced, then the game is frustrating. Learning math is similar. Our challenge as teachers is to find a way to make math challenging without it feeling overwhelming. And yes it can be challenging and fun. Most students learn best through social interaction. We need to provide opportunities for students to struggle individually with an interesting problem, share with a small group their thinking and try to move forward in their understanding of the problem, and then share as a class the different processes and solutions. Then math can be challenging and interesting (fun.)
  2. Jeffrey Topp • I think the problem is more fundamental than math, the challenge is getting students to look for challenges and see conquering those challenges as being fun because ultimately life is about finding out what we are made of. I have always been good in math but received poor grades in high school and didn’t learn anything until I realized that the challenge of solving difficult problems was actually fun. Once I realized that, everything fell into place.Math is a great venue to teach this concept because, frankly, thinking is challenging. As a country, though, we are getting lazy and rather than accepting that there are students who won’t spend the time thinking we change the material to require more memorizing, or process following. This hurts everybody.
  3. Sheldon Dan • I don’t know if math should be “easy,” but it should be understandable. I have taught developmental math at a community college in Memphis, and one of my goals is to help people understand a subject that many fear, especially my students who have not been in a math class in many years. Therefore, my concern is more for them to know why they are doing something as well as how to do it. “Fun” is not really a consideration, and I don’t think it should be. I think if the concepts can be taught in an interesting way, say by the use of manipulatives, that is a bonus, but we can’t lose sight of the fact that there are some things in math which will not be “fun” and they are still necessary in our classes.
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Can having fun and learning math coexist?

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I wrote a little post titled Mathematics is not easy  to challenge the teachers I work with to rethink the way they teach mathematics. I  shared the post in LinkedIn and it generated intelligent discussions and ideas about mathematics and teaching mathematics from the community Math, Math Education, Math Culture. I think we can learn a lot from the well-thought of comments and reactions from the community. There are so far 224 comments. Let me share the comments and reactions related to the existence of ‘fun’ in mathematics. Before you read the comments and reactions, I suggest you first read the post Math is not easy.

  1. Certainly not a waste of time. Making it fun and easy has great benefits, such as increased love for math overall, stronger levels of confidence, and an encouragement toward number sense – Steve Kleinrichert
  2. So are you suggesting that we just tell our students on day one that “this is not going to be a fun class. This material is going to be difficult and you probably won’t understand it!”…..
    And what is your “fact” based on? “Math is not an easy subject”… With proper techniques & approaches, math can be as easy as any other subject. And “… not easy to learn it” depends on the student, teacher & learning goals.
    Also, if you are suggesting that teachers accept the “fact”, how will that make it better for anyone, students or teachers? – Scott Taylor.
  3. There’s a big difference between making math fun by putting in external wastes of time that obscure the math, and making it fun *because* of the mathematics and the interesting problems within. I agree that we shouldn’t do the former, which I think is what Erlina is saying. But we should try to choose problems that are fun and engaging through their mathematical content. – Daniel Zaharopol.
  4. So much mathematics can be learnt through a playful interaction with the problem. In fact, we often colour our students perception of what we expect by using words like problem when puzzle or dilemma would be just as suitable. A sense of fun is, in my experience, essential as it prevents students from giving up and writing off the ability that they have already. – Mike Chittenden
  5. Making mathematics easy is the goal of every diligent math teacher. Like perfection, the ideal may be unattainable but the pursuit is worthy. – Charles Ashbacher
  6. I also disagree with you. If teachers stopped trying to make math fun I think math would get even more boring. There is so much beauty and mystery. I asked my students what they thought math was and they gave some very interesting answers. Since I am teaching in China none of my students said stupid or not fun. There is fun to be had and interesting things to talk about. – Dominique Lomax
  7. My mathematics teachers did not try to make mathematics fun, yet I found that mathematics is interesting in itself. – Ng Foo Keong (Sophus)
  8. Math and any other subject can be difficult / boring / not fun depending on our (teachers) goals. I have never pretended that I educate new generation of mathematicians. My goal is to develop the way of mathematical thinking through delivering strong basis of mathematical concepts. Can it be fun? Maybe. Is it difficult? For whom: teachers or students? It depends on…. Lots of parameters. – Bess Ostrovsky

More related insights in The fun in learning mathematics is in the challenge.