Tag: teaching mathematics

Posted in Geogebra, Math blogs

Math and Multimedia Carnival # 17 will be hosted here

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Hello bloggers. It’s carnival time again. The 17th edition of Math and Multimedia Carnival will be over here at Mathematics for Teaching. It will go live on 28th November 2011.

You are most welcome to share your posts about mathematics and the teaching and learning of it. You can submit your posts in Math and Multimedia Blog Carnival #17 .

Wondering what on earth a math carnival is? Check out these out: (No, not Zac Efron, the carnivals). But if you want to see Zac, check out the dvd 17  Again.

Please Like, Tweet, and Share so more bloggers will know. Thanks.

Posted in What is mathematics

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.

Posted in Teaching mathematics

Making generalizations in mathematics

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Making generalizations is fundamental to mathematics. Developing the skill of making generalizations and making it part of the students’ mental disposition or  habits of mind in learning and dealing with mathematics is one of the important goals of mathematics education.  Making generalizations is a skill, vital in the functioning of society. It is one of the reasons why mathematics is in the curriculum. Learning mathematics (if taught properly) is the best context for developing the skill of making generalizations.

What is generalization?

There are three meanings attached to generalization from the literature. The first is as a synonym for abstraction. That is, the process of generalization is the process of “finding and singling out [of properties] in a whole class of similar objects. In this sense it is a synonym for abstraction (click here to read my post about abstraction). The second meaning includes extension (empirical or mathematical) of existing concept  or a mathematical invention. Perhaps the most famous example of the latter is the invention of non-euclidean geometry. The third meaning defines generalization in terms of its product. If the product of abstraction is a concept, the product of generalization is a statement relating the concepts, that is, a theorem.

For further discussion on these meanings, read Michael Mitchelmore paper The role of abstraction and generalization in the development of mathematical knowledge. For discussion about the importance of generalization and some example of giving emphasis to it in teaching algebra, the book Approaches to Algebra – Perspectives for Research and Teaching is highly recommended. There is a chapter about making generalizations and with sample tasks that help promote this attitude.

Sample lessons

Mathematical investigations and open-ended problem solving tasks are ways of engaging students in making generalizations. The following posts describes lessons of this type:

  1. Sorting number expressions
  2. Lesson study: Teaching subtraction of integers
  3. Math investigation lesson: polygons and algebraic expressions
  4. Polygons and teaching operations on algebraic expressions

Of course it is not just the type of tasks or the design of the lesson but also the classroom environment that will help promote making generalization and make it part of classroom culture. Students will need a classroom environment that allows them time for exploration and reinvention. They will need an environment where a questioning attitude is promoted: “Does that always work?” ,”How do I know it works”? They will need an environment that accords respect for their ideas, simple or differing they may be.

Posted in Algebra, Mathematics education

When is a math problem a problem?

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One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.