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.
Posted in Elementary School Math, Number Sense

Are negative numbers less than zero?

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I found this interesting article about negative numbers. It’s a quote from the paper  titled The  extension of the natural number domain to the integers in the transition from arithmetic to algebra by Aurora Gallardo. The quote was transcribed from the article Negative by D’Alembert (1717-1738) for Diderot Encyclopedia.

In order to be able to determine the whole notion, we must see, first, that those so called negative quantities, and mistakenly assumed as below Zero, are quite often represented by true quantities, as in Geometry where the negative lines are no different from the positive ones, if not by their position relative to some other line or common point. See CURVE. Therefore, we may readily infer that the negative quantities found in calculation are, indeed, true quantities, but they are true in a different sense than previously assumed. For instance, assume we are trying to determine the value of a number x which, added to 100, gives 50, Algebra tells us that: x + 100 = 50, and that: x = –50, showing that the quantity x is equal to 50, and that instead of being added to 100, it must be subtracted from that number. Consequently, the problem should have been stated in the following way: Find the quantity x which, subtracted from 100, gives 50. Thus, we would have: 100 – x = 50, and x = 50. The negative form for x would then no longer exist. Thus, the negative quantities really show the calculation of positive quantities assumed in a wrong position. The minus sign found in front of a quantity is meant to rectify and correct a mistake in the hypothesis, as clearly shown by the above example. (quoted in Glaeser, 1981, 323–324)

Interesting, isn’t it? Numbers are abstract ideas. They get their meanings from the context we apply them to. Of course from the school mathematics point of view we cannot start with this idea.

Here are the different meanings of the negative number that students should know before they leave sixth grade: 1) it is the result of subtraction when a bigger number is taken away from a smaller number; 2) it is the opposite of a counting/ natural number; 3) that when added to its opposite counting number results to zero; and 4) it represents the position of a point to the left of zero.

Likewise for the minus sign which indicates subtraction. Subtraction has three meanings: take away, find the difference, and inverse operation of addition. For further explanation read the post What exactly are we doing when we subtract?

Posted in Elementary School Math, Teaching mathematics

What are the uses of examples in teaching mathematics?

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Giving examples, sometimes tons of them, is not an uncommon practice in teaching mathematics. How do we use examples? When do we use them?  In his paper, The  purpose, design, and use of examples in the teaching of elementary mathematics, Tim Rowland considers the different purposes for which teachers use example in mathematics teaching and examine how well these examples were achieving the objective of the lesson. He classified the use of examples in two types – deductive and inductive.

Types of examples

Examples are used deductively when they are given as ‘exercises’. These examples are usually given after teaching a particular procedure. The initial purpose is to assist retention by repetition of procedure and then eventually for students to develop fluency with it. It is hoped that through working with these examples, new awareness and new understanding of the preocedure and the concepts involved will be created (I’m not sure if many teachers do something to make this explicit). In using examples for this purpose, the teachers should not just haphazardly give examples. For instance, practice examples on subtraction by decomposition ought to include some possibilities for zeros in the minuend. For practice in subtracting integers, the range of examples should include all the possible cases such as minuend and subtrahend both positive; minuend and subtrahend both negative with minuend greater than subtrahend and vice versa, etc.

The second type of examples is done more inductively. Here, examples are used to teach a particular concept. Their role in concept development is to provoke or facilitate abstraction. The teacher’s  choice of examples for the purpose of abstraction reflects his/her awareness of the nature of the concept and the category of things included in it, which of these categories may be considered exceptional and the dimensions of possible variation within a particular category. In other words, teachers must not only give examples but give nonexamples of the concept as well.

Sequencing examples

It is not only the example but also the sequence that they are given that affect the kind of mathematics that is learned. Rowland reports in his paper a Grade 1 lesson about numbers that add up to 10. The teacher asked “If we have nine, how may more to make 10?”. The subsequent examples after 9 are as follows: 8, 5, 7, 4, 10, 8, 2, 1, 7, 3. This looks like random examples but in the analysis of Rowland it was not. The teacher had a purpose in each example. It was not random.

  • 8: the teacher knows that the pupils usually uses the strategy of counting up so they will have success here
  • 5: this will bring up the strategy of a well-known double – doubling being a key strategy for mental calculation
  • 7: same as in 8 but this time, pupils have to count up a little bit further
  • 4: for the more able students
  • 10: to point to the fact that zero is also a number which can be added to another number
  • 8: strange to repeat an example but the teacher used this to ask the pupil who answered 2 “If I’ve got 2, how many more do I need to make 10?” which was the next example.
  • 2: here the teacher said based on previous interaction “2 add to 8, 8 add to 2, it’s the same thing (commutative property and counting up from larger number)
  • 1: the teacher did not ask how many more to make 10 as this will trigger counting up but instead related it to 2 and 8 to make obvious the efficiency of the strategy of counting up from a bigger number and perhaps to make the children be aware of commutativity.
  • 7 and 3: to reinforce the strategies made explicit in using 8 and 2 as examples.
Let us be us more conscious of the kind of examples we give to our students in teaching mathematics.

 

Posted in Geogebra

Pathways to mathematical understanding using GeoGebra

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You may want to check-out the first-ever book about the use of GeoGebra on the teaching and learning of mathematics: Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra by Ligguo Bu and Robert Schoen.

Supported by new developments in model-centered learning and instruction, the chapters in this book move beyond the traditional views of mathematics and mathematics teaching, providing theoretical perspectives and examples of practice for enhancing students’ mathematical understanding through mathematical and didactical modeling.

Designed specifically for teaching mathematics, GeoGebra integrates dynamic multiple representations in a conceptually rich learning environment that supports the exploration, construction, and evaluation of mathematical models and simulations. The open source nature of GeoGebra has led to a growing international community of mathematicians, teacher educators, and classroom teachers who seek to tackle the challenges and complexity of mathematics education through a grassroots initiative using instructional innovations.

The chapters cover six themes: 1) the history, philosophy, and theory behind GeoGebra, 2) dynamic models and simulations, 3) problem solving and attitude change, 4) GeoGebra as a cognitive and didactical tool, 5) curricular challenges and initiatives, 6) equity and sustainability in technology use. This book should be of interest to mathematics educators, mathematicians, and graduate students in STEM education and instructional technologies.

STEM – Science, Technology, Engineering, Mathematics

Wikipedia on model-centered instruction:

The model-centered instruction was developed by Andre Gibbons. It is based on the assumption that the purpose of instruction is to help learners construct knowledge about objects and events in their environment. In the field of cognitive psychology, theorists assert that knowledge is represented and stored in human memory as dynamic, networked structures generally known as schema or mental models. This concept of mental models was incorporated by Gibbons into the theory of model-centered instruction. This theory is based on the assumption that learners construct mental models as they process information they have acquired through observations of or interactions with objects, events, and environments. Instructional designers can assist learners by (a) helping them focus attention on specific information about an object, event, or environment and (b) initiating events or activities designed to trigger learning processes.

I’m not sure if the book cites research cases that show how using Geogebra or interacting with applets help students build those mental models. It would be interesting if somebody will really do a study on this.