Tag: teaching mathematics

Posted in Algebra

Teaching irrational numbers – break it to me gently

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Numbers generally emerged from the practical need to express measurement. From counting numbers to whole numbers, to the set of integers, and to the rational numbers, we have always been able to use numbers to express measures. Up to the set of rational numbers, mathematics is practical, numbers are useful and easy to make sense of. But what about the irrational numbers? You can tell by the name how it shook the rational mind of the early Greeks.


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Unlike rationals that emerged out of practical need, irrational numbers emerged out of theoretical need of mathematics for logical consistency. It could therefore be a little hard for students to make sense of and hard for teachers to teach. Surds, \pi, and e are not only difficult to work with, they are also difficult to understand conceptually.

It is not surprising that some textbooks, teaching guides, and lesson plans uses the following stunts to introduce irrational numbers:

After discussing how terminating decimal numbers and repeating decimal numbers are rational, you can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERS.

What’s wrong with this? Nothing, except that it doesn’t make sense to students. It assumes that students understand the real number system and that the set of real numbers can be divided into two sets – rational and irrational. But, students have yet to learn these.

Some start with definitions:

Rational numbers are all numbers of the form  \frac{p}{q} where p and q are integers and q \neq 0. Irrational numbers are all the numbers that cannot be expressed in the form of \frac{p}{q} where p and q are integers.

How would we convince a student that there is indeed a number that cannot be expressed as a quotient of two integers or that there is a number that cannot be divided by another number not equal to zero? It’s not a very good idea but even if we tell them that \sqrt{2} is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.

Before introducing irrational numbers, students should be given tasks that raises the possibility of the existence of a number other than rational numbers. Another way is to let them realize that the set of rational numbers cannot represent the measures of all line segments. Tasks that would help them get a sense of infinitude of numbers will also help. The idea is to prepare the garden well before planting. Read my post on why I think it is bad practice to teach a mathematical concept via its definition.

Posted in Geometry

Problem Solving Involving Quadrilaterals

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‘To understand mathematics is to make connections.’ This is one of the central ideas in current reforms in mathematics teaching. Every question, every task a teacher prepares in his/her math classes should contribute towards strengthening the connections among concepts. There are many ways of doing this. In this post I will share one of the ways this can be done: Use the same context for different problems.

The following are some of the problems that can be formulated based on quadrilateral BADF.  You can pose these problems to your class but the best way is to simply show the diagram to the students then ask them to formulate the problems themselves.

quadrilateral

Problem #1. What is the area of the quadrilateral? Show different methods.

The solution to this question depends on the grade level of students. The one shown below can be done by a Grade 5 or 6 student. Continue reading “Problem Solving Involving Quadrilaterals” →

Posted in Algebra, High school mathematics

Properties of equality – do you need them to solve equations?

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Should students learn the properties of equality before we ask them to solve equations in one variable? Would it be too hard for them to solve the equation 2(x + 7) = 4x without knowing the properties of equality?

In the grades, pupils learn to find equivalent ways of expressing a number. For example 8 can be written as 4 + 4, 3 + 5, 4 x 2, 10 – 2. Now, what has the pupils previous experience of expressing numbers in different ways got to do with solving equations in one variable?

Let us take this problem. What value of x will make the statement 2(x-5) = 20 true?. The strategy is to express the terms in equivalent forms.

2(x-5) = 20 can be expressed as 2(x-5) = 2(10).

2(x-5) = 2(10) implies (x – 5) = 10

x-5 = 10 can be expresses as x-5 = 15 – 5. Thus x = 15.

This way of thinking can be used to solve the equation 2(x + 7) = 4x.

2(x+7) = 2(2x)

=>    (x+7) = 2x

=>    x + 7 = x + x

=>    x = 7.

Of course not all equations can be solved by this method efficiently.   So you may asked ‘why not teach them the properties of equality first before asking them to solve equations like these?’  Here are some benefits of asking students to solve equations first before teaching the properties of equality:

1.  It makes students focus on the structure of the equation. Noticing equivalent structure is very useful in doing mathematics.

2.  It makes the equations like 3x = 18, x + 15 = 5, which are used to introduce how the properties are applied, problems for babies.

3. It is easier to do mentally. Try solving equations using the properties of equality mentally so you’ll know what I mean.

4. I hope you also notice that the technique has similarities for proving identities.

So when do we teach the properties of equality? In my opinion, after the students have been exposed to this way of solving and thinking.

Here’s on how to introduce the properties of equality via problem solving.

Posted in Curriculum Reform

What is mathematical literacy?

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Defining mathematical literacy

The Program for International Student Assessment (PISA) of the OECD describes mathematical literacy as:

“an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen” (OECD,1999).

Mathematical literacy therefore involves more than executing mathematical procedures and possessions of basic knowledge that would allow a citizen to get by. Mathematical literacy is mathematical knowledge, methods, and processes applied in various contexts in insightful and reflective ways. According to de Lange, mathematical literacy is the overarching literacy that includes numeracy, quantitative literacy and spatial literacy. Each of these type of literacy empowers the individual in making sense of and understanding aspects of the world and his/her experiences.

De Lange’s tree structure of mathematical literacy.Spatial literacy empowers an individual to understand the three-dimensional world in which he/she lives and move. This necessitates understanding of properties of objects, the relative positions of objects and its effect on one’s visual perception, the creation of all kinds of three-dimensional paths and routes, navigational practices, etc. Numeracy is the ability to handle numbers and data in order to evaluate statements regarding problems and situations that needs mental processing and estimating real-world context. Quantitative literacy expands numeracy to include use of mathematics in dealing with change, quantitative relationships and uncertainties. Click here for deLange’s paper on this topic.

Implications to curriculum and instruction

To identify and understand the role that mathematics plays in the world is to be literate about mathematics and its applications. This means that individuals need to have an understanding of its core concepts, tools of inquiry, methods and structure.

To be able use mathematics in ways that meet the needs of one’s life as a constructive, concerned, and reflective citizen necessitates learning mathematics that is not isolated from the students’ experiences.

To be able to use mathematics to make well-founded judgment demands learning experiences that would engage students in problem solving and investigation as these would equip them to use mathematics to represent, communicate, and reason, to make decisions and to participate creatively and productively in the functioning of society.

These show that mathematical literacy requires learning mathematical concepts and principles that would be applicable to the individual and society’s life and activities; equip individuals the necessary skills in using mathematics to reason and make decisions; enable individuals to get a sense of the nature and power of the discipline in order to understand its role in the world.

To teach mathematical literacy, curriculum and instruction should therefore include these 3 R’s:

  • Relevant mathematical concepts, principles and procedures
  • Real-life context which can be investigated and modeled mathematically
  • Rich mathematical tasks that fosters conceptual understanding and development of skills and habits of mind

Check out these great books on mathematical literacy: