Category: Elementary School Math

Posted in Elementary School Math, Number Sense

What are fractions and what does it mean to understand them?

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Negative numbers, the irrationals, and imaginary numbers are not that easy to make sense of for many students. But this is something understandable. One only needs to check-out the histories of these numbers. The mathematicians themselves took a long time to accept and make sense of them. But fractions? How can something so natural, useful, and so much a part of our everyday life be so difficult? Didn’t we learn what’s half  before we even learn to count to 10? I’m sure this was true even with our brother cavemen. So how come the sight of a fraction enough to scare the wits out of many of our pupils and yes, adults, too?

Fractions are used to represent seemingly unrelated mathematical concepts and this is what makes these numbers not easy to make sense of and work with. In mathematics, fractions are used to represent a:

  1. Part-whole relationship – the fraction 2/3 represents a part of a whole, two parts of three equal parts;
  2. Quotient – 2/3 means 2 divided by 3;
  3. Ratio – as in two parts to three parts; and
  4. Measure – as in measure of position, e.g, 2/3 represents the position of a point on a number line.

Of these four, it is the part-whole relationship that dominates textbooks. For many this conception is what they all know about fractions. While it is also the easiest of the four to make sense of, students requires series of learning activities to fully understand part-whole relationship . Crucial to this notion is the ability to partition a continuous quantity or a set of discrete objects into equal sized parts. Below are sample tasks to teach/assess this understanding. They call for visualizing skills.

Of course understanding fractions involve more than just being able to use them in representing quantities in different contexts. There’s the notion of fraction equivalence, which is one of the most important mathematical ideas in the primary school mathematics and a major difficulty. This difficulty is ascribed to the multiplicative nature of this concept. There’s the notion of comparison of fraction which includes finding the order relation between two fractions. And if your students are having a hard time on comparing fractions you can check their understanding of equivalence of two fractions. It could be the culprit. And let’s not forget the operations on fractions. An understanding of the procedure for adding, subtracting, multiplying, and dividing of fractions depends on students’ depth of understanding of the different ways fractions are conceived, on the way fractions are used to represent quantities, on the idea of equivalent fractions, and on order relation between fractions, and  many others such as the meaning of the operations themselves.

A study has been conducted categorizing students levels of conception of fractions, at least up to addition operation. Just click on the link to read the summary.

 

Posted in Elementary School Math, Geometry

Angles are not that easy to see

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Like most numbers, geometric objects such as angles, are abstraction from properties of real objects and quantities. For example, the idea of “two-ness” can be abstracted from real objects such as two apples, two chairs, two goats, etc. It will not take along for a learner to figure out what the idea of two means. Abstracting angles from real objects this way is not as easy as one might think it is.

Look around you and find something that to you looks like an angle. Chances are you would identify corners as forming an angle.  That’s easy because you see two sides meeting at a corner. But doesn’t the door also forms an angle when you open it?  But where is the other side? How about turning the door knob? Doesn’t it form an angle also? But where are the two sides there? It doesn’t even have a corner!

Mitchelmore and White (2000) of Australian Catholic University conducted a study of 2nd, 4th, 6th and 8th grade students understanding and difficulty about angles.  They found that students do not readily incorporate ‘turning’ in their idea of angles. They found that it is the line (or arms) of angle  which are the key to students identifying angles in different physical situation. Their study showed the easiest angles for students to learn are the two-line angles. These are angles in which both arms are visible such as corners of geometrical figures, corners of rooms, blades of a pair of scissors. The second group of angles are the one-line angles. In these angles, only one arm is visible. The other line must be imagined or remembered. Examples are the angles formed by an opening in a door, a hand of a clock and sloping of roofs. The most difficult for the students to identify are the no-line angles in which neither arms of the angles are visible. Examples include the turning wheel and spinning ball.

One can be said to have an understanding of the concept of angle if he/she can recognize all these types of angles in physical objects and is able to see that they all share the same property: they all consist of two linear parts (even if they are not visible) and they cross or meet at a point and that the relative inclination of the two parts has some significance – it defines the sharpness of the corner or the their openness.

So what is the implication of these to teaching? The most obvious is the importance of exposing students to as many different physical situation that can be represented by angles. Starting with the definition an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle and then drawing the angle figure on the board is certainly the most ineffective strategy the teacher  can do to teach students about angles.

 

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.