Tag: teaching principles

Posted in Number Sense

Math Knowledge for Teaching Addition

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This post is the second in the series of post about the Math Knowledge for Teaching (MKT) where I present task/lesson that teachers and interested readers of this blog can discuss. The first is about Tangents to Curves, a Year 12 lesson. This second post is for young learners.

The task

How many small cubes make up this shape?

cubes

This is a pretty simple task.  Any Grade 1 pupil will have no difficulty giving the correct answer. All they need to do is to count the cubes. Yesterday, in one my workshop with teachers about lesson study, we viewed a Japanese lesson using the same task but was used in such away that children will learn not just counting.

The lesson

Before this lesson the class already learned that putting together concept and the symbol + and =.

The pupils were given small cubes to play with on their tables. After a minute, the 2x2x2 cube was shown on the TV screen and the teacher asked the class to predict how many small cubes make-up the shape. Some used their cubes to make a similar shape without the teacher encouragement to do so. The cubes were only there to help those who might have trouble imagining the bigger cube were some parts are not shown. The pupils counted the visible cubes one-by-one and then those not seen in the drawing (a drawing of the cube is posted on the board). But, the teacher was not just after the answer 8, he was after the learners’ counting strategy. So he asked: Can you use the + sign to show us your counting strategy? Some of the students answers were: 4+4 = 8, 2+2+2+2 = 8, 6+1+1=8. But, the teacher was not only after this, he wanted the class to realize that this number expressions may have come from a different way of looking at the cube. He started with those who wrote 4+4 to show the class how this counting was done. There were two different strategies: halving the cube vertically and the other horizontally which the students demonstrated using the cubes. All throughout the teacher was asking the class, “Can you follow the thinking? “Do you have a different idea?” “Who has another idea?”

After the summarizing the different ideas of the pupils in the first task, the teacher gave the second task:

What is your idea for counting the small cubes in this shape? Show your idea in numbers and symbols.

cubes

The shape was projected on the TV screen as the teacher rotated the shapes. The pupils came-up with different combinations of visible and not visible cubes like 7+3 = 10, 4+6 = 10, etc. They were invited to explain these expressions and their thinking using the drawing on the board. The teacher did not have any difficulty getting the answer he wanted from the pupils: “We already know that this shape (the big cube) is 8 so we just add 2  (8+2 = 10).

Questions for Teachers Discussion/Reflection:
  1. What about numbers will the pupils learn in the lesson?
  2. What is the role of technology and visuals in this lesson?
  3. What about mathematics is given emphasis in the lesson?
  4. What mathematics teaching and learning principles underpin the design of the lesson?

Remember this quote from George Polya: What the teacher says in the classroom is not unimportant, but what the students think is a thousand times more important.

math knowledge

For further reading:

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (Studies in Mathematical Thinking and Learning Series)

Posted in Algebra, Graphs and Functions, High school mathematics

What is an inverse function?

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In mathematics, the inverse function is a function that undoes another function. For example,  given the function f(x) = 2x. If you input a into the function f, the output is 2a. The inverse function of  f(x) is the function g(x) such that if you input 2a into g(x) its output is a. Now what is g(x) equal to? How does its graph look like? Is the inverse of a function also a function? These are the basic questions students need to answer about inverse function.

How to teach the inverse function
Functions and their inverses

The idea of inverse function can be taught deductively by starting with its definition then asking students to find the equation of the inverse function by switching the x and y in the original function then expressing the equation in the form y = f(x). This is an approach I will not do of course as I always like my students to discover things for themselves and see and express relationships in all three representations: numerical (ordered pairs or table of values), geometrical (graphs) and symbolic (equation) representations.

In teaching the inverse function it is important for students to realize that not all function have an inverse that is also a function, that the graph of the inverse of a function is a reflection along the line y = x, and that the inverse function does not necessarily belong to the same family as the given function.

The concept of inverse function is usually taught to introduce the logarithmic function as inverse of exponential function. Important ideas about inverse function such as those I mentioned are not usually given much attention. Perhaps teachers are too excited to do the logarithmic functions.

I suggest the following sequence for teaching inverse. I’m sure many teachers and textbooks also do it this way. What I may just be pointing out is the reason behind the sequence. I also developed three worksheets using GeoGebra. The worksheet is interactive so that students will be able to make sense of inverse of function on their own.

Start with linear function. Its inverse is also a function and it’s easy for students to figure out that all they need to do is to switch the x‘s and y‘s then solve for y to find the equation. You may need to see the inverse of linear function activity so you can make sense of what I am saying.

The next activity should now involve a quadratic function. The purpose of this activity is to create cognitive conflict as it’s inverse is not a function. The domain needs to be restricted in order to get an inverse that is also a function. Depending on your class, the algebraic part (finding equation of the inverse) can be done later but it’s important for the students at this point to see the graph of the inverse of a quadratic to convince them that indeed it is not a function. Click the link to open the activity inverse of quadratic functions.

The third activity will be the inverse of exponential function. By this time students will be more careful in assuming that the inverse of a function is always a function. Except this time it is! It is also one-to-one just like linear, but it’s equation in y belong to a new family of function – the logarithmic function. Click the link for the activity on inverse of  exponential functions.

Teaching principles

There are at least three math teaching principles illustrated in the suggested lesson sequencing for teaching the inverse function and introducing logarithmic function.

  1. Connecting with previously learned concepts. Start with something that students can already do but in a different context. In the above examples they are already familiar with linear function and they already know how to find its equation.
  2. Creating cognitive conflict. The purpose is to challenge possible assumptions and expose possible misconceptions.
  3. Making connections. Mathematics is only understood and hence powerful when there is a rich and strong connections among related concepts, representations, and procedures.

You may find the Precalculus: Functions and Graphs a good reference.