The first of these videos is on using graphing software for multiple representations.
The video features an example of me using graphing software to highlight the link between graphical, algebraic and numerical representations so students can understand the ideas behind differentiation.
Questions for reflectionThe video suggests three questions for reflection:
- What topics would you use graphing software for?
- What are the advantages of using prepared files?
- What questioning strategies are effective when using graphing software?
What topics would you use graphing software for?When teaching A level Maths, especially Pure, I can't think of a single topic where using software to display multiple representations won't enhance students' understanding. This video shows how graphical, and numerical representations can be used to show connections for differentiation but similar could be used for coordinate geometry, trigonometry, series, integration, vectors and almost all other topics.
When using multiple representations the numerical representation is often considered as much as the link between algebraic and graphical but it is a very powerful representation for students to hang their understanding on. A table of values is very concrete and easily understandable for students. When first meeting graphs or functions this will often be the representation that students start with and so referring back to it in more advanced topics will help them appreciate how these topics have be constructed by extending ideas they are familiar with.
What are the advantages of using prepared files?For teachers who are not as confident with using software it can be very reassuring to use a file that has been prepared in advanced. If you know that there are only one or two sliders or points on the screen that you need to move, and everything else will be displayed, then there is less chance that it will go wrong or that the software will display something strange. Although I am now confident to produce files "on the fly" with software, building them up as I go along, I did not start out with this confidence. For me, using prepared files was a necessary first step in building my confidence in using technology in front of a class of students and I would recommend this to other teachers starting out using technology.
There are still cases where I use prepared files now. This is mainly when I want to emphasise a certain concept and the construction would take too long, and be an unnecessary distraction, to do it live.
What questioning strategies are effective when using graphing software?One of the best questions to ask, especially when an object is moved in dynamic software, is "Describe how this object is moving and explain why is is moving in this way". The connections between the representations are often best expressed in terms of how a change in one representation will affect a change in another one. In this example here I used the change in the gradient of the tangent to the curve as the point moved along y=x² and y=x³. For the latter of these it helped the students see that the gradient would always be positive (or 0) and that this was consistent with an algebraic version of the derivative dy/dx=3x².
As a teacher it's a common to hear the observation that you understand something better if you have to explain it yourself. "Describe how this object is moving and explain why is is moving in this way" is an opportunity for the students to engage in the activity of explain a mathematical relationship which can improve their understanding. This question works especially well because an object is moving - it feels very natural when observing the motion of an object to want to explain why it is moving in a certain way.
An alternative way to ask a similar question is to tell student that you are going to change something and ask them to describe how the objects will move before you do so. You can then check the answer. Here's one to try:
- Plot y=x² and y=bx+1.
- Describe how the midpoint of the points of intersection of the two curves will move as b is varied.