Research Lesson on Graphs of Linear Functions

A group of teachers in Kranji Secondary School, as part of Ministry of Education Partner Schools Programme, are exploring ways to make the 21st Century Competencies a reality in the classroom.

Yesterday, I had the good fortune to facilitate a post-lesson discussion for a mathematics lesson on graphs of linear functions.

The lesson was to use technology to help students learn how a linear graph changes as the values of m and of c in y = mx + c change.

I learn something through my observations. I would like to share two of them here. I also learn many things by listening to teachers who shared their observations. Again, I will share two of them here.

Observation 1: Language. The six boys I observed used word like "goes upwards" and "tilt" to describe the graph. Three of them described the graph as "moves upwards". Even the more careful student wrote "when looked from left to right, slants upwards". I learn that students tend to be at a decriptive level in the learning of graphs. If they stay at this level, the main purpose of learning graphs is lost. We learn graphs so that we have a good way to represent relationships between quantities - so that we are able to make predictions or generalizations, among other things. Why didn't any of these students use language that are indicative of their appreciation that it is about relationships? I wonder if any students said or wrote when x increases, y increases when they describe a graph of y = mx + c when m > 0?

Reflecting on the I See.I Think. I Wonder. tool, the use of language indicates that the six students are at I See. Can a teacher move them to the next level by asking: Why do you think the shape of the graph is such?

I wonder how the graph looks like if the value of c increases / decreases / remain unchanged but the value of m becomes positive.

Note to self: Try to learn in oter research lessons if consistent use of a thinking routine such as I See. I Think. I Wonder. will affect students' use of language which is reflective of their level of understanding.

Observation 2: Technology. I was surprised that 14-year olds who are so adept at figuring out all things technological took some time to figure out a fairly straight forward software such as the graphing software they were using.

I See students struggling to figure out a relatively simple software. I Think this is unusual given that I have seen many kids and definitely teenagers capable of figuring out things on a cell phone or on a laptop screen. I Think too much instructions on how to use the software can be not a good thing. I Wonder if it is sufficient have the students open the software and tell them, please show the graph of y = 2x + 1 using the software; go figure out.

What I Learn by Listening to Other Observers (1) The class was entire engaged but even when some student is interested to pursue a point further, the partner was not entirely interested. Even when two students had contradicting findings, they were happy to let it go without questioning their findings. I learnt after listening to three observations that as a teacher I need to distinguish between superficial engagement and deep engagement.

(2) One observer was wondering about what if we use a "more progressive" worksheet. Combining what she shared and my own observation, I learn the limitation of the use of a standard worksheets in a 21st CC classroom.

I am interested what she envision a "more progressive" worksheet would look like.

What would I do if I were to teach the same lesson? I would start by asking the students to log on to Ace Learning and go to I.Tools. I would then say go figure out with the help of your partner how to show the graph of y = x + 2. Now, what if you change the value of the coefficient of x (the number in front of x) to other positive values? What do you see? Why? I wonder what happens if the coefficient of x is negative. What do you see? Why is the value of y decreasing when the value of x goes up?

Now work with your friends to (1) see what happens when the value of the constant changes (2) think of an explanation for what you see.

To conclude the lesson, I wonder why the graphs are always straight lines in this lesson. I wonder how the values of y changes when the value of x increases if it is not y = x + 1 but instead it is y = x^2 + 1 that is what if the x is now the square of x? I wonder what will happen? Is it right? Someone says the value of y increases more for the same increase in x. Can you show using the graph.

I wonder what happen if the equation is y = 2^x + 1?