When presented with a topic to cover, there are two dominant ends of the Math-Ed spectrum. First, you have the transmission approach which carefully selects examples that represent the questions of that type that will be encountered in homework packages and on unit exams. The teacher can predict exactly what the class will look like, and students have very little control. Second, you have the open-ended approach where students are given a leading question and formulate ideas in order to solve it. In this process, the students may end up wherever their minds (and motivation) take them. The teacher has almost no clue where the class is going, and the students are given ultimate control. If a teacher is presented with these two juxtaposed methods, they will wisely choose control.
In my class, I work to find a happy mix between the two. I focus on designing "tasks" that give students control over certain learnings, but are focused enough that I can also appease the rigid timeline in the front of my curricular documents. The tasks do not require a large amount of creativity, but a willingness to see opportunities to step back and hand control to the students. It involves me being willing to guess where students will go, and ask leading questions to push them further. Every topic becomes an opportunity for me to grow as a teacher, because I don't know where students will go, what misconceptions they will challenge me with, and what discoveries they will make. (By the way, I hate using the word "discovery" because it has become the whipping boy in every staff room conversation. For some reason, teachers doubt their students' ability to make connections without explicit instruction.)
The following lesson is an example of viewing the classroom slightly differently. The original idea was not mine, but as one of my Education professors once told me, "Teachers make excellent pirates." The original lesson plan came from Great Maths Teaching Ideas. They often tweet lessons with interesting connections to nature, sports, and society.
The original lesson was framed under the framework of an "unusual way to teach plotting straight line graphs" by examining the linear function of cricket chirps and temperature. Other that that creative context, there was little difference between this problem and those found in textbooks. They created a worksheet where students were given a function, a table of values, and a grid to plot on. They were then asked a series of questions. A creative, real-world situation doesn't necessarily constitute a change in teacher thinking.
I took the context and blossomed its potential to include a variety of pathways and afford opportunities for students to practice multiple skills in a cooperative setting. Because of the close ties to science, I chose to place the problem in a lab setting. For background on mathematics labs see "Merit To Mathematics Labs"--an earlier post on this blog.
Linear Functions Lab
Students are separated into groups of two or three and handed out two handouts. The first is a copy of the "lab report" that looks very similar to the handout from the original lesson, and the second is a copy of a graph that converts degrees Fahrenheit to degrees Celsius. The image I used is below:
They are not given the function which creates this graph for good reason. It only provides another opportunity for a students to create the function once they work with function notation.
Once all students are settled, I call their attention to the screen at the front of the room and play the following video clip from the hit show "The Big Bang Theory". It provides an engaging start to the lab.
When the clip is done, I show the wikipedia page for the Snowy Tree Cricket. The bottom of the page provides the linear relationship between the "chirps every 15 seconds" and "temperature in Fahrenheit". Students choose variables, label axes, and then we create the linear function as a group. suggestions are taken and interpreted until we land that if c = "Chirps per 15 seconds" and F = Temperature (Degrees F) then:
F = c + 37
This is a fairly simple relationship, and students are set out to graphing it with a table of values. Students have had experience with graphing functions before. I insert this lab directly after I have introduced the idea of slope, but before the idea of "y-intercept". As students work, I circulate and ask questions like:
What's the rate of change?
Should the point be connected? (this is an interesting one)
What is the Domain and Range of the function?
What's the temperature if the cricket isn't chirping? Can you know for sure?
When students become comfortable with the first graph, I initiate new learning by preying on their unfamiliarity of the Fahrenheit system. I ask them to switch from Fahrenheit to Celsius to make the graph more relevant. The only tool they are given is the conversion graph--they are not given a function. They are left to devise their own action plan. Should the scale be changed? Maybe the graph is exactly the same? Do we need a new function? How can we make one? What are our new variables? Which axis should represent each?
This second part of the lab provides stratification for students that need it. Some students may struggle with the first plotting, and it gives extra time for them to learn it while providing an extension for those upper level students. It practices reading graphs and interpreting their results. Students will come out with two straight line graphs with unique slopes and y-intercepts. At this point, I ask the students to find the slope of the new graph and then compare them. Which temperature increases faster as the cricket chirps more? We then examine the "b" value in the equation. How does the constant translate onto the graph? We look back at their table of inputs to see what is going on. I might choose to grab a blank piece of graph paper and draw a random line on it. Can we follow the pattern to write a function to represent the new line?
If students really excel, this a chance to introduce composition of functions. The teacher must be prepared to move along with groups at their own, individual pace. Some may rocket through the graphing but get stuck with the conversion graph. Others may need assistance with basic algebra in the first table of values.
Assessment is two-fold. Students are required to hand in their lab sheet and graph stapled together with their names on it. Also, they fill out a quick self and peer assessment on how their classmates contributed. Most of the assessment are not surprising because I can gauge the feeling as I circulate from group to group. I make sure that I leave 10 minutes at the end of class. Five is used to de-brief the many learning styles that I saw around the class. I detail various strategies so students get the sense of the diversity. It also highlights hard work and increases motivation. The last five is spent on assessment and other logistical efforts such as hand-in and final questions.
The context of the problem is creatively posed, but that is not what makes it full of rich learning. It takes a quick shift in teacher perspective to get the most out of creative ideas. Their task is pointed but autonomous. Here students work to expand on prior skills and create new understanding. We are working toward the connection between slope and y-intercept of a graph and the values of "m" and "b" in y=mx+b. A class like this creates an anchor lesson where I can always look back on and say, "remember the crickets". Students dive in and create their own understanding through active learning. You don't have to be the most creative teacher in the world to allow students to proceed on their own, you just need a little foresight as to where they may go.