I was recently in a school where an Algebra 2 teacher was lamenting how painful it was to allow his students to use calculators in class and on tests. His diatribe reminded me of my stand on calculators in the classroom when NCTM first recommended their use in the late 1980s. I questioned why I should allow students to use a calculator to multiply 3/4 × 1/2 when, as high schoolers, they should be able to multiply fractions without a calculator. In order for technology to be a tool of learning, what I really had to learn was what it means to “do mathematics.”
This scenario brought to mind a lesson I recently presented on radioactive decay. As I prepared for the lesson, I remembered a tutoring experience I had just had on this same topic. The teacher of the student I was tutoring had presented the topic by listing very similar formulas for exponential growth and decay on the board:
Growth y = aebx, b> 0
Decay y = ae-bx, b>0
The teacher then worked some examples for the class and assigned the homework. The student’s focus was on doing the computation without understanding what the variables in the equations represented. Meanwhile, I wanted the lesson to emphasize the concept, not the computation.
I found the lesson I was looking for in Lesson 5.1 from Discovering Advanced Algebra. The lesson starts with all students standing and rolling a die. Anyone who rolls a 1 must sit down. The process is repeated until fewer than 3 people are left standing. The data is then collected in a table of how many people remain standing after each roll. The points from the table are then plotted and the data is analyzed to determine what kind of sequence it most resembles. After identifying the starting value and common ratio, students write an explicit formula for the data:
y = u0rx
The graphing calculator is then used to graph this explicit formula on the same screen as the data points from the table. Thanks to the investigation, the variables have a concrete meaning for students. And thanks to the technology, different representations of the data are meaningfully connected.
Following a colleague’s suggestion, I then repeated the activity using Fathom. Having done the investigation concretely, students were able to use the technology to simulate the die roll, generate the data in a table, graph the points, generate an equation, and superimpose the graph of the function over the plot of the points. Instead of spending time rolling dice and plotting points, students could analyze the shape of the graph generated by the data and describe it as an exponential function. The function can be further explored to explain half-life and how that would relate to doubling in exponential growth.
One of the Common Core Standards for Mathematical Practice is “Use appropriate tools strategically.” The standard elaborates to say that mathematically proficient students:
- Consider the available tools when solving a mathematical problem. (e.g. pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software)
- Know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data.
- Use technological tools to explore and deepen their understanding of concepts.
The question is not whether to use technology but what technology is appropriate for the concept to be developed. If the objective is to learn to multiply fractions, I still would not use a calculator. If the objective is to learn how to plot points, I would not use a graphing calculator. If, however, the objective is to model real world growth and decay situations, recognize that the function models growth when b > 1 and decay when b < 1, and evaluate an exponential function using either explicit equations or graphical methods, then the lesson is greatly enhanced by the use of technology.
Technology plays such a dominant role in our society. The extensive use of graphical and statistical data in the media sets a higher standard of quantitative literacy. And new demands of the workplace place a greater value on the analytical skills of mathematics than on the mechanical skills of computation. Data is ever-changing, and it is imperative that students know how to utilize the technologies that are readily available in order to analyze it.
I think back to my pre-technology days when I, too, wrote the formula on the board, did a few examples, and then assigned the homework. How I wish I could issue a recall on all those students that I cheated out of truly understanding the mathematics.