On a whim I have decided to clean out my digital closet, when a thought occurred to me to look through my old lecture preparations from the various levels of secondary mathematics I have taught over the years. Coming into the profession in the middle of standardized testing and Explicit Direct Instruction (EDI), I look at the creations I used and marvel at how sterile and uninviting they must have been to the kids in my classroom. Foregoing the personal saga of how that makes me feel and what I think about all of that, I have elected to challenge myself with the lens of hindsight and the recent opportunities I have had in professional development (PD) to create lessons that I would consider giving today. My challenge is to make it compelling enough for students to want to learn, to make it “relevant” (a term taking on a new meaning for me), and cover the same material. In undertaking this process, I thought I would start where I started teaching, with algebra, and move through the course work from there – keep in mind, part of me wants to give up before I started because the folks at Mathematics Vision Project (MVP) have already created a wonderful curriculum, but I want to try it anyway. So here it goes, my first attempt.
The first chapter we covered in algebra was a review of arithmetic, covering the basic properties of numbers in base ten. Each lesson was scripted for the students and the students were only required to write the examples given in the lecture. Each section had notes like these
During a recent TED talk I was watching an idea hit me about using the Fibonacci sequence as a natural place to start this unit. Imagining an introduction to this unit with a low threshold, where the only requirement is for students to be brave enough to trust their gut. I imagine starting the unit off one of two ways, either using pictures of real world objects where this sequence shows up, or starting with a string of numbers in order from the sequence running by until the kids get so bored and tired they ask something about what’s the point of each of these numbers…I don’t know which method would grab their attention more, I don’t know which method would allow for the kids to ask about whether or not there is a pattern in the numbers they are seeing, but I do see this as a great place to start, a place where all the learners can access the material.
The reason I chose this as a starting unit follows directly as we investigate the patterns within the Fibonacci sequence. For example, the sequence is generated by adding the previous two terms in the sequence, so addition of like terms is covered. For subtraction, we could define another sequence and either subtract the terms component-wise, or we could ask what sequence would be generated if we subtracted the previous two terms, which would then cover the concepts of what happens when you have a difference between two negative terms. The investigation of division would be a straight forward check on the division of the last term by the preceding term which generates the golden ratio, and we could talk about convergence of a sequence to a particular value. We could cover multiplication by thinking of the terms as unit squares and placing them next to each other, which also leads to a discussion about area and perimeter. Then we could get really crazy and square the terms of the sums and notice how this interacts with a total sum, or look at the cubes of each term and their sum and determine how those patterns emerge…even investigating these would lead to great discussion as we make sense of the patterns that arise.
With this one sequence we would be able to cover each of the sections in the first unit, with the cohesion of a single focus used in multiple representations and this frameworks also introduces area, perimeter, pattern recognition, multiple representations of data, and convergence. All of these things fall out of looking at a single sequence and asking questions, seeking patterns, and making sense of problems. Moreover, since the sequence is discrete and may be made finite, students have a concrete example upon which to stand. As I see this unfold in my mind’s eye, I see how each unit could be displayed, prompted, and differentiated for each of the students. I will need to clean up this process and provide some examples of what I am thinking to make it clear as you read this, but off the cuff does this seem to follow a more cohesive unit that the EDI lessons shown above? What would make for better modifications and adjustments to cover the material of this first unit? Is there a more engaging representation out there?