Introducing logarithms

Having not taught logarithms for about 10 years, I looked through several blog posts detailing how to best introduce this topic to students, given how much less experience they've had with logs versus other types of functions. The main idea that I wanted to reinforce for students is that logs are defined as an inverse of exponential functions and to continue connecting them back to features of those functions, which they have had much more experience manipulating, graphing, and applying. As usual, Mimi had a lesson that resonated with me. I liked her formulation of a log equation as a form of a question.

From her blog:

But I wanted to add more emphasis on its derivation as an inverse. So I stole liberally from the CME Algebra 2 lesson on introducing logarithms. We started with a discussion of one-to-one functions (I was surprised to hear that they had never heard of this term before) to know if we could even go into the land of inverses. Then, once we had established that exponential functions were indeed one-to-one and did a quick summary of inverse functions, with which they were already familiar, I had them create a table of value for the function $y=2^{x}$ and then create a table for the inverse function, which we were going to call $L_{2}(x)$. We then started playing around with this function... What would it look like when graphed? Why? Is it also one-to-one? Why or why not? Can we evaluate it for certain values of x? Are there any restrictions on its domain or range? Why?

I really liked how this lesson went. It seemed to build a good deal of intuition for logarithmic functions that I felt had been missing for me when I'd taught it in the past. There was a feeling of exploring a new function, but not one that had been plopped down from the sky and was completely mysterious. There was definitely enough newness to keep it interesting, but not so much that it was overwhelming. I liked how the CME lesson didn't even call the function a logarithm, just L. The less bogging down in new terminology, the better, I say.

I showed students how to evaluate logarithms of any base using technology (both Wolfram-Alpha and Desmos are friendly to different bases) and then gave them the base change formula so that they could solve application problems now rather than waiting until we had developed more of the properties and derived the formula. We'll derive it soon, but I think that it will be more motivating now that they know why it's useful and I wanted them to be able to apply logarithms on day 1 without fending off tons of properties coming at them from left and right.

I think that it was also helpful that while we were working with exponential functions and equations, I would routinely throw in questions that required figuring out the value of the exponent, which students were forced to solve via guess and check and every time, I would mention that soon, we will be able to find these values more precisely with logs. Again, I like the idea of removing the mystery and foreignness of a new concept as much as possible... previewing and embedding it within more familiar concepts makes it less scary and connects it to prior knowledge when it is finally encountered.

Here's my actual handout:

Day 5 - Logarithms

I do regret that I didn't do a project in this unit, but we've been doing tons of projects and at this point, content coverage needs to take a bit of a priority. Also, assigning projects requires me to grade the previously assigned projects and well, yeah, let's just say that I'm a teensy bit behind.

Update:

Just saw that Henry Picciotto wrote a recent blog post explaining his approach to teaching logarithms as super-scientific notation. I would like to rework my lesson plan to include both approaches for next time.

SBG: Assessing Mathematical Practices

In an earlier post, I wrote about the challenges of giving meaningful feedback without using grades as motivation. But now, I am thinking about the challenges when grades are part of the picture. Over the summer, we put together a set of mathematical practices (a mix of aspects unique to our school, Common Core, Park School's Mathematical Habits of Mind, and work done by Avery (his post on Mathematical Habits is here), who teaches 5th and 6th grade Math at our school). The five main categories are

• Investigate, Explore, and Play
• Represent
• Reason
• Communicate
• Growth Mindset

Within each one are four sub-categories and four "levels:" Emerging, Developing, Strong, and Leading. 

Mathematical Practices 14-15

This template formed the backbone of my feedback this year. Every chapter has its own content objectives, but the practices continue the entire year and are consistently being used by all of the Upper School math teachers.

In order for this to be useful to students, most assignments had a content component, which was assessed separately, and a practices component. No assignment included all of the practices, but each one included at least a few. I would let students know in advance which practices I would be assessing with a particular assignment/project. Sometimes, there would be a self-assessment component first ("highlight the level you think you have demonstrated for each practice assessed on this assignment and give evidence for your conclusion."). They would get back a rubric with the appropriate cells highlighted, along with comments and suggestions for improvement.

Pros: very specific and detailed feedback, it was very clear to students how highly the practices were being valued, they understood them better as they continued to self-assess and get feedback on them over time, they began incorporating the language of the practices in their overall reflections on the course and in their work for the class, and they demonstrated progress and growth over time

Cons: there are so many sub-categories and so much detail that it took a while before students were really clear as to what each one meant, it took an almost unreasonably long period of time for me to do each assessment and justify each rating, compiling all of this in a non-formulaic way for a final semester grade was a Herculean effort that I don't know that I can ever undertake again, and the sheer volume of feedback that this resulted in for students, families, and advisors was overwhelming and therefore not practical in the long run

The main change that I think we will make for next year is to eliminate the sub-categories. They can be there in the background if we want to make reference to specific aspects of each practice, but always including each one is just too much. I would also like to build in more time for students to revise their work and make improvements on a project they've gotten back rather than waiting for the next project or paper in order to improve. I'm currently in discussions with other 10th grade teachers to use the last week of the school year as a time for students to put together a portfolio that will include one paper or project from Math, History, English, Science, and World Language from earlier in the year, but revised and improved to incorporate the feedback and learning that has taken place since then. I would love for revision and iteration to be a regular part of the learning cycle in all of our math classes and for feedback to be a step along the way, not the end.

I also need more regular ways to give feedback to students on practices that are not always assessed on projects, such as ones having to do with their growth mindset and collaboration and contribution towards class. There is so much already to plan, assess, and give feedback on that this one definitely slips through the cracks. But I keep reminding myself that if I want something to be a vital part of the class and for students to make progress on it, I need to regularly assess it, give feedback on it, provide explicit instruction on how to improve it, and ample opportunity to revise and iterate and apply it again and again. It makes sense to me that quality is way better than quantity here. Decreasing the number of practices, but assessing them more often and with depth, clear feedback, and explicit instruction and mentoring of students to move them along the spectrum is much better than spreading myself thin.