## Logs and Decibels

Cross Posted from Hilbert’s Hotel

On Wednesday I started a lesson that I thought would be more engaging than “do these problems and keep going”… and the students complained MORE than usual!  Ugh, what a depressing day.

Well, now it is 2 days later, and I’ve had some time to reflect on what happened.  Let me explain what I did, and how I think I could have improved, although I would appreciate feedback as to more ways for me to improve.

The lesson was about Logarithms and came after my students had learned about how to solve for logarithms, and how to add, subtract, and multiply the logarithmic function.  I had actually created the lesson back in graduate school 3 years ago and was excited to actually get to use it (last year I forgot about this lesson).  In involves having students use the equation for decibels, which I provide below, and finding sound pressure (p) when given decibels (dB) and visa versa.

$dB = 10 \times \log ( \frac{p^2}{(2 \times 10^{-5})^2})$

I actually started the lesson by giving the students three areas of their lives where logarithms (and logarithmic scales) are used: earthquakes, magnitude of stars (you can see the stars really well out here in NM), and measuring sound volume.  I then allowed them to choose one of those three areas, and had them briefly research in class the following three points/questions:

1. Find one equation involving logarithms for the situation, and explain the variables in the equation.
2. Compare a few of the items on a scale and explain “how many times more” one of these things is than the other using the equation (e.g. “how many times brighter is Venus than the North Star and where do they fall on the scale?”)
3. Explain why logarithms (and a logarithmic scale) is used in this situation.

The students researched and presented, and all was smooth sailing up to this point.  Many of them were getting the point I was trying to make: that if you didn’t have a logarithmic scale, the numbers would be ridiculous to use and would be difficult to compare.  Instead, because scientists use a logarithmic scale, many people don’t understand when these are reported in the news (comparing a magnitude 4 earthquake to a magnitude 8 earthquake… the second one isn’t twice as big–it’s 10^4 times bigger!).

At this point, I decided to give “my presentation” on decibels (this was the least selected topic, and, in my opinion, one of the most interesting).

So as you can see above, I gave them the equations and showed a cool scale comparing decibels.  After this, I gave them the following worksheet (just pages 1 and 2) to fill out.

Note: I would NOT recommend stealing this worksheet without heavy editing!

Now note that this was my advanced Precalculus class, so I was trying to let them figure out the majority of this on their own–like Dan Meyer says “Be less helpful.”

Well, not only were students hopelessly lost, but they started complaining about the activity, which is rare for that class.  In my eyes, they were complaining about an activity which drew the math closer to the “real world” and gave purpose to what we were doing.  As one of the students commented “you guys always complain and ask ‘why are we doing this’ and this is why, and now you’re asking ‘why are we doing this?’ again??”

Well, I left the classroom rather discouraged and frustrated because I had expected a “better than average” lesson at worst, and was hoping that the students would really get into it, but instead I found a classroom of frustrated, tired, bored, and unengaged (is that being redundant?) students.

So what went wrong?

I want to start by think about something I read on a few people’s blogs, who had failed lessons and reflected on them.  The common theme between these was that the lesson was a great idea because it applied the content to the math very well, but it lacked a hook.

So I started looking for ways to improve that and I want to pause and quote a post from Dan Meyer’s blog:

Both of these things interested me, but the line from there to a classroom modeling task forces me to ask myself:

1. What question would lead to that interesting knowledge?
2. Is there some way I can provoke that question visually?
3. Could a student guess at that question?
4. What information would a student need to answer that question?
5. What mathematical tools would a student need to answer that question?
6. Is there some way to confirm the answer visually?

So the next time you see something that’s simultaneously a) interesting to you and b) mathematical, try running through those questions above and see how they’d play out. In the meantime, you can check out my specific answers above.

Those are all very, very important questions if you are designing a lesson for students to be engaged and interested because you think the topic is interesting.  As Evan Weinberg said:

I need to be a lot more aware of the level of my own excitement around activity in comparison to that of the students.

So how can I improve the hook in this particular lesson?

Well, for starters, I could not start by giving them the equation.  Could they find the equation above simply given a list of sounds, their respective sound pressures and decibel levels?  Possibly and possibly not, but now we’ll never know.  Either way, they would have a much more intuitive grasp of what the equation means.  Instead, I found myself explaining the equation and the parts of the equation several times over–I probably wasted more time than I would have if they had struggled with it!

This lesson also enlightened me to a difference between good struggling and bad struggling, which I previously did not know existed.  Students struggling to find “the right place to plug in the number you gave them into the equation you gave them” is not good struggling.  Students working to find a relationship between numbers and working to have a rigorous conversation about it is better.  Students struggling to see what that relationship means and how they can use that relationship for future problems is even better.

I thought I had I created a pretty neat extension activity, if I can just find the right hook and present it better (this class is still working their way through this worksheet) I might have a chance to redeem this activity for the students and for myself.  As for answering Dan’s questions above, no, I do not think there is an easy way to verify this visually.  If the students were elite musicians, I could possibly verify our finds through experiments and having the students bring their instruments into class.  Even with speakers, I’m not sure students (nor I) could hear the differences in decibels acutely enough to know whether our calculations were correct.

I will also not discount the effect of students’ lives into the situation.  I spoke with one particularly frustrated student later and found out that this student was having a “bad 3 days outside of class”, for which she and I apologized to each other for our respective lack of thoughtfulness.  You can never discount the baggage that many students bring to your room on a daily basis.  However, I will not use that as an excuse not to improve this lesson which clearly could have been better handled.

Please let me know if you have a better idea on this lesson or if you have a better idea for a lesson plan using logarithms (I suppose I haven’t had time to search many sources for one of these).