How rigorous mathematics should be taught

Following up on the discussion on f(t) about how to teach log laws, I'd like to share the primary school video I mentioned in the comments.
Actually, there are at least two videos, and this brilliantness is produced by always brilliant Deborah Loewenberg Ball.


1. Sean numbers*: Ball improvises a lesson about even and odd numbers, wherein her third-graders derive the precise definitions. 


2. Betsy's conjecture: wherein third-graders explore proof. 
You can also find transcripts and teacher notes for these videos. 


I first read about them in this NYT article, which contains many other ideas worth thinking about.


*Side note: I LOVE how Ball engages her students by giving students' names to conjectures and numbers! 

My intro to Logarithms

This year was the first time I taught logarithms, and since logarithms is a tricky subject for many students (it was for me, for many years) I wanted to make sure to get it right from the start. In my school, it is very common that students decide to switch to the lower level math when they encounter logarithms. I was determined that this not happen this year.

There were several awesome resources I found when browsing the blogosphere, including James Tanton's approach of making a riddle and changing the name. I also loved Dan Greene's idea of having a symbol, L, instead of the word log. 

However, the idea that really struck me was in a comment by Mr H to JD2718's post on logarithms. Mr H suggested that we start with logarithmic tables, and go from there, and that's kinda what I did.

Table of Bases and Powers

This is how the class actually went:

1. Give the class a few exponential equations which they can solve by converting to the same base. Intermix with these a few equations which cannot be solved that way. The students must graph or be stuck. 
2. When a sufficient number of students are bewildered, give them the table above and ask them without further explanations if perhaps this table can be of some help. Let them figure out how to read the table and use it. Most of my students were able to figure it out quite quickly. 
3. Ask students: what are you doing, how would you explain it to someone who missed today's class? Think pair share one minute per step. 
4. Divide whiteboard in two, on one side write Logarithms and on the other write Square roots.  Compare and contrast different features, like how both are something you do to something else,   how square roots and squares are inverses, logarithms and bases are inverses (they "undo" each other if students haven't done inverse functions yet) and how they are used in specific examples.
5. Ask students to return to table and solve 10^x = 50. Show them that calculator can do it as well, but that for now base 10 problems are the only ones calc is useful for. 
6. Give a few more exercises using table and calculator. 
7. Show youtube logarithm song.

Homework: students received a link to a google-form to answer some questions about logarithms, so that I could check their conceptual understanding. Only about half of the students did this but their answers showed good understanding. 

Next class, I started out by giving students this handout:

Using Logs for Effective Graphing

The students were quite impressed with this new tool to fit different size data on the same scale, and were even more intrigued once I showed them the xkcd graphs depth and height. 

I also showed them pictures of the richter scale such as this one and students were intrigued and showed good understanding such as "Look, the Chile earth quake was about 10^3.5 times stronger than the Hiroshima bomb!". Music to my ears. 

Then I introduced the laws of logarithms. I butchered that one. Check out Kate Nowak's recent post for a killer idea instead.

Next class, we looked at logarithmic functions. I started out by giving students a tiny review of inverses ("they undo each other, reflected around y=x, x and y are reversed") and then gave them this worksheet. 

Logarithmic Functions

I really don't see the need to go into detail with log functions so I didn't. 

Natural logarithm was introduced as inverse of natural exponential function and little else was done with it. 

Test results were decent, no one treated logs as an object, and no one switched to the lower level. :)

Scratch

A friend today told me about Scratch as a possible fun and simplish move towards that Conrad Wolfram vision of computational math in schools. I'm currently too groggy from post-surgery morphine to dig in and create any teaching material using scratch right now.  Has anyone used this for teaching yet?