Finding the "Hidden Wows"

I was working with some middle school teachers-to-be who are noticing some of the aspects of algebra we tend to take for granted after years of practice and application, and they are noticing how difficult it can be to anticipate the strategies and struggles of students who are just learning algebra.
 
One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the Hidden Wows.

Suppose I am preparing a lesson for sixth graders to introduce problems of this form:

Solve: ax = b

 My planning might start out sounding a bit like this:

When will I use this? (Laws of exponents?)

The Question: 

I received the following email from my Mom the other day. (She's been a middle school teacher since the early '90s). She wrote:

Jon, I received The Question today from one of my 7th graders today: "When am I ever going to need to know how to do this???" He is working on zero and negative exponents and had to solve problems like this:

Write an equivalent expression for 

The answer was

He understands how to do the problems--just wants to know why he needs to know, why it's not a waste of his time.

My Response:

Good questions! I wrote a blog post about the matter... let me know what you tell him. I'd love to hear how he responds.

http://profjonh.blogspot.com/2014/05/when-will-i-use-this-laws-of-exponents.html

That Blog Post (AKA, This Blog Post):

My colleague answers that question this way:

I don't know when or if you will ever need this particular concept. It depends on what you do with your life and what technological advances are made in the future. But you know what you will need to be able to do, regardless? You will need to problem solve. You will need to think critically (reason and prove). You will need to be able to communicate quantitative thinking to others. You will need to use representations to support your thinking and share your thinking. And you will need to make connections in order to consolidate your understanding. Mathematics is a discipline that provides opportunities to practice and strengthen all of these skills. So, as we solve for x, I want you to monitor your thinking because that's what's really important.

(Read his full blog post at: http://deltascape.blogspot.com/2012/05/when-will-we-ever-use-this.html)

He's talking about the Process Standards from NCTM (2000), which are now reflected in CCSS Standards for Mathematical Practice:

But just in case that argument doesn't satisfy your young math skeptic, you might want to share the following list of websites with him (see below). It turns out rational exponents are at the heart of a great many useful scientific formulas, not to mention a lot of really cool (and really beautiful) mathematics!

Good luck!
- Jon

Seven applications of rational exponents:

• Chemistry:
     http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/The_Rate_Law
• Modern Physics (Albert Einstein):
     http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Mass-velocity_relationship
• Classical Physics (Isaac Newton, born on Christmas Day, 1642!):
     http://www.logrog.net/users/lkovach/physics/Universal%20Gravitation/Days%201-3.pdf
• Rocket Science:
     http://exploration.grc.nasa.gov/education/rocket/isentrop.html
• Astronomy:
     http://scioly.org/wiki/images/c/c6/Formula_Sheet.pdf
• Wall Street investment trading:
     http://fr.accuracy.com/visuels/28t-1271338388.jpg
• Renewable Energy (Wind):
     http://www.raeng.org.uk/education/diploma/maths/pdf/exemplars_advanced/23_wind_turbine.pdf

Installing Cabinet Hardware (or, Fun with Fractions)

So I was installing cabinet hardware today...

...and some math happened! See, I had measured one drawer to be 17+3/8 inches wide. To place the pull correctly, I needed to find the center line of the drawer. So I needed to divide 17 3/8 by 2... not exactly compatible numbers.

Let's look at a few of my options.

Option 1: Use a calculator. 

My laptop is sitting right here. It has a built-in calculator:

Is there a problem here?

Is there a problem here?

from Doug Fisher's Michigan Reading Association Presentation (via delta_dc)

A student in my W14 teacher-assisting seminar raised this question:

If the [desirable] Japanese lesson style* is all about posing meaningful problems and allowing students to explore them, and if the proper role of the teacher is to lend perspective and support in those investigations, then why are we taught to use gradual release of responsibility?
  * we might substitute problem-based learning, or 3 act lessons, or active inquiry, or...

Then today (4/9/14) I read this on Twitter from @ZPMath.

Recognizing Our Implicit Assumptions

I'm reading The Teaching Gap with my EDI 331 students (again) this semester. Every time I read it, I find something different that stands out for me. First, it was the perspective I gained by comparing "typical" lessons from Japan, Germany, and the U.S. The second time, it was focus on improving teaching rather than improving teachers, which coincided with all the voices in 2012-13 clamoring for attracting better, smarter, more biz-savvy people into the teaching profession, because--the voices loudly proclaimed--the teachers we have now are just not cutting it.

This time around, what struck me was the discussion of the culturally embedded assumptions of what it means to "teach, learn, and do mathematics".

Some of my students have been tweeting about that:

Reading #TTG ch6 has me thinking bout how teachers can effectively promote worthwhile change in stnd script #ed331 pic.twitter.com/YiSQA6LxuP — Dominic Taylor (@DominicTaylor11) March 7, 2014

With that context, let's get on with the post.