We're studying quadratics....and even the name usually strikes fear in the heart of the most competent adult. I didn't want it to be that way for my math students. I wrote a good lesson plan and then I let students help me modify it---they "taught" me how to teach them better thru the interaction and feedback we gave to each other during the learning process.

**I create a scaffolding technique, but students helped me add, delete and amend it until it worked for the way they think.**

Even if you don't teach math, this works...because I've done things like it in science and social studies. I guess I just had forgotten about using it. So the trick is to create a means for scaffolding and then let students help you fine-tune it...meeting their needs. In addition, you'll learn so much about the topic that would never occur to you otherwise.

We started off slowly using the geometric area model. Our textbook tells us that Greek mathematicians used this method as long ago as 300BC. I think that sort of impresses students that they are following in the steps of ancience Greeks. We were also able to pair this ancient technique with modern-day technology using Geometer's Sketchpad and Algebra-Tiles sketch that was available in the software library.

From there we started moving into a more symbolically version of finding the "x" solutions, figuring out what this means in real life and how to even use an old kindergarten Valentine making technique. It wasn't easy but they hung in there.

Here's what I learned.....

If I divided my SmartBoard into two areas: one side represented what the problem we were studying and the other side represented the thinking someone would need to do in order to solve the problem. This helped tremendously and as we worked problem after problem, they helped me refine my thinking list.

- Look for GCF
- Look for a letter that could be factored
- Find both factors(it's a mulitplication expression after all)--usually we write something like this to remind us ----> a product=factor * factor
- Solve for zero---finding both "x"s
- Set those "x"s equal to the x-intercepts
- If you need to find a min or max, find the line of symmetry--we would say "this is like when you folded the paper in two and cut out your Valentine heart " and we always did the hand motions!
- Use the line to find the "y" of that max or min

So all of this is on one side and then they use the other side to solve the equation. It's scaffolding and it helps them ingrain the process in their brain. I'm not sure they realized how much they helped me think about their thinking, but student feedback helped me zero in on what they needed me to "think aloud" for them. Throughout the unit, you would see students able to stop look at the process list. They could perform this procedure independently.

To enrich the lesson, we did a one day mini-lesson that showed them the quadratic formula. It ties what the ancient Greeks did to another pretty old mathematician....Francois Viete. He was the French mathematician that published this formula way back in the late 1500s. Here's the second thing I learned...they couldn't use this formula on problems unless they were in the standard form of the quadratic. Again I could scaffold this by simply writing the standard form and then helping them use

I can imagine them in high school...thinking all this was foolishness. And it will be then. But right now, where they are developmentally in building up their quadratic muscles, it is perfect.

At a Mind, Brain and Education Session at Harvard in 2007, where there were 92 mostly teachers plus a varied assortment such as myself, a Prof. asked the audience to raise their hands if they had used the process of solving a quadratic equation in the last six months.

When just four hands went up, he asked the audience why we were teaching this. It bears thinking. In a world where there are more smartphones than toothbrushes, the issue needs consideration. Are we teaching quadratic equation solutons because questions like this are easier to mark, or do they offer as the author has said some "quadratic muscles". I guess as long as there is time in the syllabus and there can be some value associated with it, it is worth doing.

Another point of view could be that for most of us who do not aspire to be Professors of Maths, there must be a lot in the math syllabus that could be removed and replaced by curriculum requirements that demand student ability to model a real world problem, use the most appropriate computation methods to solve it - whether it is Mathcad or the many open source solutions, and then assess the sensitivity of the solution to errors in the data used. There is plenty more of real world maths that could be added to this.

Posted by: samar | May 10, 2013 at 01:01 PM

Dear Samar,

Thanks for your comment. I think this is a perfect moment in learning where the real world does intersect with the need to know. As my students would tell you, they have found hundreds of places where quadratic relationships govern how the real world works.

For them, realizing that different kinds of things behave in the different ways was enlightening. We just had a conversation yesterday in class about the kinds of things that behave in a linear fashion, the kinds of things that behave in an exponential fashion and now the kinds of things that behave in a quadratic fashion. They were amazed and the language of math is helping them describe and understand the world better.

Probably few of them will be Math Professors, but I think this touch of how math informs their understanding of rollercoasters, shoot a basketball, lofting a soccer ball over the face of the goal, compares to compounding interest rates or how fast yeast grows in our science experiments is something that will help them throughout their lifetime.

I know I can still quote little snippets of poems that I was made to memorize when I was in elementary school....and suddenly they pop in my head and give me a smile or a little comfort. I think this is the same way.

Posted by: Marsha | May 10, 2013 at 01:53 PM

Dear Marsha

Thanks for your enriching reply.

I could not agree with you more in that Mathematics helps us to understand the world around us. The issue really is of how we compute the mathematics. Currently it appears the focus is on computation through traditional means. This of course limits the amount of time we can spend exploring real life issues with our students.

I would suggest two methods for exploration but before doing so, I would like to honor your committment to making maths meaningful to your students by developing this intensity of interest in the subject yourself.

One complement to traditional calculation as opposed to an alternative would be the use of system dynamics models. Diana Fisher is a school teacher who has done wonderful work in this area. You may care to look at the following paper to get some idea of its application to the issue of linear, nonlinear and quadratic relationships:

http://www.clexchange.org/ftp/documents/Implementation/IM1997-07IntegrationSDMath.pdf

Another complement would be to use the Open Source language R. This of course requires developing some sort of programming ability so it could be a later initiative. When it comes to modeling real world phenomenon it is a serious contender.

I have recently had the pleasure of making a mathematical model of how apartments in my city generate wet and dry waste, and have spent many years working with predictive models for large ships. However, I can not solve a differential equation by hand. Indeed I have never felt the need to do so since leaving school.

The real issue is whether we spend our time trying to laboriously solve different types of equations or we spend that time using Tracker or even the multiple 3D sensors in our smartphones to explore motion. The emerging professors can go the whole hog on the hand calculations but for the rest of us, understanding the beauty of Maths is liable to be more stimulating.

Incidentally something all forward thinking educators like yourself should watch: http://www.pbs.org/wnet/ted-talks-education/video/full-program/

Thank you for the conversation. I hope it helps make the education system a tiny bit better worldwide.

Posted by: samar | May 10, 2013 at 09:38 PM

Dear Samar,

Thanks again for your response. WOW...it gives me so much to read and investigate. It will take some time for me to go thru all of it.

But I can tell you this. I already believe in using models...mostly because of the power they give to students in thinking about things. We are heavy users of Geometer's Sketchpad....using it to learn all about slope, rate of change, parallel and perpendicular lines, how to find points on a circle using Pythagoras, etc etc etc. So "R" seems like a natural extension.

Here's where I get a little stuck, to be honest. My students are just at the beginning of Algebra....they arrive in August not understanding linear equations. It takes weeks of discussion, experimenting and practicing for them to become fully proficient in "knowing" positive/negative coefficients, y-intercepts etc. Much more quickly they can rotely identify these things, but to truly understand and for them to feel confident.

My challenge is to take students from that starting point all the way thru using quadratics here at the end of the course. It's a wonderful journey and they work hard to make that progress.....and we do so much more than focus on the computational proficiency. We develop number sense, properties of math sense and even a student's ability to use key vocabulary words.

The question I always have to answer is how to make each and every step logical and confidence building. I can't use a tool, no matter how wonderful, if it doesn't add to that end goal. So I agree with you wholeheartedly. I also thank you for your interest and suggestions.

Posted by: Marsha | May 11, 2013 at 08:26 AM