Mathematical Modeling and Real-Life Data

Today in Calculus, we looked at three different types of regression equations that can be used to model data. Linear regression was first. The students measured each others’ forearm lengths and their heights. You can have a good conversation about which should be independent and which dependent. We decided the forearm lengths were independent, and height was dependent. We put the data into our TI-84 Lists, determined appropriate window parameters, and plotted the data. Then we used the LinReg feature to graph a line of best fit:

LinReg

The next regression equation we investigated was the quadratic one. For this one, I used the old favorite, the water fountain stream:

Fountain

We pasted it into GeoGebra and put some points on the path of the water. Using those points and the QuadReg feature of the TI-84, we came up with an excellent model:

QuadReg

Finally, we used the sunrise and sunset times of the fifteenth day of each month in Nashville, TN to see how a sinusoidal function can model data. (I grabbed the data from the US Naval Observatory.) Students converted the times that were in hours:minutes to decimal hours (16 + 56/60) – (6+57/60). We entered the months (January = 1, February = 2, etc.) and length of daylight into the calculator’s lists and used SinReg to determine the sinusoidal function that best fits the data. If time permits, it’s a nice exercise for the students to figure out the amplitude, period, phase shift, and translation themselves, and compare their function to the calculator’s.

SinReg

This set of exercises took about 50 minutes, and the girls got pretty excited when they saw how closely their functions matched the data, and how they can use them to determine other points.

 

 

I Stink, You Stink, We All Stink At Math

The 2014/15 school year is underway, and one of my goals is to write a post at least once a week. This week, we’re taking a look at an article from the New York Times that has caused a bit of a stir among math teachers: Why Do Americans Stink at Math? In it, author Elizabeth Green laments the failure of American teachers to embrace and properly implement needed reforms.

When I began teaching in the late ’80s, I remember the big splash NCTM made with their Principles To Actions. A Japanese teacher, Takeshi Matuyama, read about them and began applying them to his teaching with great success. However, when he visited the United States, he was surprised and disappointed to find that American teachers were still teaching math the same way it had been taught 100 years earlier.

Now there is a heated discussion over the Common Core math reforms. Current NCTM president Diane Briars has weighed in with a letter supporting Common Core, and tying it to the earlier Principles. Will teaching reform take hold this time, or will we continue to use 19th century techniques?

Green’s article explains the obstacles to implementing true reform, and she does an excellent job outlining why Japan’s math teachers are so effective. The main obstacle to reform in the United States is inertia – most teachers teach the way they were taught, so there is a built-in resistance to change. Another obstacle is inadequate training. A one-day workshop on the Common Core is not enough to equip teacher to radically change their methods. As Green writes, “In the hands of unprepared teachers, alternative algorithms are worse than just teaching them [students] standard algorithms.”

Why is Japanese math instruction so effective? The primary reason is Japanese math teachers are always reviewing their lessons, especially with peers and mentors. They invite other teachers to observe their classes, and then they go over ways they can improve. Their lessons are much more student-centered as well. Green compares American teaching style to Japanese as “I, We, You” vs. “You, Y’all, We”.

In the “I, We, You” model, the teacher works an example, then asks the students to work a few in class, and then assigns a bunch of homework problems based on those examples. In the “You, Y’all, We” model, the teacher posts a Problem of the Day that students wrestle with. Through their struggle to solve it – first individually, then in peer groups – the math concept is introduced in a way that encourages students to discover it on their own. “Almost half of Japanese students’ time was spent doing work that the researchers termed ‘invent/think'”.

There is a growing movement among teachers to embrace change. Dan Meyer is one of the most prominent voices encouraging teachers to move to a student-centered, active learning. Once you begin to connect with these innovative educators, you will get very excited about the future of math instruction in America.

If you’ve read this far, I encourage you to read the entire article. Even if you aren’t a math teacher, it has some valuable advice to teachers of any subject. For me, the take-away quote is this:

We will have to come to see math not as a list of rules to be memorized but as a way of looking at the world that really makes sense.

 

The Future of Coding? Check out Wolfram Language

Earlier this year, I taught a mini-course in computer coding, and it rekindled my interest in that science. I just came across a video where Stephen Wolfram (of Mathematica fame) previews a new language he’s been developing for 30 years. Here’s the official description:

Designed for the new generation of programmers, the Wolfram Language has a vast depth of built-in algorithms and knowledge, all automatically accessible through its elegant unified symbolic language. Scalable for programs from tiny to huge, with immediate deployment locally and in the cloud, the Wolfram Language builds on clear principles—and 25+ years of development—to create what promises to be the world’s most productive programming language.

It looks to be relatively easy to use, while incredibly powerful. In fact, I’ve never seen anything like it. From what I can tell, in many cases the coder can simply type what he or she would like to see, and Wolfram Language converts the text into code:

Wolfram1

 

It links to the cloud, and data from WolframAlpha or any other site can be incorporated into it.

The video is 13 minutes long, but well worth your time if you’re interested at all in the future of coding:

Here is a link to the reference guide to Wolfram Language.

Winning and Losing

Thaddeus Wert:

An excellent reflection by the Director of our Upper School on winning and losing, and the lessons to be gained.

Originally posted on Bears Repeating:

Part of growing up is learning how to play the game. When teaching a young child how to play a card or board game, we usually begin with games of luck. With no real skill involved in these games, a child is able to discover, in a non-threatening environment, that she sometimes wins and sometimes loses. The goal is not as much about memorizing the rules of the game as it is about learning how to behave when we win or lose.

We have seen children, young adults, and older adults win and lose well and not so well.   Winning gracefully is always the easiest, but even that can become challenging when it happens too often in a short period of time. Coaches never want to be the undefeated team going into the finals in a tournament. Feelings of overconfidence can quickly and quietly soften our drive and resolve.   Let’s…

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