Chanllenging Students, Challenging Educators

This summer Apple offered a number of short webinars for educators in their Apple Summer Semester Series.  One of the sessions focused on Mathematics and featured Dan Meyer, a high school math teacher who is attempting to turn mathematics education upside down.  His main focus is selling math to students in a way that is more engaging than the traditional mathematics text.  To do this he poses traditional math problems in non-traditional ways.  Typically by staging videos that get students questioning and inspiring them to look for answers on their own. He also proposes that this format for introducing problems increases the level of challenge for students.

As I think about what I want my future classroom to look like, a number of the themes Dan emphasizes keep coming up.  For example, I want students to learn to question and know what information is necessary to answer their question, and then know how to go about finding the answers to their questions.  I want students to feel confident in their answers, specifically, to recognize when their answers are reasonable and when they clearly are not reasonable.  I want students to understand that math is powerful and can really help them understand how the world works.

This style of teaching, however, I find very unsettling.  If the level of challenge for students is increased, by default, the level of challenge is also increased for teachers. I was not taught in this manner, as a student the instruction I received was overwhelmingly of the skill and drill variety.  I can write beautiful mathematical proofs, and I can solve equations, but I do not feel confident as I watch these videos that I would know how to start some of these problems.  If I don't know how to begin how can I help students see where to begin?  

With that concern in mind, I remembered reading about Thinking Mathematically by Mason, Burton and Stacy.  The intended purpose of this book is to help develop problem solving skills.  So I have ordered this book and in between my regular coursework, I am going to begin working through this book to strengthen these skills. Disclaimer: the problems in Thinking Mathematically are not presented in the raw way that Dan recommends for true exploration, however, I figure as long as I am interested that is more than can be said of high school students faced with a traditional mathematics textbook.  I am hoping that if I can learn to be more confident in approaching this type of problem then maybe I will be able to introduce these sorts of problems into my classroom successfully. 

5 Must Have High School Math iPad Apps

Image Source: Tom Raftery

I understand that in this economy, not every school will be able to afford iPads for every classroom. However, I think that having at least one iPad in each mathematics classroom should be a goal for all schools.  The iPad offers quick reference, and review information in a format that is accessible to all students.  Specifically, the iPad is a great tool for students who are struggling with a concept or topic and need an extra push to grasp the ideas.

Here are a few of what I consider to be the most useful math apps for the high school classroom:

1. Math Ref  Cost: FREE

Math Ref is an easy to use math reference guide, browse by discipline or use the search feature, this app is useful for students studying Algebra, Geometry, Trigonometry, Calculus and more.  Each topic gives an explanation and graphics, as well as examples and relations to other disciplines.

2. Evernote Cost: Free

At first glance this may seem an unusual choice for an app in a mathematics classroom.  However, if you have one student who is an excellent note taker you can assign that student the responsibility of taking notes with Evernote.  Evernote gives you the option of recording the lecture while taking notes so the notes are synced with the discussion. The person taking notes can draw diagrams as well as take written notes.  If an ill student misses class, they can quickly catch up on what was covered while they were gone.

3.  Quick Graph Cost: Free

The Quick Graph app allows the user to graph in both 2D and 3D.  The user is able to enter multiple formulas and quickly see how the graph changes as the equation changes.  This app has a built in mathematical keyboard making entering formulas fast and efficient.

4. Sudoku Cost: Free

This is the best of the Sudoku games available on the iPad.  There are multiple levels of play: easy, medium, hard, and expert.  This version allows students to get hints, make notes, and can be set to show incorrect answers.  Sudoku is an excellent logic building tool and this app makes it fun and easy.

5.  Wolfram Alpha Cost: $1.99

The reason I love this app is because of its data features.  Students can quickly access data on just about any question they can dream up.  This app provides the foundation for data analysis and statistical review.  It will also compute complex calculations and display the answers in a variety of ways.  This is a great tool for examining how the math disciplines are related.

Math Without Computation?

Everyday professionals, who are not mathematicians, solve difficult mathematical problems without ever making a single calculation.  From calculating trends in the stock market to predicting population changes, these professionals find mathematical answers to interesting questions without spending hours or days making calculations by hand.

According to Conrad Wolfram, head of the mathematical lab at Wolfram Research (the developers of the computational engine, WolframAlpha) as computers become more and more sophisticated at prediction models and complex calculations, the professional's work becomes more about asking the right questions and understanding the answers provided by the mathematical computations, than actually performing calculations.

In this TED Talk, Conrad Wolfram explores the idea that we should be teaching students to think like mathematicians instead of teaching them to be computers.



Although this idea seems extreme on the surface, I think there are valid reasons for focusing our educational energies on thinking skills rather than computational skills.  It is true that understanding the basic computations and mathematical relationships is important for mathematicians who will continue advancing mathematical research.  However, most students will not become mathematicians; they will become professionals who use math to answer complicated questions everyday. And for those students who do wish to become mathematicians, many university mathematics programs begin with the basic skills and math theory.  Knowing how to approach a problem and knowing what questions might provide answers to a problem are the skills needed by most professionals.

I think that it is vitally important that students know how to use the tools available to find the answers to their questions, and how to understand the answers provided by those tools.  If even students in the elementary grades with the use of computer models, can analyze complex calculus problems, why wouldn't we want to give them that opportunity?  Isn't our goal as educators to prepare students to be effective, productive, contributing members of society?  I propose that we will not meet that goal if we do not teach our students how to search out the answers to their questions using the technology available to them, rather than asking them to spend valuable time making calculations that an effective, productive professional currently would handle with a computer.


Image Source: Marcus Mo