Posted in Algebra

Guest Post: Mega Math Challenge

Only a few days remain to register for Moody’s Mega Math Challenge

math challengeSo what’s the best way to teach mathematics to the fast-paced, multi-tasking young people of the information age? Crunching numbers with pencil and paper and poring over the pages of conventional textbooks just doesn’t cut it with this tech-savvy generation, so used to instant gratification!

Today’s math teachers, for their part, are finding new ways to make the subject relevant and engaging for their pupils, ultimately helping them use mathematics to solve everyday problems. There are many ways to stimulate the children of the digital revolution but perhaps none capture the spirit of applied mathematics like Moody’s Mega Math (M3) Challenge (http://m3challenge.siam.org/about/), which allows students to do rather than just read, memorize, or calculate.

The M3 Challenge is a free applied math competition for high school students that connects textbook and classroom learning to the “real world” by simulating the genuine and practical issues we face as a society and in our daily lives. Teachers who coach M3 Challenge participants realize the contest’s potential to educate students in math modeling. And while teacher involvement is critical – they register and prepare teams of juniors and seniors – the responsibility of developing a viable solution paper by the end of Challenge day (and possibly winning a share of the $115,000 in total scholarships) falls squarely on the shoulders of the thousands of students who participate each year.

Challenge Champion

“You have to let the kids do their thing. I try not to direct, I try not to drive,” Ellen Leblanc, an experienced coach from New Jersey’s High Technology High School, shared. “Initially, and prior to the Challenge weekend, the students and I do a little bit of brainstorming: what could the Challenge problem be this year? If the question were “X,” how would you approach it and what is important? Beyond that, you have to leave it up to the students,” she said.

With so much technology at their fingertips, high schoolers in 2013 are used to doing more than just reading and answering textbook problems. Some have the benefit of being offered math modeling classes at their high school, some experience technology-based lessons in their classrooms, and others use the skills in their math toolboxes for extracurricular activities.

“This is really the only competition in the nation where kids come together and have this day-long charrette in a high-performance work team that is so similar to what we do in industry. To have that experience as a high school junior or senior really opens their eyes to what a career in a math-related field can be like. It is incredibly influential,” explained Mary Redford, team coach from Nashoba Regional High School in Massachusetts.

Registration must be completed by each team’s teacher-coach by Feb 22 at 6:00 p.m. EST. It is both quick and easy and there are no fees whatsoever. Register now at http://m3challenge.siam.org/participate/.

Posted in Algebra, Calculus

8 Different Ways to Think of the Derivative

In his paper The Transition to Formal Thinking in Mathematics, David Tall presents W.P. Thurston’s seven different ways to think of the derivative:

  1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
  2. Symbolic: the derivative of x^n is nx^n?1, the derivative of sin(x) is cos(x), the derivative of f ? g is f ? ? g ? g? , etc.
  3. Logical: f ?(x) = d if and only if for every ? there is a ? such that when 0 <|?x|< ? , then slope
  4. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
  5. Rate: the instantaneous speed of f (t) , when t is time.
  6. Approximation: The derivative of a function is the best linear approximation to the function near a point.
  7. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. (Thurston, 1994.)

David Tall argued that the list excluded the global concept of local straightness so he added added the eighth that he believes that other 7 can be built.

8. Embodied: the (changing) slope of the graph itself.

In the same paper, David Tall presents a learning framework for derivative based on his Three Worlds of Mathematics Framework.

understanding derivative

Posted in Mathematics education

Top 5 Best Math Education Sites and Blogs

Math teachers serious at improving their craft should find a wealth of resources in the following math education sites:

1. The Klein Project blog is a collection of vignettes written for secondary school mathematics teacher. The blog is unique in the sense that unlike other blogs for teachers, “the vignette is not about pedagogy, but inspires good teaching. It is not about curriculum, but it challenges teachers to reconsider what they teach. It is not a resource for classroom use, but source of inspiration upon which teachers can draw. The goal is to refresh and enrich teachers’ mathematical knowledge.” Each vignette starts with something with which the teacher is familiar and then move towards a greater understanding of the subject through a piece of interesting mathematics. It will ultimately illustrate a key principle of mathematics.

Here is a list of interesting vignettes from the blog:

2. The NCETM Portal contains excellent resources and support tools for math teachers continuing professional development.

My personal favorite in the portal is their collection of research study modules. I also highly recommend the Personal Learning section which includes the Professional Learning Framework, Self-evaluation Tools, as well as a Personal Learning Space for anyone registered with the NCETM, which is free. You can use these self-evaluation tools to check your and your understanding of the mathematics you are teaching and to explore ideas on how to develop your practice. Click How confident are you to teach mathematics for sample questions.

NCETM Math Teaching

3. NRich – is a collection of resources for teachers, students, and parents. It is hosted by the University of Cambridge. I love this site because it promote learning mathematics through problem solving. The following description about their resources in the Teaching Guide page should be enough make you signup to them. It’s free!

At NRICH we believe that:
  • Our activities can provoke mathematical thinking.
  • Students can learn by exploring, noticing and discussing.
  • This can lead to conjecturing, explaining, generalising, convincing and proof.
  • In a classroom, the students’ role is to focus on the mathematics while the teacher focusses on the learners.
  • The teacher should aim to do for students only what they cannot yet do for themselves.

4. Math Education Podcast  is a collection of interviews with mathematics education researchers about their recent studies. This is hosted by Samuel Otten of the University of Missouri. For math education students and researchers, this site is for you.

5. The Math Forum @ Drexel – offers a wealth of problems and puzzles, online mentoring, research, team problem solving, and professional development. The site need no introduction. Their most popular service is Ask Dr. Math.

Posted in Mathematics education

Levels of Problem Solving Skills

Here is one way of describing students levels of problem solving skills in mathematics. I call them levels of problem solving skills rather than process of reflective abstraction as described in the original paper. As math teachers it is important that we are aware of our students learning trajectory in problem solving so we can properly help them move into the next level.problem solving

Level 1 – Recognition

Students at this level have the ability to recognize characteristics of a previously solved problem in a new situation and believe that one can do again what one did before. Solvers operating at this level would not be able to anticipate sources of difficulty and would be surprised by complications that might occur as they attempted their solution. A student operating at this level would not be able to mentally run-through a solution method in order to confirm or reject its usefulness.

Level 2 – Re-presentation

Students at this level are able to run through a problem mentally and are able to anticipate potential sources of difficulty and promise. Solvers who operate at this level are more flexible in their thinking and are not only able to recognize similarities between problems, they are also able to notice the differences that might cause them difficulty if they tried to repeat a previously used method of solution. Such solvers could imagine using the methods and could even imagine some of the problems they might encounter but could not take the results as a given. At this level, the subject would be unable to think about potential methods of solution and the anticipated results of such activity.

Level 3 – Structural abstraction

Students at this level evaluates solution prospects based on mental run-throughs of potential methods as well as methods that have been used before. They are able to discern the characteristics that are necessary to solve the problem and are able to evaluate the merits of a solution method based on these characteristics. This level evidences considerable flexibility of thought.

Level 4 – Structural awareness

A solver operating at this level is able to anticipate the results of potential activity without having to complete a mental run-through of the solution activity. The problem structure created by the solver has become an object of reflection. The student is able to consider such structures as objects and is able to make judgments about them without resorting to physically or mentally representing methods of solution.

The levels of problem solving skills described above indicate that as solvers attain the higher levels they become increasingly flexible in their thinking. This framework is from the dissertation of Cifarelli but I read it from the paper The roles of reification and reflective abstraction in the development of abstract thought: Transitions from arithmetic to algebra by Tracy Goodson-Espy. Educational Studies in Mathematics 36: 219–245, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

You may also be interested on Levels of understanding of function in equation form based on my own research on understanding function.

Image Credit: vidoons.com/how-it-works