Bringing Reality into the Classroom

In recent years, I have run sessions on campus and made high school visits to work interactively with students on the mathematics of certain kinds of fractals. Through worksheets, discussions, and computer demonstrations, the students can understand the main mathematical ideas. It works well for several reasons: the ideas are presented in an accessible way, there are pictures, there is a "puzzle" nature to things like the "fractal dimension," we end up working on applications with real images (a photo of the class), and (as they write in feedback) I think they find the whole thing fun. I would like to share with you the topics that the students, their teachers, and I work through, as well as the worksheets and computer demonstrations/programs that have evolved over time. My goal is to convey to you the fun, wondrous, and accessible nature of the material.

Multiple representations of concepts enhance student’s learning and understanding and increase student’s engagement.  In this workshop we will explore how the use of GSP in the math classroom, enables students to explore and learn on their own the meaning of several concepts that seem much more difficult than they really are, and to take their learning into their own hands. The participant will have a chance to observe, see, and maybe even practice some GSP applications.  Fractals and their applications in the world around us, iterative sequences, Fibonacci sequences, Rene Descartes Curve Sketching Devices, etc, will be such examples.  We will also be discussing some time-saving techniques with GSP such as: iterations, animations etc. 

Many students struggle with math because of its abstractness.  As a result, they rely on mechanical memorization rather than conceptual understanding to learn math.  In this talk, we illustrate some creative ideas on how to make abstract math intuitive, vivid, and real.

Mathematical models are used to investigate and make predictions about real
world phenomenon. Mathematical models have advanced knowledge in many fields (i.e. biology, chemistry, geography, epidemiology, immunology, etc) and have aided in the development of new tools used to simplify and optimize social,
industrial and health processes. In this session we will review the modelling
process and discuss some advancements due to modelling in biology, medicine and public health.

Mathematics contests engage students in mathematical reasoning and problem
solving. They can be used to measure performance and mathematical ability of
individuals or teams, and can also be used to provide opportunities for students or teams to apply mathematical tools and concepts to real world problems. This session will include an overview of some high school mathematics and mathematical modelling contests, and a discussion on the benefits of providing these contests in the high school setting.

Within the grade 11 Functions course, MCR3U, the use of computer technologies ensure that concepts of variable and function can become rich explorations where finding solutions to satisfy equations becomes the outcome of a student’s understanding of material rather than the ultimate goal.
 
Digital image processing provides a dynamic exploration of the relationship established by functions, including the roles of variables, domain and range. In this manner functions are no longer abstract objects that students test against vertical lines, instead, functions become meaningful representations that influence real-world occurrences.
 
To this end, we have created a consolidation activity which will lead students through an investigation exploring how the concepts of function, domain, range and inverse apply to digital imaging.

Our mini unit is a four-lesson sequence designed to introduce college-level students to basic matrix operations through image processing. The unit is part of the Math 19, Finite Mathematics Course, offered by St. Lawrence College in Kingston. Math 19 is selected by students interested in medical and business fields. Community college learners demand practical up-to-date learning experiences. Medical image processing offers an authentic application of basic matrix theory. Lessons in the unit include: Introduction to MATLAB, Types of Medical Imaging (X-Ray, CAT-Scan, MRI, and ultrasound), Matrix Subtraction and Addition of Medical Images, and Improving Visibility of Medical Images Using Logarithmic and Contrast Transforms. Several classroom- ready activities and games are included such as spot-the-difference and a hospital walkthrough activity. Ethical issues of misdiagnosis are also discussed.

This lesson plan is designed for college students. It aims at introducing to students an interesting application of the matrices in representing the digital images. Taking into consideration a certain level of difficulty shown by students  in operating with matrices, the goal of this lesson plan is inviting students to make simple arithmetic operations with digital images, as an ‘easy to be memorized method’ of  doing the same operations with matrices. This will help students for a better visualization of the results via fun activities in digital image processing. some of the activities include: representing of digital images by matrices, adding and subtracting matrices, finding the inverse of a given matrix, finding the transposed of a matrix and understand how this transformation operates to the digital images, as well as representing the linear combination of matrices in terms of digital images.