Blog Post #6

The.Secret.Life.of.Chaos. by costello74

This week's learning module had two main focuses, they are, connecting mathematics with the world and using technology to teach mathematics.  I will address each focus in this respective order.

Connecting mathematics with the world is important for establishing a practical relationship between the sometimes arduous work of mathematics and its inherent relationship with our common experiences. Developing an understanding of this relationship is important because it helps to foster in students an appreciation for the value of mathematics and its far reaching applicability.  Above is a BBC documentary called "The Secret life of Chaos."  This video explore the multifaceted ways in which mathematics is deeply rooted in the world we inhabit.  The kinds of examples it explores in this documentary provide valuable insights into how the world functions and how mathematics can be used to help us understand the nature of those functions.  I think that bringing to light some these more profound ideas to student awareness is a positive step in reinforcing an understanding of the awesome nature and broad applicability of mathematical concepts.  I believe that creating an educational environment that would allow students to study these kinds of concepts in more detail would not only increase student engagement with mathematics but would also enlighten their understanding of the world by engrossing them in some of the underlying concepts that define it.

The other focus in this week's learning module was the use of gaming applications to help teach students mathematical concepts.  Gaming application are a fun and engaging way to get students interested in mathematics.  By approaching mathematics as a form of 'play,' students are more likely to enjoy mathematics as something desirable to interact with.  Good gaming apps also have the added benefit of offering students various levels of engagement.  They are able to challenge students who are more advanced or provide scaffolding opportunities by lowering the difficulty for students that are struggling.  Offering students a variety of gaming apps that focus on a big idea also allows for a level of choice that should increase student engagement with mathematics.  Finally, gaming apps often provide students will useful visual aids when demonstrating various mathematical problems.  These visual aids create a diverse learning environment where the differentiation of instruction can optimize student learning preferences.

Blog Post #5

Youcubed - Stanford University - Visual Math Image

       This week's online learning module included a focus on using visual images.  In this post I will discuss the benefits of using visual images when teaching or learning mathematics.  In particular, I will discuss the importance of using visual images and their significance for differentiating instruction.

       It is important for both teaching practice and student learning that differentiated instruction be undertaken in the classroom.  Incorporating the use of visual images can help achieve this latter end in various ways for mathematics.  Perhaps the most obvious way visual images help students learn a variety of mathematical concepts is that they offer an alternative representation of a math problem or question.  In most cases, mathematical problems are posed to students using numbers.  Numbers have an abstract quality that can be alienating for some students.  For instance, sometimes students have trouble understanding the value that numbers represent or have difficulty connecting numbers to their 'real world' applications.  Using visual images to represent math problems or their subsequent solutions offers students a means to understanding mathematics that is simultaneously less abstract than numbers and often a closer reflection of the 'real world' around us.  Furthermore, visual images are naturally a tremendous benefit for visual learners.  Providing a means for visual learners preferred method of acquiring knowledge creates a bridge to understanding that is not always available in the traditional format of teaching mathematics.

       While visual images are beneficial to student learning they should also be incorporated into teaching assessment.  Differentiated instruction usually focuses on how to offer a variety of learning opportunities for students.  But it is sometimes overlooked in terms of its applications for assessment.  Allowing students to use visual images in mathematics to demonstrate their solutions to problems should be adopted as a valid form of submitting work for assessment.  What is important is that students are able to demonstrate their understanding of a subject, as oppose to demanding students demonstrate their understanding in only one particular chosen format.  That is, good pedagogy strives for student comprehension in all of its potential forms.  Bad pedagogy demands that students limit their potential for understanding by narrowing their opportunities for learning and stifling their potential for creativity.  Since the use of visual images in mathematics diversifies student learning and assessment potential it should be incorporated as valid mathematical format and tool for twenty-first century teaching.

 

Blog Post #4

Study Group - Tiger Bookstore Blog

        This week in the online module of the course a variety of issues relating to mathematics were addressed.  Some of these issues included connections in mathematics, reasoning and mathematics, and flexibility with mathematics.  However, while I found all of the aforementioned topics to be of interest the subject of this blog will be on the importance of study groups.  In one of Jo Boaler's online videos she expounds the advantages of study groups and the benefits they offer students.  She argues that evidence indicates that students of mathematics that partake in study groups tend to have significant advantages over students that study independently and that the grades attained by those that engage in study groups exceed those of other students.  Boaler points to an experiment done by the mathematician Uri Treisman that supports her argument and helps to demonstrate the value of study groups for students.  One advantage of the study group is it provides the ability for students to collaborate with each other.  Collaboration allows for students to discuss particular problems and to use their reasoning skills collectively in their attempts to find solutions.  Collaboration also allows for multiple approaches to a given problem, this in turn provides a showcase for diverse methodology creating an environment with greater accessibility for student comprehension.  In other words, the ability to discuss mathematical problems with peers offers significant advantages over attempting to tackle problems individually in isolation.

       In my view, and taking the perspective of an educator, I think forming study groups would be a beneficial facet for strengthening student comprehension.  It would also seem that if this is true for mathematics it should also be true of other academic subjects as well.  Incorporating study groups in the classroom environment should help engage students in the kinds of collaborative behaviours that Boaler and company argue strongly benefit learning.  Therefore, organizing study groups for students should become a natural part of any classroom and for any subject.  Significantly, it is the role of the teacher to help organize their students' study groups and to help guide them into effective methods of group collaboration.  If successful, teaching students these kinds of collaborative skills should provide them with a tool they can use as they approach higher education and more complex problems.

       

Blog Post #3

Michael Jordan "Failure" - Nike Commercial - 1997

 Mistakes and Speed in Mathematics

       This week's online class focused on what might be considered the relationship between pedagogical practice of mathematics and pedagogical ethics.  The focus here is not necessarily on what teachers teach (I.e. content) but on how teachers teach (I.e. methodology).  In a series of videos Jo Boaler and other academics argue for a more enlightened approach to the way teachers instruct classes on mathematics.  Two premises in particular are put forward, they are, roughly, "mistakes should be viewed as an opportunity to grow and develop our understanding" and "being fast at mathematics is not a necessary condition for being good at mathematics."  I will expand on these two positions respectively.
       In my opinion the claim that mistakes should be viewed as an opportunity to grow and develop our understanding is a sound principle for any academic setting.  This kind of attitude towards learning demonstrates a determination for improvement.  Significantly, teachers plays an important role of instilling this kind of attitude in their classrooms.  This is achieved by creating an environment that does not punish or ridicule mistakes but rather inspects and ameliorates them.  Such integrity in the classroom can only be achieved if the instructor is able to foster welcoming and open mentality.  Approaching mistakes as an opportunity as oppose to a catastrophe is no easy feat and will require a psychological shift in the mindset of both teacher and student.
       The other premise presented in this week's module states that being fast at mathematics is not a necessary condition for being good at mathematics.  As argued in the previous paragraph, instilling this kind of mentality in the classroom is largely dependent on the pedagogical approach an instructor adopts.  Traditionally, being fast at something was often associated with being competent at something.  However, according to new research espoused by Jo Boaler and others, being fast at mathematics is not a necessary conditions for being good at mathematics.  The term "good" here demarcates competence and understanding.  Boaler and others argue that deeper thinking requires time and patience.  Furthermore, it is an indicator of an extensively holistic understanding of a broader framework.  It follows that students requiring extended time to analyze a given problem is not necessarily an indicator that they can not grasp the solution.  Rather, time should not be a major factor when it comes to the intellectual engagement with a particular problem.  It is the engagement itself that is important and the learning development that it galvanizes.
       In closing, I believe that both of the aforementioned premises seem to be sound principles for teachers to adopt in their classrooms, and not only for the teaching of mathematics.  Allowing students to explore their mistakes without the fear of sanction and allotting them sufficient time to process their thoughts are ideals worthy of a virtuous pedagogical practice.

Blog Post #2

Growth and Underachievement - Schneider, David

For my second blog post I would like to discuss both the growth mind set and what is referred to by educators as the "minds on" portion of a teaching lesson.  I will address both issues respectively.

The primary focus of this week's class has been on what is referred to as the "growth mindset".  After doing a little research I found what I think is a good article about the subject by The Glossary of Education Reform entitled Growth Mindset.  The article identifies Carol Dweck as the primary developer of the theory.  She is quoted as stating:“In a growth mindset, people believe that their most basic abilities can be developed through dedication and hard work—brains and talent are just the starting point. This view creates a love of learning and a resilience that is essential for great accomplishment.” This stands in opposition to what she calls a "fixed mindset."  She explains, “In a fixed mindset, people believe their basic qualities, like their intelligence or talent, are simply fixed traits. They spend their time documenting their intelligence or talent instead of developing them. They also believe that talent alone creates success—without effort.”  Significantly, it is import for educators to foster the ideals of the growth mindset in their students while simultaneously steering them away from the debilitating qualities of a fixed mindset.  The former provides students with the opportunity to develop and prosper, while the latter seems to offer only stagnation and even decline.

While the ideals proclaimed by believers of the growth mindset model are inspiring, this theory can also be used to help influence other pedagogical practices beyond idealism.  What I am referring to in this instance is the aspect of teaching often referred to as the "minds on" portion of a lesson.  Loosely speaking, "minds on" is a kind of 'warm up' activity teachers use prior to starting the main portion of a particular lesson.  In class we discussed many ways in which these activities can be done effectively and to encourage aspects of a growth mindset.  Spending time as a class looking at a question collectively and allowing students to decipher their own solutions allows for a low pressure environment that fosters ingenuity and creativity.  Catering to a more free and less rigid approach to mathematics definitely appeals to me personally and I will be sure to incorporate aspects of the growth mindset when organizing my lesson plans in the future.

Blog Post #1

Mandelbrot Set

Introduction

Hello and welcome to my mathematics blog site!  My name is Corey Padgett and I am currently a teacher's candidate at Brock University, Ontario, Canada.  The following blog posts will be used to track my progress as I develop the fundamental and pedagogical skills required to be an effective and valued teacher of mathematics.  To achieve this latter goal my posts will discuss content covered during class, class assignments, and other relevant mathematical curiosities or interests that I have encountered throughout my time in this course.  I hope you enjoy what it has to offer.  Now let's buckle-up and get ready for the ride!

Week 1

Like most first days of school time was spent in class on administrative work.  These tasks included expounding the class syllabus, student expectations, and required assignments.  Doubtlessly, this latter work is necessary for organizing and clarifying class criteria, but, unfortunately, it does not provide interesting content for a blog post.  Fortunately, however, our class was exposed to a math based game that I found to be quite stimulating.  The game I am referring to is called "Game About Squares," I will now proceed to discuss its significance for teaching and learning mathematics.

Game About Squares can be categorized as what is often called "gamification." "Gamification" is loosely defined as an interactive method for learning a particular skill or content knowledge through means of typical gaming techniques.  In other words, gamification attempts to increase student interest in learning by incorporating and combining the engaging aspects of game play with academic content.  In my opinion, Game About Squares is an ideal example of gamification.  While it is probably easiest to actually play the game to understand what it is, I will briefly try to explain the goals of Game About Squares here.  Basically, the goal of the game is to get squares to cover circles of the same colour.  This seems simple enough, however, as the game progresses multiple squares of various colours are introduced and the squares begin to move in multiple, often inconvenient, directions.  These aspects create a challenging but fun and engaging game for students to interact with and, importantly, learn from.

Gamification is an important academic tool to use when developing lesson plans for student learning.  Significantly, students become more engaged when learning is interactive, progressively challenging, and, perhaps most importantly, fun.  If done properly, gamification has the qualities to achieve these latter goals effectively.  While Game About Squares focuses primarily on learning and developing spatial reasoning skills, there are numerous math games available for almost any possible learning goal a teacher may require.  Since in my experience gamification is an effective means for getting students interested and actively learning I believe it is a good way to introduce a unit, and should always be readily available for teachers of mathematics, especially when student interest seems to be waning.  Therefore, in terms of engaging students and developing academic skill sets gamification is perhaps one of the most effective methods for achieving these latter ends.