Weekly Report & Reflection Week #11

http://bit.ly/1NjSUvk

       This week I will be writing about how I have used my knowledge gained in math class and applied this experience in an interdisciplinary project.  To be more precise, I used my knowledge of mathematics to develop a unit plan for mathematics.  This unit plan was a collaborative assignment for my assessment course.  My colleges and I choose to focus our assignment on the grade eight mathematics curriculum expectations for circles.  We divided up the unit into five separate lessons.  

       First, we decided to developed a lesson plan for assessment for learning.  This required developing a lesson that would assess student's previous learning.  To do so we proposed to test students on their knowledge and ability to apply formulas for measuring the area and perimeter of rectangles.  If students are proficient in these latter capacities it seemed appropriate to move on to measuring circles.  The second lesson plan required students to measure and apply the formula for calculating the area of a circle, while the third lesson plan required students to measure and apply the formula for calculating the circumference of a circle.  Lessons two and three are perhaps the most crucial parts of the unit.  If students are able to comprehend how to calculate the area and circumference of a circles successfully and consistently, they should be able to complete the rest of the unit without too much difficulty.  It was for this reason that extra time would be allotted to lesson two and three depending on how students in class progressed as a whole.

       Lessons four and five were intended to be a little more fun and interactive for students.  Lesson four focused on a group project.  For this project students are asked to complete a scavenger hunt.  Their goal was to go around the class and the school measuring different circles for either their circumference or their area.  Lesson five centered around an independent project.  This project required students to discover and record measurements for both the area and circumference of three different circles outside of school.

       I thought organizing a unit plan was a great way to see long term instructional planning coincide with long term student learning.  I feel like I have not yet received enough experience lesson planning with a long term strategy in mind until this opportunity.  I think that this will be a useful in the future and especially in term of mathematics where it is vital that students comprehend fundamentals before moving on to larger projects.  I think this project would have been a lot more difficult if I had not had the experience gained from my mathematics course this semester.

Weekly Report & Reflection Week #10

Julia E. Diggins, String, Straightedge, and Shadow Viking Press, New York , 1965. (Illustrations by Corydon Bell).

       This week I have decided to talk about the digital image word problem.  The project required all students to take a picture of something fascinating and to create a mathematics problem based on that image. I think the idea for the project is quite interesting.  In particular, I think it is great way to attempt to differentiate instruction for mathematics.  Too often mathematics is taught and understood in a theoretical and abstract manner.  With the digital image we are able to show the practical application of mathematics and really establish a solid practical use for mathematical applications.  I think developing a good practical understanding for mathematics is an important component for keeping students interested and engaged.

      Once I learned about this project Thales using his shadow to measure the great pyramids of Giza sprung to my mind (see image above).  Thales provides an ingenious solution to solving the measurement problem as to the height of the pyramid by using his shadow.  This is analogous to the digital image project for the class.  That is, we were sent out to discover something interesting in the world, identify its mathematical components, and to create problem for which we could provide a solution.  Thales provides the quintessential example for such a problem by measuring the iconic pyramids using the proportions of shadows.
      Finally, I think if I were to use this project for my students I would probably reverse the roles.  That is, rather than find an image and create a problem from it for the students to solve, I would ask the students to find their own image and to create their own problem to solve.  I would probably provide detailed guidelines and examples depending on the grade level.  However, all in all, this was a fun project to be involved in and I would recommend doing something similar with other classes in the future.

Weekly Report & Reflection Week #9

A Proof From Euclid's Elements. http://bit.ly/1MZhNfG. Public Domain.

       This week I will be talking about my presentation on calculating the circumferences of circles.  The intention of the presentation was to provide a mock lesson for teacher candidates in my class and to hone my own skills in regards to organizing and presenting lectures in mathematics.  This presentation was done collaboratively with a peer teacher candidate.

       Altogether I thought this experience was quite positive.  Normally, I am not the biggest fan of collaborating assignments.  However, in my opinion, this collaborative assignment worked quite effectively.  Naturally, it helped that I worked well with my presenting partner.  We got off to a good start and organized ourselves without too much difficulty.  Cooperatively, we decided on which area of the curriculum expectations for measurement we would cover in our presentation.  That is, we identified the overall and specific expectations from The Ontario Curriculum: Grades 1 - 8 Mathematics that we would lecture on for our presentation.  These were: "Measurement: developing circumference...for a circle" (p. 109) and "measure the circumference, radius, and diameter of circular objects, using concrete materials (Sample Problem: Use string to measure the circumferences of different circular objects.)" (p. 113).  With this in place our next step was deciding on how we wanted to do our presentation in order to meet these latter curriculum expectations.

       In my view, this next step is where the value of doing a collaborative assignment really came to fruition.  My partner and I were able to share some thoughts and ideas about how we go about presenting the assignment to the class.  It was very helpful to have someone that could provide a different insight to the assignment and also help criticize or reinforce my own ideas (I hope this was reciprocal for my assignment partner).  After working together and brainstorming some ideas we decided on splitting our presentation in to three parts.  First, we decided to present with a lecture explaining the formula for calculating the circumference of a circle (Circumference = π × diameter = 2 × π × radius) and its applications.  With this information in place, we decided our second component would involve group work using string.  

       This second part of the presentation was a key factor for demonstrating student learning.  My presenting partner and myself knew this assignment would work based on our own experience preparing the assignment before class.  We created a handout with various sized circles on it and provided string and rulers for the class to use.  The learning goal for the class was to measure the circumference of the circles and reveal how this measurement worked with the circle's radius/diameter to calculate Pi.  String was a good manipulative to use to help differentiated instruction.  It was also a useful device for showing the relationship between the circumference of a circle as measured by the string with both the circle's radius/diameter and Pi.  With the success of this assignment we decided there was room to try something a little more unorthodox and, hopefully, enjoyable.

      For the third part of our presentation my presenting partner and myself decided to try an experiment called "Pi and Buffon's Matches" (please see Blog Post #8 for more details on this experiment).  Unfortunately, this assignment, although engaging, was not a great success.  Only one out of the three of our groups were able to come up with a calculation reasonably close to Pi.  I think if I were to try and repeat this experiment again with a class I would have to practice it many times in advance to make sure I was using an adequate amount of toothpicks with an appropriate sized paper for distribution.  While this was part of the presentation was not a great success, I feel that it has alerted me to the need for extensive preparation when trying to something a little unorthodox with a class.

       All in all, I thought the presentation went quite while.  I felt that we provided a valuable example to our associate teachers as to what a good lecture on calculating the circumferences of circles might look like.  While there is a little bit of work that still needs to be done in some areas, I would feel confident employing this style of lecture and group work assignment with a class of grade eight students.

Weekly Report & Reflection Week #8

       Today I will be talking about the above video "Pi and Buffon's Matches - Numberphile."  I found this video to be both interesting and informative.  It provides an excellent activity for students to engage in that can develop both their understanding and application of Pi.  According to The Ontario Curriculum: Mathematics Grade 1 - 8, a specific expectation for grade eight is to "measure the circumference, radius, and diameter of circular objects, using concrete materials (113)."  The match activity demonstrated in the video provides a novel opportunity for students learn these specific expectations.  In particular, the activity helps to deepen students' understanding of how Pi can be applicable in many different instances in the world.  That is, the application of Pi is not simply limited to circular objects, because, as the video shows, the degrees of the rotational axis of a match stick (and similar objects) can be accounted for using Pi.
       While some of the mathematical calculations in this video go far beyond the curriculum expectations for grade eight, the value of the activity is that it should hopefully make students more aware of the pervasive applications of mathematics in their everyday life.  Beyond this, the activity seems like it would be a fun change from simply teaching students about Pi using a textbook.  Getting students engaged in math requires making activities fun and interesting.  I do think this activity has some limitations and would most likely require a supplementary lesson to expand students' knowledge of Pi.  However, I think it would be an excellent activity to begin a unit on measuring the areas and circumferences of circles, as it has the potential to really get students interested in the application of mathematics with the world around them.

Weekly Report & Reflection Week #7

      For this week's blog post I decided to write about the math game website I've been exploring called Cool Math Games.  Cool Math Games is a valuable math gaming site because of the extensive variety it offers to students in terms of both game play and mathematical content.  The site organizes games by mathematical subject, such as, "numbers," "logic," "strategy," etc.  Each of these categories offers dozens of games for students to choose from.  The site is also useful because not only do the games vary in terms of content but they also vary in terms of difficulty.  You can find games that help students learn the basics of addition and subtraction, suitable for primary grades.  But you can also find games that are more advanced, for example, on fractions, multiplication, and division, appropriate for the early junior level. The games all vary in quality of game play and, significantly, in terms of the quality of their learning experience.  For this latter reason it would be up to instructors to test the games for the actual value they offer students relevant to their learning the curriculum requirements.  However, that being said, this site could be offered as an incentive to students.  For example, let's say a student has finished and handed in their homework, they can play Cool Math Games for 10 minutes at the end of class for a reward, or, if the student has finished all their in-class work correctly they may go on the site, etc.
       While the educational experience of Cool Math Games will vary with each particular game, the site does offer some engaging material.  In my opinion, the primary value of this kind of site is not so much in the learning experience students receive, but, rather, it is the interest in math that it helps to develop.  Too often students are turned off by mathematics because they find the content boring or believe it inapplicable to their everyday lives.  Sites like Cool Math Games assist instructors in showing students the kind of mathematics that can be simultaneously enjoyable, interesting, and challenging.  Having these games geared towards younger students is therefore beneficial, as they are more impressionable and will begin to form opinions about mathematics at this stage in their education.  Finally, it is not only important that instructors teach students the required curriculum expectations, as it is important for future learning that they also foster interest in the subject as well.  Sites like Cool Math Games can do precisely that.

TapToLearn, 2015

Weekly Report & Reflection Week #6

Proportions

       This past week my class covered a until lesson on proportions.  I wanted a bit of refresher on the subject so I went to the Math Antics channel on YouTube.  Lately, I've been using Math Antics quite frequently.  It has been a useful instructional tool, helping me to brush up on my mathematical knowledge and skills.  The strength of Math Antics as a learning tool is the way they present each problem.  The math lessons they provide are very clear, slowly delivered, and gradually progress in difficulty.  For example, in the video on proportions above the instructor first provides a reminder of what fractions and ratios are before moving on to define proportions.  His first lesson on proportions explains the technique of cross multiplying to find an unknown value.  This lesson attempts to be as simple as possible, allowing students to gain a basic grasp of how to find unknown values in proportional problems.  Following this, the second lesson is a little more difficult.  Applying proportions in a practical way, the instructor uses the example of a map of Hawaii to show it's proportional relation with the actual island of Hawaii.  This lesson proceeds to explain how to find an unknown value by using division.  Overall, what is presented is a clear and concise lesson on how to understand proportions and how to calculate unknown values in proportional problems.

       Learning proportions is a requirement for grade eight in the The Ontario Curriculum, Grades: 1-8 Mathematics.  A specific requirement of the curriculum states that students should be able to "solve problems involving proportions, using concrete materials, drawings, and variables" (Ontario Curriculum, 112).  In the Math Antics video supplied above the instructor provides a good example of this specific curriculum requirement when he employees the use of a map.  Using the map to calculate proportions is a good practical lesson that can be used with students.  

       For example, a fun exercise might require students to break up in to small groups.  Each group is given a map with a list of cities on it.  Group members are to imagine they are a flight crew travelling to each of these cities listed in the specific order or "flight plan."  Using what they know about proportions each group must calculate the total number of kilometers traveled by their flight crew over the course of their entire journey.  Each group could have different flight plans or even maps of different proportional sizes.  The group that calculates the correct total amount of kilometers they have traveled wins a prize.  If there is time left after this activity student could be asked to estimate how far away they think a particular city is (e.g. how far do you think Hamilton, Ontario is from Vancouver, British Columbia? Etc.).  Students can then measure and use their working knowledge of proportions to find out which estimation is the closest.  First to find the calculation correctly and closest estimation gets a prize.

     In closing, I will add a few final thoughts about Math Antics and it's potential instructional value.  In particular, one possible use for instructional videos like Math Antics is it's potential to be used in the flipped classroom model.  Students can view this video prior to class, devoting more in-class time to questions and, for instance, the flight plan activity example I have provided.  By employing the flipped classroom model and using sources with quality videos like Math Antics, instructors can devote more their time to organizing meaningful activities and helping students that are struggling with comprehension.

Weekly Report & Reflection Week #5

I have decided to dedicate my blog post this week to our mathematics unit on Great Games.  I just finished the requirements for this assignment (posting on three games and commenting on two posts about games), so I thought it would be appropriate to write a few words on what I have taken from this experience.  In general, I found the unit to be quite rewarding.  Its intent is to help incorporate gamification (applying elements of game playing to activities) with educational teaching strategies.  I believe gamification can work as a valuable tool for Twenty-First Century instructors.  In my opinion, its greatest value is to provide students with educational material that is engaging, entertaining, and interactive.  The games that I experienced tended to be free to users and available online.  Some were simple and basic (for example, check out the game Puppy Chase), while others were complex and required a lot of investment (for example, check out the game Prodigy).  Significantly, the games cover a variety of subjects that relate to the specific requirements set out in  The Ontario Curriculum Mathematics Grade 1-8.  That is, there are games available for additional and subtraction, integers, fractions, multiplication, division, and so on.  Furthermore, it is possible to find games with various subject matter and levels of difficulty that would be appropriate for a wide range of age groups.

I think that incorporating gamification with mathematical educational is a wise instructional strategy.  Sadly, students often complain that learning mathematics is boring and disengaging.  However, I believe gamification can help to rectify these often hastily assumed opinions.  Gamification has the ability to get students interested in learning by challenging them in creative ways.  Furthermore, if the games offered to students are good enough it is likely they will invest a lot time in them.  It is often difficult to get students to invest a lot of time in to further developing their understanding of foundational knowledge.  Gamification may be a technique that can help to improve this problem.  Realistically, I do not think gamification is a kind of magic bullet for solving educational instruction, but I do think that when students enjoy what they are learning they desire to learn more. Therefore gamification can act as a great facilitator for increasing student interests while simultaneously improving their basic skills and knowledge.

clipartsheep.com (public domain)