So What?

Towers, J., Martin, L. C., & Heater, B. (2013). Teaching and learning mathematics in the

collective. The Journal of Mathematical Behavior, 32(3), 424-433.

Purpose of article (What was studied/discussed): 
"In this paper we analyze and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented.” The article explores teaching and learning in highly interactive classrooms, which is something not all teachers and students are used to, especially in mathematical classrooms. The article also focuses on the role of the teacher in inquiry-oriented classrooms, which again dramatically differs from the original role of the teacher because they now act as a facilitator, rather than a lecturer. Finally, the article discusses the idea of positioning the work as a way of being with mathematics, in hopes that math will actually mean something to the students rather than just something else they have to learn. Overall, the authors claimed, “We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective.”

Important terms: Enactivism; Group learning; Group cognition

Results:
The authors focused on the idea of enactivism because it is in the interaction between the learner and environment that learning happens, not because of the learning environment (including the teacher) or the learning him/herself. The study then was based on the nature of collective mathematical understanding, which evolves from interaction and woven ideas. Data was collected in two high school classrooms (taught by the same teacher) in a single high school in a Canadian city.  "Our analysis reveals significant insights into the ways in which a collective might be orchestrated in the high-school setting."  The authors then clustered their findings into two groups: "(1) those that address the (teaching) structures necessary for initiating and sustaining such a collective, and (2) those that reveal the kind of relationship with mathematics that is fostered in this environment."

So What?

Should students primarily be in or out of their seats while learning?
The authors primarily focus on the set up of the classroom and were analyzed throughout the study. "Most days, the students are offered a problem and encouraged to get out of their desks to work on the many whiteboards.  Considerable numbers of whiteboard pens are made available to them and they jostle for position, some students writing on the board, others offering suggestions about what to write or draw.  Students often add to one another's drawings, or erase all or parts of a drawing someone else has created."  Personally, I love the idea of having students out of their seats and presenting their work to the rest of the class because this fosters higher-order thinking, questioning, and good discussion.  Do you think this classroom environment is effective or would it simply cause chaos and disturbance for other students?

Is the planning worth the product?

As a new teacher, I am certainly aware of how demanding this job is because we have to constantly be prepared, in ways that are primarily beneficial for our students. Do we have time to really design a plan of which fosters group work and collaboration or should we just stick to the basics to save time? “Planning for teaching and learning in this kind of space requires a preparation that goes well beyond the textbook and teachers’ guide. Sharon plans for teaching in a flexible manner, starting with the curriculum guide to ‘locate’ herself for the unit, drawing on a range of resources or creating her own problems and activities, and always reflecting on the previous lesson to determine where to begin the next.” I know that this type of classroom is more meaningful and engaging, but is all the planning worth the final outcomes?

Do you agree with the idea that “what is not prohibited is permitted”?
What can we interpret about a way of being with mathematics in a classroom like the one the authors described? “Students move freely around the room once the task is initiated.  There are natural limits but there is no densely woven, blocking, and stifling system of rules.” The authors stress on the idea that mathematical problem solving should not be something students need to struggle with on their own, but rather make it a public process and discuss with others. This type of interaction certainly calls for such freedom of movement around the classroom. In this type of classroom students are encouraged to look at other classmates’ work, by analyzing, questioning and challenging other students’ solutions. Do you agree with this idea of learning or do you consider it cheating? 

Text Set Collection #2

Topic: Trigonometric Functions

Books:

"Trigonometry For Dummies"

Grade: 9-12

Trigonometry For Dummies is a friendly guide for students who fear trig by explaining the basic concepts , offering lots of easy-to-grasp example problems, while adding a humorous tone throughout the entire book. I also chose this book as a resource because it successfully explains the "why" of trigonometry by using real-world examples that highlight the value of trigonometry in a variety of careers.




"The Humongous Book of Trigonometry Problems"

Grade: 9-12

The author created this book to offer, not only a collection of problems, but also notes in the margins, adding missing steps and simplifying concepts and solutions. I thoroughly enjoyed this aspect of the book because students will now have the opportunity to attempt the problems with the security of no longer being left to wonder where a particular answer came form.



"Trigonometry Demystified"

Grade: 9-12

The ongoing problem with mathematics is that students are never offered an explanation of where the concepts came from but this book unravels many topics' fundamental concepts and theories at the student's own pace.  This book eases students into trig and offers them detailed examples make it easy to understand the material, and end-of-chapter assessments to help reinforce key ideas.



"Trigonometry Success In 20 Minutes a Day"

Grade: 9-12

"Trigonometry Success in 20 Minutes a Day helps students quickly and painlessly acquire practical skills in trigonometry through a series of simple, 20-step lesson plans."  The authors of this book thoroughly explain to their readers all of the basic concepts of trigonometry and offer them hundreds of practice problems for students to attempt on their own to master the material.


Websites:

http://www.regentsprep.org/regents/math/algtrig/ATT1/trigsix.htm

Grade: 9-12

"The goal of this nonprofit site is to help high school students meet the New York State Regents requirements in English, Mathematics, Science, and Social Studies." The website offers students wonderful lessons, resources, and practice problems regarding the six trigonometric functions to help them master the key concepts of trigonometry.

http://www.clarku.edu/~djoyce/trig/functions.html

Grade: high school/college

Clark University's website offers students a "Dave's Short Trig Course," which simply brakes down the trigonometric functions by using key vocabulary, interactive videos, and in-depth explanations regarding the history of trigonometry.



http://www.purplemath.com/modules/grphtrig.htm

Grade: 9-12

"Purplemath contains practical trig lessons demonstrating useful techniques and pointing out common errors.  Lessons are written with the struggling student in mind."  I chose this website as a resource for trigonometric functions because it contains helpful trigonometry lessons, quizzes, worksheets, and free online tutoring to help students gain self-confidence.
Purplemath created a wonderful webpage for students to reference while graphing trigonometric functions, which includes examples with amplitude, vertical shift, and phase shift.





http://www.mathopenref.com/trigfunctions.html

Grade: 9-12

The Math Open Reference Project is a free interactive math textbook on the web, initially covering high-school geometry. These interactive tools and animations, provides an engaging way to learn and explore the subject because teachers will now have new ways to teach, and provide students with a new way to learn that is fun and engaging.

http://tutorial.math.lamar.edu/Classes/CalcI/TrigFcns.aspx

Grade: 10-12

Paul's Online Math Notes is a free website that includes math tutorials and notes pertaining to Algebra, all the way through Calculus III.  I chose this website as a resource because I frequently use it to answer any questions I have. Paul's online notes provide students with simple explanations clarified a different way.  These notes are very well organized, allowing students to quickly see the connections between different topics by simply scrolling through his notes. 




https://www.khanacademy.org/math/trigonometry/basic-trigonometry

Grade: 9-12

Khan Academy is an organization with the goal of changing education for the better by providing a free world-class education for anyone, anywhere, anytime. Khan Academy's materials and resources are available to everyone (teachers, students, parents) completely free of charge.  The Trigonometry page provides students with many YouTube videos that they can watch, as well as interactive notes for them to reference.  These videos reinvent education and allows the opportunity for more instruction for the students.



http://www.sparknotes.com/math/

Grade: 9-12

SparkNotes is known for containing "today's most popular study guides."  I chose SparkNotes as a resource because it offers study guides and discussion forums for many mathematical subject.  I was thrilled to find out that this site offers detailed, educational study guides for mathematics because I believe that so many students already use SparkNotes to improve their understanding of other topics. The website is extremely easy to understand and navigate, allowing any student to study and practice at home on their own!



http://www.mathsisfun.com/algebra/trigonometry.html

Grade: K-12

I absolutely love Math Is Fun and frequently use it on my own time for my own benefit! It is an excellent resource for students of all ages, teachers, and even parents because is mathematically beneficial and easy to navigate.  On this website trigonometry is easily explained for all to understand and offers multiple puzzles, games, quizzes, and interactive worksheets.



http://www.sosmath.com/trig/Trig2/trig2/trig2.html

Grade: high school/college

S.O.S. MATH is a free online resource of math review material for students, ranging from Algebra to Differential Equations. It's a wonderful resource for high school students, because it offers help with homework and quickly refreshes their prior-knowledge of many key concepts. S.O.S. Math is filled with more than 2,500 pages of mathematical resources and easy to understand explanations, including an in-depth analysis of trigonometry.



http://www.coolmath.com/reference/triangles-trigonmetry.html

Grade: K-12

Cool Math is a fun and interactive website for those who are confused, bored, hate math, or simply can't get enough of it!  Wether students just want to play games to enhance their understanding or read about a concept that they can't seem to master, this site offers every type of student an opportunity to explore and find a love for mathematics.  I absolutely love this website and often reference to it while teaching.



http://www.mathtutor.ac.uk/trigonometry

Grade: 7-12

Math Tutor was created by a group of teachers, mathematicians and new media producers from the Universities of Leeds, Lougborough and Coventry to create a mathematics tutorial site regarding seven different math topics, including trigonometry.  This site offers interactive videos for students to watch, text for them to read, a diagnostic test for students to complete, and many exercises for them to really master trigonometry at their own pace.

Reflection #11

Sinatra: Teaching Learners to Think, Read, and Write More Effectively in Content Subjects

"The method of teaching learners to think, read, and write more effectively described in this article is a teaching-learning construct that incorporates text structure, concept or cognitive mapping, and a shift in teaching style that helps learners achieve the reading-writing connection in content subjects."  The article then goes onto discuss how schools can implement a plan like the Teaching-Learning Construct.  Teachers initially have to model to their students what they are expecting from them, especially whole-class instruction to explain and organize the content information of the current topic.  Once students have completed a few model mapping lessons, the teacher can move to "guided practice," in which students are practicing the strategy with the help and guidance of the teacher.  Finally, once students have had to opportunity to work together they can then complete these mapping strategies on their own by incorporating higher-level thinking and planning.

I believe it is extremely important for teachers to remember how valuable a picture or visual representation can be for students' academic success.  "Because concept maps portray model plans of text structure, students formulate mental plans of comprehending and composing as they read and write."  Through implementing this type of strategy in the classroom, teachers are capitalizing on the "big idea" and allowing students to discover it on their own, which gives the material much more meaning.  This style of teaching can certainly be applied to by content area because it is extremely important in mathematics for teachers to understand that this different teaching style, "involves moving from a didactic, direct-instruction teaching style to one that is more indirect and calls for student participation." Giving students the opportunity to really explore a topic by completing hands-on and authentic activities, they are able to gain a deep understanding of the concepts and ideas. Mathematics teachers could definitely use a classification map in geometry because the main topic is classified or categorized into different subclasses and/or by relationships of different examples (i.e. triangles).