Text set 2: Pascal's Triangle

Research papers:

• Pascal's Triangle; The Many Tricks Held Within by Mary Daughtry, 2010

A research paper looking into the different uses for Pascal's Triangle extending to probability
http://education.uncc.edu/cstem/sites/education.uncc.edu.cstem/files/media/SV/2010/ME/Mary%20Margaret%20Daughtry-%20Pascal's%20Triangle-%20The%20Many%20Tricks%20H.pdf

Informative sites

• Montclair; Pascal's Triangle

This is a cool informative site.  There are links on the left that guide you through the article starting with an introduction, patterns, then applications.
http://pages.csam.montclair.edu/~kazimir/conclusion.html

• Mathisfun.com

This site is chock full of examples and explanations on how Pascal's Triangle works
http://www.mathsisfun.com/pascals-triangle.html

• Binomial Theorem and Pascal's Triangle

This is a stepped out introduction to pascals triangle and even explains in detail how to find the nth terms coefficient.

http://mathematics.laerd.com/maths/binomial-theorem-intro.php
Books:

• Pascal's Arithmetical Triangle by A. W. F. Edwards 

Looks into the historical development of Pascal's Triangle and further explains the how, what, and why's

Articles:

• Exploring Pascal's Triangle by Tom Davis

An article" intertwining pedagogical advice with mathematics" leading students into discovery
http://www.geometer.org/mathcircles/pascal.pdf

• Encyclopedia Brittanica; Pascal's Triangle by William Hosch

A brief history and background of Pascal's Triangle including a brief description of how the arrays work
http://www.britannica.com/EBchecked/topic/445453/Pascals-triangle

• Mathematical Association of America Patterns in Pascal's Triangle by Kathleen Shannon and Michael Bardzell

A deeper look into the many patterns and gems that Pascal's Triangle illuminates
http://www.maa.org/publications/periodicals/loci/joma/patterns-in-pascals-triangle-with-a-twist-introduction

• Mathematics in School: v36 no2 Pascal's Pupils by Paul Stephenson

This article addresses the importance of student discovery and that our students may even discover things we are unaware of.  Suggests a method to use with Pascal's triangle to assist in enabling discovery

• Mathematics in School: v 26 no2 Spinning Beyond Pascal's Triangle by Chris Grover

This article takes Pascal's Triangle and applies it to probabilities, showing yet another useful take on the Triangle.

Lessons:

• Ezine Pascal's Triangle and Binomial Expansion by Alec Shute

An explanation of how to expand binomials using Pascal's Theorem.  Although this is not technically a lesson, I felt it fit better under this category.
http://ezinearticles.com/?Pascals-Triangle-and-the-Binomial-Expansion&id=7297721

Number patterns in Pascal's Triangle by Ulysses Harrison  This is a lesson plan used to help students recognize the number patterns found in Pascal's Triangle

http://mypages.iit.edu/~smile/ma9408.html

• Teacher's Institute: Exploring Pascal's Triangle and other Recursive Patterns by Carole Bersani

Gives an introduction and brief historical connection as well as 3 lesson plans to teach Pascal's Triangle by recognizing patterns
http://www.cfep.uci.edu/uci-sati/faculty/carole_bersani_full.html

Study skills

I am realizing more and more everyday that students in high school lack the necessary study skills to be successful.  We as teachers are not only teaching our content specific area; math, foreign language, science, etc., however are also teaching reading & writing skills, and study skills.  The readings for this week provided some practical techniques to present to our students to help them master the material. Two techniques in particular that stood out to me included the split-verbatim technique and the I-search paper.  I used a modified version of the split verbatim technique while in college; while I didn't divide my paper in half for note-taking, I did review my notes and make comments are rewrite something in my own words.  I think this technique would be very beneficial for students in a highschool am class, because sometimes you need to rewrite things in a context you understand.  The I-paper technique, I was truly intrigued by.  I feel students would be more motivated to complete a research paper if they are able to step outside the use of traditional research methods.  I also feel students would have a greater appreciation for the topic as they go and do field research.  I'm not sure how I would use this last one in a math context, unless I asked the students to research real world applications of a given topic...then, they may be able to do a little field research asking various professionals to describe the mathematics they use in their profession.

Web Review 2

http://www.regentsprep.org/Regents/math/algtrig/Games/algebra2games.htm

I did a search for Algebra 2 interactive resources and came across the Regents Test prep.

The target age group is High school students, Sophomore-Senior.

This could be a resource site for both students and teachers.  The Algebra 2 portion gave interactive games related to evaluating functions, quadratic functions, simplifying radical expression, and solving absolute value inequalities and graphs.

I did not note any cross curricular connections.

The site is free of ads and does not cost anything to access (BONUS!!)

When I navigated to the home page of the site, there were a multitude of practice examples for many different content areas; math, science, and U.S. history.

I don't find that adult supervision would be required to utilize this site.

Motivating readers

  In reading the article on motivating readers, I found some very interesting information to help me understand my oldest daughters lack of desire for reading and to help my youngest daughter develop better reading skills.  I can relate with the authors findings that students generally rated themselves low in reading ability on the survey yet during interviews they stated many types of texts that they enjoyed reading.  My oldest daughter claims to dislike reading, yet I find her reading many types of texts on a regular basis; researching topics online, cookbooks, and two of her favorite series, Baby Mouse and The Homework Machine.  I find myself consistently puzzled at her claim that she dislikes reading.  I have reason to believe that she claims to dislike reading when it becomes something she has to do, rather than when she wants to.  My youngest daughter struggles with reading comprehension and recollection.  Based on the article, if I provide both my daughters with text that I know they are interested in, then I can not only improve comprehension, but increase the frequency of their reading.
 As teachers, especially in a world filled with resources; both electronic and print, we need to access a variety of resources to give students options to find text that interests them.  It's not that students do not like reading however they do not like what they are reading.  The next big question is how can we get students more interested in academic reading?  I think finding articles with grade appropriate, academic language related to students interests would be a start.  Also, the authors mention reading aloud to students to encourage academic reading comprehension.  Overall, I think the article helped me to make sense of current reading issues I have in my own home, and as a teacher helped me to understand why student's claim they do not like reading.

Complex Numbers Text Set

 I have selected the following texts as I believe, together, they provide a well rounded approach to teaching students complex numbers.  There are example lesson plans, additional resources for students and teachers, and techniques used in the development of lessons.  I hope you find the texts useful.
 

• Project Maths Development Team (2010) www.projectmaths.com.ie/documents/teachers/tl_complex_numbers.pdf

          Teaching and Learning Plans: Complex Number Operations Leaving Certificate Syllabus
          ARTICLE Reading Level: 9.4
          A unit plan on complex numbers that describes and breaks down the prior knowledge students                     need as well as learning objectives.  Lessons are presented sequentially with objectives and types of
          assessments used.
                                                                                                                                                                   

• Excursions in the History of Mathematics By Israel Kleiner (2012)         

          Ch12. History of Complex Numbers with a Moral for Teachers (p261-272)
          BOOK Reading Level: 12.7
          This chapter gives a history of the birth of complex numbers into our number system and how it grew
          and evolved over time.  Also includes an evaluation of the importance of teachers understanding the 
          history behind mathematics and developing an appreciation of mathematics that is visible to their 
          students.
                                                                                                                                                                   

• Complex Numbers http://www.mathsisfun.com/numbers/complex-numbers.html  

          WEBSITE Reading Level: 5.1
          A nice supplement to teacher instruction.  Gives students a step by step guide on operations involving 
          complex numbers plus gives many examples.
                                                                                                                                                                   

• Complex Roots http://www.purplemath.com/modules/complex.htm 

          WEBSITE Reading Level: 7.1
           Similar to Math is Fun, Purple Math is good to supplement teacher instruction at home.  It provides a
           tutoring video as well as.  Teaches students why we encounter complex numbers and what they are
          good for and gives a lot of interactive examples.
                                                                                                                                                                     

• Complex Numbers www.mathwarehouse.com/algebra/complexnumber         

         WEBSITE Reading Level: NA
          A series of topics on complex number operations with interactive examples.  Easy to navigate; a good 
         tool to use in the classroom or as extra help for students.
                                                                                                                                                                   

• Algebra II www.sparknotes.com/math/algebra2/factoring/section4

          STUDY GUIDES Reading Level: 10.5
          Discusses the roots of factoring using the quadratic formula and the discriminant.  Further discusses 
          how the discriminant gives us the type and number of solutions and gives students a preview of how it 
          would look graphically.
                                                                                                                                                                   

• Differentiation in Practice:  A Resource Guide for Differentiating Curriculum Grades 9-12 By Carol Ann Tomlinson and Cindy A Strickland (2005) Ch 7

          BOOK Reading Level: 9
          This book breaks learning topics into chapters and gives a detailed description of how the lesson is 
          taught including prior knowledge students will need and any scaffolding that may be needed.  What I 
          thought was really cool was the authors provided a commentary alongside the lessons sharing their 
          experience while giving the lesson.
                                                                                                                                                                     

• Graphing Complex Solutions to Quadratics By James Tarton (2009)

          ESSAY Reading Level: 5.7
          A simple step by step process in graphing quadratics with complex roots.  Give visual representations
                                                                                                                                                                     

• Situation: Complex Roots in Conjugate Pairs By Sarah Major (2013).  UGA Center for Proficiency in Teaching Mathematics

         RESEARCH ESSAY Reading Level: 12
         The essay addresses the question, "Do all complex solutions appear as conjugate pairs...?"  Major
         observed a classroom where this question came up and divulged 6 mathematical foci in considering 
         when complex conjugates occur.  I found this particularly interesting as students ask questions such 
        as these that we need to be able to explain appropriately.
                                                                                                                                                                   

• Finding Complex Roots: Can You Trust Your Calculator By Barbara Ciesla & John Watson (2006).  Mathematics Teacher Vol 99 Issue 5 pg 366

         ARTICLE Reading Level: 12
         Examines when the textbook answer is different than what a TI-83 gives you.  It goes over techniques
         to clear up and overcome common assumptions when this occurs. 
                                                                                                                                                                      

• Rational Expressions: A Math Teacher By Michael Pershan  rationalexpressions.blogspot.com/2012/12/teaching-complex-numbers.html

          BLOG Reading Level: 6.8
          Here, Mr. Pershan was trying to develop a unique way to introduce complex numbers to his students 
          by building a foundation allowing his students to discover operations on complex numbers.
                                                                                                                                                                      

• Teaching the Complex Numbers By Emily R. Grosholz (2013) pg 62-73.  Journal of Humanistic Mathematics Vol 3 No.1

         ARTICLE Reading Level: 8.1
         Discusses a brief history into the learning and discovery of complex numbers and further discusses the
         many modes of representation needed to teach our students about complex numbers.
                                                                                                                                                                      

• www.regentsprep.org/regents/math/algtrig/ATE3/quadcompractice.htm

         WEBSITE Reading Level: 10
         Provides a series of interactive practice problems with stepped out solutions involving quadratics with
         complex roots.
                                                                                                                                                                   

• Complex Numbers in Teacher Education: Connecting Mathematics and Pedagogy. Math Forum 

         ARTICLE Reading Level: 14
         This article discuss how complex numbers are disappearing and the importance of having this structure 
         in the number system as it links to many areas of mathematics.  The article also ties in the importance of
         pedagogy in delivering the topic related to complex numbers.
                                                                                                                                                                      

• Cool Math 

         WEBSITE Reading Level: k-12
         Complex Numbers
         The site has two different areas when looking at complex numbers; one for teachers and one for 
         students.  For the student, the site explains what imaginary numbers are and breaks it down into easy
         to understand.  For the teacher, the site offers helpful resources and suggestions as well as support.

Building Vocabulary

This weeks readings expanded the depth of my understanding in regards to the complexities and levels of vocabulary.  I have always known there were different types of vocabulary such as conversational and academic vocabulary and more recently content vocabulary, but to see it broke down even further enabled me to visualize the many levels of vocabulary we use everyday. The article by Brommley further explains the importance of vocabulary and language and how we are able to expand our daily vocabulary through connections with what we know and how vocabulary helps with comprehension, fluency and academic achievement.  Knowing what and how to teach vocabulary to our students is important. The article by Baumann and Graves discussed several techniques in which teachers can use to help students learn new vocabulary.  Being a pre-service math teacher, I appreciated the inclusion of symbols in the classification scheme, as we read symbols every day to determine what to do next.  In teaching vocabulary, I found Marzano and Pickering's method the most beneficial.  The technique was illustrated in a very methodical manner; selecting vocabulary based on instruction and arranging the vocab in a respective way.  I am quickly beginning to realize the importance of reading in all content areas..and even more so, helping students develop a rich, colorful vocabulary filled with conversational, literary, and academic language.

Strategies for Developing Vocabulary

I have had a difficult time relating or seeing how to apply the strategies in the last two weeks readings to mathematics.  I can totally see how they could be used in other content areas however find only a few strategies that would be beneficial in my content area.  The two strategies I think that would apply best in a mathematical sense are the personal glossary and clues and questions, both from the BBR chapter.  My students could definitely benefit from a personal glossary of mathematical terms that also incorporate clues and questions as well.  I think I would format the vocabulary entries with, of course, the vocab word first followed by a definition and then a formula or mathematical label.  I would then request below the definition, 3-4 clues or questions to help students identify key elements.  This is just a rough sketch of what it could possibly look like and would probably need to be fine tuned. 

Overall, I think the techniques and strategies listed would be beneficial for varying content areas.  I did become frustrated however when the two readings overlapped as much as they did…not too many new techniques.