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.

Reading Strategies for Struggling Readers and Critical Reading

  This weeks readings by Daniel & Zemelman (D&Z) and Bean, Baldwin & Readence (BBR), gave some truly practical techniques that I can apply in a mathematics content area.  Looking first at D & Z, they discussed techniques that can be used to help the struggling reader.  There were four strategies they mentioned, relationships, model reading, build engagement, and promote self monitoring, which can be tailored to my content area.  Building supportive relationships in your classroom is a give in.  That is the number one thing that is agreed upon across the board and should begin the first day of class and nurtured throughout the year.  Thinking more specifically about reading mathematics, the latter 3 techniques are quite beneficial.  Word problems seem to always be troublesome for students.  If I model my thinking while I go through the problem, students get to hear and see the strategies I implement while pulling out and applying important information.  In using the self-monitoring technique, I think the sticky notes are a great idea.  While giving a lesson, I would have students place a sticky tab in their notes, anywhere they think they don't understand something, then I would have them get into their groups after the lesson to help each other alleviate confusion.
I had a difficult time finding strategies that would be beneficial in a mathematical context in the BBR readings.  A lot of the strategies they used would be great in something such as English, Science, or World History.  Did anybody find any techniques that they would find useful from the BBR readings?

Holding Text

Depending on the purpose of the text, how the text is written, and reading ability, reading can be a challenge at times; especially remembering important information from a text.  You may be thinking, “How can it be challenging?”  Consider reading a dry, boring, informational text.  Think about the amount of times you had to stop and reread the paragraph you just finished because you did not understand or simply could not recall what you just read.  In academia, students are stuck reading dry, informational texts all the time.  I can relate to this as before I had found a technique to remember text, I had found myself reading and rereading text because I was not retaining the information.  There are a large number of excellent techniques students can use to make notes to “hold their thinking” or help them to remember important content.  Taking into account that I am a pre-service mathematics instructor, some techniques I would apply in my classroom include;

Sticky Notes- having kids write questions they do not understand and stick them near the passage or concept in the book

KWL- I would approach this technique in two different ways 1) as a chalk talk and 2) as an individual journal entry

Graphic Organizers-I would use these to help students make connections among concepts.