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.

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.

Teaching the Text

I absolutely loved the readings this week on textbook’s and other resources. I completely agree with the author in that our textbooks are too big and crammed with too much information that we teacher’s are supposed to cover. Kids aren’t learning and understanding anything…they are merely surviving. Teachers are suffering as well, because they are no longer teaching for understanding but teaching to meet state standards and test scores. I can relate to one of the articles where it was mentioned that although their daughter received exemplary grades, when they got to college they realized they did not understand a thing. The same occurred to me. Now I was not valedictorian of my class, however graduated with honors and when I got to college, after the first two semesters, I realized I truly did not understand anything…I merely learned enough information to survive high school and score well on exams and the SAT. Textbooks are not designed to be an easy or enjoyable read but as the author’s state a reference book, filled with facts, definitions, charts, and diagrams. If we are to really teach our kids, they need to understand the material and have enough understanding to challenge ideas and think critically about those ideas; otherwise we are just teaching them to survive. There are some good alternative resources available that enable teacher’s to get their students to dive deeper into the material. Teacher’s have the power to deviate from the textbook and with careful planning and research, can locate materials that will enhance student understanding and knowledge and make it something memorable.

Reading Strategies

This week’s readings were very important in that we all teach reading and that we are not just teaching reading, but content specific reading. Furthermore we are teaching students how to read to learn. It is important, in my opinion, during the first day of class, to introduce students to the textbook. Students can then familiarize themselves with the text and what different symbols mean. Something else we all need to be aware of is that we are masters of the content and we have seen this material time and time again and need to stop and pause that our students may be looking at this material as if it were Greek. In some of my post teaching reflections I have noted that I need to lower my expectations of students abilities. Most have never seen this stuff before. In overcoming this, some techniques I saw a useful in teaching text involved brainstorming, anticipation guides, exit tickets, and vocabulary word sorts. These types of activities help to activate students thinking and get them thinking critically about the topic. I actually used the brainstorming technique when we introduced the topic on lines. I asked students to tell me everything they knew about lines. This technique allowed students to prepare their brains for the upcoming material and allowed me to see what knowledge they already had. Techniques I am eager to try out include the vocabulary word sorts. If I make this as a pre-lesson assignment, then students will be at an advantage when we start the next lesson because they will be familiar with new terms.

Web Review

Algebra2bits.com

I found this website to be a great resource for teachers to use in their classrooms. It provides lessons, worksheets and interactive games.

•The target age group is 14-16, freshmen-junior

•This is a resource site for teachers with interactive games, lessons, and worksheets. The website is appropriate for the classroom. Due to subscription limitations, I could not thoroughly research the website aside from the examples they provided.

Content connections that I noted included economics, science, and reading.

•Given the limitations of access, for what I could work with and access, the site seems appropriatly interactive.

•The site appears to be free of ads. Since you have to pay for a subscription to the site, I see no reason to have ads (most free sites utilize ads for income).

• The website has a multitude of resources for varying levels of mathematics that include supplemental materials, lessons, and interactive games.

•There are different subscription areas that you sign up for. For example, if you are teaching algebra and geometry, there are two different subscripion areas that you must sign up for. Subscriptions cost $39.95 per a year. They do offer group discounts and discounts to returning subscribers.

•Yes the site requires you to sign up and pay to use.

•The site explicitly states for teachers or parents to use with their students. I'm pretty sure adult supervision would not be required however an adult would be required to gain access the materials.

Teaching reading for content

In the 3rd chapter of, Do I Really Have to Teach Reading, the idea of learning to read in different content areas is presented. I can see how teacher's of more application based content areas such as science and math could become frustrated with the idea of teaching reading. The reality of it though is that we are teaching students how to read Math or Science, which is quite different from reading English or US History. At first, when considering teaching students to read in Math, I initially thought, "why should I teach students to read...won't they learn that in English?" However it's not English reading I am teaching them but Mathematical reading. How an English expert reads a word problem and how a Math expert reads a word problem are completely different? In teaching students how to read Math, they will learn how to make sense of the content, vocabulary, theorems, graphs, and how to read and break down a word problem. These are just a few examples of how I can teach students to read Math. In chapter 4 they discuss how one can use multiple texts to help with struggling readers. For the most part, students in high school who struggle with reading generally have a difficult time due to not understanding vocabulary or having a lack of background knowledge or the material is simply uninteresting. Providing students with varying texts that match their reading abilities enables them to grasp the content required of them and hopefully increase their reading abilities in the long run. Tovani makes a valid point that students shut down and essentially stop reading when they struggle to understand something or the reading is very dry and boring. I can relate to this for when I read something required of me and it is boring, I become easily distracted and have to re-read the text two and sometimes three times over. I also tend to shut down if the text is filled with technical words...I find medical journals and law journals particularly difficult to read and try to skim through them, skipping over the technical terms hoping to make sense of the content. I agree with her strongly that if we don't make additional text available to students we are only hindering their learning and passing the problem along to someone else.

Reflection #1

REED Reflection

I have a strong belief that children are a product of their environment.  In reading, The Early Catastrophe, my beliefs became solidified with the research conducted by Hart and Risley.  We have a rising problem in our country and as stated by Risley and Hart, the sheer magnitude of intervention needed to make changes to current trends needs to start now before we reach a point of no return.  In considering intervention, I liked what Fitzgerald and Graves discussed in Reading Supports for All; scaffolded reading experiences.  Although the idea is great for content areas that require a large amount of textual reading, being such that I am a pre-service math teacher I find it hard to consider using scaffolded reading experiences.  The chapter on Building a Community of Learners, in Subject Matters,( p. 173) I find having students read a mathematical process and write down their ideas on how the process works (pre-reading) could be a good stepping stone in scaffolded reading experiences.  From that point questions could be developed and students could critique and justify their reasoning with one another.  Then students could read into the material further to gain a better understanding (during reading).  After the have read the actual text they could discuss their findings (post reading). 

The most important aspect to any learning experience is the classroom community.  In the text, Building a Community of Learners, Daniels and Zemelman discuss the absolute importance of developing a positive community of learning.  If we are to expect any form of intervention to work, I believe it all starts in the classroom.  Students need to know that they are in a safe, non-judgmental environment, where their opinions are valued, where it is okay to make mistakes and inquiry is encouraged.  Positive learning experiences occur in a safe learning environment within an encouraging classroom community