Scaffolding a Comprehension Lesson on the Applications of Sequences and Series Using a Complex Text

About this text:

Text_1

Click here to view this text!

Summary:

This is a print blog post by a group of students from IIT Madras (the Indian Institute of Technology Madras – a public engineering and research institution in India) working under the NSS-IITM (a group working on projects dedicated to the education and service of others).   They present and explain numerous sequences and series that are some of the most well-known number sets in mathematics.  They begin explaining what the Fibonacci numbers are (the most famous set of the list presented).  They then discuss where we see terms of this number sequence in nature and its connections to the Golden Ratio.  They also mention some of the interesting mathematical properties the Fibonacci Numbers display.  The next sequences they explain are the Figurate Number Sequences, Lazy Caterer’s Sequence, Magic Square Series, Catalan Number Series, and the Look and Say Sequence.  They end mentioning some special series in other fields outside of mathematics; however, they do not explain these at all.

Complexity:

First, this text is at about an eleventh grade reading level.  Second, the writing style of this text is more relaxed and informal.  It is not written in a narrative format, but it is written towards high school students.  There are pictures or diagrams to represent the number sequence being explained which helps bring the complexity of the text down.  Also, the writers always write out the sequence along with showing the summation notation for the series, also keeping the complexity down.  The series are shown in two different ways.  Also, the structure of the text makes it very easy to follow.  The different sequences and series are clearly separated and the introduction paragraph sets the reader up nicely for the text to follow.  There is also a lot of white space so the reading isn’t so dense and difficult to read.  All of these things bring the complexity down.  Third, the reader does have to have a decent amount of prior knowledge to understand this text.  They must understand sequences and series, what they are and how they are set up/shown, and some general areas of study in math.  This definitely makes the text more complex.  The topics covered are very interesting though and something students can easily see so their motivation to read the text and curiosity will be high making the complexity low.

Citation:

IIT Madras. “Famous Mathematical Sequences and Series.” Web blog post. Edu-Blog. NSS-IITM, 16 Apr. 2013. Web. 24 July 2014. <http://edublognss.wordpress.com/2013/04/16/famous-mathematical-sequences-and-series/&gt;.

Context to use this text in:

This text would be used for high school algebra I or II students.  Therefore, depending on the student, the reader could be anywhere from a ninth through twelfth grade student.

Within the curriculum of algebra, there is always a chapter or part of a chapter devoted to learning sequences and series.  Textbooks typically cover what they are, the different mathematical representations of them, how to go between these different representations, and some simple, just-scratching-the-surface real world examples.  This text would then be explored at the end of this chapter or sections within the chapter, possibly even after the formal assessment on the entire chapter.  Hypothetically, I would devote a day to wrap up this concept and make sense of why it is included in our curriculum because it always seemed to me as if it were an afterthought or even “filler” concept.  We would take a day as a class to explore the applications and importance of sequences and series outside of the classroom and in the real world.  The day following reading this text would include any last comments or questions on this topic and we would begin our next chapter or unit of exploration in the curriculum.

Guiding questions to establish purpose for this text:

What are the sets of numbers being discussed?  What is the rule between terms for each of these sets of numbers?  Why are they, and sequences and series in general, important?

Lesson plan for this text:

 

What teachers do …

What students do …

Before Reading Text

Introduce TED talk video 

Click here to view this video!

  • “We have just finished learning the basic concepts of sequences and series.”
  • “Now let’s explore why we learned sequences and series by watching a video on one particularly famous and powerful example of sequences and series.”

 

Play TED talk video

Students will observe the TED talk video and write down three places they’ve observed the Fibonacci Sequence (two can be from the video).  Be specific!

Introduce “Nature by Numbers” 

Click here to view this video!

  • “The speaker says and begins to show these terms in this sequence appearing all over nature.”
  • “Let’s look at some more examples of where this number set appears in nature.”
  • “You will not understand everything in the video, but that’s okay!  It’s a cool video and really shows the power of number patterns and their abundance in nature.”

 

Play “Nature by Numbers” video

Students will observe “Nature by Numbers” thinking about the magnitude of this sequence of numbers.  There is no real activity here.  It is more of a “wow” moment for the students.

Introduce Vocabulary Activity

Vocabulary Activity Handout

  • “We are going to be reading a text on other well-known and important sequences and series.  This vocabulary is used in it so it is important you understand these words to understand the reading.”
  • Give the instructions

 

Circulate the room while the students are working, looking for large ideas the students are missing, misconceptions, or if they have questions.

  • Students will work on the vocabulary activity independently to fill in as many boxes as they can.
  • Students will work on the vocabulary activity in groups of three pooling their knowledge to help everyone in the group fill in the description boxes they didn’t know.

Lead the class in a whole class discussion on any last questions or concepts they are unsure about.  The objective is for all students to have some understanding now of all the terms by pooling everyone’s knowledge.

Students will ask questions about the last terms that were still confusing to the students even after their group conversations.

Introduce the Text

Reading Guide Handout

  • Give the guiding reading questions
  • Preview the text with the students (looking at the structure and visuals)
  • Preview the reading guide with the students (Note: they must only read the sections of the text in the reading guide)

Students will preview the reading and guide with the teacher.

While Reading Text

Circulate the room while students are reading looking for large ideas students are missing in the reading and so students can ask questions if they have them quietly.

Students will read the text while filling in the reading chart.

After Reading Text

Circulate the room while discussing with partners listening for large misconceptions and answering student’s questions when needed.

Students will discuss the chart with a partner focusing on the following two questions:

  • What was one series that completely confused you?
  • What was one series that you really liked or intrigued you?

Have a couple students share out their responses as one large group (the whole class) and clear up any misconceptions or questions.

Students will share out to the whole class their responses to the pair and share.

Give Admit Slip for the next day

  • “Look up one of these series and find one example of where it is found or used in the world that hasn’t already been presented today.”

 

Possible Extension: Have students create their own sequence based on a pattern they encounter on a regular basis.  Describe the pattern and where it is found.  Then, represent this pattern as a sequence of numbers and describe the rule governing the sequence.

Citations:

Benjamin, Arthur.  “The magic of Fibonacci numbers.”  TEDGlobal 2013.  TED Conferences, LLC.  June 2013.  Conference Presentation.

IIT Madras. “Famous Mathematical Sequences and Series.” Web blog post. Edu-Blog. NSS-IITM, 16 Apr. 2013. Web. 24 July 2014. <http://edublognss.wordpress.com/2013/04/16/famous-mathematical-sequences-and-series/&gt;.

Vila, Cristobal.  Eterea.  Web.  30 July, 2014.

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A Text Set on Sequences and Series

Below I present a text set of six texts related to sequences and series.  During a unit covering sequences and series, I would sprinkle these texts in to supplement student comprehension.  I would use text #4 to introduce the entire unit on sequences and series.  Then, I would move on to the first couple of lessons in the unit.  When students have some background knowledge and now must begin to move between the different representations of series they learned, I would use text #5 with the activity I describe below.  We would then finish the content of the unit and probably do a formal assessment.  With the theory and conceptual part of our discussion on sequences and series complete, we would do a day or two exploring why these are important.  We would begin with texts #1 and #2 on day one.  They present some famous and important examples of sequences and series.  Then, on day two, we would use texts #3 and #6 covering a final example.  There are some other possible arrangements of the texts, which I briefly discuss below.  Compiling a text set on a topic really opened my eyes to how to begin to really develop a lesson for your students and help them gain a bigger picture of the world around them.  This is only the tip of the iceberg and I can’t wait to continue to learn and dig through materials!


 

1.  “Famous Mathematical Sequences and Series” by Edu-Blog

Text_1

Click here to view this text!

  • Summary: This is a print blog post by a group of students from IIT Madras (the Indian Institute of Technology Madras – a public engineering and research institution in India) working under the NSS-IITM (a group working on projects dedicated to the education and service of others).   They present and explain numerous sequences and series that are some of the most well-known number sets.  They begin explaining the Fibonacci numbers (the most famous set of the list presented).  They then discuss where we see terms of this number sequence in nature and its connections to the Golden Ratio.  They also mention some of the interesting mathematical properties the Fibonacci Numbers display.  The next sequences they explain are the Figurate Number Sequences, Lazy Caterer’s Sequence, Magic Square Series, Catalan Number Series, and the Look and Say Sequence.  They end mentioning some special series in other fields outside of mathematics; however, they do not explain these at all.
  • Complexity: First, this text is at about an eleventh grade reading level.  Second, the writing style of this text is more relaxed and informal.  It is not written in a narrative format, but it is written towards high school students.  There are pictures or diagrams to represent the number sequence being explained which helps bring the complexity of the text down.  Also, the writers always write out the sequence along with showing the summation notation for the series, also keeping the complexity down.  The series are shown in two different ways.  Also, the structure of the text makes it very easy to follow.  The different sequences and series are clearly separated and the introduction paragraph sets the reader up nicely for the text to follow.  There is also a lot of white space so the reading isn’t so dense and difficult to read.  All of these things bring the complexity down.  Third, the reader does have to have a decent amount of prior knowledge to understand this text.  They must understand sequences and series, what they are and how they are set up/shown, and some general areas of study in math.  This definitely makes the text more complex.  The topics covered are very interesting though and something students can easily see so their motivation to read the text and curiosity will be high making the complexity low.
  • When to use this text:  I would use this text at the end of a unit on sequences and series.  It shows some great applications of sequences and series in the real world.  It is the answer to the “why” question so many students ponder.  Not only does this show students that an abstract concept in math has some pretty cool and important applications, but it also will spike their interest about math because these applications are interesting to read about and not so abstract or dense with theory.
  • Connecting this to the students: When my students read this text, I want them to be thinking about all of these applications in their life.  I want them to be connecting this back to themselves.  That is the purpose of this reading is showing the students that this topic in math is in their own lives.  I want them to be thinking about where the examples that the text talks about show up in their own lives and if they can think of any other examples not mentioned.
  • Citation:

IIT Madras. “Famous Mathematical Sequences and Series.” Web blog post. Edu-Blog. NSS-IITM, 16 Apr. 2013. Web. 24 July 2014. <http://edublognss.wordpress.com/2013/04/16/famous-mathematical-sequences-and-series/&gt;.


 

2.  “Polygonal and Other Figurate Numbers” by Encyclopedia Britannica

Text_2

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  • Summary: This text is a print text in the Encyclopedia Britannica discussing figurate numbers.  They begin by discussing why figurate numbers are used and why mathematicians began investigating numbers in this fashion.  They go on to discuss the different types of figurate numbers including square numbers, oblong numbers, gnomons, and triangular numbers.  They also show the numerical and pictorial representations for all of these number sets.
  • Complexity: First, this text is at about a twelfth grade reading level definitely making it a more complex text for high school students to tackle.  Second, the structure of this text is less complex.  There is decent spacing between different discussions.  There are pictures to go along with almost every concept being discussed.  The introduction does a nice job of setting up the text.  The author is conscious about making connections between the different topics so none of them stand alone.  They all connect with one another.  All of these factors really bring the complexity of the text down, but the amount of academic vocabulary really increases the complexity.  Every word has an important meaning and purpose for the text so it must be read slowly and numerous times to understand the meaning of the text.  It is a shorter text which brings the complexity down; however, those meaty words can be difficult for some students to chew on.
  • When to use this text: The purpose of this text would be a continuation of the last text.  I would use it in a very similar manner, at the end of a unit on sequences and series.  It supplements the first text very nicely because it is an extension on what figurate numbers actually are.  It goes into more depth on one of the topics the first text briefly touched on.
  • Connecting this to the students: While my students are reading this text, I want them to be working alongside the text and drawing out the number sets as they discuss them to deepen their comprehension of the text.  There are illustrations to go along with the text to help them see how to do this, but then actually reproducing the sequence will ensure they are reading carefully and connecting with the text.
  • Citation:

Schaaf, William L. “Polygonal and Figurate Numbers.” Encyclopaedia Britannica.  Encyclopaedia Britannica, Inc., 11 May 2006. Web. 24 July 2014. <http://www.britannica.com/EBchecked/topic/422300/number-game/27910/Polygonal-and-other-figurate-numbers&gt;.


 

3.  “Lucas number” by Wikipedia

Text_3(2)

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  • Summary: This is a print text by Wikipedia on the Lucas numbers.  This is a rather short entry for Wikipedia.  Further, I would not have my students read the entire entry as it is very dense.  I would just cover the introduction, history, and definition of this particular example with them.  This text goes over yet another commonly seen sequence of numbers, the Lucas numbers.  It explains who came up with them and what this sequence of numbers looks like.  It also references the Fibonacci numbers often and this set’s connections with the Fibonacci Numbers.
  • Complexity: First, this text is at about the level that a freshman in college would be reading.  Second, the structure of this text has some elements making it less complex.  The relationships between the Lucas numbers, Fibonacci numbers, Golden Ratio, and Wythoff Array are clearly stated.  The student can easily see how these concepts interact with one another.  Also, there are clear titles separating the text into logical sections making it easier to read, and the text is not very lengthy.  Some elements making the text more complex include: there are no concrete examples of where the Lucas numbers appear in the real world and they do not introduce all the concepts they are connecting together (like the Fibonacci numbers) one at a time.  Further, when they do reference other large concepts, they do not explain them at all.  They’ve created a hyperlink to those concepts’ pages which is helpful, but not nearly as helpful if they gave a brief description there at the time of learning about this new concept.  They assume that either you know the concept or can go looking for it in other places.  Third, students must have a lot of prior knowledge to access this text.  The most prior knowledge yet out of all the texts.  This definitely makes it a more complex text.  However, it is another very interesting, short text motivating students to continue reading and push through.
  • When to use this text: As above, I would use this text after the other two.  It requires background knowledge on sequences and series that we would cover during our unit in class and then knowledge on other sequences we would cover with the first and second texts in this set.  This text, then, is just another way to teach students about the many applications of this topic in math and provide them with another sequence of numbers to peak their interest into the ever surprising world of math.
  • Connecting this to the students: Now that the students have read two other texts on this application topic, as they read this third text, I would push them to consider how many other sequences of numbers exist, especially ones that show special properties in the real/natural world.  I would have them see if they could create their own sequence of numbers displaying a certain pattern they see around them and explain what they are displaying.
  • Citation:

“Lucas number.”  Wikipedia.  http://en.wikipedia.org/wiki/Lucas_number.


 

4.  “The Magic of Fibonacci numbers” By TED Talks

Text_4

Click here to view this text!

  • Summary: This is a digital media text, more specifically a video.  Arthur Benjamin, a professor of math and “Mathemagician”, gives a TED talk on the importance of going beyond mathematics in a textbook with a teacher lecturing.  He emphasizes the importance of exploring math and all the beauty within it.  He does this through a discussion of the Fibonacci numbers.  He begins by explaining the importance of thinking about math creatively and inspirationally, rather than just as more meaningless calculations.  He then goes into what the Fibonacci numbers are and where they appear in the natural world.  He ends with a discussion on the interesting and rather “mind blowing” mathematical properties the Fibonacci numbers possess.
  • Complexity: First, the auditory aspect of this video is at about a seventh grade reading level.  Second, the tone Benjamin uses is very conversational.  He doesn’t speak very quickly and uses a lot of great inflection.   It feels as if you could be having a conversation with him right there one-on-one.  This definitely makes the video less complex.  He pairs his talk with a visual presentation in the background.  His presentation is extremely visually appealing.  He uses colors and explains most of the pictures and examples shown behind him.  It isn’t too chaotic or full making it very easy to follow and understand.  He also provides concrete examples for students to grasp a hold of.  The video is also under ten minutes keeping student’s attention and increasing their motivation to watch the video.  All of these factors keep the complexity of the video low even though the topic being discussed is rather complex itself.  Third, students really do not need any prior knowledge to understand this video.  The information provided is basic enough that they needn’t know anything even regarding sequences or series.  This brings the complexity way down.  Also, their interest will be rather high since it is a video rather than a print reading and a TED talk which most students really enjoy watching.
  • When to use this text: I would use this text to introduce the unit on sequences and series.  It shows students an application of the concept they are going to be learning and its importance in the real world.  The video is short and not too in-depth making it a good introductory video.  You could also use it to introduce the exploration on applications of sequences and series like the Fibonacci numbers, Lucas numbers, etc. for the same reasons.
  • Connecting this to the students: While students watch this video, I want them to be thinking how this application is in fact an application of sequences and series.  Where are sequences and series applied here?
  • Citation:

Benjamin, Arthur.  “The magic of Fibonacci numbers.”  TEDGlobal 2013.  TED Conferences, LLC.  June 2013.  Conference Presentation.


 

5.  “Series and Sigma Notation” By Cool Math

Text_5(2)

Text_5

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  • Summary:  This is a digital media text, more specifically an infographic.  This image explains all the pieces and parts of the special sigma notation when we represent a series using it.
  • Complexity: First, there are no large pieces of text to evaluate the quantitative complexity of this infographic, but there is still some information to gain just from simply investigating the words they use.  The language used is not basic math terms, but they also aren’t too difficult to either.  Students with a basic knowledge of series and the parts of them will have no problem understanding the terms used to explain the notation.  The blurbs next to each piece of the notation are short and sweet which doesn’t make this so complex.  Second, the structure of the text is very good.  Each piece of the notation has its own description.  The author uses colors to help the reader differentiate between the different parts.  There is a concrete example given to see exactly what translates to what in the notation too.  The author also doesn’t get too wordy.  All of these elements make this text not too complex.  Third, students do not need much prior knowledge to understand this information.  It is typical to see in some textbooks.  Also, because there is color and a picture, students won’t see this as too complex of a text.
  • When to use this text: The purpose of this text is to teach students early on in the unit on sequences and series the typical notation used to communicate series.  This is a great reference tool for students as they will often need to move between the two representations of series as a list of numbers and then with using sigma.  As a teacher, I would begin with an example of two representations for a series, a list of numbers and the sigma representation, and then ask them to find any connections between the two representations.  Then we would walk through what each part of the sigma notation means together.  This infographic is a great summary of that mini-activity.  It has all the important information students need to know about sigma notation.
  • Connecting this to the students: While students are examining this infographic, they would be looking for the pieces they’ve already identified in our mini-activity.  They would find and solidify the information they knew prior to me teaching the lesson and then look for the information they weren’t completely correct on during the activity to amend and understand the notation.  They would be doing personal evaluations on their knowledge of the notation.
  • Citation:

“Sequences & Series Lesson 3 – Series and Sigma Notation (page 2 of 6).” Coolmath.com.  n.d.  Web.  24 July 2013. http://www.coolmath.com/algebra/19-sequences-series/03-series-sigma-notation-02.htm.


 

6.  Polygonal Numbers Demonstration By Wolfram Demonstrations Project

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  • Summary: This is an interactive online activity for students to explore what figurate numbers are, what they look like, and how they can be manipulated.  This demonstration along with all the other Wolfram Demonstrations were produced by Stephen Wolfram.  It is an extension of the Wolfram world.

Text_6

  • Complexity: First, the little word blurb below the activity is at the collegiate level for reading.  Other than that, there are no words in the activity.  Second, the demonstration is very easy to use.  There are few buttons to push or sliders to move.  Only the essentials are there.  There are two sliders and that is all making it very easy for a student to use on their own.  The picture resulting from the two values you set the sliders to is very easy to understand and uses colors and numbers to help the user read and understand the resulting figure.  There is absolutely no irrelevant information or even extra images presented to confuse the reader keeping the complexity low.  Third, students do need some prior knowledge on what figurate numbers are to understand what the demonstration is showing them, but after a quick introduction, this is a perfect tool to really show students what figurate numbers are.  This allows them to explore them.  The student interest is very high then because they are actually interacting with the material and filling in the blanks to the little information they have currently on figurate numbers.  There is nothing they have to read or watch.  They get to play!
  • When to use this text: I would use this text, as I discussed above, as an extension to learning about figurate numbers, an application of sequences and series, the unit being covered.  This will really help students to understand what this odd sequence of numbers are.  This gives students the visual to go along with the print text explaining the concept.  This would be paired with texts one and two.
  • Connecting this to the students: While the students are working with this demonstration, I simply want them to test their understanding of what figurate numbers are.  I want them to understand what they are.  I want them to be able to explain them using words or pictures.
  • Citation:

Weisstein, Eric W. & Wolfram, Stephen.   “Polygonal Numbers.”  Wolfram Demonstrations Project.  Wolfram Demonstrations Project and CDF Technology.  Web.  24 July 2014.

Extra Funny

Visualizing the Fibonacci Numbers via Digital Story

Click here to view my visualization of the Fibonacci Numbers!

This task of representing some of the information I’ve found on my topic in a visual manner rather than through print was rather difficult for me.  It took me a very long time and lots of digging into examples to figure out what I wanted to do, what I could do (due to my technological idiocy at times), and what fit well with the information I wanted to communicate.  Finding a good balance between these three variables was really difficult.  But, when I did finally settle on digital storytelling, the project moved right along.

I chose digital story telling because I wanted to walk with my students as they explored the Fibonacci Spiral and numbers in nature.  I knew a picture would not be nearly as effective as a video for showing this.  I was aiming much more for a visual and spoken explanation of Fibonacci and his work rather than a pictorial.

My Process of Creating the Video:

I began by looking at what parts of the information I’d found I wanted to display in my visualization.  Then, I wrote out a kind-of script for what I would voice over the video and wrote in breaks to show where I would turn my page in my notebook.  This was probably the longest part of the process because I had to think of a logical way to introduce my information since the meat of my information is about an application of a certain math concept.  You need some background knowledge to understand the Fibonacci sequence.  Then, I wrote out a visual story board for what I would write on each page of my notebook and gathered the objects I needed for my video (ex: an artichoke, a pineapple, different plants and flowers, apples, bananas, pens, paper, etc.).

I then filmed all of the visual pieces and put them onto my computer.  I cut and pasted the pieces of video together in Windows Movie Maker to form a logical and flowing video that moved along.  Then, I narrated over the video to explain the visual things I was doing.

I literally matched image (visual video) with print (voicing over the video) in my eyes.  I have given students two different ways of taking in information in one video which I really like doing.

My Understanding Now of this Topic and Teaching:

This process of visualizing a concept really deepened my understanding of what the Fibonacci sequence is in ways I never could have gotten from reading in many ways.  The main way being that it is easy to read about an example of the author exploring the spirals of a pineapple or looking for the Fibonacci numbers in plants, but when I actually had to look for examples myself and then find the numbers within the plants, I found a whole new appreciation for the prevalence of these numbers in nature!  It really sunk in when I looked for examples and I began to see them all around me, especially with the spirals.

Another way this project deepened my understanding is that it is very easy to read about an example of where Fibonacci’s Numbers appear or how his spiral is created, but it is much more difficult if you are asked to reproduce this.  It took me awhile to truly understand how the spiral actually appears in fruits and vegetables, but when I did, it was very clear to me.

I think this is similar to having students read a chapter or blurb of notes and then asking them to reproduce the experiment or example they read about.  Teachers do this all the time (especially in math classrooms)!  Now, after completing this project, I believe that representing information for students in more than one way, or even better having them represent that information in another way, is more important than ever.

Where I Am, and Where I Want To Go

My current knowledge of sequences and series feels incomplete to say the least.  I know there are dramatic holes in my knowledge on this concept from the rudimentary level through the advanced material on it.  I do understand summation notation to a point.  I generally understand how series are set up.  I understand how to take a sequence of numbers and write it as a series and then go back the other way.  Beyond that, my knowledge is very limited.  I learned, but no longer truly remember, the tests for when sequences and series converge and diverge, and I do not remember all the different types of sequences and series.  Finally, I do know a few examples and applications of sequences and series, but I don’t even really understand those even.

Therefore, I don’t know:

  • the difference between sequences and series,
  • the ins and the outs of the notation,
  • the tests for convergence and divergence,
  • the specific different types, and,
  • most importantly, the applications for sequences and series.

With these things in mind and my focus on teaching high school students who are still so young in their math careers, I will be looking to answer the following questions through my research on this topic.

  • What is the difference between sequences and series?
  • What is the relationship between a list of numbers and the summation representation for that set?  What does every piece of the summation notation mean?
  • Are there strategies to finding the summation notation which represents a list of numbers?
  • What are some easier and more complex applications within math (ex: they are used as the basis of calculus and integrals) and outside of math?  I want to be able to understand numerous examples to present to my students in a manner they will understand.

I will begin to look for the answers to these questions and classroom friendly texts concerning my topic in the following resources:

  • Calculus: Early Transcendentals by James Stewart

This is the calculus text I used when I went through the calculus series.  It will give me a good refresher and background knowledge on sequences and series.  I will not necessarily use this for information to tell my students.  This will be used for me to further my own knowledge of the content.

  • High School Algebra Textbook

This text will be used to help me see what the students are learning about sequences and series and how they are learning it.  Then, I will be able to supplement using my further knowledge of the subject and finding appropriate readings to further student’s learning.

The Common Core State Standards will help me to see exactly what the students must know and how far I should push them in a more complex understanding of sequences and series.  I have been known to take a concept or idea and run too far with it so this will help keep my inquiry focused where it should be.

This link gives information on the Fibonacci sequence.  The information is definitely more complex and aimed for a more advanced math reader like myself, but it is a great application that the students will be empowered to hear about since they’ve heard of it before.  They will now feel even more knowledgeable knowing what it actually is.

This TED talk is about mathematics in general and also on the Fibonacci sequence.  This is another great resource and application of sequences and series.  It also has a great message about mathematics and why we teach it.

  • Math through the Ages: A Gentle History for Teachers and Others by William P. Berlinghoff and Fernando Q. Gouvea

This book discusses the history of mathematics.  More particularly, it discusses Fibonacci and his development of his numbers.  I think it is important to give students the whole picture when you are giving them an application of something you are doing in class.  This will give them the historical background behind this number set.

About this Blog

Hey there!  My name is Kate, and I am a secondary math education student at UW-Milwaukee getting ready to enter into my first round of student teaching this fall.   Math has always been one of my strengths, but not necessarily one of my passions.  Growing up I was convinced I was going to be the first female president, and then later I was going to climb Mount Everest.  Clearly every little girl’s dreams for when they grew up.  I liked the idea of the challenge and to push myself further than I’d ever gone before.  Then, when it came down to really thinking about what I was going to do with the rest of my life, teaching seemed like the obvious answer to me based on my past.  I would play school all the time with my little sister.  (I was the teacher of course.)  I loved helping others in school, and I was good at it.  However, I didn’t immediately jump to math.  I initially wanted to be a history teacher.  Then I decided I wanted to be a psychology teacher.  I found my love and passion for teaching math when I began to tutor my peers in math at the end of high school.  I realized that I could help make all of the madness and confusion that is often math seem not so scary and conquerable to others.  This feeling of leading others to understand math was something I could never let go of.  Seeing a student have that “ah ha” moment is why I am going to be a math teacher.

Therefore, throughout this blog, I am going to explore a concept in mathematics that has always confused me a little and likes to sneak up all the time in math.  I will be exploring sequences and series in math at a more rudimentary level where high school students will be working with them and not at the calculus level.  I was never formally introduced to sequences and series which is why I think I never really understood them.  My teachers always just assumed that we had learned the basics of them in the previous class.  In addition, and possibly more importantly, I will be exploring applications of sequences and series which I really was never exposed to.  One particular example of sequences I will be looking into is the Fibonacci Numbers.  These are a sequence of numbers which shows up all over the world and is tied together with the Golden Ratio.

 

The Fibonacci Numbers show up all over nature.  The number of petals on a flower is typically a Fibonacci Number.

 

I want to show students the applications of sequences and series and just how important they are because then I believe they will have an easier and more enjoyable time learning about them then I did along with many other math students I know.  If you show a student why something matters, they are more likely to work towards understanding it.

Sequences and series always seemed to me to be one of the abandoned topics in math.  Nobody every truly dove into them with me as a student.  I want my students to see the power and importance of sequences and series and that will begin with my knowledge and exploration of the topic.