Thursday, July 23, 2020

Blog Post #4


Blog Post 4

In my class, I have 35 high school juniors. The racial demographics of the school is 12% white. Unfortunately, my class doesn’t reflect that, it’s 20% white and more than 60% of the students are male. It is unfortunate that even at the age of 16, we are having a hard time diversifying STEM fields. Each of the students have already demonstrated that they have the intelligence and the drive to be successful in math. All they need now is for someone to believe in them and help them to get excited about the power of mathematics. Our subject is going to be complex numbers. These students already have been introduced to polynomial equations. They know how to graph a quadratic equation, but to this point, they’ve been told that the square root of a negative number doesn’t exists.  This week, I get to blow their minds.


Video




Summary:  This video presents the idea of complex numbers with a cartoon drawing walking on a number line.

To introduce the idea, I want to start with a video that presents the concept of complex numbers with a visualization of rotational analysis. The quantitative analysis of this video by storytoolz was 11.3. I disagree with this rating. At 11.3, that would imply that the material is at the mid to high end of my class. However, because of the accompanying visual images, I feel that it is well within their zone of proximal development. Quantitatively, the text is moderately complex. According to the “Text Complexity” chart, the sentences are simple and the language is literal.  The purpose of the video is explicitly stated. The reason it is not slightly complex is due to the knowledge demands. The students will need to be familiar with terms ‘rotation’, ‘vector’, as well as ‘multiplicative inverse’. I feel this makes the knowledge demand to be very complex. The speech is too fast for ELLs, but there are subtitles.  The playback speed is able to be varied.
The purpose of this video is to introduce a method of visualizing complex numbers that doesn’t depend upon a Euclidean circle. After watching the video, the students should have an opportunity to discuss what they have seen. Questions surrounding what a multiplicative inverse is? How does it relate to the topic of complex numbers? What did the video author mean when he said, “Calculating with imaginary numbers is the mathematical equivalent to believing in fairies.”?


Print



Summary: The article interprets portions of the story “Alice in Wonderland” as a parody of the confusion over the concept of complex numbers.

This text has a quantitative complexity according to Storytooz of 13.8. The longest sentence in the sample submitted was 44 words. That length can be difficult to manage for students. Qualitatively complexity is also very complex. It relies upon knowledge of a story written more than a century ago. There are few graphics to help the concept. The connection between ideas is not intuitive and would probably require two readings to understand what the author was intending.
I would choose to give a copy of this text to the students for them to read at their leisure and I would encourage them to read through the text. I feel it offers a chance for enrichment of the activity that might inspire some of the students.  However, for the portion dealt with in class, I would only use the text insert 2/3rd of the way through the article.  This insert graphically describes the issue of complex numbers.




According to storytoolz, this text is also very complex. It was rated at 15.9 due to sentence length. Qualitatively it has very complex vocabulary and subject matter. But, the graphic and other text features make it moderately complex.
The visualization presents a mathematical principle in a way that underscores the controversy surrounding imaginary numbers. They seem unreal, like believing in fairies.

Print
The Crest of the Peacock: Non-European Roots of Mathematics, by George Gheverghese Joseph. Princeton Publishing Press. ISBN 978-0-691-13526-7

Summary: p. 382 – 385  Indian mathematicians in the 8th century documented mathematical proof for what western mathematics would call the Fundamental Theorem of Algebra. Baudhayana’s solution was one that would ultimately show the basis for what is now called complex numbers.


Culturally relevant
This text is qualitatively complex and quantitatively very complex. Storytoolz gave it an average rating of 15.6. This is accurate. The passage could be read aloud and then students could put the information into their own words.  By including information about lesser known mathematicians, we can dispel the myth that only white men can be good at math.


Video


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Summary:  This video presents the concept of complex number with similar information given in the first video. The instructor introduces the cyclic nature of complex numbers and ties the concept in with the circle presented in most math textbooks.

According to Storytoolz, the quantitative complexity is 12.8, fully 1/3 of the sentences were longer than 32 words. While this is an issue with reading on paper, I believe that because this is oral, with written captions, the quantitative complexity should be lower.  The qualitative complexity for text structure is very complex. Without the images, it would be extremely difficult to follow the material. Both language features and purpose I believe are quantitatively moderately complex. The sentences are long, but the words are short. The information in each sentence is presented graphically as well as orally to help with the comprehension. The knowledge demands are complex. This does require an understanding of the concepts of multiplicative rotation.

Wrapping it up

The first video introduced a concept of how to imagine complex numbers without having to understand why it appears that they form a circle. As the class moves from that initial image, they are told why imaginary numbers are so important. The texts show that principles of continuity and the Fundamental Theorem of Algebra depend upon the use of complex numbers.  In order to draw in the reluctant student, I included two print text that ground the information with the learning of a non-western mathematician and a connection to a fictional story some might find interesting.  Up to this point, the complexity of the tasks required by the student are limited to the first two levels of Blooms Taxonomy. The students will be remembering facts and enhancing their understanding. The last video restates similar material to the information presented in the first video with one difference. It ties the information in to the cyclic nature of i . Because there is no new material in this video, it is appropriate to expect that the complexity of the task can be increased.  It is at this point that a student could be expected to compute equations with complex numbers. This would involve the next two levels of Blooms taxonomy, applying and analyzing.




Sunday, July 19, 2020

Blog post #3


Imaginary Numbers Aren’t Pretend.
Here is the link for my visualization. 

When I started this project, I knew that it would be difficult to take what many people refer to as imaginary and represent it in a format that made sense. The concept of complex numbers is taught in Algebra 2. Common Core standards recommend 
 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number”  


    To me, this standard seems less than ideal. There are many high school students who successfully complete mathematics without a solid understanding of what complex numbers are, and why they are critical to mathematics.  My own informal survey showed that often times students only knew complex numbers had “something to do with the square root of -1.” 

     This is a topic that I am passionate about. Modern computer graphics to render the images that appear 3 dimensional require vectors using complex numbers.  Computing the emulates artificial intelligence relies upon complex numbers. MRI imaging technology requires complex numbers. If our students think that complex numbers are just make believe, they will not be able to keep up in highly technical fields. 
     
    My target audience is high school students who are currently in or have completed Algebra 2. I also wanted to make the presentation helpful for all of the people who (like me a few years back) think that complex numbers are imaginary and unimportant.

    Presenting the information in a graphical way was very challenging for me. I spent quite a bit of time on Mathematica and Wolfram Alpha, but I wasn’t able to make the visual representation clear. Ultimately, I used a graphic that had been designed by Welsh Labs 

    Initially, I wanted to demonstrate the complex plane with intersecting circles. By using a parabola instead, the mathematics was more straight forward. The process of recording the information was where I learned the most in this project. Each slide walks the student through the process of visualizing the complex plane. Initially, I was making the entire thing too complicated. Because the software strongly encourages each recording to be less than ten seconds, I had to rephrase each point and trim it down until it was as simple as I could make it.  I think that makes the presentation better. Initially I was confusing the audience with too much information.


Friday, July 10, 2020

Blog Post #2

Not complex numbers can do imaginary things too

Complex numbers are extremely important for modern computing. They are used in the algorithm that helps determine the Amazon “You might like” suggestions. Complex numbers are used to make MRI scans readable. They are used in the mapping software on your phone.  But, most people still call them imaginary as if they were a convenient myth, like Santa or the Tooth Fairy.

I currently know that complex numbers are useful in many emerging technologies. I’d like to know how they are incorporated into that.  I know that complex numbers were controversial in the 1800’s. The mathematics community was pretty divided for quite some time on what they were and how to quantify them. I want to know how that became resolved. I know that complex numbers are not representable on a number line. But, I wonder if there is a better visualization for them, that makes imaginary numbers seem more “real”.

Some of my sources are: "Teaching 
 “Teaching the Complex Numbers: What History and Philosophy of Mathematics Suggest” Journal of Humanistic Mathematics, https://scholarship.claremont.edu/jhm/vol3iss1/6
  
 “At the Intersection of Mathematics and Humor: Lewis Carroll’s “Alices” and Symbolical Algebra” https://www.jstor.org/stable/3826762?seq=1&cid=pdf 


There is a series on YouTube called “Imaginary Numbers Are Real” https://www.youtube.com/channel/UConVfxXodg78Tzh5nNu85Ew

“The story of Mathematics” https://www.storyofmathematics.com/19th.html

Thursday, July 02, 2020

Blog post #1


Hello everyone!
I am nearly done with my degree in Secondary Math Education from the University of Wisconsin, Milwaukee. 

Not too long ago, I finished raising these amazing individuals.  I looked around and realized I wasn't really happy with the world that they were inheriting, so I decided to go out and change the world.

I am committed to changing the racial and socioeconomic injustices in the United States.  

Recently, I discovered an organization called Abolitionist Teaching Network.  They offer a number of resources for teachers to help them create an anti racist environment for the students to learn.  

Tuesday, October 09, 2007

Monday Blues

Monday morning. God, I hate Monday's. No, that isn't right. I don't hate Monday...oh, it's morning I hate. I don't even hate mornings, if they could just come more slowly. Why do mornings have to jump up without warning. Demanding attention like a some little curtain climber. And they are so loud. (I meant mornings are loud, but actually, curtain climbers are loud too.)

This Monday was louder than most. Dinner was crowing by 6. (Dinner is a rooster, who we thought was a hen, until he developed some odd habits) The neighbor rode his Harley to work. And some street worker decided the church parking lot adjacent to my yard needed to be patched. But, as a quick thinking pain in the asphalt repairman, he left the truck in reverse for 20 minutes allowing all of us to enjoy the serenade of it's warning tone.

Nauseated and crabby as a constipated New Englander, I lay on the couch, wanting to vanish.

The phone rang. It was Judie. Judie is awesome. She's one of those women who manages to balance everything and still has time to take in 20 neighborhood kids and give them a shoulder, a friend and a mom. Honest, she has 20 extras. On top of her own 10 children.

Judie doesn't call on a Monday morning unless she needs something. She knows me too well. She knows the pains of virtual schooling. She is my mentor.

"Karen," she says "I'm in the hospital."
It takes me a millisecond to remember. She wasn't feeling well. She's been so tired lately. She was going in to the doctor, sometime....when was that....
"are you sitting down" she continues
...we talked just a few weeks ago. We laughed. I just got the tickets for the ballet. Why is she asking me to sit down?...
"It's leukemia. "

I honestly don't know what came next. Did I ask about Mark? Or the kids? Was I at least kind and gentle? Did I show compassion? I don't remember. Somewhere in the conversation she told me the odds are against her. Somewhere in the conversation she made me laugh. I wish it had been the other way around. Somewhere in the conversation my Monday blues seemed so small in comparison.