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

An alternative introduction to Imaginary numbers

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

  Alices Adventures in Algebra

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 2/3^rd 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 8^th 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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 Imaginary numbers are Real , Welsh Labs

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.