edu:topics:complex:start [Simonweb wikinotes]

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[[mteach:mtm4:topic]] |
[[https://docs.google.com/document/d/1w-epDeIDy9vcTxKUj_DxknepcGQx98_wK4oO27pI0nc/edit#|google-doc]]
with Atlas Huang, Han Lee | [[http://edu.simonwise.net/topics/complex/]]

<quote>
Students work in a group of 2 or 3 to conduct an in-depth study of the teaching and learning of a selected topic identified from the subject outline content list

Students will research the issues around the teaching of a particular topic in mathematics and develop a set of resources that could be used by teachers in teaching that topic. The resources should be more than just a list of web pages or exercises from a textbook. The idea of this assessment task is that your research will guide you in designing resources or the appropriate use of other’s resources to enhance students’ understanding and help students deal with the possible misconceptions.


You will submit a 1000 word report outlining the research you have used in designing your resources including links to the resources. Your report could be in the form of a webpage.

Soundness of argument including illustration of key points with examples and citations from relevant literature 30%


Balance of participation of colleagues 20%

Clarity of communication 20%


Soundness of organisational, presentation and communication skills 30%
</quote>

======Complex Numbers topics page======
[[intro-lesson]] | [[context-numbers]] | [[history-complex]] | [[applications]] | [[booklist]] | [[references]]

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<h5>A report on research into teaching the topic Complex Numbers, within the NSW syllabus</h5>

<p>A review of the syllabus documents (NESA 2017), linked above, reveals a topic which focuses
on the abstract. No specific applications outside pure mathematics are mentioned.

This presents some challenges, but also the opportunity to consider numbers and mathematical structures
in a more formal way, while reviewing some of the foundations. Introducing the new number 
<strong><em>i</em></strong>
and the remarkable way such a simple idea can open huge new vistas in mathematics can be a path to
expanding students' ideas about what mathematics is and what mathematical thinking involves at its
best and most exciting.

<p>What the syllabus outlines is the algebraic, geometric and trigonometric aspects of complex numbers. Their
power is they unify these areas which have been somewhat distinct till this stage in the student's journey through mathematics. There have been opportunities
where they intersect. Pythagoras Theorem keeps cropping up through the years in new contexts. Quadratics
and cubics relate to area and space. These two topics are perhaps the most ancient serious mathematics,
developed well over 1000 years before Pythagoras and classical Greek culture as formal, written ideas.
We have collected some ideas and resources helping to illuminate this background, this journey through
mathematics, and perhaps to reflect on not just our own journeys but the
collective journey that has led us to this point over the millenia, see the 'Numbers' link below.


<p>Perhaps we can see the invention of counting (millenia earlier than that) as the first
mathematics (Everett 2017). Or perhaps the geometry which is clearly possible without
numbers at all. Here we offer some reading, including a fascinating examination
of the last two statements, as a way to help leverage this topic for the purposes outlined above. See
the 'Books' link below for material suitable for teachers as background, or for offering to students at
various levels of mathematical understanding. This material is very relevant to teachers and students
seeking to place Complex Numbers as the next enhancement of that number system, which we have invented to help
us build models and describe the world. Not the last one, but it is the last we will meet formally in
high school. At this stage they are the keystone that links the strands, but that is a very formal thing, and it was not the original motivation. It was something discovered later.

<p>The principle that meaningful lessons can foster positive attitudes, improve lesson enjoyment and
enhance student learning (Whittle, 2007) is, quite properly, at the heart of much lesson planning.
The challenge here is how to place Complex Numbers in a way that is meaningful in applied mathematics,
apart from the pure mathematical ways outlined above. They have remained part of mathematics (despite
the doubts of the first mathematicians who used them) for almost 500 years now. This suggests they must have
some practical applications, contemporary usage on the cutting edge of physics aside (that application is amazing, but tough (Penrose, 2004)). We looked at where else they are used, see the 'Applications' link below for a variety of
resources and commentary on the outcome. Many of the places that complex numbers help solve certain
algebraic situations require too much setting up and too much mathematical knowledge from beyond this
syllabus. Johnson (2014) suggests that complex numbers are normally used when some quantity has a
magnitude as well as a phase, and gives practical examples in physics and engineering which include
electrical circuits, vibrations and resonances, optical wave interference and quantum mechanical wave
functions.

<p> We propose introducing the topic with solutions to cubic equations, Bagni (2010) has looked
closely at the advantages of this approach. See the
'Intro-lesson' link below for details. Can we find
an approach that offers some practical purpose, some motivation for "just making it up" other than
stepping back and saying "isn't it beautiful". The introduction of <strong><em>i</em></strong> is hard.
We (teachers) have hammered the impossibility of taking the square
root of a negative number over and over again. It was the rule. This may have been a mistake, but we have. 
But in this respect our students are just like the mathematicians of 1530s.

<p>Bagni (2000) considered this, and suggests that the historical order could inform our teaching. He
tested this on students. The quadratic setting, with the final answer as <strong><em>i</em></strong>
just hits rejection, this is not allowed, <strong><em>i</em></strong> is not the solution, there is
<strong>no</strong> solution. But going via a complex number to arrive at a real
solution, a solution that can be seen to be correct independently, is much more believable. The
historical narrative shows the resistance, Bagni confirms the effect on students. He proposes using
the historical artefacts of Bombelli's publication that first clearly laid out the system.

<p>The history of our development of difficult ideas can help understand the motivations behind them, and
the most practical ideas driving them. As mathematicians we have often refined and justified the formal
structures with the benefit of hindsight, and in some ways this makes the concepts clearer.
But it can obscure the
relationships to the wider world that were the primary reason for the invention, or at least for persisting
with it beyond the joy of recognising something elegant and compelling in its own right. How to approach
teaching calculus can be considered in this respect. Should we first look at limits and the 19th century
formal derivations, or should we start with the original motivation, considering calculus
first as the language for describing change over time, and especially for understanding the motion of
objects under the influence of gravity? To understand better the
place of complex numbers we researched their historical development, and the outcome and resources are shared
under the 'History' link below.

<p>Penrose (2004), Lockhart (2017, 2002), Schrödinger (1944),
Gleick (1987) and many
more all consider, in their very different ways, this relationship
between the beauty of the pure idea and the way it builds the language we use to describe and understand
our world. <em>Time</em> (Butler & Messel 1965) has been in my bookshelf for 40 years, a series of lectures
and activities for a summer camp at Sydney University for "year 4" (now 10 or 11) high school students. The
purpose of the camp was to inspire, with a connection between the beauty of the mathematical ideas and the surprising but very real outcomes they describe. In their case the challenge was time and relativity. They
included quite a range of activities and ideas not directly related to the topic.

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====more====
This topic, Complex Numbers, arrives when our students (who are 
extension 2) have much of the
background to start to put the whole beautiful structure together. For some the elegant formality of
the formal constructions they may study as undergraduates, if they take the pure mathematics path,
will provide that. Some of the reading we have proposed could help them get a glipse of this earier.

But for others there needs to be something else.

----

perhaps re the lesson connecting to Argand Plane, use:

<quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote>
There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Gottingen. Gauss introduced the term complex number

<quote>If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and
√ −1, instead of being
called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.
</quote></quote>

----

<quote></quote>http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf<quote>
According to [B. L. van der Waerden, A History of Algebra, Springer Verlag, NY 1985.],

"Cardano was the first to introduce complex numbers a + √−b into algebra, but had misgivings about it."

<quote></quote>In Chapter 37 of Ars Magna the following problem is posed:<quote>
"To divide 10 in two parts, the product of which is 40”.

It is clear that this case is impossible. Nevertheless, we shall work thus:

We divide 10 into two equal parts, making each 5.

These we square, making 25.

Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book leaving a remainder of -15,

the square root of which added to or subtracted from 5 gives parts the product of which is 40.

These will be 5 + √−15 and 5 − √−15.

Putting aside the mental tortures involved, multiply 5 + √−15 and 5 −√−15

making 25 − (−15) which is +15. Hence this product is 40.
</quote></quote>

----

... from pp2 to top of numbers ... that the ancient pyth, cubis and quadratics are linked here with complex

Then consider Everett, Lockhart for the creation of numbers 'so far'.

Consider Shrödinger, Gleick, Gallileo and finally Penrose for narratives of science, mathematics and
the excitment of the journey. Penrose is about complex numbers but only for a few. Yet keep him in
mind ... one of my fellow pure maths majors was there, in part, because he was given a copy in school.

Consider Weeks, if you want something leading up to Penrose, or really building the spatial imagination
that will help in this topic. Consider how all these ideas were developed, the stories are very real, they
can make the mathematics come alive.

So ... for some we have them in the formality, providing we do not neglect that. For others, I hope the
historical journey will help justify. Next we looked into the practical, the concrete applications.

<p>way by revealling
the way we built these structures we might find ways to help teach them. It is very relevant to consider
with Complex Numbers, as with Calculus

<p>And these stories, especially the more recent ones where the narrative of discovery is known, can make
great reading ... for the teacher (perhaps arming them with some nice resposes to curly questions),
and for our students. Mathematics is creative, it is exploration, imagination, discovery and above all
"just making it up" ... in a very special way of course.

<p>Here we propose an introductory lesson.

<p>After we have introduced the new number, as an invention, with the suggestion that this new invention
did prove usefull ... so it is stillbaround centuries later.

<p>As a teacher consider some of the historical background. Use it to build on the "just make it up"



====notes====
[[edu:resources:schrodinger-life]] | [[edu:resources:gleick-chaos]] | galileo |
copernicus | [[edu:resources:penrose-physics]]

... a rough outline of the overview:

(I see everything we have found so far linking from here ... see the site outline, so lesson exerpts
and I have abparagraph or so onbeach book I have mentioned)
====A report on research into teaching the topic Complex Numbers, within the NSW syllabus====

A review of the syllabus document reveals a topic which focusses on the abstract, there are not any
specific applications mentioned.

This presents some challenges, and opportunities.

What the syllabus outlines is algebraic, geometric and trigonometric aspects of complex numbers. Their
power is they unify what, till this stage in the students journey through mathematics, had been
somewhat distinct areas.

Generally geometry is one topic, algebra another, split into their strands. There have been opportunities
where they intersect, Pythagorus theorem keeps cropping up through the
years with triangles, and into algebra. Quadratics and cubics relate to area and space.

The history of our development of these ideas can help understand the relationships, and by revealling
the way we built these structures we might find ways to help teach them.

And these stories, especially the more recent ones where the narrative of discovery is known, can make
great reading ... for the teacher (perhaps arming them with some nice resposes to curly questions),
and for our students. Mathematics is creative, it is exploration, imagination, discovery and above all
"just making it up" ... in a very special way of course.

And this topic, complex numbers, arrives when our students (who are extension 2) have much of the
background to start to put the whole beautiful structure together. For some the elegant formality of
the formal constructions they may study as undergraduates, if they take the pure mathematics path,
can provide that. But for others there needs to be something else.

Complex numbers, at this stage are the keystone that links the strands, but that is a very formal thing,
and it was not the original motivation. It is something discovered later.

The introduction of 'i' is hard.

We have hammered the impossibility of taking the square root of a negative number overnand over agian.
This may have been a mistake, but we have. But in this respectbwe are just like the mathemeticians of 1530s.

Bagni considered this, and suggests that the historical order could inform or teaching. He tested this on
students. The quadratic setting, with the final answer as 'i' just hits rejection, this is not allowed,
'i' is not the solution ... there is no solution. But going via a complex number to arrive at a real
solution, a solution that can be seen to ge correct independently, is much more believable. The
historical narrative shows the resistance, Bagni confirms the effect on students. He proposes using
the historical artefacts of Borrelli's publication that first clearly laid out the system.

Here we propose an introductory lesson.

After we have introduced the new number, as an invention, with the suggestion that this new invention
did prove usefull ... so it is stillbaround centuries later.

As a teacher consider some of the historical background. Use it to build on the "just make it up"

Then consider Everett, Lockhart for the creation of numbers 'so far'.

Consider Shrödinger, Gleick, Gallileo and finally Penrose for narratives of science, mathematics and
the excitment of the journey. Penrose is about complex numbers but only for a few. Yet keep him in
mind ... one of my fellow pure maths majors was there, in part, because he was given a copy in school.

Consider Weeks, if you want something leading up to Penrose, or really building the spatial imagination
that will help in this topic. Consider how all these ideas were developed, the stories are very real, they
can maie the mathematics come alive.

So ... for some we have them in the formality, providing we do not neglect that. For others, I hope the
historical journey will help justify. Next we looked into the practical, the concrete applications.

Contemporary usage on the cutting edge of physics aside ... that stuff is amazing, but tough.

We looked at were else it is used.

This video lays out the linkage to another step in our exquisite structure ... but matrices are for
another year, now is not the moment to make anothe great leap of invention! (in a better world, perhaps
a syllabus less obsessed with 17th and 18th century mechanics, we might have less calculus and more
modern mathematics in mathematics advanced, maybe even a little something in stage 5)

A lot of the places that complex numbers solve certain algebraic situations requires to much setting up,
too much from outside the syllabus. We have started with the cubic, and these were some of the original
motivations, but mostly they are rather esoteric.

But there are practical science applications.

A lesson can be created by pulling suitable tasks out of a method working with mechanical vibrations.
We would show this video first, then consider one or two parts as an exercise in compkex numbers.

A lesson could also be derived from some of the calculations involved in electrical circuits.
The power of this area is that complex numbers convert a difficult feedback problem involving
differential equations into a much more straightfoward problem in geomerty, within the Argand plane.




====site outline====
initial proposal … what have I left out?
  * overview … note “outline research used” is the report required, with links to resources
      * 1000 words for the “report” as a website is VERY short ... but must be a 1000wd report on research done, with links to resources.
    * issues
      * getting over years of destructive drill: “you cannot take square root of negative number”
      * finding an accessible use-case outside the classroom
    * opportunities (a part of the use-cases, but more ‘pure’)
      * linking algebra, geometry and trigonometry in a way nothing else can
        * complex plane is the cartesian plane **and more**
        * then the unit circle, viewed in the complex plane, opens up an immense vista
      * the equation … <html><span style="font-size:150%;">e<sup>iπ</sup>+1=0</span></html> … links every core number (it in a sense defines complex indices … but I need to think about this statement further, it must be introduced exactly in the right way, at the right time, probably best as we start to look at the unit circle just after multiplication as add angles, multiply lengths and an exploration of many roots of 1 and -1 … in a careful extension of indices … this is a tricky jump)
    * resources … these are the “links” from the “report”
      * lesson excerpt 1
      * lesson excerpt 2
      * lesson excerpt ….
      * Borrelli’s explanation, images, translation, modern form
      * The story of the invention, disbelief, struggle, personal conflict
      * Something re modern physics, something reasonably accessible
      * …...
      * A set of links for the more adventurous, more advanced
        * Penrose … “only for the very game!”
        * ???
    * syllabus links
    * previous knowledge
      * the starting point
      * consideration of the order in the year… esp wrt vectors 
    * where it leads
      * these are extension 2, many will do undergrad maths, a summary of what is next
    * academic reference list
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