The Ghostly Presence of Famous Mathematicians

Feda

Niyaz

Mathew

“Math Projects” Assignment

The Ghostly Presence of Famous Mathematicians

http://www.biographybase.com/biography/Al_Khwarizmi.html

http://www.muslimheritage.com/topics/default.cfm?ArticleID=631

Mohammed Ibn-Musa al-Khwarizmi was born around 780 and died around 850. Not much is known about his life. Yet, during the period of Islamic Dynasty in which Al-Khwarizmi lived, an academy was established in Baghdad that is called the House of Wisdom. In that academy, many scholars preserved most of the Greek mathematics and science that eventually led to the stimulation and reinforcement of learning in Europe.

Alkhwarizmi was one of these scholars. He was known as the Father of Algebra and as the mathematician who brought the concept of zero to the Western world

He founded the concept of the algorithm in mathematics and the word "algorithm" is an English translation of his name. He also made major contributions to the fields of algebra, trigonometry, astronomy, geography and cartography.

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He developed algebra in linear and quadratic equations. His most famous - Hisab al-jabr w'al-muqabala, which means "Science of the completion and the balancing”, is from which we got the name for algebra itself. The Al-jabr wa'l-muqabala began with a discussion of solving equations of first and second degree

The process takes place by using the two operations of al-jabr and al-muqabala.

Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples,

"al-jabr" transforms x² = 40 x - 4 x² into 5 x² = 40 x.

The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.

For example, "al-muqabala" reduces 50 + 3 x + x² = 29 + 10 x to 21 + x² = 7 x

As we see here, the term algebra is only one half of what this process should be called. It was supposed to be called algebra and almuqabala.

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Another intervention was introducing the Hindu-Arabic numerals ( 1, 2, 3, 4, 5, 6, 7, 8, 9) to medieval Europe. The Hindu-Arabic numerals and the place value of numbers were introduced in 500s AD. After around 20 years, Alkhwarizmi, wrote about it in his book but also included an explanation of the use of zero in the same meaning we use these days and which was still confusing to people at that point of history.

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He also used a geometric proof to solve the equation x² + 10 x + 39= 0 by completing the square. He begins with a square of side x, which therefore represents an area x² (Fig 1). To the square he adds 10x and this is done by adding to the four sides of the square four rectangles each of width 10/4 and length x (Fig 2). Fig 2 has area x² + 10 x but from the above equation that is equal to 39. To complete the outside big square he adds four little squares of area ⁵/₂ × ⁵/₂ = ²⁵/₄ each. As a result, the outside big square in Fig 3 has an area of 4 × ²⁵/₄ + 39 = 25 + 39 = 64. Which means that the side of the outside big square is equal to 8. But the same side is of length ⁵/₂ + x + ⁵/₂ so x + 5 = 8, and hence x = 3.

2) Strengths:

- This project is well designed for those students who enjoy the imaginative process of artistic creation. (S)

- It is also helpful for the students to get a historical perspective of mathematics. (S)

- It gives the students the opportunity to learn who were the mathematicians that invented certain topics, hence giving them a human connection to topics learned in mathematics. (T, S)

- The end result of the figure is much better than a poster of some research topic. (T, S)

Weakness:

- The project may take too long, and it may cost the students some money, if they

are not able to use the materials provided for them in their art classes. (S)

- The end result of building a figure may be nice, but it is not so interactive. (T, S)

- The students would have to make the figure and in addition have to do extra research to find out more about the life of the mathematician. It may be too much work. (S)

- With a life-size model, it would be hard to transport the figure and the class space may be limited. (S)

- The project may not involve a lot of mathematical content for the student to learn. (T, S)

- This also may be hard to mark from the teachers view. (T)

T – Teacher’s perspective

S – Student’s perspective

3) After looking at the strengths and weaknesses of the given project we have decided to modify it from building a model of a famous mathematician to conducting an interview with a famous mathematician.

Students would work in pairs to research a famous mathematician of their choice. They would research and then present the information found to the rest of the class in a short 5-10 min interview. While conducting the interview students would pose as an interviewer and the mathematician in question. The student posing as the mathematician would be strongly encouraged to dress up as the mathematician.

As well as the interview students would be expected to make a poster with a brief summary/description of the interview/mathematician researched, which would be posted in the classroom. The poster should contain a graphic of the famous mathematician. Both the poster and the interview should answer basic questions such as what era was the mathematician alive and what were some of their most famous/well known contributions to the world of mathematics?

This project can be done at any high school grade level, however the higher the grade level the more in depth and detail the interview/poster would need to be. The purpose of this project is to acknowledge the mathematicians that have formed the mathematics that we use today and to add a human touch to the math classroom by adding some decorations to the walls. Students will be marked on the information contained in both their interview and on their poster, as well as their creativity and their presentation (a rubric would need to be developed that would be appropriate for each grade level that this project would done at).

The Project’s Marking Rubric

Students work	Mark distribution	Marking
Interview	Organization, presentation and participation of both students	5 Marks
Poster	Layout and attractiveness	5 Marks
Creativity	Bringing new and exceptional ideas	5 marks
Informative content	For both the poster and the interview. Information about the mathematician life (5 marks) Information about the mathematician work (5 marks)	10 marks
Final score		Out of 20 Marks

Response to creativity, flexibility, adaptivity, and strategy use in mathematics

I found the contents of this article to be closely linked with the concept of which we discussed earlier in the term, about relational and instrumental learning. Except that these three categories of creativity, flexibility, and adaptivity are all very hard to teach directly, and they are not so much a teaching style that we have to adopt, but rather skills we as teachers are trying to cultivate in our students. Realizing this, I can’t but help to ask the question, can we teach students to be creative, flexible, and adaptive? Certainly, we can try, there is always hope of course. However, it is almost as if the students have to refine these three skills through active experience.

Creativity springs from inspiration, just like the example the author used in the being of the article, of how Ferit had this epiphany to add 100 and then subtract in order to complete the operation, but imaging the scenario of the teacher prompting the students to come up with the Ferit strategy, all that the teacher could possibly ask is, “Can anyone think of another way of completing this operation, other than the conventional way that we just used?” And at this point, the spot light is on the students and if they are actively thinking about it then someone (in this case Ferit) would be able to come up with the creative answer, there is no other way for the teacher to prompt the answer he/she would want. Or perhaps, we can also let the students get in groups and discuss how they would come up with a creative alternative approach. But ultimately, it is the students who have to actively think in order to reach the desired creative solution. Certainly, after obtaining the desired solution, it would be necessary for the teacher to encourage this type of thinking that Ferit used. I think it is rather difficult to teach these skills, but nonetheless, we can encourage these virtues.

It almost seems that students’ personality or characteristics are manifested when problem solving. Being flexible when solving problems for example is probably easier for someone who is flexible by nature, as a result they can appreciate and attempt other ways of solving a problem. On the other hand, someone who is stubborn or not flexible by nature might just stick to one way of doing a problem, and as a result not explore the other possible ways of solving a problem. This perhaps is also true for adaptivity. But this is not to say that someone who is stubborn can’t solve a problem in a flexible way. But I was thinking that personality does somehow affect how we attempt problems.

Anyway, these we my random thoughts for this article…

Word Problem Analysis

A tree casts a shadow 8m long. At the same time a 2-m wall casts a shadow 1.6m long.

a) Sketch a diagram.
b) What is the height of the tree?
From the grade 9 Math Makes Sense text book, p352

Questions to consider:

1. Is it practical?
It is somewhat practical.

2. Is it memorable?
Not really.

3. Is it solvable?
Yes it is.

4. Can it be interpreted in more than one way?
Yes, the tree and wall might be close enough that one can ask if the shadow of the tree is partially on the wall, it doesn’t mention anything about the shadow being on a flat surface, like the ground for example.

5. Is it strange in any way?
No it’s not extremely strange, but it’s a rather dry problem.

6. How would you extend the problem?
To make the problem somewhat memorable, it would be nice to involve some human aspect to the problem. For example a younger brother and an older brother standing next to each other on a sunny day, and the younger brother wants to know how tall his brother is.

Two Column Problem Solving

Two Column Problem Solving for Cut-Quad-triangle Problem

Click on the link to see the problem I attempted to solve.

Niyaz, Paul, and Michelle were solving this problem. We didn't have a chance to talk about it much together, but our solutions are different...

*If the link above doesn't work, please let me know. Thanks.

Practicum Stories

Story One

One eye opener for me was how tiring teaching was, and I didn’t even get to teach a full load yet! I finally understood what all my high school teachers were telling me when I was a student, “Don’t become a teacher, it’s time and energy consuming…” Despite not listening to my teachers’ advice, I enjoyed my practicum and I felt like I was in my element. It didn’t feel awkward, except for getting to know the new environment. It was something I couldn’t wait to do, so these past two weeks were very self confirming in what I always wanted to do – teach.

Although it was tiring at times, my days seemed to always end with an uncontrollable smirk on my face as I was walking away from the school. I realized that my day was well spent and how fulfilled I was with what I was doing throughout the day. The exciting part about teaching is not what you are teaching but who you are teaching, and I found out that students are the pulse of the world. They are the ones bringing new ideas and energy into the world, so spending my time with them and guiding them in whatever way I could during my day was a fruitful experience. And of course there was a lot to reflect on and think, how would I have done that differently? Overall, I can’t imagine myself doing something else other than teaching for now.

Story Two

Another realization of teaching during my practicum was that being a teacher involved being involved with pretty much all of society in some way. I was very conscious of my actions even outside of school as I was walking down the streets in the neighborhood. Not only was I more conscious about my image as I was walking down the streets, but I felt that as a teacher you are dealing with pretty much all the baggage students bring to the classroom, and unfortunately not all of it is good baggage. The whole community seems to somehow manifest its character in the school setting. It’s a fascinating environment to be in, at times overwhelming, but nonetheless exciting. The school seems to be a different place every day; you never know what might just arise in your classrooms that you will have to address. That uncertainty adds excitement and calls for caution at times, this realization was unexpected to me, but I was glad to come across this realization now rather than later.