Ghostly Presence of Famous Mathematicians

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

By Feda, Niyaz, and Mathew

 

Reflection on Creativity, flexibility, adaptivity, and strategy use in mathematics by Christoph Selter

I found this article to be simply reiterating and reinforcing the idea of relational versus instrumental learning and teaching. The discussion about creative, flexible and adaptive problem solving, rather than routine problem solving and using problem solving techniques that are suggested, i.e. "use the compensation approach to solve the following..." extends what we have previously discussed about relational and instrumental understanding. By telling students what technique to use to solve a particular problem is limiting to their mathematical creativity on one hand, but on the other it can force them to think in a different way than they may have otherwise and get them to see another way of doing a problem. I don't, however, see this as teaching flexibility, creativity and adaptivity. I see this as being helpful for some students, in particular those who struggle with recognizing what method to use, in which case I think that it is clear that these students do not have a solid conceptual understanding of the materials yet and thus need more help. I feel that if a student can recognize a method, or be able to modify a method, that will help them get to a correct solution then they should be encouraged, and given the freedom, to use it. This is, as I believe, the most important thing.

Something else that I found to be interesting that was mentioned in this article, although not very surprising, was that students with good prior knowledge had a harder time and were less successful in being adaptive than students that had less prior knowledge. This begs the question then as to how important previous knowledge really is when teaching for good relational and conceptual understanding? A method of teaching that I found myself adopting during my short practicum was an exploratory teaching/learning method where students are strongly encouraged to attempt questions with hardly any formal instruction but rather a little bit of guidance towards the direction in which they should be going.

Problems with Problems

Math Power 11 page 512, question 37

A satellite is in an orbit 1000 km above the surface of the Earth. A receiving dish is located so that the directions from the satellite to the dish and from the satellite to the centre of the Earth make an angle of 27º. If a signal from the satellite travels at 3 x 10^8 m/s, how long does it take to reach the dish, to the nearest thousandth of a second? Assume that the radius of the Earth is 6370 km.

In analyzing the above question it seems to be a reasonably good question. It has good and memorable imagery of a satellite orbiting the Earth, along with a nice diagram, not shown here. This imagery is not necessarily the most practical, but it does the job and is not too distracting for the most part, as the question is not about the satellite orbiting. That being said the wording of the problem is not as clear or concise as it could be. In the question we are given all the information required to solve the problem but it is given awkwardly. If it were not for the diagram it may become somewhat unclear as to what the set up should look like. The question gives the angle made by the dish, the satellite and the centre of the Earth in a confusing way, instead it could just read something like, "When the satellite makes an angle of 27º with the receiving dish and the centre of the Earth it sends a signal to the dish." This would be a better way to give the necessary information and the reader wouldn't become confused about "the directions" and also, on a more literal note, the question states that the satellite is orbiting, implying that it is moving and thus the angle would be changing with time, some students could be very distracted by this inaccuracy.

Two-Column Problem Solving

Short Practicum Stories

In the past two weeks I had the chance to observe, learn from, and get to know two teachers, the teachers that I will be working with later this school year. In this short time that we have just spent together I can see that we are going to become good friends and that I will learn a lot from each of them. Something that they both mentioned to me at separate times is that they are "in the business of edutainment", and that this is something that I should seriously think about.

As someone that is new to teaching I am quite concerned about being correct in the content that I am teaching and to be clear. As a result of this I find myself being rather monotonous from time to time as am giving a lesson, so to hear my sponsor teachers tell me that they are in the business of "edutainment" is probably something that is greatly needed. I consider myself to be a funny person and to have a really good sense of humour, but for some reason or another that part of me has a hard time finding its way out when I am giving a lesson. For this reason, among many others, I am greatly looking forward to working with and observing and learning from these teachers more and more.

I can already see that I have learned how to be more animated in the class from my sponsor teachers. For example this past Friday I came to school dressed up in a costume for Halloween, this already made it easier to act the part. As I became more and more comfortable with the students the more I was able to get into character, which by the way was Chef Gordon Ramsay, and the students really responded well to it, I was able to get more answers to questions and more questions themselves from the students and therefore was able to give a much nicer lesson where students were more actively involved in their own learning. Perhaps, Chef Ramsay will have to make a few more appearances throughout the school year.