On Shoulders of Giants: Historical Math Cubes

One of my favorite topics in math is actually HISTORY.

All too often we treat MATHEMATICS like some monolithic and pure subject, devoid of the quirks of human nature.

But as Spock (Star Trek) so clearly stated, "We are essentially irrational beings." And mathematics is a part of it all.

Each year I have my 7th graders do a mini research project I call "On the Shoulders of Giants" (a quote attributed to Isaac Newton). They are asked to research an important historical figure in mathematics and present this information on a 5 inch cube. I do not tell them how to construct this cube so that they have to muddle through the measurements and design. When all the cubes are completed we stack them up in class. It is a fun way to present information.

This year I had my university students do this project. Here are some samples:

They brought their cubes to class and without following a real script, each present the info to the group. It was interesting to hear how much they had found out and been impressed by their historical figure. The feedback I received was that the project was quite a bit harder than they had anticipated. In particular, it was challenging to create a sturdy 5 inch cube and then condense the information to the 6 faces in meaningful ways. One student found herself trying to layout the required information in a logical manner on the faces, which gave her a new way to think about what she had learned and wanted to present.

Overall, a successful project.

Platonic Solids will be covered tonight in my university class

Platonic solids were named after Plato, who was one of the first philosophers to be struck by their beauty and rarity. But Plato did more than admire them: he made them the center of his theory of the universe. 

Plato believed that the world was composed entirely of four elements: fire, air, water, and earth. He was one of the originators of atomic theory, believing that each of the elements was made up of tiny fundamental particles. The shapes that he chose for the elements were the Platonic solids. 

In Plato's system, the tetrahedron was the shape of fire, perhaps because of its sharp edges. 

The octahedron was air. 

Water was made up of icosahedra, which are the most smooth and round of the Platonic solids. 

And the earth consisted of cubes, which are solid and sturdy. 

This analysis left one solid unmatched: the dodecahedron. Plato decided that the it was the symbol of the "quintessence," writing, "God used this solid for the whole universe, embroidering figures on it." 

Plato's description of the universe made a deep impression on his disciples, but it failed to satisfy his most illustrious student, Aristotle. 

Aristotle reasoned that if the elements came in the forms of the Platonic solids, then each of the Platonic solids should stack together, leaving no holes, since for example water is smooth and continuous, with no gaps. But, Aristotle pointed out, the only Platonic solids that can fill space without gaps are the cube and the tetrahedron, hence the other solids cannot possibly be the foundation for the elements. His argument struck his followers as so cogent that the atomic theory was discarded, to be ignored for centuries.

Aristotle's analysis contained a famous error: the tetrahedron does not fill space without gaps. 

Incredibly, Aristotle's mistake was not discovered for more than 17 centuries. Aristotle was so highly esteemed by his followers that they confined themselves to trying to calculate how many tetrahedra would fit around one corner in space, rather than considering the possibility that the great man was mistaken.

Funny how one guy could have SUCH in impact on history.

Teaching Adults

I am currently teaching a math class to prospective teachers. The topics are measurement, geometry and probability. I have no real idea how all three of these topics got placed into one semester course, but I do like the constructivist approach this college class takes in its presentation.

I have taught this class easily 10 times over almost as many years. For the first couple of years I stayed very true to the syllabus that was given to me. I am not a university professor and I did not necessarily feel I had a role in changing the content of the course.

However, with the passage of time, a growth in my professional confidence and the fact that the creators of the course have since retired, I have made some significant alterations to the content. I have rearranged the activities to place measurement of angles before the investigation of polygons and polyhedrons. I have also included a strong dose of Pythagorean theorem so show the students that it is much, much more than a simple collection of letter to the second power. Finally, I have significantly beefed up the probability section to include many different and varied experiments based primarily on materials I got from the NCTM. For the most part, I think I have taken a fine course and made it somewhat more rational.

This course was designed for 15 to 20 students. The homeworks are writing intensive. There is no textbook. Classroom activities were designed for group sharing and discussion. This semester I have 37 students! This is a huge, huge increase and it is due to the fact that the university has had to make some severe cutbacks in course offerings. Since this course is required for several different degrees, it is often oversubscribed. But this year is a crazy one.

The funny thing, though, is that there is a sense of well being in the class. I see happy and above all, grateful students when I walk into the crowded room. There are only two men in the course and nearly everyone there is under 24 years of age. This evening the two men were absent and I got to thinking how this class could have been in some all girls school. Then I was wondering about the positive and potentially negative impacts of this gender segregation on math learning.

Wondering, but no answers. With 37 students, I worry I won't have the time to ever really think about this question again.

This evening's class was about the sum of interior angles of polygons. I defined the terms and then passed out some pre-cut triangles to the group. I then asked them to rip off the corners (the angles) and rearrange them to see that no matter what type of triangle you had, the three angles lined up evenly on a line. This demonstrated that the sum of interior angles = 180 degrees. I know this does not "prove"it in a formal sense, and I said so in class, but for a group of elementary teacher, it is a very simple and powerful demonstration.

We spent the rest of the class making our own tangrams from a blank piece of paper. We followed directions I had found years ago to fold and rip paper to come up the the seven classic tangram pieces. All along, I was having a conversation with them about similarity and congruence; of quadrilaterals and equilaterals; and rectangles vs parallelograms. To the end the evening the students used protractors to measure the angles of the polygons we created, then tried to reassemble the pieces into the original square sheet of paper I had given them.

My ultimate goal is to teach a rigorous math class in fun and intriguing ways. I want to infect these soon to be teachers with a sense that math can be mysterious yet explainable, creative yet powerful, and above all, to set aside their own math-phobic habits (if they have them) and set out to excite a new generation of students.

My Math Methods Syllabus

Syllabus:   EED 784

Curriculum and Instruction in Mathematics (CLAD Emphasis)

This course covers methods and materials for teaching mathematics to linguistically and culturally diverse elementary school students. This includes a review of content of mathematics curriculum, classroom organization, assessment, and guided experiences in schools.

Course Objectives

         • Review the content of elementary education math curriculum

         • Learn new and interesting ways to approach arithmetic concepts and      skills

         • Consider central question: How does a mathematician think and act?

Course Materials

         Binder for Handouts

         Notebook or laptop computer for note taking

         Basic function calculator, ruler and protractor

         NO TEXTBOOK!

Grading Policy (or how to get a A in this class):

         Grading is based on the following criteria

Attendance and Participation	25%
Homework	25%
Math Observation Logs	25%
Final Exam of Math Skills and Concepts	25%

Attendance and Participation:

         The main learning in this course comes from participation in the activities in class, discussing them with peers and reflection. Since we will only be meeting 12 times, missing any class or even part of a class will be detrimental to you development as a teacher.

         Because emergencies occur, missing one class will not affect your final grade. Missing more will be problematic. Avoid this. Tardiness has the same impact. Two late arrivals will count as a absence.

Homework:

         There will be weekly homework. Expect it to take from 30-60 minutes. It will take several forms: reflection, investigation and/or review of materials or methods. Just like attendance, late work is problematic. Avoid it.

Math Observation Logs:

         I am assuming that most of you are working in school settings already. You will be required to keep a math observation log describing the math activities you observe in these classes. These logs need to be typed in a Word document and turned in at midterm and final term. Two entries per week are expected, more welcome.

         In addition to these logs, you will be responsible for typing up notes for the class once this semester. These will be shared with your colleagues and will also count as part of your grade. We will have a sign up sheet in class to determine when you will be responsible for these notes.

Final Exam:

         Since this class reviews general elementary math curriculum, there will be a final exam in which you will demonstrate understanding of the skills and concepts covered in class.

Note Taking:

         Since this course does not have a textbook, note taking becomes more important. Each class session will have one “official” note taker, but you should consider yourself the best single source of information. In addition, I place a high premium of neat and orderly work, so I expect the logs and homework to be typed and correctly formatted. For this reason (and many more) feel free to bring a laptop to class and use it. Hopefully you have access to the Internet for our math investigations and research.  Just resist the temptations of Facebook, emails and their likes.

Some Personal Teaching News

This semester, in addition to my existing, full time job as a 7th and 8th grade math teacher in an independent school, I will be teaching two courses at our local university.
Crazy as it may seem, I feel that the university jobs give me a little equilibrium that I struggle to find when overly focussed on adolescents in a private school setting
The first university course is called: Concepts in Geometry, measurement and probability. It is geared towards teachers getting their supplementary credential, but lots of students take it to satisfy their final math requirement for a liberal studies major. I have taught this course many times over 9 years and rather enjoy it. I have been following a syllabus and course work given to me many years ago by my university mentors, but I am not so beholden to all the materials anymore. This course is offered through the Math department, though, which is a more conservative, traditional environment than the education department (where these courses were developed).
The second university course I will be teaching is a basic elementary math course for the teacher credentialing program. I am very excited by this course. First, I get to fully develop it, following certain loose guidelines. I substituted for this class last semester and found it very fulfilling to bring my years of experience and ideas to a whole new group of teachers (or soon to be teachers). I am also excited to be affiliated with the education department, where I feel my real vocation to be. I would love to see how I could parlay this connection into some sort of teaching position and student teacher supervision. 


But I digress. The course starts in two weeks. It meets once a week for three hours. This is the structure I am planning to use:

Math Warm Up

Problem Solving

Main Activity

Math Game

Reflection

Homework

The guiding question for the semester will be:

How do we act and behave like mathematicians?

Class #1 January 25: History of Math and Math Education

Introductions

Picture representing our feelings about math class

History of Math

Story of ONE

one anecdote from video you want to remember

Class #2  February 3:  What is mathematics and why do we study them?

Class #3  February 10:  Place value

Class #4  February 17: Addition

Class #5  February 24:  Subtraction

Class #6  March 3:  Multiplication

Class #7  March 10:  Multiplication/Memorization:

Class #8  March 17:  Division

Class #9  March 24: Fractions

Class #10  April 14:  Decimals and percents

Class #11  April 21: Basic Geometry: area, perimeter, angles, polygons

Class #12  April 28:  Statistics and Probability

Class # 13  May 5:  Measurement 

-units of measurement

-linear, area

-perimeter vs area

-volume

May 12 Final Exam

What do you think? What should I change?

Math 565: Concepts of Geometry, Measurement, and Probability

An email I just sent out to the students signed up for this course starting January 2010:

Math 565 was originally designed for fewer students than the 30 currently enrolled. It is organized around group activities, often with manipulatives or materials. We are to be located in Thorton 425, which I will have to look at in terms of space. There will be an expectation for group work in class, so attendance will be kept.

I wanted to make a few things clear from the start:

This type of math class is much more about the conceptual understanding of the topics, alternative ways of thinking about math, and JUSTIFICATION. It is different from other math courses because it is not a formula or problem set driven class. That has been confusing to some people in the past. 

1.  This is NOT a lecture course. There will be NO textbook. There will be handouts, homework problem sets and a requirement to take notes. My STRONG recommendation is that you have a three hole binder for the materials and that you devise a good system of keeping everything organized.

2.  All homework assignments will be on the iLearn website through SFSU. If you are not currently familiar with this, you will need to become so. 

3.  All homework assignments will need to be typed and clearly organized. If you explain something via a diagram, it has to be clearly and cleanly presented inside the homeworks. Calculations that are hard to type need to be hand written  clearly, cleanly and organized. Part of your homework grade will be based on these requirements.  No late work will be accepted as it becomes way too complex and unruly.

4.  There will be 10 homework assignments and the best 9 grades will be used towards your final grade (this means that one homework will be dropped).

5.  Because I am not a professor at State, I don't have an office or office hours. But I do take great care in supporting my students through this course. Email to this address is the best way to ask a question. I also meet with students @ other times and places as needed.

My understanding is that the iLearn section for our class will open up in January. Once it is available, I will post the syllabus. 

In case you are wondering, my day job is a 7th and 8th grade math teacher. I have taught for 20+ years and have been a 565 instructor for the past 7 years. I love this course, find it challenging and hope to transmit this enthusiasm to you. Feel free to contact me before class starts with any questions:  Happy Holidays!

yours

Glenn Kenyon






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