Category: Teaching
Published: Friday, 05 June 2015

This is a course offered by the Math Department at Indiana University. The course's intended audience are pre-service teachers that will teach at the elementary school level. This is a -somewhat- rigorous course on the foundations of all mathematics: the whole numbers and their arithmetic. The course also covers integers and rational numbers with their respective operations. The idea is for students to be able to understand why the math works and explain them to others. The textbook used for this class is Elementary Mathematics for Teachers by Parker and Baldridge that follows the Singapore Math method. The book is a nice resource for elementary teachers to have. Next I will provide some resources I think are useful for this course and finally, some advice of my own -based on my experience teaching this course 3 times- before taking this class.

- Activity Notes: all prepared by Serife Sevis (some may contain errors, be careful!)

- Exam 1 (no solutions provided)
- Exam 2 (no solutions provided)
- Exam 3 - [Solutions]

- Divisibility Criteria
- Russian Algorithm prepared by Justin Cyr
- Clock Congruence prepared by Sisi Tang
- Final Exam (from a different but similar class)

The goal of the course is that students build upon their previous arithmetic foundation to understand why the fundamental operations (addition, multiplication, subtraction and division) on numbers (whole numbers, integers and rational numbers) work, and be able to intuitively explain them to others. And here in lies the biggest obstacle for students in this class: a weak arithmetical foundation. Even though students know -in principle- how to add, subtract, multiply and divide whole numbers/integers/fractions, some do not have a practical knowledge, i.e., the kind of knowledge that would allow them to do these operations in a heartbeat.

Since students lack this practical knowledge, some of the more advance (and interesting!) material, such as elementary number theoretical proofs (for example, show that the sum of two even numbers is even) gets lost in the way. This is a clear example of Mathematics as a subject that builds upon itself where previous material is essential to grasp more advance material. If students cannot add properly, it is nearly impossible to fully grasp the beauty of elementary proofs or even be able to understand the algebra needed for said proofs.

My advice: make sure you have a clear understanding of the elementary operations before enrolling in this course. And by that I mean: throw away the calculator and start doing the math in your head. In particular, you should be able to add fractions (I found this to be the more challenging of the operations). The course is intended for you to understand why the operations work and be able to explain them to others. However, there is rarely time to go over the mechanics. You have to take care of that before class. This is a really fun class that sheds light on many things we do routinely in our daily lives. The only way to get the full out of the class is to come prepare for it!

Let me finish by giving you an example of what I mean by coming prepared to class. Suppose you want to get your driver's license. As you may know, you have to do both a theoretical and a practical exam in order to obtain a driver's license. In the practical exam you actually get to drive a car with an approved instructor sitting next to you, taking notes on how you perform. Ok, so while driving the car you would certainly not ask the instructor: "should I stop at this red light?". You are suppose to know what to do -the mechanics- when presented with various driving signals. Instead, you could ask -and maybe you wouldn't, but just for the sake of argument- why the light is red and not blue or any other color. There is a reason why the light is red and not any other color (click here if you are interested). Asking whether or not to stop in your driver's test would be the equivalent of asking why 1/2 + 1/3 = 5/6 in our math class. This you should know and be second nature (like stoping at a red light!). Instead, you could ask why is it that we need a common denominator to add fractions and we can't just add top and bottom like 1/2 + 1/3 = (1+1)/(2+3)?

In terms of the proofs we do in T101, a common question is "why 2**k** + 2**l** = 2(**k**+**l**)"? (here k and l are whole numbers). You should know the distributive property (even if you do not know the name "distributive" that is ok), and in a heartbeat know that **a**(**b**+**c**) = **ab**+**ac**. Immediately. The question "why 2**k** + 2**l** = 2(**k**+**l**)"? is another equivalent to "should I stop at this red light?" in your driving test. If you find yourself asking these type of questions, and by that I mean questions about the result of an arithmetical operation, then you are not prepared for class. Instead, an interesting question would be why the distributive property works the way it does and not in some other way. Why do we have **a**(**b**+**c**) = **ad** + **ac** and not **a**(**b**+**c**) = **ab** + **c**, for instance. (Hint: think of the area of a rectangle with base **a** and height **b**+**c**, now subdivide the rectangle into 4 smaller rectangles). If you find yourself thinking these questions then you are on the right track! The beauty of this course is that after it, you should know why elementary math works and even more, be able to explain it and draw pictures accordingly. Pretty cool stuff!