In a previous post we began to discuss the idea of a basis in mathematics. The examples given in that post are finite-dimensional vector spaces, and in this post we are going to generalize them by giving some examples of infinite-dimensional vector spaces. But before we do this, let’s play with some motivating examples not … Read more
One of the most important tools that mathematicians and scientists use to cope with the daunting complexity of the world goes by the name of reductionism. That is, one first identifies the key parts of a complex system, then one strives to understand the parts, and finally one strives to understand how the parts fit … Read more
A good question and some interesting answers at MathOverflow. Hat-tip to Kareem Carr. (This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)
Once upon a time, an instructor (whom I shall call Professor “A”) went on sabbatical leave. As a result, another instructor (whom I shall call part-time instructor “b”) was called upon to teach Linear Algebra III, which was normally taught by Professor “A.” All 16 of the students who attended Linear Algebra III had successfully … Read more
My basic attitude towards memorization in mathematics education is to memorize the absolute minimum, but memorize that minimum perfectly. Part of a mathematics teacher’s job, in my view, is to guide students to understand what this “minimum” is, and then encourage them to memorize it, helping them to find effective means for memorization. Effective means, … Read more
I wrote about the power of abstraction earlier, and I just came across a beautiful passage on the same subject by one of my favourite authors, the prolific and master expositor, John Stillwell (see also here). It’s taken from the preface to Elements of Algebra: Geometry, Numbers, Equations, Springer1994: Algebra is abstract mathematics — let … Read more
Thomas Wolf, who is a professor in the Brock University Mathematics Department, has created a wonderful series of mathematics contests for elementary-school students: the Caribou Mathematics Competition. Contests are available at each of three Grade levels: Grades 3/4, Grades 5/6, and Grades 7/8. The next contest date is 16 February 2011, so if you are … Read more
The more you understand, the less you have to memorize. A good example is trigonometric identities, of which there are quite a number. Should a student memorize trigonometric identities? Well, at first, it is probably wise to memorize a few of them. Part of a teacher’s job is to help students identify what is essential … Read more
My father-in-law runs a fruit farm, and made an interesting observation one day about weather prediction: If the weather report says 40% chance of rain, it never rains. If it says 50%, then maybe it rains, maybe not. But if it says 70%, then it rains for sure. This may seem self-contradictory, but it raises … Read more
One of the obstacles to learning in mathematics and physics is the fact that there are many closely related concepts, although logically distinct. Additionally, the same structures (logical or mathematical) occur over and over again in our mathematical models of the world. For both reasons, the same word is sometimes used to mean several different … Read more