Repetition in math class

Practice makes perfect! or Perfect Practice makes Perfect!

These, and others, are comments I hear as to why we need practice in math classes.  This "practice" can be seen by worksheets, flash cards, and multiplication tables.  I, however, disagree with this notion.

The problem occurs when we look at what the students are supposedly practicing when they complete the above tasks. 

These tasks promote the idea of efficiency over understanding.  I remember back to when I used to administer "Mad Minutes" (basic questions which have to be completed in a minute or less, and their mark was based on only the ones which were answered correctly.  If a student did not answer it, or did so and was incorrect, this student would lose marks).  On multiple occasions, I saw students writing down numbers which made no sense, simply due to the pressure put on them during this timed exam. 

Of course, how can students complete deep math questions if they don't understand their basics?  Well I have a great story against this...and the main character in this story is me!  I still to this day, 30 years old with a bachelor degree in Mathematics, and 3 courses left in my Masters of Mathematics, cannot recite the multiplication tables.  I struggle deeply with my 7 and 8 times tables.  Does this make me a weak math student?  Does this imply I will not be able to answer deep questions?  I would hope not one person would answer yes to either of these questions.  However, the way I used to assess math, through repetition, would never allow myself to succeed in my own courses. 

If it was not for my own mother, who strongly refused to use flash cards at home, I would probably have grown up hating mathematics.

As a math teacher I needed to understand that the beauty of mathematics does not come from memorization of basic facts, but instead the use of basic facts to solve problems which a person may encounter on a daily basis.  Does understanding basic facts allow for students to solve problems quicker? Of course, but should we judge the quality of answer solely based on the time given?

I have given tasks to my students, some of which are upcoming blog posts, where students have chosen to complete multiple questions, of similar types, to come to a conclusion.  The difference in these tasks, however, is the word "choice".  If we allow students to decide how many problems he/she needs to solve, to demonstrate higher level thinking, then I guarantee your students will start to see the beauty of mathematics as a wonderful, sometimes chaotic, subject which is not limited to solving petty details.

Prime numbers and Cicada

Lost on how to find meaningful ways to introduce prime numbers?

Why don't we look at Cicadas 

These are winged insects that evolved around 1.8 million years ago.  What is interesting is that their life cycles follow prime numbers.  They emerge, mate and die quickly during the spring of either the 13th or 17th year.    On these such years, they build an exit tunnel where millions quickly exit and overcome any predators, such as birds, by using their vast numbers to their advantage.

Why every 13 and 17 years?

This way, it will be impossible for any other life cycle to line up with the life cycle of the Cicada.

For example, lets suppose they emerged every 12th year.  This would have animals with life cycles of 2, 3, 4, and 6 years to line up perfectly every time they emerged.

The life cycle being a prime number is, what scientists predict, what has been their greatest strength in their survival for over 1 million years.