EMTH Reflection: Understanding Math

In this reflection I will be taking a look at the assigned reading for this week and give my thoughts on a particular section of the reading based on my feelings and personal experiences.  This reading is from the textbook teaching Mathematics through Problem Solving Grades 6-12 written by Harold Schoen and Randall Charles.

In the first chapter, it gives two reasons why understanding math is important.  The first one is: “[…] understanding a topic ensures that everything one knows about the topic will be useful.  One will remember things when one needs them and will use them flexibly to handle new situations.”  I agree completely with that statement.  For instance, different concepts that are learnt in math one year can come up again many years later.  For instance, in grade four I learnt how to do long division.  We did enough examples that I understood the concept and even though I could use a calculator for long division for most grades after that, I still knew how to do it.  Then, in my first year of university long division came up again but instead of just numbers, there were polynomials.  So, I remembered how to do long division but I had to put my prior knowledge to use in a new situation.  This example demonstrates what the first reason of why understanding math is important.  That is just one example though, the book gives a different one but there are many more than just those.

The second reason is the following: “[…] to set understanding as a central goal.”  I agree with the second reason because I understand how it feels to actually understand something compared to just memorizing a concept or barely getting by on a concept.  When I am learning a new difficult math topic and having troubles with some of the questions from the new topic, it is frustrating.  But then, when I work through the questions and finally come to understand the questions and topic, it is very satisfying.  I have been in situations in Chemistry or Physics though when, in some cases, I have just memorized what to do without really understanding the concept.  It is nowhere near as satisfying then understanding a concept.  In general, it feels a lot better when I can talk confidently and fully understand different concepts than just memorize what to do and keep being frustrated.  Therefore I agree why this is the second reason they give in why understanding math is important.

This was part of the reading really stood out to me because I could recall on my own previous experiences and make connections to what the main points they were trying to get across.

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EMTH Reflection: Folding a Piece of Paper

This week,  in a second year course I’m taking called EMTH, we covered a couple word problems in class.  There was an example called the Paper Strip that we did.  It is from the textbook Thinking Mathematically written by J. Mason, L. Burton, and K. Stacey.  The question is the following:

“Imagine a long thin strip of paper stretched out in front of you, left to right.  Imagine taking the ends in your hands and placing the right hand on top of the left.  Now press the strip flap so that it is folded in half and has a crease.  Repeat the whole operation on the new strip two more times.   How many creases are there?  How many creases will there be if the operation is repeated ten times in total?”

We were then asked to do the problem.  To complete the problem I decided to take a piece of loose leaf and test the question out using the paper because I find that using paper as a visual helps me.  I found out the following pattern from using the piece of loose leaf:

# Of Folds:       1          2          3          4          5

# Of Creases: 1          2          4          8          16

The pattern is that the number of creases is multiplied by two with every fold.  So, to solve the question I added 1+2+4+8+…+256+523=1023.  So, if the operation was repeated ten times, there would be a total of 1023 creases.

I then started to remember that I did this sort of problem in high school.  In Pre-Calculus 20 I remember doing a unit on geometric and arithmetic sequences and series.  I then looked at the Saskatchewan Curriculum online and looked at the outcome and indicators that are involved with the unit to see where this problem would fit under.  It would be under the Outcome for P20.10 which is: Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.  This problem is an example of a geometric series and so I would use this problem in the classroom as a problem for my students before introducing geometric series to them.  I would have my students do the problem to see how my students would get their answer and after discussing how some students got their solutions I would introduce geometric series.  Then after teaching them that there is a formula for these types of problems I would have them do a similar problem and apply the formula.

I really enjoyed doing this problem in class this week because I was able to connect this problem to things I learned in high school and then was able to figure out how I would use this problem in a future lesson plan.