Response to Selter's article on "Creativity, flexibility, adaptivity, and strategy use in mathematics"

In this article, the author focuses on three ways of teaching and learning mathematics:

• Creativity: is the ability to invent new or modify known strategies
• Flexibility: is the ability to switch between different strategies
• Adaptivity: is the ability to use appropriate strategies the individual has creatively developed or flexibly selected

The content of this article is quite important to us as educators. Especially the results of a number of conducted experiments on students in elementary are fascinating. Through the results of those experiments, we can learn a great deal about student learning mathematics. Which in term will help us to teach mathematics more effectively. Teaching creativity, flexibility and adaptivity in mathematics for young students is not just to prepare our students to the challenging of high level academic in university, but also to provide our students practical skills of problem solving in their future working environment.

To my opinion, Mathematics is the foundation of all sciences, not just because of the mathematical knowledge as prior requirements, but also the logical thinking which are obtained through the process of learning. Knowing how to apply creativity, flexibility and adaptivity in learning process students will have a deep understanding what they learn and be able to obtain high level academic achievement.  

Problem Solving Assignment From "Thinking Mathematically" book

Problem Solving Assignment From the book ‘Thinking Mathematically’

Group members: Meghan, Eddie, Hung Dang 

Problem “31” page # 179

Description: Two players alternately name a number from 1, 2, 3, 4 or 5. The first player to bring the combined total of all the numbers announced to 31 wins. What is the best number to announce if you go first?

Analysis:

31	30	29	28	27	26	25	24	23	22	21	20	19
W						L						L

18	17	16	15	14	13	12	11	10	9	8	7	6	5	4	3	2	1
					L						L						L

Starting from the number 31, I notice that if I can force the other player announces his number when the total announced numbers at 25 then I can win the game. From 25, the maximum number he can reach is 30,  the smallest number he can reach is 26, within the range from 26 to 30, I can reach to the winning number 31 by announcing any number from 1 to 5

Using the same strategy, if I can force the other player announce his number with the total equal to 19, from 19 he can reach from 20 to 24 then I can announce my number to bring the total to 25

In general, if any player announces numbers when the total are 25, 19, 13, 7 will lose the game or in general 6n + 1 

Strategy

So the best strategy to play the game if I go first is starting with 1 because the other player can not bring the total up to 7, the maximum number he can reach is 6. And then my next move is to bring the total to 7

Modified Problem

We can modify to obtain another problem such that: "Two players play a game which has the rule as following: there are 27 balls, each player can pick up 1, 2 or 3 balls alternately. Who picks up the last ball will lose the game. How do you play to win the game?”

Comments: 

This problem is interesting and the key point to win the game is to find out numbers which are called 'losing numbers' in order to come up with the winning strategy. Those losing numbers will be written as a pattern. Those problem look complicated at first but if we understand one particular problem then we can easily solve another ones

Problem Solving - Unclear problem

From Math Power 10, Section 3.9, Factoring topic, page 130, question# 21

Question: 
Factor if possible 9x^2 - 15x - 4, the answer in the text book is not possible  

To my opinion, the question is unclear when students are asked to find the solution with integers only. Actually, we can factor this expression, as long as the determinant b^2 - 4ac > 0, the corresponding equation has 2 distinct real roots, let call them x1 and x2 then we can factor the expression as   a(x - x1)(x - x2).

In this case the determinant is (-15)^2 - 4(9)(-4) > 0, so the corresponding equation has two different real roots, they are irrational numbers, so we can factor this expression as 9(x - x1)(x - x2)

This section, to my opinion, gives students a misleading concept about factoring a quadratic expressions since the irrational roots of quadratic expressions are ignored in the process of factoring.

My Practicum Story

My Practicum Story

If someone asks me what is the most important day during my practicum then I would say that is the day that I do my teaching lesson. And until now ..I still don’t believe that I didn’t show up in the class room at the day I supposed to give my very first math lesson in my teaching career, of course if I still want to pursuit this career. This event was just like a wake up call to my conscience, being a teacher is just not standing in front of the class room and teach a lesson. Teaching is a lot more than that, the lesson I have learned is to prepare well in advance, be responsible to what you do and be a good model

And here is my full story …

John Oliver high school is my practicum school, because I have two teaching subject areas: Computer Science and Math so I have two SA’s (Sponsor Teacher) and my schedule during the practicum was set as following: the first two blocks I went to my IT sponsor teacher class, the last two block after lunch I went to my Math sponsor teacher class room. Unless I went to another class to observe another teacher, otherwise I sat the my sponsor class room and tried to help students as much as I could during activities in class. Things went well in almost two week practicum until the day I supposed to give a Math lesson in grade 10 class. That was the day that I was waiting for a long time because math is my favorite subject and I was eager to do the best I could during that class

Somehow, in my mind I thought that I would teach in the third block as usual because that was the time I supposed to spend with my math sponsor teacher…But in fact, my lesson was in the second block. Which meant I showed up in the class when the lesson was over, and of course, the sponsor teacher had to do the job which was mine and she didn’t prepare for that..

What happened next? A lot of explaining going on ..saying sorry …and I was really upset about myself