"One Two Three Infinity…." George Gamow, 1988, p. 10

My solution for "The Dishes Problem" is as below:

1. The least common multiple of 2, 3, and 4 is 12. I will calculate  how many dishes are used for 12 people. 

2. There are 13 dishes needed (6 rice dishes, 4 broth dishes and 3 meat dishes) for 12 people.

3. There are five 13s in 65, so 12 times 5, there are totally 60 people enjoy the meal. 

Examples, puzzles and histories of mathematics from diverse culture will make Mathematics engaging and fun. Students could learn mathematics and different cultures at the same time. Moreover, they can learn multiple ways to solve the similar problems. 

When I read the book " One Two Three Infinity…." (George Gamow, 1988) I am fascinated by the stories introduced inside. The hand-drawings, photos, characters and plots attract us to read them (please see the figure above ). There is another classic story from Sun Zi Suan Jing which is called " Chicken and rabbits in a cage". It is very interesting. I found almost all kids like to try to solve it. They draw the graphs and learn the principles, and they enjoy it very much. I believe our students will like these puzzles and examples and  benefit from them. 

Project (Sep 28) ---Mathematics & Art

 

This piece of artwork was created by Emily Cannon, and our group selected her works as our project. The inspiration comes from the idea of combining fractals. The necklace and earrings show the famous Sierpinski triangle, which is a typical fractal. The detailed information about Sierpinski triangle is introduced in Megan ( our project group member)'s Blog. In this post, I will focus on square fractal which inspired by Emily's another work shown in below photo. 

This pair of earrings depict two generators of fractals and the generators of their two possible combined fractals. Fractal is a never-ending pattern, and no matter how far you zoom in you will always see the same structure. With this property, I created an artwork called Fractal Rose.  

I used square instead of triangle. Two forms of artistic expression were used---Drawing and Origami. Origami, a Japanese term, is the art of paper folding. By using Origami technique I can transform a 2-D paper sheet into a 3-d paper sculpture to make my rose more vivid.

The outermost square is the initiator. The generator is to connect the mid point of each side of the initiator. Then another square is formed inside. Then we could repeat this process as many time as we can. Similarly operation for Origami. I keep folding layer by layer in the same way. I made 5 layers for my rose. 

What does this project bring to me and why is it important for us to feel the beauty of mathematics in the format of art? As Emily said, " Mathematics and art can combine to make beautiful, elegant, and meaningful pieces! The patterns created by mathematics are often quite lovely and aesthetically pleasing. Additionally, by interpreting mathematics in different forms and arrangements, we can share mathematics with a wide variety of people and learn more about the mathematics behind our art pieces. " I couldn't agree more with her. 

This is the Box fractal. I think we could use it as an in-class activity in high school when students learn the exponential function. 

I use Box Fractal to show them how fast this things can grow and how to generate an exponential function from tables. This activity could be done in small groups. The questions could be:

--How many black boxes in step n, find Nn?

--If the side length of the box in step 0 is 1, please find the side length of each box in step n, Ln?

--The area of the box in step 0 is 1 (1x1),what is the total area of black boxes in step 1, and in step 20?

Firstly, I will invite student to find the pattern of Box fractal. Then I will suggest students to make table, help them to determine independent variable and dependent variable, and encourage them to draw graph.  Finally, they can find the exponential function for this fractal, so they can find the number of boxes in any term. Through this activity students can get an overall understanding on exponential function, and help them to review the prior knowledge they learned  and the knowledge they are learning.

Entry Slip (Sep 25th)---Mathematical understanding and multiple representations

After reading the article " The Role of Representation(s) in Developing Mathematical Understanding", I understand that mathematical representation is a very important tool for understanding and communication, and it can help students to better understand mathematical concepts. I am convinced by authors' argument about the mutual influence between the two forms of representations. The external representation such as number, graphs, equations and charts could help students to gradually build of mental images, and then understand mathematics concepts. On the other hand, internal representation such as mental image and idea could help students  to solve problems through externalizing of mental images. As stated in the article " The development of students' mathematical understanding and representational thinking requires the combination of multiple representations as well as the interaction of both internal and external representations".(p.126)

In this article, authors introduce several external and internal mathematical representations. For example, numbers and sets, blocks and symbols. I think one of the mathematical representations not included in this article is number line. Using number line could be helpful in fostering children's number sense especially when we teach children addition and subtraction. By moving around on the line, students can have the mental image of how many increased or decreased. And number line is convenient to be used in the daily life. When students use ruler, they can use the knowledge of number line to make measurements. When they walk on the street, they can use the idea of number line to estimate the journey. 

In the Math classroom, teachers need to try to use multiple representations to try to facilitate students' understanding of mathematical concepts, and "emerge out of [students]' interaction with the social and material settings of activity. (p.124)

Entry slip (Sep 23)----- My Favorite Teachers and Letter from My Future Students

My favorite teachers

From grade 1 to grade 12 there were six teachers taught me mathematics. My favorite Math teacher is my grade 6 Math teacher. She is my role model. Before grade 6, I was not very good at Math. Since she became our Math teacher, she tried different ways to help students better understand the beauty and power of mathematics. I still remember she used water and 3D shape models to teach us volume of all kinds of 3D shapes and their relationships.  Intrigued by the passion and energy in her voice tones and gestures, I gradually became interested in mathematics. She made me realize that I could be good at Math. 

She was always looking for ways to encourage her students and not hesitated to affirm their every little progress. Her encouragements convinced me that I can be good at mathematics. Since then, I never give up on any math question. I appreciated her help and feel lucky to have her at grade 6. 

My second favorite Math teachers is my grade 10-12 Algebra teacher. My algebra teacher is a man who have  good sense of humor. Even today, I still remember the scene where we were amused by him. His blackboard work was well organized and neat. And more importantly, he was caring about us. He talked with us about our future, health and life, and gave us many good advises. He was like our mentor more than a Math teacher. 

I think I don't have a least favorite math teacher. My least favorite teacher is in other subject. Every math teacher helped me developed my math interest and proficiency. If I am a Math teacher, I will try to involve my students in the lessons and help them to build confidence on this subject like my first favorite Math teacher did to me. 

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Dear Mrs. Zhang, 

         I am ___ in your grade 10 class. I am writing to thank you for  your encouragement and support last year. 

         I appreciate that you always be there to support us before and after each class. Thank you for your patience and understanding. 

         Now, I could help my classmates on their Math homework. I joined a Math club as a peer Math tutor. Without your help I wouldn't have confidence on Math. 

        Wish you all the best!

_____

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Dear Mrs. Zhang,

     I am _____ in your grade 12 class. Recently I am under high pressure to the Math tests. 

     I think I have many questions about what we are learning, but I don't know how to start asking. Sometimes when you walk around in the classroom I feel distracted, and especially when you stop at my position trying to offer help I feel so embarrassing. 

     I need your understanding and help. 

_________

 

Exit slip (Sep 21)----- Group Discussion of the Skemp article

For the first point, one group concluded that "earlier in life it doesn't make sense to do relational". I partially agree with it. Based on my personal experience, I think relational teaching is also needed for some young students. Building connections between knowledge they know and don't know could help those students learn new concepts easier and faster. I do like the conclusion of another group on this point, which is " to integrate relational and instrumental understanding together". Teachers need to observe and analysis students' performance  and determine the adequate  way of teaching, relational or instrumental. 

For the third point, I agree that students could overcome the "Math anxiety" which comes from home through continually being encouraged in the classroom. Teachers play a very important role to help students to recognize and extend their potential in Math, and teach them to be tenacious and courageous to face failures and difficulties. That is what Math is really about, face it, fail it, face it, fail it,…, until succeed.

Math puzzle: The locker problem (Sep 21)

 

A school has 1,000 students and 1,000 lockers. On the first day of school, all the lockers are open.

• Student #1 closes each locker.
• Student #2 opens each second locker.
• Student #3 changes the state of each third locker (i.e. opens them if they are closed, or closes them if they are open)
• ...and so on, until all 1,000 students have had their turn.

After all 1,000 lockers are done, which lockers are open? Which are closed? Why?

To solve this problem, the mathematical thinking I used include: identify assumption, looking for pattern, looking for connection, identify appropriate knowledge and reasoning. 

Problem solving strategies I used include: drawing graph, guessing and checking, working backward.  

Solution: 

1.    Drawing a chart is a good start.

This is my first draft. Using pencil to draw could save your papers, since it is easy to make mistakes.

In this chart, "circles" stand for opened, "crosses" stand for closed.

Then I just follow the rules to complete the chart.

 

2.    Observation.

When I was drawing, I found two things from this chart.

·        Circles and crosses in the upper triangle shows the final state of each locker. Since Nth student will only start to change the states from the Nth locker.

·        I found 1st, 4th, 9th lockers will be closed at the end. I guessed (making assumption) there is something related to perfect square numbers, but I was not sure. I needed to draw a larger chart, then try to prove it.

 

 3.  Draw second chart.

I drew a neat larger chart in order to make sure that I could start to think about this problem from perfect square numbers. I stopped drawing my second chart at 16, another perfect square number.

   4. Build connection ( This step normally take more time.)

I tried to:

 - Find patterns from the chart. Failed, since I couldn't find any pattern in these circles and crosses.

 - Try to think backwards from perfect square numbers. Failed, since I couldn't see any difference from the chart between perfect square numbers and other numbers.

 Then go back to problem, and change direction.

when the state is changed? By checking the 16th row, I found the state of this locker is changed at 2, 4, 8 , and 16. That's right, 2, 4, and 8 are factors of 16, and that makes sense that student #2, students #4, and students # 8 will change the state of the 16th locker.

The state of Nth locker will be changed by those students whose number is the factor of N.  

Since there are only two types of the state of the locker, we just need to know whether the amount of factors of N is even or odd. (Reasoning)

We know perfect square numbers have odd number of factors, and other numbers have even number of factors (factor pairs). (Identify appropriate knowledge)

Then the answer is:  The lockers with perfect square numbers are closed, the rest of the lockers are opened.

First Reading (Sep 9)--Skemp on two approaches to teaching and learning mathematics

 

In this article, Skemp mentions two types of understanding in terms of Mathematics teaching and learning. They are "relational understanding" and "instrumental understanding". I personally prefer relational understanding in learning Mathematics, and I believe with "relational understanding" students could learn how to apply the knowledge and theories in different cases, and the methods of determining and solving problems. However, I know in some circumstances, instrumental understanding is also necessary. For example, if students could understand instrumentally Sine is the ratio of the opposite side of a right triangle to the hypotenuse, they could solve the problem quickly and reliably.  

 

When I was reading the article, three parts of the article resonate me:

1. " The other mis-match, in which pupils are trying to understand relationally but the teaching makes this impossible, can be a more damaging one."

 

Skemp uses "damage" to describe this situation. I think the biggest risk in this case is the children's curiosity will be gradually killed, and they will get lost on the way of pursuing knowledge. In addition, for some children, they need to know the reason why first, then remember, otherwise they cannot remember and solve the problem. I used to help a child with long division. He couldn't do it until I explained to him why we need to start from highest digit, and why we need to do subtraction.

 

2. The example about set. The teacher asked the class to pay attention to write their answers exactly like the example in the book.

 

This is the wrong way of using examples in teaching. Teachers use examples to help students better understand individual concepts. Encouraging students to learn examples could make the teaching more relevant and more understandable to the students. Examples could foster students' relational understanding. How to best use examples is critical. If a teacher asks the students to write their answers exactly like an example, I am sure that students could not get higher level of understanding on the concept behind this example. It's because they cannot connect the information in this example with what they already learned or experienced.  

 

3. "Difficulty of assessment of whether a person understands relationally or instrumentally."

 

Skemp suggests " In a teaching situation, talking with the pupil is almost certainly the best way to find out" whether he understands relationally or instrumentally, " but in a class of over 30, it may be difficult to find the time"(p.12) I totally agree with his suggestion. Communication is always a good choice to understand our students. On the other and, students' understanding could transfer from instrumentally to relationally. We need to give them more time to digest what they have just learnt.