2-column solution problem_one sum

Here is the solution I constructed for the two-column solution problem.

The question is called One Sum, from John Mason's Thinking Mathematically on page 208.

Dave Hewitt: Arbitrary and Necessary

1) What does Hewitt mean by "arbitrary' and "necessary"? How do you decide, for a particular lesson, what is arbitrary and what necessary?

2) How might this idea influence how you plan your lessons, and particularly, how you decide "Who does the math" in your math class?

By Hewitt’s definitions, names and labels are normally arbitrary to students as the reasons things are named and labeled are because of conventions we inherit from ancestors. Names are things teachers need to inform students so that they can share the same information in communication. On the contrary, necessary things are usually workable and understandable by students as students do not have to be informed and can be figured out by some reasons. 

What Hewitt indicated was necessary to me as I agree as well that there are things in mathematics that are “given” and provided in advance for use and discussion. The list on page 4 indicates that there are things that are arbitrary in mathematics. Names, and the meanings of these names, need to be memorised for future usage. To me, what are needed to memorize are the bounded relationships that a name is a “short cut” for things with some given properties.

I always believe mathematicians and mathematics users are lazy in the sense that they tend to use a name to indicate things with similar attributes. Although a name can be changed to some other names, the meaning of the name is not changeable. Thus for arbitrary it means the relationships: both the name and its meaning. 

In my class, I strongly believe it is important to show students what are arbitrary and what are necessary. However, I need to be very careful as some necessary knowledge is arbitrary for some students. In the article Hewitt considered this as this is not the right time to teach these students the knowledge. To me this means that necessity and arbitrary can be internally transformed based on students’ backgrounds and mathematical levels. If a class is full of arbitrary, I may not be surprised why students do not like this class. To me, even though introducing arbitraries is essential, minimizing the amount of arbitrary in class is crucial to maximize students’ learning outcomes.

 

Reflection on Dave Hewitt's teaching video

The counting numbers strategy is wonderful! Dave first knocked the blackboard to make noise to attract students' attention. He then start to count "1" with a knock, "2" with a knock right to the previous knock, and so on. After counting "7", he started to knock the next one without counting the number. The next moment was amazing: students started to count "8" with him. He then knocked the next one and students responsed by "9". After several knocks and responses, he orally called "dot, dot, dot" and knocked the wall with calling "20". Students responsed the next knock with "21". Dave repeated the game until he moved to the back of the classroom and said "1,000,000". He started to knock backward and students quickly figured out the trick. When he knocked a place left to the zero, students started to called "-1". Then Dave started to knock multiple places and waited for response. Students were able to response correctly as well. Then this is the moment he started to introduce plus and minus integers.

Dave provided students long pauses after questioning, and actively performed to engage students. Even if some students had their hands raised, Dave still waited a few more seconds for the whole class to digest the information and make sure everyone has been on the right track. This is a strategy I need to use in my class.

SNAP fair?

Could you, and would you, run a SNAP Math Fair in your practicum high school? Why/ Why not? If you can imagine doing so, how would you adapt the Math Fair to your school and classes, and why?

The purpose of a math fair is to let students feel the beauty of mathematics. To do so it engages students in mathematics with extra-curricular materials that are interesting (not boring) and participatory, encourages students to solve problems in a non- or low-competitive level, and gives students opportunities to present their findings.

I would, and I could run a SNAP math fair in Churchill Secondary. However, I may encourage students to include some problems from pokers, card games, board games, chess, GO, Sanguosha, etc. Most of the math problems in the booklet are math-related, but they are strategic games as well. For some people feeling these problems are tooooo mathematical, games with more strategic features can be used.

By the way, technology can be used in the fair as well. 

So far I am not quite sure how I would run the fair. I will observe more fairs and then prepare my own. But no doubt, a math/strategic game fair is a good idea to involve students.





MOA Math Fair

 Several interesting things I like to mention about the math fair in MOA.

First, students were enthusiastic. Sissi and I tried all the problems with students' invitations. 

Second, problems were well-stated and mathematically enlightful, although some problems were mathematically identical under different scenarios.  
(a) One interesting problem was to find a "winning strategy" of a game: 
      Two players in turn take one or two ships away from the ocean. There are 6 ships in the ocean. Whoever takes the last ship away wins the game. 
      Another similar game was to remove trees from land. 
      All the two games shared the same mathematical concepts. It required students to think backward and understand some number theory. The senarios were interesting as well: the ocean one can be considered as an "environmental" topic, while the second one reflected the terrible result of over-lumbering.
(b) Another interesting problem is a changing-position senario. There are seven places placing along a line (consider it is a number axis from -3 to 3, namely, -3, -2, -1, 0, 1, 2, 3). Three birds occupy -3, -2, -1, while three ducks take places of 1, 2, 3. Birds and ducks want to switch positions, but a bird can only move to a larger number, while a duck can only move to a smaller number. A place cannot be occupied by a bird and a duck at the same time. All the animals can move to an empty adjacent place, or jump above another animal to the next empty place.
Studetns mentioned that this problem had taken them three days to be solved. And there was another similar problem but different animals.
(c) There are 9 objects with identical shape. One of them is over weight. Here is a scale that one can measure if the weights on both sides are equal or not, but it cannot measure the exact value of weight. How many times can you find the over-weight object?
The smallest number of trials should be 2, but the student serving this problem believed it to be 3. The good thing is that I only did similar problems on book and in mind, but I have never tried it with my hands! Wonderful!

 All in all, students were all engaged and the math fair was fun! Enjoy the happiness of using/learning mathematics. This is the main goal.