Lesson plan_ introduction of probability

Lesson Plan for Micro Teaching

Shan, Amandeep, and Pari

Subject: Probability of independent event	Lesson Number: 1 of 1
Grade: Grade 8	Time: 15 minutes
Big Idea for the Lesson: Finding probability of independent event.
PLOs:   (SWBAT)                 Data Analysis: critique ways in which data is presented            Chance and Uncertainty: to solve problems involving the probability of independent events
Objectives:             Students are expected to learn about how to find the probability of an event by using the definition of probability.
Materials:            Paper, pencils, fair 6-sided dice (* 24), laptops.
Hook: (about 2 min)             Showing the 1-die situation using MIT Scratch/EXCEL              Introducing the pattern that all numbers increased together. Development: (about 3 min)       Teacher-led:       -    Demonstrate sample space, outcome, event, probability, experiment, and trial. (2 min)                   Class activity: (about 9 min)              Invite students (in pairs) to explore the 2-die situation and record the statistics. (4 min)              Collect group results and draw the histogram (frequency vs possible outcomes). (3)               Solve the posed question. (2 min) Assessment:               Students would be assessed on their participation and work throughout the lesson Closing: (about 1 min):                Students would be advised to write down key points of the lesson on their notebook.

Micro teaching reflection_probability

Here is the statistics on audience' reflections. The scores of each item are low in average with large standard deviation. Basically this means that our teaching was terrible......

Here is my personal reflection (my own reflection is not in the statistics above).

I believe the Excel and Scratch demonstration of what probability is is  interesting and may engage students. This is also mentioned in the comments from audience.

All in all we did not fully explain all we wanted to teach. The materials about probability was not organized well enough that some topics were off-tracked. For example, we should not have explained sample space through talking about set......

Our last activity was not organized well enough. Apparently we three did not make the activity clear enough.

We need to make good improvement in closing, assessment, and future usage. This is mainly because of lack of time and emotional control with such a tough situation.

John Mason: Questioning in Mathematics Education

1) Do Mason's ideas might connect with inquiry-based learning in secondary school mathematics? (And why or why not?)
2) How might Mason's ideas about questions in math class be incorporated into your unit planning for your long practicum?

    John Mason in his article Questioning in Mathematics Education argues the necessity of challenging students in class teaching. Students gain experience when they are stuck and are able to observe situations like “what to do when [they] get stuck” and “what to do when they don’t know what to do”. I personally believe this is a good article demonstrating how to ask an inquiry question in class.

    These challenging moments usually come with teachers’ in-class questions. Mason believes there are two different ways of asking question. The first way is asking as telling: “the teacher has something come to mind and then asks a question which is intended to direct or focus student attention on what has come to mind”. Students may directly response by guessing “what is in the teacher’s mind about this problem” but do not consider what the problem truly is and how to solve the problem. Some examples of this type of questions may be: “what do we do with our rulers”, “what does the definition tell”, “what is the next procedure”, etc. Always, a teacher should realize that a question of this type is asking when students response with “inappropriate or unexpected [replies]” which are different from an “expected, even an intended, answer in the teacher’s mind”.  The second way is called “asking as enquiring” (inquiring?). It appears when the “[teacher] asking does not know the answer and is presumably seeking that answer”. Different from the previous type of asking “listening for an expected response”, this type of asking emphasizes on listening to what students are saying/doing. The differences are that the teacher’s responses may be right or wrong in the “asking as telling” type, while the teacher’s responses can be interested with various reasons in the “asking as enquiry” type. Through this asking type, students are encouraged to share and reveal their thinking by being asked “how do you know”, “will that always be the case”, “what else might that be the case”, etc. Students are encouraged to see/analyze/rethink/discuss their responses from multiple perspectives (interesting in this/that way) rather than from the right/wrong perspectives. 

    Mason even mentions some other ways that can encourage students to share their thoughts in class. Instead of asking questions, students can construct mathematical problems for themselves. They can also construct examples of mathematical objects meeting various constraints. The goal of these activities is to force students to think “beyond the first (usually simple) example that comes to mind” so that they can “enrich their example space while revealing the dimensions they are aware of that can be changed, and even something about the range of permissible change in those dimensions”. Mason even thinks that “a student does not vary something that can be varied or change something in a particular way does not mean that they did not think of it, only that s/he did does not reveal it”. 

    In practice, I consider both types of asking matter and should be used in class. The first type of asking is suitable when clarification is needed, while the second type of asking should become the majority of my asking. All in all, as a teacher, my responses show my attitude toward students’ responses, and what help students are my thoughts that can enlighten them in some ways but not a cold yes/no/right/wrong answer that may limit their feelings of enlightened. Apparently, as a result, I do not want to eliminate the better results by failing to ask good questions at the beginning.

 

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.