Assignment #2: Microteaching

Group: Manjeet Mahal, Tam Tran, Kevin Greene

Microteaching: Integer Arithmetic

Math 8

Time: 15 minutes

I. Teaching Objectives:

• To demonstrate to students how to add, subtract, multiply, and divide integers using examples that are relevant to everyday life
• To have students become comfortable dealing with integers
• To have students understand how integer arithmetic works
• To engage students in the lesson

II. Learning Objectives:

• Students will be able to add, subtract, multiply, and divide integers
• Students will be able to understand how the integer rules were formulated
• Students will be able to relate addition and subtraction of integers to the thermometer
• Students will be able to visualize the process of adding and subtracting integers

III. Bridge/ Pre-test: 1 minute

Ask students which of the following numbers are integers: ½, 0.65, 10, squr(8), and -31.

IV. Participatory Activities: 10 minutes

Addition and Subtraction of Integers (4 minutes)

• Tam will start off by explaining addition and subtraction to the students by relating it to the thermometer
• Each time we add 1 cup of hot water the thermometer will go up 1°C.
• Each time we add 1 ice cube the thermometer will go down 1°C.
• During her lesson, Tam will be involving the students by asking them questions and by having volunteers demonstrate on the board using the thermometer to derive the answer. Tam will also involve students by having them discuss other ways they can arrive at the same answer.
• Examples:

1. Adding negative number: 8+(-6) = __
2. Subtracting positive number: 7-3 = __
3. Subtracting negative number: 9-(-4) = __

Instead of taking away 4 ice cubes to get 13°C, what else can we do?

How else can we rewrite 9-(-4)?

This leads to our general statement:

Taking away a negative number is the same as adding a positive number.

Eg. 9-(-4)=13 is the same as 9+4 =13

Now let’s try to solve this question by using the thermometer analogy:

4 + (-5) – (+2) + (-6)

Multiplication of Integers (3 minutes)

· Kevin will start the multiplication of integers by using the hot water and ice cubes analogy

· He will involve students by asking them questions as he explains the topic

· Positive and Positive:

· I’m going to take a container, and fill it to the top with 10 cups of hot water

§ Draw container with +10 inside

o How much hot water would I have if I had 2 such containers?

§ Wait for answer, ask how they got it, draw 2^nd container

§ (+10) + (+10) = (+20)

§ Include multiplying only if stated

o What if I had 3 such containers? 7? 12?

§ Ask how they got the answer, search for multiplying

§ ___ x (+10) = (+___)

o Now we’re going to use a container that can only hold 5 cups of water.

§ Draw container with +5 inside

o How much hot water do I have if I fill 3 of these containers?

§ How did they get the answer?

§ (+5) + (+5) + (+5) = (+15)

§ 3 x (+5) = (+15)

· Negatives:

· So what about ice? I’ll draw a bag, and you guys tell me how many ice cubes I can fit inside it.

§ Draw a bag with the number inside

o So if I had 6 of these bags, how many ice cubes would I have?

§ 6 x (-___) = (-___)

§ Ask for another number of bags

· General:

· Now I’m going to change a few things. Let’s see if you guys can get some answer

§ 4 x (-6) =

§ 9 x (-2) =

§ 5 x (+4) = Note the sign change!

§ 11 x (+12) =

o If needed still, explain how 6 is the same as +6

§ (+6) x (+10) =

§ (+10) x (-4) = Refer above if needed

§ (-2) x (+6) =

o Now let’s take a step back and see if we can notice a pattern

§ + + With reference

§ + - With reference

o So when the two signs are the same, we get a positive. When the two signs are different, we get a negative.

§ Diagram with partitions

o Who wants to guess the pattern if I put up another two signs that are the same?

§ - -

§ (-5) x (-2) =

Dividing Integers (3 minutes)

· Manjeet will discuss division of integer

· She will involve the class by having students participate in finding the solutions to the examples that she provides

· Example #1: Dividing positive numbers

· Find the value of ? for the following questions:

1. 7 × ? = 21

· ? = 3 – ask students how they got this

· Find ? by doing 21 ÷ 3 = 7

2. ? × 5 = 15

· ? = 3

· Find ? by doing 15 ÷ 5 = 3

· Example #2: Dividing Numbers that have different signs

· Find the value of ? for the following questions:

1. ? × 2 = -4

· ? = -2

· Therefore, (-2) × 2 = -4

· So, find ? by doing (-4) ÷ 2 = -2

2. (-3) × ? = 9

· ? = -3

· Find ? by doing 9 ÷ (-3) = -3

· Example #3: Dividing two negative numbers

· Find the value of ? for the following questions:

1. (-3) × ? = (-9)

· ? = 3

· Find ? by doing (-9) ÷ (-3) = 3

2. ? × (-10) = -100

· ? = (-100) ÷ (-10) = 10

· Additional Examples:

1. 81 ÷ 9 = ? ? = 9 Check: 9 × 9 =81

2. (-8) ÷ 2 = ? ? = -4 Check: (-4) × 2 = -8

3. 24 ÷ (-8) = ? ? = -3 Check: (-3) × (-8) = 24

4. (-50) ÷ (-10) = ? ? = 5 Check: 5 × (-10) = -50

V. Post-test: (3 minutes)

· Manjeet will do a little review of what has been taught by asking the students a few questions about integer arithmetic

o Give 2 solutions to the following problems:

· __ ÷ __ = -9

· __ × __ = -13

· __ - __ = -2

VI. Conclusion: (1 minute)

· Students will now have a better understanding of integers

"Adapting and Extending Secondary Mathematics Activities": pg. 106-151 - Five Interesting Quotes

1) Teacher advice for passing examination questions such as this would be to not treat the question as a real problem, but to ignore the context and look for the school mathematics" (Prestage and Perks, pg. 107).
I found this quote interesting because I think that there is no point asking students to use school mathematics to answer questions if those questions do not require math. I think that students should only be asked questions that require math to be answered. I think students find it useless to answer questions using math or a certain mathematical concept when they can answer the question without it.

2) "For most of you the questions can be answered easily (not so the pupils)" (Prestage and Perks, pg. 110).
This quote caught my attention because I think that it is very true. It is very important for the teacher to always remember that something that might seem very easy for him/her, is not easy for everyone. Teachers have to remain patient until their students understand the concept or problem. I have heard of many teachers who would get upset if students did not understand the concept/problems that the teacher thought was easy. According to me, these kinds of teachers are not very good teachers.

3) "You will need to adapt all activities for the short- and long-term needs of the individuals you are helping to learn mathematics" (Prestage and Perks, pg. 130).
I like this quote because it reinforces the ideas that teaching is student-centered. Also, teachers need to always adapt their teaching: sometimes a lesson that works for one group of students will not work for another group. Therefore, teachers have to be flexible.

4) "If we design activities that connect the mathematical content in as many different ways as possible, you can make the syllabus more manageable" (Prestage and Perks, pg. 141).
I like the quote because I think it provides a good suggestion on how to work through the curriculum in a manageable way. There is a lot of material to cover in the curriculum and connecting the different concepts is what a teacher has to do to get through it. It would be impossible to get through the entire curriculum by doing each concept on its own.

5) "Add to the columns by stealing as many ideas as you can from other people" (Prestage and Perks, pg. 143)
This quote stood out for me because it reveals that colleagues, students, parents, and others are an excellent resource for teachers - teachers can get a lot of useful ideas from others. I think it is very important for teachers to know this so that they do not think that they have to make up everything themselves.

"Adapting and Extending Secondary Mathematics Activities": If...then... Questions - pg. 64-105

1) If teachers use the domino style game to practice a mathematical concept then will students get a better understanding of that concept because the game is more interesting then just doing questions?

2) If students rely too much on technology (i.e. graphing calculators and computers) to solve problems then will they not know how to solve problems without technology? What can the teacher do to make sure this does not happen?

3) If teachers promote talking, writing, moving, listening, imagining, and reading in their math classes then will more students have success in math?

What I Found Interesting about Assignment #1 Group Presentations

1) One of the interesting things that I learned from Kevin, Noriko, and James' presentation was that the teacher thought it was easier to deviate from the curriculum in east end schools rather than in west end schools. This is the case because in west end schools parents are more concerned about their children getting the "proper" education so they can do well on tests. I thought this was interesting because it reveals that for many parents having their children be successful on exams is the most important thing.

2) I found it interesting that the student Jason's group interviewed thought that giving good lecture notes is a good quality of teachers. I found this interesting because it shows that students like having lecture notes.

3) I think it is interesting how the teacher Bali's group interviewed takes the effort to phone student's parents and tell them what is expected of their children in his/her math class and what topics their children are covering in class. I thought this was interesting because it shows how much the teacher cares about his/her students.

4) I thought it was interesting how the student Tam's group interviewed liked the teacher putting up all the steps in an example right away instead of letting the students work on it by themselves for a little while. I found this interesting because it shows that some students are just interested in getting answers correct - they like instrumental understanding.

5) I thought it was interesting how the teacher Sang's group interviewed said he did not change his teaching style until his fourth year of teaching. At first his lectures were very structured. His comment really caught my attention because I know that when I start teaching, my lectures will also be very structured. I am glad to hear that with more experience some teachers make their lectures less structured and explore a little more.

"Adapting and Extending Secondary Mathematics Activities" Discussion Questions for pg. 31-63

October 3, 2007

1) What are some of the benefits and limitations for the "removing and adding constraints" strategy for students? Justify your answers.

2) What are some of the benefits and limitations for "removing and adding constraints" strategy for teachers? Justify your answers.

3) Is the "changing the task" strategy referred to in chapter 5 an effective way to promote deeper understanding of mathematical concepts for students? How or how not.

4) Will you use the "changing a task" strategy when you start teaching? Why or why not.

Assignment #1: My Response to Conversation with Teacher and Student

How can I keep students motivated in math class? How can I make my math class creative and interesting, while still following the curriculum? What are some of the students’ views about math? These are some of the questions that I have been pondering over for the last few months. Having a conversation with a math teacher and student gave me some answers to these questions. However, even though the conversations Tom and I had with a teacher and student were very informative, I do not think I will ever get complete answers to these questions because the way math is taught and students attitudes towards math are always changing and every teacher and every student has a different view about math and how to teach it. However, I feel that this assignment was useful in letting us get a sense of how some teachers go about teaching math and what feelings students have towards math. Having these ideas will be very beneficial to us when we start our practicum.
We had a conversation with a math teacher who has only been teaching for a few years. However, I found this to be very beneficial because I felt that she could relate to us and the situation we are in now better than a more experienced teacher because she went through the same program not too long ago herself. I really liked the teacher’s idea of only lecturing for about 10-20 minutes at the beginning of class to introduce a new topic and then leaving the rest of the period for the students to practice what they learned by either doing questions or more creative assignments. I really liked this lecturing method and I want to try it myself because I think that students will not get very bored because the lecture time is not very long. I personally think that lecturing is important because, from my experience as a student and a volunteer, I found that students are more capable of understanding a topic/concept and how to work with that topic/concept when a teacher explains it to them, rather than if they have to read the textbook and learn it themselves. My grade 8 math teacher made us learn everything ourselves – he never lectured. The entire class found this very difficult, and the only reason I did good in grade 8 math was because my dad taught me everything. Moreover, I find the activities that she does with her students to be creative and interesting for the students, and I think that when I am doing my practicum I will do some of these activities with my class. Some of her activities include putting students in groups or pairs and having them do questions which they will either present to the class or teach the class. I think by doing these kinds of activities students get a deeper understanding of the topic because they have to teach the question to the class, and I think that a person really has to understand a topic to teach it to the class. I also agree with the teacher that telling students that math is used a lot in the “real world” and without math students limit some of their opportunities in university is a good way to motivate them. Overall, I found the conversation with the math teacher to be very informative.
The student that we had a conversation with is a lot like some of the students in my high school math classes: he is not very interested in math. I found that having a conversation with a student who is not interested in math was very beneficial because it gave us an understanding of what some of these students think of math and what recommendations they have for math class. I was not very surprised by the student’s response to how he would change the way math is taught. He had an answer to this question that I heard a lot during high school and while I was volunteering: remove all the material you do not use after you graduate from the curriculum. I do not think that the answer that he gave to this question is realistic at all because teachers have to teach what is in the curriculum. I know that when I start teaching I am going to have many students in my class who feel this way, but I will just have to motivate them as much as I can. However, I did find it interesting that this student acknowledged the fact that math is a major part of everyone’s life, such as Elaine Simmt’s article mentioned. I think that it is good that even if students do not like math, they still know that it is an important part of life and it increases their opportunities in university. I was most surprised by the student’s response to the use of technology. In an age where many youths cannot live without technology, this student said that math should be taught using the least amount of technology. He believes that students should not rely too much on calculators and computers because this will result in people forgetting how to do simple calculations. I found that students who I graduated with were less able to do simple calculations, such as multiplication, in their heads because they relied too much on calculators. I am glad to hear that there are now “no calculator” sections to tests to make sure that students are still able to do math in their heads.
Prior to my conversation with a math teacher and student I knew that I had to be a creative teacher who could motivate students because not all students are interested in math. The conversation with the math teacher gave me some good ideas of what kind of activities to do with students to keep their interest and to get them to really understand the topic. Whereas, the conversation with the student gave me a chance to come face to face with some of the feelings that students who are not very keen in math have about the subject. I feel that no matter how much I try there will always be students who detest math and complain about it in class; however, I have to help these students be as successful as they can in the subject.

Assignment #1: Group Summary of Conversation

Tom and I had a conversation with a math teacher who has been teaching for approximately three years. We felt that it would be beneficial to meet with a teacher who has not been teaching for a very long time because we believed a “younger” teacher would be able to relate to us and the questions we had. Overall, we found the conversation with the teacher to be very educational because we learned a lot about the different kinds of activities we can do to keep students interested and how to make sure our students are successful.
The teacher we met with, who we will call Miss D, mentioned to us that she emphasizes both instrumental and relational understanding. Miss D determines what understanding to use based on the concept that she is teaching. She told us that relational understanding is important because it helps students understand the importance of a concept – and if they do not understand the importance then they will not be interested in the concept. Instrumental understanding is also important because it gives the students the capacity to retain the information that is being taught.
Furthermore, Miss D explained to us that introducing a new concept with a 10 to 20 minute lecture followed by review games and hands-on activities is very effective. Some of the activities that she has her students do are questions from the textbook, working in groups to answer questions which will be presented to the class, and having students teach problems to the class. She finds that these activities keep most of the students interested in math. Moreover, Miss D told us that a good way to convey the relevance of math to students and to further their interest is by relating math to life outside of the classrooms. Furthermore, Miss D indicated that it is very important for the teacher to provide extra help for students whenever they need it. Overall, we really enjoyed the conversation that we had with Miss D.

Manjeet and I talked with a student who had recently completed grade 12 math. The result was interesting – as his general attitude towards math is anything but “keener”, we got a realistic version of how math is seen through a typical math student’s eyes. We asked him a few questions that mainly covered the topics of ‘how it should be taught’ and ‘relevance’.
His ideal change to the math curriculum would be to cut out all the material that you don’t use in the future once you graduate. He added that the best part of Math 12 is the graphing, because it is more common sense than anything; conics, on the other hand, is very useless after high school. This response was surprising, as conics had been mentioned the day before during class as a necessary precursor to calculus. It would seem that it is more of a specialty topic for those who want to pursue university level math, but not suited for the average math student concerned with real-life applications.
Our student goes on to discuss the relevance of math to him, saying that math is used every day in our lives and it’s important to be able to figure out problems using math (i.e. car loan interest, measurements for baking); however, if the math is not applicable to real life, then it becomes boring.
We also asked his views on using technology to teach math – he interpreted it as the use of calculators and computers to solve math equations, while we were also inquiring about illustrating/teaching through technology; nevertheless, his response was heartening: “math should be taught with the least amount of technology as possible, otherwise we wouldn’t know what to do if we didn’t have a calculator or computer. Plus if we relied on computers everyone would someday not know how to do a simple calculation.”
While our student’s responses weren’t all that shocking, it was very interesting to hear his take on what math topics were useful. It is good to hear that even though he is not passionate about math, it still holds some significance for him as preparation for real life.