Final Comments

Thanks for a great semester Susan. I had a lot of fun in MAED 314A. The party was a lot of fun and Thank you very much for the gifts - I love my math gloves. Have a wonderful holiday.

Final Reflection

The Way My Thinking has Developed over the Term

After reading all the entries in my blog I did not see a very big change in my style of writing or in my thinking. From the beginning of the term to the end, my writing and thinking has been fairly consistent. I included my honest opinion in everything that I have had to write all semester. However, after reading over my entire blog, I realized that I have learned a lot in MAED 314A. From our first blog assignment I learned that there exist two kinds of understanding: relational and instrumental. Following the reading of the article and the class discussions we have had regarding this article, I realized that incorporating relational understanding in the classroom is very important. I think that students can only develop a true understanding of the mathematical concept they are being taught if they have a relational understanding of the concept. Nevertheless, after doing the microteaching in assignment #2, I realized that it is very difficult to always teach everything using relational understanding. A balance has to exist between relational and instrumental understanding.

After reading through the peer and self assessments for both microteaching assignments we had, I realized that I did a fairly good job. Generally, I got fairly good feedback from my peers. However, from my self-assessments I realized that there are some things I need to work on. I think that I have to continue working on talking a little slower.

Additionally, after reading my writings for the numerous articles we read for class, I learned that it is important to incorporate a wide range of activities into the classroom. Constantly lecturing and giving students questions from the textbook is not a good way to engage students in math class. Many students find math boring; therefore, it is the teacher’s responsibility to get these students interested in math by introducing them to fun, engaging math activities. One such activity that I thought was very interesting and can make math more creative is writing math poems. The math poem I posted to the blog is the first poem on math I ever wrote in my entire life. Although, I do not think it is a great poem, I had a lot of fun writing it. I think that I am going to do an activity like this in my math class during my long practicum.

Although I do not think my writing and thinking changed from the beginning of the semester to the end, I really enjoyed going back and reading what I wrote on my blog. Rereading everything made me remember all the interesting things we learned during this class. One thing that has changed for me from the beginning of the semester to now is that I am more aware of how to make math fun for students. Therefore, I guess my thinking did change a bit from the beginning of the semester to the end: I became more aware of the importance of making math fun and how to go about doing this. I will try many of the activities we talked about in class or read about in the articles during my teaching career.

Thinking Mathematically: Pages 1-25

Fictional “Case Study” relating to Mason’s Strategies

Seth is a young man in my Principles of Math 10 class. He is a nice boy, but he is sure having a great deal of problems understanding the topics that are discussed in class. Seth attentively listens and writes down notes during class. Furthermore, he works hard during the class on his homework; however, he cannot solve any of the problems. I help him as much as I can during class and he is constantly coming in at lunch and breaks for help. He understands the problems when I go through them with him, but he cannot do them by himself. Recently, I have read a book by John Mason with Leone Burton and Kaye Stacey called Thinking Mathematically. I got a lot of ideas from the first chapter of this book and I think by using some of Mason’s ideas I can help Seth improve in his math. I like Mason’s idea of RUBRIC writing. I think that this will really help Seth in math because by writing “STUCK,” “AHA,” “CHECK,” and “REFLECT” in the margins of his notebook he will be able to see what exactly is causing him problems, what he understands, and what he should revisit. I think by being able to categorize where he is having problems and what he understands, both Seth and I will know what areas in math he needs to spend more time on and what areas of math he comprehends. In addition to RUBRIC writing, I think Mason’s strategy of specialization and generalization will also help Seth improve his math skills. With the specialization strategy Seth will be able to randomly choose examples to get a feel for the problem that he is working on. I think by “playing” around with these different examples Seth will develop an understanding for the problem that he is working on. Additionally, I think that the generalization strategy, which involves detecting patterns, will also help Seth improve his math skills. Once students see patterns many of them start understanding what is going on – hopefully, Seth finds that patterns help him. Moreover I will have Seth constantly write himself notes because by doing this I think that he will not forget any of the ideas that he is thinking about and this will help him solve his problems. I think this will help Seth because many times Seth can verbally say what some of the latter steps in the solution are but when trying to think of the steps that come before these ones, he forgets the latter steps. Overall, I cannot wait to have Seth try some of Mason’s strategies because I think that they will really help him in his math. Seth is a bright young man and I think these strategies will help him express his mathematics better; therefore, improving his success in math.

Math Write-up and Poem

Math Poetry

Time Writing:

Divide:

When I hear the word divide I think about cutting some object, such as pizza, into pieces for a number of different people. This word also results in the image of splitting people into groups to do group work popping into my head. The division symbol also pops into my mind when I think of the word divide. I think back to the days when I learned long division. Unlike my friends, I did not mind doing long division. The word divide also reminds me of the night before my History of Math final in university. I had to learn this new way of dividing. I think it was a method of dividing from the Arabic mathematics. I spent hours trying to figure out how to divide using this method of division. I was very fortunate that my dad knew how to divide using this method. It took me awhile to get this method, but I finally got it. I was so happy to see a question using this method of division on the exam because I knew exactly how to do it.

Zero:

When I think of the word zero, the number zero pops into my mind. I like multiplying numbers by zero because it is easy to figure out the answer. I like when numbers or problems are all to the zero exponent because these answers are very easy to figure out. Also, when I hear the word zero I think of the zero laundry detergent box that used to be on the shelf in my laundry room when I was little. Zero soap is excellent at taking out stains in clothes. I also enjoy saying the word zero because it sounds neat. If a person really thinks about it, zero represents nothing. Temperature freezes at zero degrees Celsius. The word zero rhymes with hero, and Zero to Hero was the name of the first major research paper I ever wrote. This research paper was about how Elvis Presley became the King of Rock and Roll and I wrote it in English 11. It is my most favourite paper I ever wrote.

Poem on “Division of Zero”:

Dividing by Nothing

Division – splitting items among a number of people

Zero – an empty space that represents nothing

What happens when these two words are brought together?

You get division by zero!

But wait, this means splitting a number up into no groups.

How is this possible?

I do not understand.

How do I deal with this problem? Is there a solution?

There is a solution – it is not possible!

Oh, I see, I understand - it all makes so much more sense now.

Teacher-ly Comments on Math Poetry

Math poetry is something that I have never heard of before. I think that it is very interesting and brings out a different side of math. Math poetry takes away the emotionless that many people think exists in math. I thought it was very interesting to write a poem about division by zero. At first, I found this assignment to be very difficult because I could not remember how to write a poem – I have not written a poem since high school and writing poetry was not my favourite part of English. However, once I started writing I found that writing a poem was not extremely difficult.

I think that there exist many positive aspects to having high school students write poems about math. By doing this kind of assignment math students can see that math can be a creative subject – like many other subjects. Students also get a chance to express their feelings about a particular mathematical topic with math poetry. Another positive aspect of math poetry is that it gives students a break from just answering questions and it can result in students enjoying math a great deal more. Math poetry can also help students, who have difficulties in math, express their ideas about math or a particular math topic.

Although math poetry has many strengths, it also has a few weaknesses. Some students might not be very good at writing poems. Hence, these students would find this activity to be very difficult. Moreover, if students are having difficulties in English and math then they would find this activity to be very difficult. Furthermore, some students might not learn a great deal from math poetry; therefore, it might just be a waste of time to do. However, although there are some negative aspects to math poetry, I would love to try it in my math class at least once because it might be very fun for the students and it might help them understand the math more.

I think that math poetry can be used in any grade: grade 8 to 12. I think that students must first know how to write a poem before they are told to just write a poem in math class. However, I do not think that this would be a very big problem because many students have to take an English class from grade 8 to 12. I think that it is possible for students to write a math poem on any topic in the math 8 to 12 curriculum. Therefore, I think that it is possible to use math poetry in any math class in high school and I think that it would be very interesting to try.

Short Practicum Experience

Practicum Experience

Best Practicum Story

My best practicum story occurred on Halloween – the day I taught Math 8 for the first time. I was very nervous teaching this class because my faculty advisor was observing me and I was not sure how the students would behave because it was Halloween. Additionally, I was really nervous because it was my first day using the computer tablet. I practiced writing on the computer tablet the day before, but I was still very nervous. However, once I told the students to take their notebooks out and stop talking, my nervousness subsided and I did not even notice my faculty advisor sitting at the back of the classroom writing down notes. I was successfully able to give the students their notes on mean, median, and mode by having them participate by asking them questions and having them solve examples that I asked them. Moreover, the students were very enthusiastic about what they were learning and they really enjoyed the lesson. I also did a little class activity with the students. In this activity the goal was to determine the mean, median, and mode of the shoe sizes in the class. I had the students come up and mark off their shoe size on the computer tablet. This little task made the students very happy because they were able to write on the computer tablet. For the rest of the hour the students worked on the homework or studied for the test that they were going to have during the next portion of the class. After the students finished their tests we watched Transformers for the rest of the class. Additionally, I was very happy that afternoon because I received excellent comments from my faculty advisor for my lesson. Nothing too crazy happened during this lesson, but it was still a very memorable lesson for me because it was my first time teaching a Math 8 class and both the students and I had a lot of fun during this lesson.

Changes in my teaching as a result of the practicum

Overall, I really enjoyed the short practicum and I found it to be a very educational experience. It was very nice to actually work with high school students and see their personalities – instead of just sitting in the classroom and hearing about how high school students act and think. Once I was in the classroom teaching, I realized that I had to be very clear when giving students instructions. I could not just assume that the students would understand what I was saying, even if I thought I was telling the students something they would already know. Therefore, I learned that I have to be very clear and direct when giving students, no matter what grade they are in, explanations and instructions. Furthermore, after the practicum I learned that I have to be very commanding. I realized that I cannot just ask students to do things; instead I have to demand that they perform certain tasks. I found it very difficult to be commanding; however, as a started to teach more and more classes, I was able to become more commanding. Moreover, I think that during my long practicum I will be able to do this better because I will be teaching more often. In general, I experienced good changes in my teaching as a result of the experiences that I encountered during my short practicum.

Dave Hewitt Article Assignment: Questions 1-2

“Mathematical Fluency: The Nature of Practice and the Role of Subordination”

Question #1:

Throughout his article Dave Hewitt argues that practice is not always the best way to learn a mathematical concept. He believes that learning math by just doing practice questions is not a very good way to retain knowledge. Practicing math by doing question after question helps students remember the concept for a short period; therefore, this type of practice helps students do well on tests. However, this type of practice does not help the students remember the concept after an extended period of time. Hence, students always need to review math concepts they learned during previous years at the beginning of a school year because they are not retaining the information in their long-term memory. Furthermore, Hewitt argues that giving students more questions to do and having them practice a task more often is not practical nor does it help them remember the concept. Additionally, Hewitt argues that children do not become good walkers by just practicing walking on flat surfaces. They become good walkers by practicing walking up stairs and trying to run.

Overall, I agree with some of Hewitt’s argument, but I disagree with others. I think that just practicing something for a short period of time will not allow a person to retain what is being practiced in his/her long-term memory. I think that, with only a little practice, a person can retain information in his/her long-term memory if he/she has a “deep” understanding of what is being practiced. However, unlike Hewitt, I believe that if a person practices something a lot or for a very long time he/she is retaining the information in his/her long-term memory. For example, I think a person will excel at hockey if he/she starts practicing at a young age and practices a lot. However, similarly to Hewitt, I think it is not always practical to do hundreds of questions for practice. Therefore, I think that to retain mathematical knowledge for a long period of time, a person had to have a deep understanding of the topic.

Question #2:

According to Hewitt, subordination is an excellent tool in learning and math learning. Hewitt argues that through subordination a person can learn a skill so well that when he/she carries out the skill he/she does not have to pay much attention to what he/she is doing. Hewitt believes that tools and methods are remembered faster and a better understanding exists of them if they are practiced through subordination to another task. Therefore, concepts and methods that are learned through subordination of another task are retained longer than if the concept or method is the focus of attention. Therefore, math students should be taught mathematical topics and concepts through subordination. According to Hewitt, if math is taught through subordination then the student will remember the concept for a longer period of time and not constantly need to review the concept.

I found Hewitt’s argument about subordination to be very interesting. Before reading this article I had never heard of subordination being used to teach a concept or topic. I think learning a concept/topic through subordination can help a person retain the information for a long period of time. I really liked the way Hewitt used subordination to teach his students linear equations. I would really like to use the method that he used to teach my students about linear equations because I think by using this method students will get a better understanding of linear equations and how to solve them. Moreover, I feel that through the use of subordination in learning, students are not just using memorization – which is a good thing. Overall, I agree with what Hewitt has to say about subordination.

Assignment #2: Microteaching Peer- and Self-Assessment

Microteaching: Integer Arithmetic

Peer- Assessment:

Overall, from our peers, Tam, Kevin, and I received very good comments for our microteaching on integer arithmetic. In general, most of our peers liked the temperature analogy with ice cubes and hot water that was used to explain addition, subtraction, and multiplication of integers. The students, who we presented to, also thought that we spoke clearly and loud enough. Many of them also said that we were encouraging during our lesson. Additionally, many students felt that our overall lesson was very well organized.

However, some students did mention that we needed to work on a few areas. One student mentioned that we should try to think of another way to explain that

Negative # × Negative # = Positive # instead of using patterns. Moreover, many students mentioned that we needed to find better ways to extend mathematical ideas such as providing better activities to allow students to use higher-level thinking and to provide enrichment activities for the keenest students. Nevertheless, almost all the students felt that this was very difficult to do because the 15 minutes we were all given to do this microteaching was not enough time. Hence, as a group we received very good comments, but there are some areas that we need to work on.

For my individual portion of the lesson, I explained how to do division with integers. In general, I received very good reviews from my peers on my lesson. Many of my peers felt that I spoke in a good voice and the verbal and visual communications that I used were clear. Many students felt that I was enthusiastic and cheerful. Also, one student who reviewed me felt that I did a good job including humour into the lesson. One area that I was told to work on was to talk a little slower. One student felt that I talked a little fast; however, the student was still able to understand what I was trying to teach. My peers felt that my lesson and the mathematical ideas used in the lesson were clear. However, I was told to make sure that I clearly explain what I am going to be discussing and teaching before I jump into the lesson. Generally, students were actively engaged in learning during the lesson and they felt that I showed that learners’ active engagement was valued. One student suggested that I pick on some quiet students to answer questions to make sure everyone in the class is involved in the lesson. However, this student understood that this was difficult to do because we were not given very much time. Some of the students felt I did not have a lot of variety in the activities that I offered the learners. Overall, I received fairly good reviews for offering activities that opened opportunities for mathematical inquiry and for involving higher-order thinking in the lesson. However, some students felt that I could have done a better job providing extension questions to provide chances for enrichment for the keenest students. Therefore, I received some very good reviews on my portion of the lesson and I found the suggestions that I was given to be very useful in helping me improve my teaching.

Self –Assessment:

In general, I feel that our lesson on integer arithmetic went very well. Our lesson was organized and clear and each of us had an equal amount of time to present our part of the lesson. I thought we also did a good job keeping track of the time to make sure that we did not run out of time. Moreover, I felt that Tam, Kevin, and I used very clear verbal and visual communications. We talked clearly and loud enough. I thought the temperature analogy that was used to explain addition, subtraction, and multiplication of integers was very helpful and allowed students to have a better understanding of integer arithmetic. According to me, the learners were actively engaged in learning throughout the lesson and we showed that the learners’ active engagement was valued. We did this by interacting with the students throughout the lesson by constantly asking them how to solve the problems we presented, along with other questions. Since I was the last person in the group to present, extending mathematical ideas was my responsibility. I think that in the most part the questions that I asked students during the post-test opened opportunities for mathematical inquiry, allowed for higher-order thinking in the lesson, and provided some chances for enrichment for the keener students. Hence, I think that, overall, the three of us did a good job teaching integer arithmetic in this 15 minute microteaching.

Although I feel that, overall, our microteaching on integer arithmetic went very well, I think that there are some areas that I need to work on. Even though I spoke clearly and loud enough, I think that I spoke a little too fast and went through each example too fast. Also I feel that as a group we did not provide a wide variety of activities for learners. I think that the answers that we asked the students were very good, but, perhaps, the learners would have been more engaged in the lesson if they had a wider range of activities to work on during the lesson. Moreover, like one of the students who reviewed me mentioned, I feel that I did not do a good enough job to make sure everyone was involved in the lesson. I have to make sure that when I am in the classroom I ask the “quiet” students questions to make sure that they feel involved in the class/lesson and to know if they are understanding what is being taught or not. I found this microteaching to be helpful because it made me realize what some of the things I have to make sure I work on before I start teaching.

General Comment:

Although I found this microteaching assignment to be very helpful in making me realize what I have to work on before I start teaching, I do not think that we were given enough time to include extension activities in the microteaching. I found it very difficult to teach the learners the concept, provide them interesting activities, and provide enrichment and higher-order thinking activities. As a result of this, my group was not able to include a wide range of activities in the lesson. We wanted to make sure the students got an understanding of integer arithmetic and in doing this we ran out of time to include a wide range of student participation. If we had more time we would have got the students much more involved in the lesson, instead of just asking them questions. If I had more time I would have liked to play some sort of review game with the learners and perhaps even have them come to the board to answer questions. However, I think that the questions we constantly asked the learners throughout the lesson did involve them during the 15 minutes that we had. Additionally, I think that if I had more time to present I would not have spoken as fast as I did and I would have made sure that even the quieter students were involved in lesson by answering questions. Furthermore, it was difficult to teach our peers the concepts that students in grade 8 are finding difficult. While I was teaching my peers, I was expecting them to get the questions that I asked them correct. Therefore, through this microteaching, I was not able to get a good sense of whether or not the analogies that we used would be helpful in reviewing integer arithmetic with grade 8 students. However, a teacher always has to change the way he/she teaches because every student is different and every class from year to year is different. Although 15 minutes was not enough time to do the microteaching and we were “teaching” students who already knew the topic, I still found it to be somewhat helpful. Through this microteaching, I got comfortable explaining a topic that I will have to explain to high school students. Additionally, I got a lot of ideas of how to explain other high school math concepts to students in interesting ways from the groups that I observed.

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