Saturday, October 13, 2007

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.

Thursday, October 11, 2007

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 2nd 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

Friday, October 5, 2007

"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.

Wednesday, October 3, 2007

"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.

Tuesday, October 2, 2007

"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.