I think this exercise is useful in class as an introduction or reflection exercise to get the students to start thinking and relating math topics/ideas.
Strengths:
- Easy to brainstorm ideas and possibly allow different ideas to be linked together
- Encourages free and active thinking
- Allows students a chance to reflect on current or previously learned ideas
- A different activity – might interest and engage more students especially those whom dislike math may see math in a different light
Weaknesses:
- Students can easily get off topic
- Time constraint on this exercise may give students pressure and prevent a thorough investigation in thinking about a math topic
- Some students may feel bored and left out because they could not think of anything
Tuesday, October 20, 2009
Division by Zero Poem
Multiplication of any number by 0 is fine.
What about division by 0?
0 divided by a non-zero number is fine,
Because we get 0.
A number divided by 0 and we get undefined.
Oh ooo, something is not quite right.
What do we do now?
Take a look at limits????
What about division by 0?
0 divided by a non-zero number is fine,
Because we get 0.
A number divided by 0 and we get undefined.
Oh ooo, something is not quite right.
What do we do now?
Take a look at limits????
Free-Write on Divide and Zero
Divide:
- Splitting things into equal sizes of groups or each person gets the same amount of “something”
- Opposite of multiplication
- A math concept
- Has real life applications for splitting the cost of food or accommodation costs etc between two or more people
- Dividing a fraction by 1 gives the reciprocal
- Can’t divide by 0
Zero:
- A number
- Oval-shaped or a circle
- 0 is in grades and prices, for example 80% and $10
- 0 is on the number line
- Cant divide a number by 0
- Multiplication of a number by 0 is always 0
- Reminds of me eggs
- No one wants a 0 on exams
- Neither positive or negative
- We want lots of 0s behind digit(s) 1-9 in our bank account balance
- Splitting things into equal sizes of groups or each person gets the same amount of “something”
- Opposite of multiplication
- A math concept
- Has real life applications for splitting the cost of food or accommodation costs etc between two or more people
- Dividing a fraction by 1 gives the reciprocal
- Can’t divide by 0
Zero:
- A number
- Oval-shaped or a circle
- 0 is in grades and prices, for example 80% and $10
- 0 is on the number line
- Cant divide a number by 0
- Multiplication of a number by 0 is always 0
- Reminds of me eggs
- No one wants a 0 on exams
- Neither positive or negative
- We want lots of 0s behind digit(s) 1-9 in our bank account balance
Wednesday, October 14, 2009
Reflection to Microteaching in Arithmetic Sequence
Overall, I thought the microteaching in Arithmetic Sequence went pretty well. My group received positive feedbacks. Many of our peers thought the worksheet was a good tool to engage the students in understanding the concept of arithmetic sequence as well as understanding how the formula tn = t1 + (n-1)d was derived. I think my group did a good job in the preparation work and everyone had roughly the same amount of time teaching.
However, there are definitely some things for us to improve on. One thing in particular is time management. I thought our microteaching was rushed and so we didn’t have a strong and detailed introduction and conclusion. Also I thought my group could have spent more time in explaining how the formula was derived since we may have lost some students in the process. Another thing for improvement is to incorporate more group activities. If there was more time, I would definitely let the students discuss in their groups part 3 of the worksheet. Then I would ask the students to share their discussion and go over the derivation of the formula for arithmetic sequence.
However, there are definitely some things for us to improve on. One thing in particular is time management. I thought our microteaching was rushed and so we didn’t have a strong and detailed introduction and conclusion. Also I thought my group could have spent more time in explaining how the formula was derived since we may have lost some students in the process. Another thing for improvement is to incorporate more group activities. If there was more time, I would definitely let the students discuss in their groups part 3 of the worksheet. Then I would ask the students to share their discussion and go over the derivation of the formula for arithmetic sequence.
Summary of Peer Evaluation - Mircroteaching in Arithmetic Sequence
Pros:
- Clear explanations
- Good worksheets for students to follow – step by step and helps to develop good understanding for arithmetic sequence
- Good description of concept
- Good logic
- Definitions were good
- Good preparation work and well thought out
- Class was engaged
- Good questioning skills
Cons:
- Not receptive to questions
- Should speak louder
- Lost control at times
- Should have more organized writing
Suggestions:
- Less lecturing, more activities
- Helping aids such as visuals
- Make eye-contact with everyone
By Jenny, Maryam, Candice
- Clear explanations
- Good worksheets for students to follow – step by step and helps to develop good understanding for arithmetic sequence
- Good description of concept
- Good logic
- Definitions were good
- Good preparation work and well thought out
- Class was engaged
- Good questioning skills
Cons:
- Not receptive to questions
- Should speak louder
- Lost control at times
- Should have more organized writing
Suggestions:
- Less lecturing, more activities
- Helping aids such as visuals
- Make eye-contact with everyone
By Jenny, Maryam, Candice
Sunday, October 11, 2009
Reflection to “Citizenship Education in the Context of School Mathematics”
Before reading this article, I haven’t quite thought of mathematics as having “a role in citizenship education.” However, I do agree with Simmt that most people don’t consciously realize “that there is much in our society which has been quantified” (unemployment rate, the weather forecast, smog index etc). Speaking from my own experience of learning mathematics in high school, I also agree with Simmt that math is usually taught in ways that promote “math as a set of facts, skills and processes” which problems can be solved by algorithm and systematic computational skills; accept “math as either right or wrong” without much critical thinking and understanding of math. Therefore, when I teach mathematics, I would take into consideration the suggestions Simmt made. In particular, I would encourage my students to look at the same problem from various perspectives and investigate the problem. This would “allow students with various backgrounds in math” to engage in a math activity. I would also, engage my students in mathematical conversations which may help clarify misunderstandings of math, help strengthen math concepts and/or allow students to learn from each other. Lastly, I want my students to engage in critical thinking, questioning and analyzing math problems so that they develop the potential to understand mathematics used in society and their role in shaping it.
Saturday, October 10, 2009
Mircoteaching - Arithmetic Sequence worksheet
Sequence worksheet
We have a sequence such as: 3, 7, 11, 15,19,23,27...
Part (l): Sequence Terms:
1. What is the first term? _____
2. What is the second term? _____
3. What is the difference between first and second term? _____
4. Now what is the third term? _____
A. Finding difference “d”:
1. What is the difference between second and third term? _____
2. What is difference between any consecutive numbers in the sequence? _____
Call this “d”.
3. Thus d = _____
Part (II): Relationship between each term!
Each number in the sequence is called a “term” and they are indicated as “tn”; for instance, t1 is first term, t2 is the second term, t3 is third term and so on. Now please note that 3+4=7
1. What is 3 in terms of “term”? t? _____
2. What is 7 in terms of “term”? t? _____
3. What is the relationship between t1 and t2?
Yes that's right t1 +d= t2
4. Now what is the relationship between t2 and t3? _____
5. What is the relationship between t4 and t3? _____
Part (III): Finding the nth term in the sequence/final formula:
Writing each term in terms of t1 and d:
Back to part (II) questions 4 and 5, we found out that
a) t1 +d= t2
b) t2 +d= t3
Now, using a) and b) above, we can write t3 in terms of t1 and d
(Hint: substitute t1 +d= t2 into t2 +d= t3)
t2 +d= t3 and since t1 +d= t2 ------à t1 +d+d= t3 -----à t1 +2d= t3
1. Now using the result above try to write t4 in terms of t1 and d (hint: write t4 in terms of t3 and substitute the above result for t3 and simply the work) what do you get?
2. Again using the result above do the same thing for t5, what do you get?
Finding n th term using general formula:
Writing each terms in terms of t1 and d using the same method that was shown above we see a trend! By looking at the above examples that we did you notice any trend for writing tn in terms of t1 and d?
The above question leads us to the general formula for finding nth term in the sequence. For instance in our sequence that we had in part (I) by using the general formula we can easily find 5th, 6th term, ……, 100th term and so on.
Now using the general formula found above, can you find the 20th and 10th term of the sequence?
We have a sequence such as: 3, 7, 11, 15,19,23,27...
Part (l): Sequence Terms:
1. What is the first term? _____
2. What is the second term? _____
3. What is the difference between first and second term? _____
4. Now what is the third term? _____
A. Finding difference “d”:
1. What is the difference between second and third term? _____
2. What is difference between any consecutive numbers in the sequence? _____
Call this “d”.
3. Thus d = _____
Part (II): Relationship between each term!
Each number in the sequence is called a “term” and they are indicated as “tn”; for instance, t1 is first term, t2 is the second term, t3 is third term and so on. Now please note that 3+4=7
1. What is 3 in terms of “term”? t? _____
2. What is 7 in terms of “term”? t? _____
3. What is the relationship between t1 and t2?
Yes that's right t1 +d= t2
4. Now what is the relationship between t2 and t3? _____
5. What is the relationship between t4 and t3? _____
Part (III): Finding the nth term in the sequence/final formula:
Writing each term in terms of t1 and d:
Back to part (II) questions 4 and 5, we found out that
a) t1 +d= t2
b) t2 +d= t3
Now, using a) and b) above, we can write t3 in terms of t1 and d
(Hint: substitute t1 +d= t2 into t2 +d= t3)
t2 +d= t3 and since t1 +d= t2 ------à t1 +d+d= t3 -----à t1 +2d= t3
1. Now using the result above try to write t4 in terms of t1 and d (hint: write t4 in terms of t3 and substitute the above result for t3 and simply the work) what do you get?
2. Again using the result above do the same thing for t5, what do you get?
Finding n th term using general formula:
Writing each terms in terms of t1 and d using the same method that was shown above we see a trend! By looking at the above examples that we did you notice any trend for writing tn in terms of t1 and d?
The above question leads us to the general formula for finding nth term in the sequence. For instance in our sequence that we had in part (I) by using the general formula we can easily find 5th, 6th term, ……, 100th term and so on.
Now using the general formula found above, can you find the 20th and 10th term of the sequence?
Arithmetic Sequence Microteaching
Bridge:
3, 7, 11, 15, 19, 23, 27…..
Does anyone know what is the 100th or 1000th term of this sequence?
If you don’t know, don’t worry about it. After this lesson, you all will be able to find the 100th and 1000th terms of this sequence. Actually…you can find any term you want to!
There are different types of sequences: geometric, arithmetic and other sequences. In our lesson, we will focus on arithmetic sequence.
Definition: An arithmetic sequence is a sequence where each term is formed from the preceding term by adding a constant (positive or negative)
Learning Objectives:
- Students will be able to calculate and predict terms in an arithmetic sequence where the first term and common difference are known
- Students will be able to calculate and predict terms in an arithmetic sequence where only one of the first term or common difference is known
- Students will be able to write an expression to represent general terms for an arithmetic sequence and be able to apply these expressions to solve problems
Teaching Objectives:
- To teach the students to predict and calculate the terms and common difference of an arithmetic sequence
- To engage students in classroom discussions of arithmetic sequence
- To guide students to formulate an expression for calculating the terms and common difference in an arithetmic sequence.
Pre-test:
These questions will be asked during the bridge phase:
- Does anyone know much about arithmetic sequence?
- Can anyone predict the 100th or 1000th term in the sequence?
Participation:
- Students will be encouraged to participate in class discussions and/or answer questions posed by the teacher
Post—test:
- Students will be asked to solve a challenge problem which will test them on the material just covered
Summary:
In this lesson, we taught students to write an expression for arithmetic sequence. After this lesson, students will be able to find the common difference and any term in an arithmetic sequence. However, there is more to that. Next class, we will focus on the case of calculating and predicting terms in an arithmetic sequence where both the first term and common difference are unknown. In the class after, we will introduce arithmetic series, which is the sum of a sequence. And in the near future, we will also introduce other types of sequence, such as geometric sequences.
3, 7, 11, 15, 19, 23, 27…..
Does anyone know what is the 100th or 1000th term of this sequence?
If you don’t know, don’t worry about it. After this lesson, you all will be able to find the 100th and 1000th terms of this sequence. Actually…you can find any term you want to!
There are different types of sequences: geometric, arithmetic and other sequences. In our lesson, we will focus on arithmetic sequence.
Definition: An arithmetic sequence is a sequence where each term is formed from the preceding term by adding a constant (positive or negative)
Learning Objectives:
- Students will be able to calculate and predict terms in an arithmetic sequence where the first term and common difference are known
- Students will be able to calculate and predict terms in an arithmetic sequence where only one of the first term or common difference is known
- Students will be able to write an expression to represent general terms for an arithmetic sequence and be able to apply these expressions to solve problems
Teaching Objectives:
- To teach the students to predict and calculate the terms and common difference of an arithmetic sequence
- To engage students in classroom discussions of arithmetic sequence
- To guide students to formulate an expression for calculating the terms and common difference in an arithetmic sequence.
Pre-test:
These questions will be asked during the bridge phase:
- Does anyone know much about arithmetic sequence?
- Can anyone predict the 100th or 1000th term in the sequence?
Participation:
- Students will be encouraged to participate in class discussions and/or answer questions posed by the teacher
Post—test:
- Students will be asked to solve a challenge problem which will test them on the material just covered
Summary:
In this lesson, we taught students to write an expression for arithmetic sequence. After this lesson, students will be able to find the common difference and any term in an arithmetic sequence. However, there is more to that. Next class, we will focus on the case of calculating and predicting terms in an arithmetic sequence where both the first term and common difference are unknown. In the class after, we will introduce arithmetic series, which is the sum of a sequence. And in the near future, we will also introduce other types of sequence, such as geometric sequences.
Thursday, October 8, 2009
Reflection to Art of Problem Solving – “What if Not”
How can you use some of these ideas in microteaching:
To be frankly honest, I am not too sure how I can apply the “what if not” approach in teaching all math topics. I think I can use some parts of the approach to help my students gain a better understanding in my microteaching lesson-arithmetic series. For example, given arithmetic series, I can first ask the students to predict terms in the series and calculate the sum of the series in any method they would like to use. Then engage the students in a class discussion and hopefully the students will have different methods of solving these problems. For each method, I can ask the students “what if we were not to use this method are there other ways of solving the problems?” This would be level 2a of the “What if Not” approach and the alternative methods would be level 2b. For each method, I can also pose the question (level 3) “can we apply this method and be able to calculate the sum of arithmetic series or predict terms in the series in a minute or two?” Then in the final level, together as a class we can analyze how applicable is this particular method to every arithmetic series.
Strengths and weaknesses of “What if Not” approach:
Strengths:
- Useful in facilitating thoughts and helpful to developing a more in-depth understanding a mathematical topic
- It may help to (re)connect to other mathematical topics
- It can help us to come up with creative ideas and ways of looking at a mathematical topic and perhaps, can be used as a way to engage students
Weaknesses:
- This “what if not” strategy is REALLY time-consuming
- It is challenging to implement the strategy in a classroom setting
- The list of attributes and “what if not questions” may add to students’ confusion and frustration, especially for those students who only have a basic understanding of a mathematical subject
- This strategy in the end might have a negative effect on students – for example, students may end up hating the math topic they have exhaustively examined
To be frankly honest, I am not too sure how I can apply the “what if not” approach in teaching all math topics. I think I can use some parts of the approach to help my students gain a better understanding in my microteaching lesson-arithmetic series. For example, given arithmetic series, I can first ask the students to predict terms in the series and calculate the sum of the series in any method they would like to use. Then engage the students in a class discussion and hopefully the students will have different methods of solving these problems. For each method, I can ask the students “what if we were not to use this method are there other ways of solving the problems?” This would be level 2a of the “What if Not” approach and the alternative methods would be level 2b. For each method, I can also pose the question (level 3) “can we apply this method and be able to calculate the sum of arithmetic series or predict terms in the series in a minute or two?” Then in the final level, together as a class we can analyze how applicable is this particular method to every arithmetic series.
Strengths and weaknesses of “What if Not” approach:
Strengths:
- Useful in facilitating thoughts and helpful to developing a more in-depth understanding a mathematical topic
- It may help to (re)connect to other mathematical topics
- It can help us to come up with creative ideas and ways of looking at a mathematical topic and perhaps, can be used as a way to engage students
Weaknesses:
- This “what if not” strategy is REALLY time-consuming
- It is challenging to implement the strategy in a classroom setting
- The list of attributes and “what if not questions” may add to students’ confusion and frustration, especially for those students who only have a basic understanding of a mathematical subject
- This strategy in the end might have a negative effect on students – for example, students may end up hating the math topic they have exhaustively examined
Sunday, October 4, 2009
The Art of Problem Posing - 10 Questions
1. What qualities should “good” and “bad” questions have and what are the strategies for distinguishing them?
2. In what specific ways and/or how can problem posing for students help them to better understand mathematical concepts? Evidence for this?
3. How does our past experiences and goals limit our thinking?
4. Why is it that students with prior knowledge to a certain mathematical topic have difficulties devising observational questions?
5. How can we pose problems that will engage and keep students focused on the problem presented?
6. How and in what ways do we see things differently when we pose questions?
7. What constitutes a “foolish” or “nonsensical” question?
8. What are ways to avoid imposing “a context on the situation”?
9. What is the strategy for challenging the given?
10. What are/is the most effective way for posing the “general questions” that applies to a very specific mathematical topic.
Yes, one last question.
11. How or in what ways should we ask students questions so that it is not too overwhelming for them to answer?
2. In what specific ways and/or how can problem posing for students help them to better understand mathematical concepts? Evidence for this?
3. How does our past experiences and goals limit our thinking?
4. Why is it that students with prior knowledge to a certain mathematical topic have difficulties devising observational questions?
5. How can we pose problems that will engage and keep students focused on the problem presented?
6. How and in what ways do we see things differently when we pose questions?
7. What constitutes a “foolish” or “nonsensical” question?
8. What are ways to avoid imposing “a context on the situation”?
9. What is the strategy for challenging the given?
10. What are/is the most effective way for posing the “general questions” that applies to a very specific mathematical topic.
Yes, one last question.
11. How or in what ways should we ask students questions so that it is not too overwhelming for them to answer?
Friday, October 2, 2009
10 Years Down the Road
Hi Ms. Chu,
I am writing this letter to let you know that you were the best teacher that I have ever had. There are several reasons why. They are the following:
1) Your classes are fun, creative and interesting
2) You were a great mentor
3) You were there to help the students when they needed help on homework or when they have questions from class
4) You are considerate of students
5) You always try to engage the students in the lessons
6) Your notes are clear and concise
Lastly, I just want to say that I really enjoyed your classes and I wished you were my teacher for every math class. Yours truly,
Natalie
Hi Jenny,
I just want you to know that you were my worst teacher ever in my whole entire life. Firstly, most of the time you were so bad at explaining the math concepts and left many of us confused. Secondly, you were somewhat helpful in your office hours since there were times that I was still confused after you did your share of trying to explain the concepts. Lastly, I thought your notes could be more detailed and cover a variety of problems we may see in our homework problems.
Your confused student
Comment on my hopes and worries:
In my teaching career, I want to strive to be one of the best math teachers my students will have throughout their education by fulfilling much of the qualities the first letter had addressed. My worries however, are that my explanations will not be clear enough for some students and that my teaching style may not reach out to every type of student.
I am writing this letter to let you know that you were the best teacher that I have ever had. There are several reasons why. They are the following:
1) Your classes are fun, creative and interesting
2) You were a great mentor
3) You were there to help the students when they needed help on homework or when they have questions from class
4) You are considerate of students
5) You always try to engage the students in the lessons
6) Your notes are clear and concise
Lastly, I just want to say that I really enjoyed your classes and I wished you were my teacher for every math class. Yours truly,
Natalie
Hi Jenny,
I just want you to know that you were my worst teacher ever in my whole entire life. Firstly, most of the time you were so bad at explaining the math concepts and left many of us confused. Secondly, you were somewhat helpful in your office hours since there were times that I was still confused after you did your share of trying to explain the concepts. Lastly, I thought your notes could be more detailed and cover a variety of problems we may see in our homework problems.
Your confused student
Comment on my hopes and worries:
In my teaching career, I want to strive to be one of the best math teachers my students will have throughout their education by fulfilling much of the qualities the first letter had addressed. My worries however, are that my explanations will not be clear enough for some students and that my teaching style may not reach out to every type of student.
Subscribe to:
Posts (Atom)