Unit Plan

http://docs.google.com/fileview?id=0B5xuCct142nkZjBjYzEyNzMtODNhMS00ZDBjLThhMjUtOGE1ZmNhYjY0YjUy&hl=en

Math Project

1) Project Sketch

2) Assessment: strengths/weaknesses of project, evaluation criteria, affordances & constraints, modify/adapt/extend project

3) Project idea: Divisibility test of numbers from 1 to 10 in base 10
An enrichment project

Grade Level: 10-12
Purpose: The students will gain better understanding of the divisibility of numbers in base 10 and the students will be challenged to think mathematically.
Description of activities: The students will first make conjectures about the divisibility for each of the numbers from 1 to 10. Then they will need to reason to themselves about why or why not the conjectures work.
Sources: N/A
Length of time project will take (in and out of class): 2 classes, 2 weeks
What students are required to produce: A poster or pamphlet of the students’ own conjectures on the divisibility test of numbers from 1 to 10 in base 10.
Handouts, graphics, etc.: N/A
Marking criteria: Number of solutions; Organization; Appropriate conjecture (i.e. not just saying to plug the numbers into the calculator…)

by Jenny Chu, Alan Hsueh, Sara Wang

Milkcrate Problem

Memorable Moment from Practicum

During the practicum I had the opportunity to teach a PreMath 9 lesson. The topic I was teaching was Powers. I did my lesson plan and created a worksheet for the students well in advance before the lesson. Thus, I had the opportunity to receive feedback from my SA and FA as well as revising my lesson and worksheet. My biggest concern and worry is not being able to manage the class. I also thought I may have lots of difficulty in getting one student to not talk excessively or do the worksheet without complaining loudly to the class. I was also concerned about the possibility that another student would talk rudely and give me negative attitudes. My worries and concerns quickly washed away as I was able to help the students understand the benefit of learning Powers. In my bridge activity, I had the students write out the expanded form of 2^2, 2^3, 2^4, 2^5 and 2^6 and complete the multiplication. Then I asked them, if they would always want to write out repeated multiplication expressions and the answer I get from everyone is a “no”. So no one complained or made negative remarks like “there is no point in learning this” or “this is stupid.”

My goal of the PreMath 9 lesson was to make the material easy enough that everyone can understand and learn about powers without much difficulty. Overall, my lesson and plan went well since the major difficulty students have is with the tricky problems on the worksheet. The students all worked hard throughout the class and there wasn’t excessive talking or inappropriate student behaviors. In general, my first lesson went very well and better than I had imagined. It was definitely a positive experience that I can look back at.

Reflection on Free-writing Exercise

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

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????

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

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.

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

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.

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?

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.

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

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?

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.

Reflection to Dave Hewitt’s film

There were several things that intrigued me in Dave Hewitt’s film. Firstly, I was impressed by Hewitt’s way of introducing simple algebra. To be frankly honest, I haven’t seen a math topic introduced in such a creative, interactive, fun and exciting way. Secondly, I also liked how he manages his class. In particular, I liked how he was able to engage his students throughout the lesson in an innovative way such that he was able to encourage his students to think about the same problem in different ways. Thirdly, I liked how he was able to get the whole class to participate in the lesson by getting them all to speak out the answer because it promotes the students to self-check their answer when they hear an answer very different to their own. It was also great that no student was put on the spot to feel embarrassed since the whole class was engaged in shouting out the answer. Lastly, I really liked how he taught the lesson by starting simple and gradually building up the concept and introducing x as a simple introduction to algebra. Thus, I will definitely consider incorporating some of his innovative teaching techniques into my teaching.

Reflection to Battleground Schools – Mathematics Education

Summary:
There are two main stances in mathematics education and they are the progressive and the conservative views. The progressivists’ goal is to understand mathematics relationally whereas the conservatists’ goal is to understand mathematics instrumentally through fluency.
The Progressivist movement occurred from 1910-1940 and was “led primarily by John Dewey.” This movement aimed at stirring away from the traditional instrumental learning of “meaningless memorized procedures” to focusing more at understanding “mathematics through activity and inquiry.”
The New Math movement in the 1960s was an initiative led by the School Mathematics Study Group (SMSG) and was “strongly influenced by the Bourbaki group of mathematicians in France.” The movement introduced a new curriculum “to be taught throughout the K-12 system” in an attempt to educate future scientists and intellectuals. New Math supporters “combined mainly conservative with some progressive features,” however, the movement quickly came to an end when it was “denounced in the press and popular media”.
The National Council of Teachers of Mathematics (NCTM) developed its own standards program after anticipating “imposition of national curricular standards in the US.” The NCTM was influenced by progressive and constructivists approaches. From the mid 1990s to present day, there have been continuous “battles over mathematics curricula and teaching methods.”

Response:
It was nice to be able to get some background knowledge of how mathematics education came to be today. Although, I am not in favor of any particular movement, I think there is nevertheless something for me to benefit from. In particular, I think it is beneficial for the students if they can learn math instrumentally and/or relationally; as I believe it will help to consolidate the students’ understanding when math is taught from different perspectives. Also as a future teacher, I think it is crucial that I am flexible, open-minded and well-prepared for my classes so that when a ‘new math program’ is introduced my students aren’t left confused in the dark.

Self-Reflection to Teacher and Student Interviews

In my group’s interview with two high school students, it didn’t surprise me that one student said “no one really likes math in high school.” Further, from my encounter with many university students, most people either didn’t like math or had negative experiences with math. The interviews however, did shed some light on ways students would like teachers to teach math. Both students emphasized the importance of interesting the students and making “the class a little more fun and keeps everyone from NOT falling asleep” by allowing group activities. Therefore, as a teacher candidate it’s important that I am open to students’ suggestions and try my best to create a comforting classroom environment that involves plenty of excitement and fun group activities for students to engage in.
In the interview with the math teacher, however, she said she unfortunately “wasn’t accommodating other learners especially those who have learning differences.” Therefore, as a future teacher, I shall work towards being flexible and adaptive in my teaching so that I am able to accommodate or able to reach out to the various types of learners whether it is through simplifying a complex problem into simpler parts or using real life examples to help students relate to math. Lastly, I think it is also important that I am an approachable teacher so that my students will not hesitate to come and ask me for help.

Summary to Teacher and Student Interviews

In compliance with BCCT standards, “educators will engage in career-long learning”. Similarly, as teacher candidates and future educators we will continue to learn from our professors as well as from teacher and student interviews. In this interview, we have in our group of three collaboratively created a list of nine questions we most wanted to ask high school math teachers and students. From our interview with a math teacher and two high school students, we learned of the various resources and styles of teaching that we can make available to students to facilitate their learning in math.

Our first two interviewees were high school female students in grades nine and twelve respectively. We asked these students if they have troubles doing math, what would they do and why? The grade nine student said “if I have problem[s] doing math, I will go and ask my teacher” since she felt her teacher was approachable. The second student said that she will ask her friends because “discussions with her friends are good enough”. As teacher candidates then, we must recognize that our availability and approachability for students plays an important role in helping them learn math.

We also asked our two interviewees which area of math they find difficult and how their teachers can help them learn the topic better. One student has difficulties with volumes and angles and she prefers to learn “by sitting and taking notes” whereas the other student has difficulties with trigonometry and prefers to learn math “visually.” Thus, as teacher candidates we must keep in mind that each individual’s ways of learning is as different as the colors of the rainbow.

Our forth question was what can teachers do to motivate the students to learn math. In general, both students said it was hard to motivate them in a subject they disliked. However, they suggested that their teachers can try to interest them by making “the lesson a funner [group] activity.” Ironically, this ties into our last question, in which we asked the students how they would define a good math teacher. The grade nine student described “a good math teacher [as] someone [who] makes the class a little more fun and keeps everyone from NOT falling asleep.” On the other hand, the grade twelve student said a good math teacher “shoudn’t be monotone” and should make her “feel interested in what he/she teaches.”

In our interview with the high school math teacher however, we asked her what is the hardest thing to be a good math teacher? She replied “the most difficult part of being a math teacher is getting the students to get excited about Math.” What is interesting, however, is that she told us “some students say that they were once interested in Math” Therefore, as teachers it’s important that we try our best to teach math in fun and interesting ways.

This led into our second question inquiring any teaching advice(s) that our interviewee can give to teacher candidates. In her response, she told us we should avoid getting “disillusioned with teaching especially if the students are unmotivated.”In other words, it is important for teachers to overcome any teaching difficulties or harsh criticisms that she/he may encounter in the profession by approaching these problems optimistically and continuing to learn and develop professionally. For example, if a student is distracted, we shouldn’t take it personally and lose our initiative in teaching. Rather, we can refocus students back to the math lesson by “asking the distracted student to answer the question.” Distracted students also serve as a valuable sign in how well teachers are engaging their students in the class.

Thus, we asked the interviewee which is/are the most effective techniques she has used in teaching and why? She replied “I try to use real life examples…so that the seemingly abstract concepts can be more concrete.” This method however, “works only most of the time.” The reason for this as reflected by the interviewee is that she “wasn’t accommodating other learners especially those who have learning differences.” She also noted that “it was challenging to NOT teach the way she learned the concept.” Therefore, as teacher candidates, we should be flexible and adaptive in our teaching so that we are able to accommodate most if not all our students. Another effective teaching technique the interviewee had used all the time was “simplifying a complex concept using simpler examples.”

In conclusion, the interviews with the students helped us gain insight into their expectations from their math teachers and the methods that can be used to facilitate their learning in math. Generally, a great math teacher should be flexible and open-minded in his/her teaching to create a comforting classroom environment that involves plenty of excitement and fun group activities for students to engage in.

by Jenny, Maryam, Candice

Interview with High School Math Students

1. If you have troubles doing math, what would you do and why? Would you ask help from your teacher and/or classmate or go to problem sessions? Why or why not?

Student 1 :
I usually ask friends if I have troubles with math, and vice versa. My friends and I would check math assignments together. So when somebody has a question, we’ll discuss about it together. Therefore, it’s not necessary for me to go to problem sessions afterwards because discussions with my friends are good enough.

Student 2:
If i had trouble doing math i would probably go and ask my teacher for help at the end of class. Most kids if they dont understand it they will give up so you have to encourage them to come see you or the teacher after school or at anytime for help. Another thing that i might do is get a tutor that always helps.

2. What area of math do you find the most difficult? Is there anything your teachers can do to facilitate your learning better in this area?

Student 1:
I am not good with graph/sin/cos/tan.
I find myself understanding better if I can logically understand or know why and how things (in this case, math) work. Because trigonometric function is my weakness; it would be better if I can find someone who can explain to me again and again about how sin/cos/tan work.

Student 2:
Volumes and angles seemed a little hard to me and i dont really think there is anything more the teacher can do than to teach us it. i cant remeber i finnished all my math courses so its been a year since i have been in a math class but from what i can remeber; there wasnt much the teacher could do maybe t was because i wasnt listening during the class of explaining angles and volumes i dont remeber but i think the teacher did there share.

3. How would you like to learn math. For example, using diagrams, analogies, computer games etc.

Student 1:
maybe with visuals
I’ve never learnt math with computer games so I’m not sure about that one.

Student 2:
i think the good old way of sitting in a class and listeni8ng and taking notes is the best way its just that the etacher has to make it a fun activitie with every lesson for everyone to pay attention. otherwise everyone gets bored and start fdropping out of that class or they just dont get it.. every lesson has to be a little dummed down. pretend like you are teaching 5 year olds.

4. How can your teacher motivate you to learn a subject that you are really having trouble with or just hate it altogether.

Student 1:
This is a hard question. If I don’t like something, then it is very difficult for me to deal with it all the time. I can only say to make me/myself start feeling interested about it; hopefully to find some fun in it so that I don’t dislike it so much.

Student 2:
makeing the lesson a funner activity, maybe putting us into groups

5. What are your exceptions from a good math teacher or how do you describe a good math teacher?

Student 1:
A teacher who makes me feel interested in what she/he teaches.
Or else I wouldn’t be paying too much attention about what she/he’s talking about, and that eventually results in skipping classes and/or failing tests.
Also, a good teacher to me shouldn’t be monotone while teaching class.
Even if I do like the course, it will still make me fall asleep.

Student 2:
once i had a math teacher that was a really funny guy and everyone seemed to like him alot, he tried really hard to keep us tooned in. the problem with a math class is that no one really likes math in high school well not everyone but most people. i think a good math teacher is someone that makes the class a little more fun and keeps everyone from NOT falling asleep. But that will come after years of being a teacher and being very comfortable with everyone. in the beggining you will feel a little intimidated and the kids will understand that you are new to this and you might be a little awkward. so its all good :). giveing too much homework is never a good idea try to make them learn most of everything IN CLASS. becuase knowing teenagers they barely want to touch homework.

Interview with a High School Math Teacher

1. What is the hardest thing to be a good math teacher (or teacher in general) speaking from your own experience?

Answer: I believe the most difficult part of being a math teacher is getting the students to get excited about Math. Most students I've taught are either bored with Math or just plain uninterested. What is interesting is that some students say that they were once interested in Math, but obviously, somewhere in their school life, something, or someone made them lose interest.

2. The thoughts you have about teaching, is it the same after you started teaching? What stays the same and what changed?

Answer: I used to think before I started teaching that if I try to teach any grade level in some creative way in Math, that I can get them interested again. I thought that if I use real life experiences, that I can help them relate to Math in a more realistic way. Unfortunately, I found out when I started teaching that this method, works only most of the time - still more often than not. I later found out that I wasn't accommodating other learners especially those who have learning differences. Not only was it very challenging to get these students to understand Math, but it challenging to NOT teach the way I learned the concept.

3. From your teaching experience, which is/are the most effective techniques you have used in teaching and why? Also, at which grade level is this most effective for?

Answer: I believe that one of the reasons why the students are finding Math very difficult is because they don't understand how valuable the concepts and how they can be used in real life. So whenever I teach, I try to use real life examples - applications if you will - so that the seemingly abstract concepts can be more concrete. Although, as I mentioned, this method does not work, there are still students whom I've taught who were able to understand the concept. Another method that I use all the time is simplifying a complex concept using simpler examples. For example, if a student doesn't understand how to simplify radicals, I use simpler examples such as simple fractions and use cross cancellation, etc. The steps are basically the same with the simpler examples but the numbers being used are just more complicated.

4. What teaching advice(s) or tip(s) can you give to teacher candidates? What would you do to draw back students’ attention when they become distracted?

Answer: I haven't been a teacher for that long. But I can definitely say that once I started teaching, it was easier to get disillusioned with teaching especially if the students are unmotivated. But that's exactly we should avoid. As teachers, we should aspire to continue learning so our skills don't stagnate. If we let our disillusionment get the better of us, we risk losing our drive and impetus to continue learning and continue developing professionally. When students become distracted, I usually try to refocus them by making them stop and asking the distracted student to answer the question. 5. What type of techniques do you use to teach math? Instrumental or relational?

Reflection to Robinson’s Article

After reading Robinson’s article, it made me reestablish the importance of being a flexible teacher. In particular, I liked how Robinson had assessed herself on her teaching and made changes to it. Her “changes included lecturing less, providing opportunities for higher-level student thinking, and developing classrooms norms to support student-student dialogue.” I agree with Robinson that using the Think-pair-share was a valuable tool that generated rich discussion among students and provided an opportunity for students to engage in mathematical learning. I also liked how she changed the format of her exam questions by focusing more on critical thinking, reasoning and communication skills. This gives students the opportunity to demonstrate their knowledge in math and not just on how well they can remember the procedures for solving a specific type of math question.
With all the changes geared toward helping the students to learn better, I question and doubt the value of lecturing less in class. I think restricting lecture time “to a maximum of 60 minutes per week” can be hurtful for the students in that they will not get enough instructional time to develop a good understanding in harder topics. However, I must applaud Robinson’s efforts for trying out new ways to facilitate learning in her students and for her efforts to become a better teacher.

My Most Memorable Teachers

One of my most memorable math teachers was Mr. Freisen. His writing was neat and legible and notes were clear and easy to understand. I also liked how he would give his students time every class to do homework. It was helpful because it allowed his students to seek help if necessary. I also liked how he would tell us stories that are not necessarily related to math. The stories made math classes more fun and interesting. Overall, he was a great teacher because he cared about his students’ learning. Even if you were no longer in his class, he would stay afterschool to help you learn math.

My second most memorable teacher was my university professor, Dr. Pierce. He was really passionate and enthusiastic about his teaching. He gets really excited when students are engaged and he loves it especially when students ask him challenging questions. He would always answer them with passion and excitement. He is also really helpful during his office hours. Dr. Pierce cared very much about his students’ learning because he would often stay past office hours to make sure everyone has had their chance to clarify misunderstandings and confusions.

Reflecting back to these two teachers, I guess what makes them such great teachers was that they had a broad knowledge in math that they wanted to share. They were both passionate and enthusiastic about math. They were both helpful and dedicated to making sure that their students were learning math. They are good role models to learn from and I wish to carry their good qualities into my own teaching.

My First Microteaching Assessment

My peers thought that:
- There was a clear learning objective
- Some thought the bridge was good and some don’t remember it
- There was a pretest of prior knowledge
- There was good participation
- There was a good post-test
- There was a good conclusion
- I had great patience
Some strengths of the lesson include:
- Good time management
- Fun, interesting and engaging
Some improvements for the lesson include:
- Preparing the origami paper
- Start with easier origami

Self-Assessment:
Things that went well in my lesson include:
- Good interaction and good group participation
- good introduction/pretest
If I were to teach this lesson again, I would improve by:
- Slowing down the lesson pace.
- Making sure some of the folding procedures was clearer so that everyone can see and understand
- Helping some people more
Based on my peers’ feedback I would like to make the following changes for this lesson:
- Making a stronger and more memorable bridge and pretest since some people did not recall them
- Perhaps choose an easier origami figure

Microteaching 1: Making an Origami Crane

Microteaching 1: Making an Origami Crane

Bridge:
I will begin with a brief introduction to origami. I will start by explaining what “ori” and “gami” means in Japanese. Then I will ask the students if they can guess where origami originated from. After which, I will explain that origami originated from China.

Teaching Objectives:
1) To build a classroom community.
2) To create a fun icebreaker activity.
3) To encourage students to participate in classroom activities.

Learning Objectives:
1) Students will be able to make an origami crane.
2) Students will be able to define the term origami.

Pretest:
These questions will be asked in the bridge phase.
1. Ask the students if they know what “ori” and “gami” refers to in Japanese.
2. Ask the students if they know where origami originated from.

Participatory activity:
I will demonstrate how to make an origami crane and encourage students to talk to each other and help each other if they are having problems when making the crane.

Post-test:
I will ask the students to make a second origami crane on their own without any help.

Summary:
I will go over the procedures needed to make an origami crane. After which, I will end by giving students additional origami links. The students will be able to challenge themselves at making harder origami figures at their own leisure.

Origami Links – Diagrams of Various Origami Figures
1. http://dev.origami.com/diagram.cfm
2. http://www.tinyshiny.com/How-to_Projects.php
3. http://www.oriland.com/studio/main.asp
4. http://www.origami-club.com/en/pageB.htm

Video Links – Vidoes of Various Origami Figures
1. http://www.videojug.com/tag/origami
2. http://www.pem.org/sites/origami/

Commentary on “Relational Understanding and Instrumental Understanding” by Richard R. Skemp

Skemp distinguishes between ‘relational understanding’ which is “knowing what to do and why” and ‘instrumental understanding’ which is “described as rules without reasons.”

In his opinion, “pupils whose goal is to understand instrumentally…won’t want to know all the careful groundwork…” True for some students but not for all students since it is possible to convince and engage the student in the groundwork or theory. Like he said “if people get satisfaction…[they may] try to understand relationally new material.” Skemp also claims that instrumental mathematics is usually easier to understand.” Is this the case for everyone? Each student has his/her own unique way of learning; some are visually learners, some are audio learners while others are hands-on learners. What is easier to understand depends on the student’s learning ability.
However, I agree that sometimes “relational understanding of a particular topic is too difficult” and “takes too long.” Thus, forcing teachers to turn to instrumental learning. After all, life is not without challenges. These challenges include “over-burdened syllabi” which Skemp claims, "almost all syllabi would be much better if much reduced in amount so that there would be time to teach them better.” Instead of always reducing course content, isn’t better to find an alternative means of teaching that better facilitates learning? Furthermore, I do not think it’s always fair and better to teach students less materials.

According to Skemp, relational learning will provide students with advantages such as "building up a schema” and/or increasing self-confidence since there are “ways of getting there without outside help.” I, however, think that the combination of relational and instrumental understanding will provide students with the most benefits to learning as relational understanding will cover the weaknesses of instrumental understanding and vice versa. For example, instrumental learning often follows a rigid set of instructions and not as adaptable to other situations. Relational learning however, fosters independent thinking and is more flexible for use in other situations.