Letters From Student A&B

Hi Mr X


I am so fortunate to had you as my math 10 teacher . I still remember that, I wasn’t the most bright kid in the class;and I made so many nonsense mistake in the assignment all the time. However, every assignment that I got back from you had your commons like “you are a smart boy!” or “ you can be better than Gauss!” or something like “you are a genius!!!”.
At that time, I truly believed you, so I have never given up on myself.I just got my master degree on mathematics education(secondary option). Looking back, I would never make it this far without your flatter common. Thank you again.



Best

Student A



Hi Mr X

I can't believe that I am still writing to you after left your math 12 class 10 years ago.I have to admit that you are not the best teacher I have ever had.I still remember that you kept us doing super difficult problems all the time that just turned me off as I never wanted to become a mathematician any way.Looking back, I still think that you were too unrealistic at the time and pushed me towards the direction which clearly doesn't fit me.
Best

Student B

Common:

The above two letters are fictional, however, these letters are my personal remainders when I have my own class. To be a good teacher sometimes simple comes down to always give your students a little encouragement. On the other hand ,being too strict or having unrealistic expectations and learning goal of your student definitely is not a good and effective teacher should do.

Summary and Reflection of Battleground Schools: Mathematics Education

     The battle between   Conservative mathematics education and progressive mathematics education started at 1900.As Susan states that these two rivals have philosophical differences in their dichotomies that ground opposing arguments in at least 12 areas of interest. Throughout 20^th century, there have been three major movements stand out in different period of the history in North America.

     These three movements are the progressivist for mathematics through activity and inquiry, the New Math reform movement of the 1960’s, and the so-called “Math Wars”, based on NCTM Standards reforms. Despite the philosophical differences of them, each movement has its times in the history due to the different political environment, view of nature and location of knowledge, democratization of education, and authority and obedience.

     Progressivist reform (circa 1910-1940), led by John Dewey, embraced the problematic and a degree of uncertainty as the incentive for students to form and test hypotheses and perceive patterns and relationships. On the other hand, Edward G.Begle’s “The New Math” dominated during 1960’s, which introduced some university mathematics into K-12 systems; however, it failed to meet the needs of all students. Finally, from 1990’s to present, the traditionalist and the progressivist , fueled by several special interested groups along with the media have taking the Math Wars over the NCTM Standards to the spot light of national stage in US.

     I have never had an opportunity to take a close look at the historical, political, and philosophical background of the battle until I have read Susan’s article. As a new math teacher candidate of the 21^st century, I do have my own philosophical ground on the issue of how math should be taught and what kind of math curricula should be introduced at our secondary and elementary schools. But I think that no matter how drastic the social and political environments are changing, our students should not be the only casualty in this political and philosophical battle. I urge that all parties involved  in this battle should take collaborated efforts, use some genuine ideas to find a common ground over the issue and find a balance of the two extreme( Con VS Pro) to meet the needs of every students so that our students are able to compete with their international counterparts.

TEACHER INTERVIEW REPORT(Zhi Song , Mandeep, and Feda)

Students' interview report

This interview was conducted with a three high school students from the west side of Vancouver. The three students with different age level, competency level and gender were interviewed in mathematics classrooms. Based on last year report cards, the first student H was a girl in grade 12 who was assessed as very good at math. The Second was a boy S in grade 9 who was assessed as excellent at math. The third was a boy G in grade 8 who was assessed as fair at math.

Despite the difference among these students, they all agreed about some common believes related to mathematics teaching and learning.

They all saw mathematics as useful in everyday life. However, they stressed the idea that only the basic mathematics is needed for that. They agreed that math is motivating when it makes sense and it is well understood. For these students a good teacher is the one who makes math fun to learn and explains it clearly.

An interesting point appeared through the interview was that two of the students, S who is excellent at mathematics and G who is fair, both preferred to work alone rather than in a group for two different reasons. While S thought that “working in a group would slow me down,” G commented that he liked to work “alone, because when I solve a problem it gives me the confidence that I did it without any help.” On the other side, H preferred to work in a group, because she thought that solving a problem could be a combined effort and that “ everyone is looking from a different perspective to solve the same problem.”

What was obvious from the students’ responses is that students learn differently and a teacher should be aware that different students have different needs. What a teacher could do in this situation is give the student some time to work on a problem alone and then assign them to groups. An excellent student like S will get the chance to help other students in his group after solving the problem on his own. A struggling student like G will get the chance to try to solve the problem alone and then get the help, if needed, from his friends. That might have worked well for G since when he was asked the question

Q: What do you usually do when you face a challenging problem

He answered:

G:Take some time to figure it out on my own then ask for help.

Another interesting result was related to the following question:

Would you rather have a teacher teaching you relatively easy stuff and give you an A- OR a teacher teaching you relatively difficult stuff and give you a B+

H: Depends if she is going to ignore the difficult stuff and then we miss important topics for the next year then I would rather get a B+ and learn the difficult

S: The more difficult it gets the less useful it is, so easy stuff with A-

G: I am not going to become a mathematician I only need to learn the basics so easy with A-

            For H who was preparing to go to university and knew that marks are important to get enrolled she was mature enough to realize that a good grade was not enough if that means she is going to struggle in the future by missing some important concepts. On the other hand, S and G cared about the mark the most because they had different needs and interests and they related to mathematics in a different way.

The idea of different students with different needs appear again here. Not every student wants to “become a mathematician” and not every student finds complex topics in mathematics as “useful.” Hence, when a teacher plans for a lesson she should take students’ different interests into account. A teacher should reach out to all students and make sure that everybody did understand the requirement for their grade level. Yet, at the same time, he/she has to motivate students and set high expectations so they will be pushed to do their best and not just settle with a minimum achievement. Being a good teacher is setting your students for success.

Teachers' interview report

This interview was conducted with a high school math teacher from the east end of Vancouver, where she has taught Math Essentials (Grades 8 through 10) for the past two years.  Much of what this teacher revealed in her interview confirmed what we’d expected to find in a typical math class.  In her interview, the teacher discussed how maintaining class discipline and getting her students motivated and involved in her class were often the most difficult parts of her job.  She found the students who were the most disruptive in her class often happened to be the ones who were struggling the most with the material that was being taught.  One way in which this teacher tries to address this issue is by giving all her students a clear set of goals and guidelines.  With these goals clearly defined, the students know what objective they have to work towards.  The goals may vary for each student; for some students, the goal may be to get an A as a final mark in the course, while for others it may be to simply improve their understanding of topics they hadn’t understood very well in previous years.  In each of these cases, the one thing this teacher makes sure to do is to ensure that each student is aware of the goal he or she is individually striving towards.  This way, the students can evaluate and re-evaluate themselves throughout the term or semester and reflect on how they’re doing towards reaching their goal.  The teacher found this method helped to enable students to take more of an initiative in their learning or “ownership of their own work,” as she likes to calls it.

            Another technique this teacher uses to keep her students involved in her classroom is giving her students different responsibilities.  These responsibilities can at times be academic (ie, giving out bonus assignments as a challenge) or they can be simple classroom tasks like writing the homework on the board, helping to hand out worksheets, etc.  What the teacher found with this approach was that students felt more engaged in her class and made them more comfortable to participate in class activities.

            A part of the interview that surprised us occurred when this teacher was asked which grade level she found most challenging to teach mathematics.  Her response of Math 8 was not entirely unexpected but her reasons behind this answer were interesting and something we as teacher candidates had not considered before.  The teacher found Math 8 to be more demanding to teach at times not because eighth graders usually have more energy thus require more attention; instead, the teacher found it more difficult to teach because in this grade, the teacher usually spent a lot more time teaching basic learning skills not directly related to math than she did at any other grade level.  Examples she discussed included teaching students how to write homework in their agenda, instructing them on how to take good notes, getting them to all show their work in a neat and organized manner, etc.  The teacher found teaching these skills ate into a lot of their class time, making it stressful for her to get her students through all the material in the curriculum.  We find this to be of interest because it was something we had not given any thought to until now

            In summary, we found this interview to be very informative and helpful to us as teacher candidates.  Teaching can at times be very challenging career but it is also obviously a very rewarding and enjoyable one as well.

As teacher candidates, we all will be very busy in our long practicum. From these interviews, what we have learned is that every classroom is unique, and as a student teacher, what we would like to do is: every night before go to bed, we will take 10 minutes, sit down and to really think about that whether our teaching goals have been reached and whether every student has achieved the learning goal during the day? If not, why? Then, we will take it from here. Tomorrow, it is a brand new day.

Feedbacks and reflections on microteaching

Feedbacks from colleagues

     Surprisingly, this microteaching has received lots of positive feedbacks from my colleagues. It has a  very good introduction and bridge, both learning and teaching objective are very clear, good knowledge and numbers, good communication skills, very hands-on, very animated presentation. One of my colleagues wrote that: “ can apply for a magician ‘s job!” The feedbacks are all very encouraging. Although I didn’t receive any “negative” feedback; this shows that the microteaching was very successful. I will take it as a compliment.

Personal reflections

     The reason that I had a relatively successful microteaching was that lots preparations had been done prior to the teaching. In my future teaching, I will take similar approaches. From this microteaching, I have learned that  in order to have a good and effective teaching lesson, one need to consider the following: Research and know the teaching material very well, and be the expert in the topic are that one will teach. Start the teaching plan as soon as it is assigned. Design which teaching strategy I will use. Moreover, having a backup plan ready all the time. Other factors that should be considered include the age group of the students, previous knowledge, learning skills of the students, and the diversity of the classroom.

Thoughts on Dave Hewitt‘s Teaching Video

     After watching two of Dr Dave Hewitt’s teaching videos, there are couple things that that I fell so impressed.

     First, the method of Auditory teaching and learning is fascinating; it reveals the whole new method of mathematical teaching. When I first thought that auditory teaching can only be done in the subject area of music and language. However, in this video Dr Hewitt vividly demonstrates that it can also do so well in mathematical teaching. It connects the auditory input with student’s inner thought, which is somewhat similar to the mental math teaching strategy. And it can also very effective to get the attentions of all students in the classroom.

     I really like the way that Dr Hewitt goes around the wall of the classroom, using forward and backward counting to give the students the concept of number line, and then leads to the minus number. It is very conceptual and natural. Clearly, it works really well in his grade 8 & grade 9 classes.

     . However, I think the shortcoming of this teaching strategy is that it will require all students in the class have similar level of math learning skills. For students who need to see math problem visually, and just not an auditory type of learner, then great modifications and considerations need to done in order to reach the need s of these students.



Lesson Plan Unit in Teaching a Magic Card Trick

	Content to be Covered	Time spent	Materials	Modifications
Bridges	Telling the fact that if one can perform some magic tricks can be entertaining on any group events	30 seconds	5 decks of cards
Learning Objective	To be able to observe things  in great details
Teaching Objective	To teach a group of 4-5 students to perform  a magic trick(Predict-pair/Predict- a card)
Pretest	Ask students whether they know the above trick	30 seconds
Participatory Learning	Dividing students in pairs to analysis the trick	5 minutes
Post-test	Ask each student to perform the trick in front of group of 4-5 students	3 minutes
Summary & Wrap-up	The restate the key component s of the trick ; to emphasize the learning objective	1 minute

5 Burning Questions for Math Teachers and Students

For Teachers:

1. 
   
   What do you find is the most challenging part of being a math teacher? 
2. 
   
   What in your opinion are some of the common mistakes student teachers make when they first start teaching
3. 
   
   How is your advice to make some difficult matematical concept becoming "interesting" for students? 
4. 
   
   What are the things student teachers must know or should be told before entering a class?  
5. 
   
   In a class with students of all varying degrees of math skill, how do you plan your lesson/what methods do you use so that all students are included in the learning and discussion?

For Students:

1. 
   
   What is the most embarrass moment for you  in your  math class?
2. 
   
   What kind of math teacher you dislike the most?
3. 
   
   What kind of math teacher you like the most?
4. 
   
   Would you rather given 10 easy problems or 2 very difficult problems for your math home work?
5. 
   
   Would you rather have a teacher teach you relatively easy stuff and give you an A- grade OR a teacher teach you relatively difficult stuff and give you an B+ grade?

                       Two memorable math teachers

     There were quite a few math teachers who had taught me were truly outstanding. However, among them, Mr. X and Mrs. Y had impacted my understanding of math teaching and learning the most. 　 　
     The first individual who sparked my interest in mathematics was Mr. X, my first-year calculus instructor. I was very successful in his class and received the highest grade; however, he always challenged me with complex and difficult problems on top of my regular homework. He could answer any of the math-problem at any time without hesitation. So, we called him “The God of Mathematics.” His rich knowledge in the subject area got me interested into mathematics .He would assign a couple of extra problems so that I always have something to work with. On the other hand, he would reduce the work load of some students in half if he saw them struggling through the required curriculum due to their personal hardships. Moreover, he asked me to lead a group of two or three students who had difficulties in understanding the mathematical concepts. By doing this, those students not only felt more comfortable learning from their peers; but also, I was able to reinforce my understanding and practice my mini-teaching skills. From this social context, I believe that to become a teacher, one needs to be passionate, competent, and be the expert in the subject area of teaching. However, the flexibility that Mr. X had demonstrated highlights how important it is for teachers to be sensitive and thoughtful so that they may reach their goals in teaching to a diverse body of students.
     The other individual was Mrs. Y, my first math instructor in Canada. I was in her grade 10 math class, after receiving 100% on the first midterm, she told me that if I could get the highest mark in four consecutive math courses then I would win a math scholarship. Since then I worked very hard on the following three mathematical courses and partnered with a fellow student, who was constantly thinking dropping the course due to a learning disability in mathematics. I was his free personal math tutor at the time. This not only helped him to improve his math skills but also I reinforced my own mathematical understanding. The final outcome was, I won the math scholarship and he passed all of his math courses. Not surprisingly, I was given the scholarship for excellence in math. However, at the ceremony, one thing that surprised me was the fellow student whom I tutored also received the award for “the most improved in math.” This award was sponsored by Mrs. Y. The fellow student then went on to start his apprentice career at BCIT. The award of “The most improved in math” shows that the education is not the privilege of the elite. Mrs. Y had vividly demonstrated the meanings of “education without distinction” and “excellence without distinction.”

Thoughts on “Relational Understanding and Instrumental Understanding”

     As a pre-service teacher, I think that relational understanding and instrumental understanding are not two different types of understanding but two stages of understanding. The two stages of understanding not only apply to mathematical learning but also apply to every single subject area of teaching and learning.

   I view instrumental understanding as the basic and fundamental stage of the understanding. This then leads to the ultimate stage of understanding, namely, relational understanding.

      Ideally, as educators, we want every student to have in depth relational understanding in every topic that we teach. However, there are limitations, such as under-funded public school systems, limited resources, diversity of the classroom, backwash effect of examinations, and over-burdened syllabi. In reality, the goal of relational understanding of every student in a given classroom is the biggest challenge that all math teachers are facing.

     Thus, teaching is an art, an art of juggling between what needs to be done and what actually can be done from the given resources, between rushing through over-burdened syllabi and thoroughly cover in-depth material, and between obtaining superficial instrumental understanding and the ultimate goal of relational understanding. Although public education is not always the top priority of the government; we, public educators, do need to have a life-long passion of teaching in order to fight through all the possible obstacles and not to be discouraged in reaching the ultimate goal of relational understanding.