Reflections on Fictional Letters

The exercise pointed out to me that I really want my students to value my life experience that I bring to the classroom, as well as find me very competent, patient and engaging.  My worst fear is that they look at me like I am crazy and feel I am not adding any value .... waisting their time or not appreciating them.

Fictional Letters to the Teacher

Good Student
September 29, 2020


Dear Ms. Bee,


You may not remember me, but I was in your 11 grade math class at Van High.  I am writing you to let you know the impact that you had on me then and how it has affected where I am now.  I remember meeting you the first day and thinking to myself "she is kind of old for a new teacher". In that year, we all got to know you better and learned about some of the things you had done before teaching.  What that experience taught me is that I did not have to choose my lifetime career right out of high school and that it is possible to have several careers during one's life.  Since math was my favourite subject, I went on to University to study physics.  I practiced this for a while, but have recently decided to go back to school to study law. I look forward to entering this new profession. 


I hope all is well with you, and thanks again for demonstrating to me that one can explore and do different things through their life.


--Good Student




Bad Student
September 29, 2020


Dear Ms. Bee,


You may not remember me, but I was in your 8th grade math class at Van High.  You were a new teacher at that time.  Looking back, I realize I could have probably paid more attention in class, but you did not make that easy for me. I often felt like you expected everyone to like math as much as you did.  Not everyone is interested in solving math puzzles.  In fact, I am still not interested. I appreciated that you often tried to engage us, but I saw your attempts to engage us a little pathetic.  I just wanted you to know that I am living a very happy, successful life without knowing what the Fibonacci series is.


Sincerely,


--Bad Student

Reflections on Battleground Schools: Mathematics Education

The article explains the ongoing debate in Mathematics Education stemming from two diverse views of progressive and conservative.  With its goal of understanding, the progressive view focuses on the why, provoking students to inquire and explore.  The opposing view, conservative with its goal of fluency focuses on the how, putting the onus on the teacher to deliver the appropriate curriculum.  Even though both views have a set of underlying assumptions, the ongoing battles seem more related to wider issues such as the "nature and location of knowledge", the "democratization of education" and "views on authority and obedience".


The "nature and location of knowledge" refers to where the knowledge is to come from.  The conservative view is that knowledge is in the mind of the student but must be surfaced by the teacher by applying facts.  The progressive view is that all students can acquire knowledge and that knowledge comes more from the outside world and collaborating with others than inside out via algorithms.  "Democratization of education" comes into play when considering the target audience of such views.  Within the conservative view, minimal survival skills are for many, while high level concepts are reserved for just the elite.  In the progressive view, problem solving skills should be taught to all.  Lastly, the "views on authority and obedience" are portrayed by the roles of teacher and student.  The conservative view puts the onus on the teacher for all learning and is seen as infallible.  The progressive view takes a more inclusive view and shares the responsibility of learning with the teacher and the community of students.


Although the math education debates have been ongoing, many of the battles center around three main movements: Progressive (1910 - 1940), New Math (1960's) and Math Wars (1990's onward).  The Progressive Movement argued that the current approach was meaningless, focusing on the how instead of the why.  The New Math period was a reaction to other countries producing great scientists and the desire to target learning to potential future scientists and engineers.  Math Wars came about as a belated response to the standards put forth by the NCTM.  Although well received at first, certain groups again began to question the more progressive methods in favour of a more traditional approach.


Although the two approaches are very different, I feel they are both necessary.  Exposing students to a mixed approach would be most beneficial as people all learn differently and it often takes people a while to really figure out how they learn.  I do not believe one size fits all. I do, however, believe that math courses could be more specialized earlier in the secondary curriculum to allow those students who are more interested in pursuing a career as a scientist or engineer to have more specialized instruction while not forcing all students to such curriculum.

Principles & Standards for School Mathematics - NCTM

The National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 to define the necessary building blocks of a high quality school mathematics program.  It provides a common foundation of what should be expected by teachers, students and administration.  The NCTM stresses the importance of mathematical competence for students in order to succeed in life.  Many of the principles and standards include bridging the space between concepts and practical application.

The guide contains both principles and standards.  The principles layout the fundamental beliefs of what is important in a high quality math program.  The standards are actual descriptions of what math instructors should do (process) and what students should know (content).  There are 6 main principles:

1. Equity – provide support and high expectations for ALL students
2. Curriculum – ensure a coherent, well articulated program
3. Teaching – understand what students know and need to know and support them in getting there
4. Learning – ensure understanding based on previous and new knowledge
5. Assessment – assess to help the student improve as well as help the teacher improve and adjust their teaching
6. Technology – harness the power of technology

The standards include 5 process standards and 5 content standards.  The key 5 process standards include:

1. Problem Solving: Become logical, critical thinkers 
2. Reasoning and Proof: Understand the root of the problem 
3. Communication: Express math clearly, coherently & concisely 
4. Connections: Recognize relationships in and out of math 
5. Representation: Demonstrate mathematic ideas diversely

The 5 content standards describe the progression that students should experience as it relates to particular subjects in mathematics

1. Numbers & basic operations (whole to fraction, estimations, calculations with fluency) 
2. Algebra – represent patterns and functions 
3. Geometry – analyzing shapes 
4. Measurement – basic units and systems, techniques and formulae 
5. Data Analysis and Probability – make inferences

These principles and standards should be forefront for all new and experienced teachers as it serves as the common basis for what is and should be expected for math students, teachers and administrators in North America.

Assignment #1 - Conversations with Teacher & Students

Team Members: Carly Orr, Donna Braaten & Hung Dang Le

September 24, 2010

Part I:  Student Interviews

One member of our group works in a tutoring centre.    A written survey that contained our top 5 questions was administered to about 20 students.  These students come 2-3 times a week for homework help and extra challenges.  They range from grades 6-12.    They come from both private and public schools, and different neighbourhoods in Vancouver.  The level of the students range from A to C+.   

From our surveys,  we noticed that the elementary school children gave mostly simple, and some one line answers.   The older students have more detailed answers, so our analysis is mainly based on these senior math students. 

Here is a snapshot of Student Responses:

1. Most students appreciate teachers who are patient, friendly, and care if you understand or not. Being entertaining or funny was a bonus.
2. Most students appreciate teachers who take the time to help OUTSIDE of class time. 
3. Most students (both the strong and weak ones) say they like Math, especially when they understand it, or when it is "fun". 
4. Most students want teachers who are WILLING to explain more when students need it, WILLING to offer help when kids are stuck after a lesson, WILLING to take more time to mark homework and assess more frequently to see if each student is "getting" it.
5. Most students will ask friends first when they are stuck, or postpone asking the teacher until after class, or until they have asked a friend?
6. What students find intimidating: word problems, big numbers, not understanding something, getting the "final answer" wrong (but did most of the steps right).

Based on the information gathered some reflections come to mind: 

• Math is fun (and motivating) only if you get it. Therefore as teachers, we must do whatever it takes, to help students "GET IT", both the process, and the final answer. Otherwise it is frustrating experience for both sides. 
• It is a fact that some students will need help outside of class. How far am I willing to go in extending office hours to help these students? 
• While peer-teaching is encouraged, we also want to ask: Why are students not likely to ask teachers first when they are stuck? Is it because they are inaccessible? or do teachers make the students feel "dumb"? or is the teacher not able to add any explanation that would make a difference? Maybe teachers need to make students feel safe about asking "dumb" questions. Maybe teachers need to offer more time after school. Maybe teachers need to try explaining things from another perspective (relationally?) when a students consistently still does not understand after repeated explanations? Am I willing to try another way of explaining an age-old concept?   

Part II  Teacher Interview

Ms. X is a math teacher at a high school in Vancouver not too far from UBC.  She has been teaching for about 7 years.  She was very willing to open her classroom for us to conduct an interview to provide insightful answers to our burning questions listed below:

1. How do you know whether or not students are "getting it" during class time?
2. What do you do if a student is too shy or embarrassed to ask for help?
3. How much time do you feel you need to spend on class preparation?
4. Do you have any methods, which help make a Math lesson more interesting?
5. When you mark your students’ tests, which aspect is more important: the correct number (as an answer) or the method used?

Although all of her responses were quite helpful, questions 4 and 5 revealed some particularly strong insights focusing on student engagement and assessment.

In order to make her classes more interesting, Ms. X referred to the field of “eductainment”.  She often uses humour in the classroom to keep the students engaged while also introducing them to new concepts.  Specific examples of this were as simple as putting as putting on “nerd” glasses to introducing new shapes by making the students think about what it could possibly mean (example below).  Many such pictures and graphs were placed around the classroom for the students to view. 

Ms. X also had strong opinions when it comes to assessment.  She believes that the answer as well as the approach are necessary.  More importantly, she believes in assessment for learning and not assessment of learning.  She wants to keep the student engaged and for them to remain interested.  When students start to see signs of success, it keeps them engaged.  For regular assignments, she uses a 100 – 75 – 50 – 0 scale.  If a student does not like their grade on an assignment, they may resubmit.  Tests are different and cannot be redone.   In summary, Ms. X believes teachers are put into an arbitrator role instead of coach.  She believes a student should have every opportunity to improve during the assignments with the test being the final “game”.

Lastly, Ms. X provided us with a view useful links that we can use in the classroom. In particular, Ms. X. often uses the manipulatives available on line from the National Library of Virtual Manipulatives – http://nlvm.usu.edu/ .  She also uses a popular blog with the students, FAILBlog - http://failblog.org/ and asks students to find examples of math being misused.

Feedback from MicroLesson

Most of the students in my micro-lesson felt that my topic was very clear.  They all appreciated the exercise where they had to close their eyes and envision the perfect marshmallow.  Two of the three also noted that they liked the picture of what a perfect marshmallow looked like.  2 of the 3 found that my time management could have been improved as 10 minutes seemed too long for the topic. One student liked the extra exercise of naming the mini-smore.

When I reflect on the lesson, I think the use of a faux fire and a picture of the marshmallow worked very well.  I did find that I had too much time for the topic.  In hindsight, I should have had extra exercises or topics that could have been used if needed.  I also felt that I did not deliver the "hook" to my expectations.  I was a little anxious in the beginning which made me miss the mark on that hook.  I really need to work on my hook and my timing for future lessons.

Micro Lesson using BOOPPPS

Toasting the Perfect Marshmallow


Bridge (2): 

Describe the "perfect toasted marshmallow".  Set the scenario of being around a campfire.  Close your eyes and pretend you are eating it.  Will descrbe: very hot, crisp, but not burnt, brown but not black, no cold chewy parts. Show picture of "perfect marshmallow".  Find out previous bad experiences with toasting marshmallows.

Learning Objectives (1):

- Each student will learn the techniques involved in how to toast the perfect marshmallow.

- Extra activity will be creating a mini-smore and naming it (depending on time and interest)

Teaching Objectives (0):

- Get all students involved.

- Have a least 1 student try to cook one and lets me know.

Pretest (1):

Find out previous bad experiences with toasting marshmallows.

Participatory learning (3):

Will provide hands on exercise including skewers, marshmallows and faux fire. Key concepts to cover:

- wait until flame is ready, hottest part near the bottom

- sceure the marshmallow such that it will not fall off

- if falls or catches on fire, game over

- never more than one

- start with heating middle (inside out), then top then bottom 

Post-test (1):

Will recap by asking them to point out the top 3 things to remember when cooking a marshmallow.

Summary and Wrap up (2):

- Explain how a perfect marshmallow can also serve as the basis for a mini-smore (name pending).

- Request that each student let me know when they have toasted their perfect marshmallow.

Materials:

- skewers

- marshmallow

- faux fire

- milk chocolate

- teddy grahams

Thoughts on the Dave Hewitt Video

I was very impressed by the approach taken by Dave Hewitt (secondary math teacher).  I found his style very engaging and memorable.  Two main points really resonated with me: using verbal skills to teach algebraic equations and dispelling the image of teacher being the sole arbitrator of what is right and wrong. For the purpose of this blog, I will focus on the latter.


Many students take the view that the teacher is the final judge and juror.  This could lead to a situation where students rely too much on the teacher and not enough on their own judgment.  To really encourage the students to think and to build their confidence, Dave Hewitt left it upon the students to say what was right and what was wrong.  I really liked this approach and had just heard of a similar example where a drama teacher had required that a student pass the "bird scene" before being able to audition for the school play. The teacher had not given any specific criteria of what it was to pass.  Again and again the student did the performance only to find out that she did not pass.  Each time she decided what she could improve and then improved it.  Finally, when she had felt she delivered an almost perfect performance that she was extremely proud of, she told the teacher that if he did not pass her this time, it did not matter because SHE knew that her performance was great.  He then said, you finally have confidence in what you did, hence she passed.  Although a slightly different situation, the same approach was applied of getting the students to take ownership of what they have learned and in doing so building confidence in what they have learned and applied.

Assignment #1 - Conversations with Teacher & Students

Teacher Burning Questions:

1. How do you know whether or not students are "getting it" during class time?  If a student still does not "get it" after you explain a few times what do you do?
2.  What do you do if a student is too shy or embarrassed to ask for further clarification when everyone else in the class seems to get it?
3.  How much time do you feel you need to spend on class preparation?
4.  Do you have any methods, which help make a Math lesson more interesting and easier for students to understand?
5.  When you mark your students’ tests, which aspect is more important: the correct number (as an answer) or the method which students use to obtain the answer?

Student Burning Questions

1. 1.     What intimidates you most about math?
2. What do you normally do when you don't understand something during class time?
3. What do you like most about your teachers? Least?
4.  Do you like studying Math (yes or no)? Why?
5.  What do you wish your teacher would do in order to help you study Math better?

Memorable Math Teachers

My math history dates back to 8th grade when I had my first algebra class.  I had always liked math and knew I wanted to take Calculus in grade 12.  To do this, I was on a specific path of Algebra in 8th, Geometry in 9th, AlgebraII/Trigonometry in 10th, Math Analysis in 11th and the goal, Calculus in 12th.  I did not realize it at the time but every one of my math teachers were women. Many were great, though one was not.


My favourite was my teacher for 11 and 12th grade, and the inspiration for me entering this career.  She was fairly strict and had us sit in alphabetical order.  Each morning we had daily points to confirm we did our homework but also challenge us to new problems. She was extremely organized and competent.  My best memory is learning the shape of an ellipse in which she used string and chalk across a pivot point on the chalk board to show us what an ellipse really meant. 


My least favourite teacher was 9th grade Geometry.  She was a new teacher who used to write theorems on the board verbatim with missing words that we would have to fill in.  In many cases, I found the missing words not even the key words.  To me, it was an exercise of memorization instead of learning.  I remember challenging her content on some occasions, which resulted in her dislike of me.  I do not even remember her name anymore, yet could name ALL the others.

Further Reflection on Relational Understanding

September 13, 2010
In todays class, the point of multi-representation was further illustrated by the 8x8 chessboard problem as well as dividing fractions.  I was really impressed the many methods of solving the chessboard problem, and admit some of the approaches were very easy for me and some difficult to follow.  I think that is the point of multiple approaches.  As everyone processes/takes in information differently, it is good to have alternative ways to teach the concept.  It was also very clear that just memorizing formulae is not a good approach and finding ways to visualize and really think about the question is more intriguing. 

Response to Skemp Article

200 words on response to "Relational Understanding and Instrumental Understanding" by Richard R. Skemp

September 13, 2010

This is the first time I have heard the terms relational and instrumental as it applies to understanding and mathematics.  Previously, I would have categorized the views as understanding the big picture as compared to just memorizing/knowing the formulae.  I am certainly of the belief that understanding the theory behind the equations is the best learning experience.

During my education in electrical engineering, there were many equations to learn.  In looking back at that experience, unfortunately, many of the courses did focus on instrumental understanding rather than the deeper concepts.  I do remember once of my best learning experiences was a class of Electronics.  Often in the tests, we needed to state the theory and derive the equation that would then be used to reach the final answer.  In that experience, I felt that I thoroughly understood the topic and found that very rewarding.

In an ideal learning environment, I would expect to teach the theories and foundation such that students really understand the instrumental portion.  I would see the challenges in that being (1) changing the mindset of the student to not just focus on the final answer (2) supporting resources such as textbooks and materials and lastly (3) finding the time and method to be able to assess and evaluation the relational understanding.