Response to Fictional Letters - My hopes and dreams and worst nightmares...

I hope that we can teach effectively to the majority while acknowledging the minority.   My dream is that I can make a difference to some students life and encourage them to be the best they can be.   My worst nightmare is having students think I don't know what I'm talking about, and tune-out.   I want to give every student a chance to succeed.   My hope is that I can connect at least a little bit with each student,  sometime during the term,  however small way it may be.   Ambitious?  Realistic?  Only time will tell.

Fictional Letters

The following made-up letters are an exercise to help us reflect on the influence we can have on students.

Letter #1
Dear Math Teacher,

Remember me?  I was in your Math 12 class 10 years ago.   Now I am working as a computer engineer.  I just wanted to let you know that I really appreciated the way you encouraged me in Math that year.   I still remember the day when four of us girls came to your room after school to talk about what we want to do in life.   You told us that we can achieve our dreams,  if we set our mind and worked hard at it.    You challenged us to write the Euclid Math Test.    It gave me the confidence and courage to tackle new problems,  and do well on the provincial exams.   That helped me get into UBC and into engineering school.   Now I have graduated from UBC and got my first job in telecommunications doing hardware design.   I just wanted to say THANK YOU.

Signed,
Forever Grateful Student

Letter #2
Dear Math Teacher,

I just don't get.   I never got it when you were teaching me math back in grade 11 (must have been ten years ago),  and I still don't get it.     You always doted on your favourite students,   but left the rest of us behind.    Failing your course just made me shut-down in Math.    Even though I eventually graduated from high school,   I wish I can understand math better and be better in arithmetic (ok, I know that's elementary school stuff) because working here in the warehouse at Walmart really isn't fun.    You went through your problems in class so quickly and I just couldn't follow, and you didn't notice that I was struggling.    Now when I see others doing math or computers or business getting great jobs,   I am angry.   Angry and disappointed that I did not have the chance or opportunity to go down that path.    I wish I could do it again.   But now I am busy earning a living, and so I can't.  

I'm not blaming you, I'm just letting you know how life is for a failure Math student.

Signed,
Just-Don't-Get-It Student

Reflections on "Battleground Schools: Mathematics Education".

Who would have thought that Math Education can be a political battleground!?

This article explains that indeed for the last hundred years or so in North America,  people have fought over what Math Education should be like.   Basically there are two polarized views:   "the progressive", and the "conservative" (or "traditionalist").  The "progressive" focusses on understanding, inquiry, exploratory approach.   The "conservative" is focussed on fluency, and a more authoritative and factual application of algorithms and curriculum in solving problems.

There have been three major waves in the battleground.

The Progressivist movement led primarily from 1910 to 1940 was by John Dewey challenged society to not simply know "how" to apply math to solve a problem,  but ask "why" this makes sense.   Are there other ways to do it?    He challenged students to engage in doing math, not just knowing it.   He has a passionate drive to develop Americans to be scientific and democratic thinkers rather than students who just follow rigid rules.   While adopted in some classrooms across the country, it was not widespread.

Subsequently, the New Math era in the 1960's evolved, and grew out of America's anxiety over falling behind the Soviets in science and technology.   The movement was led by university groups and later NCTM, and French mathematician Bourbaki.  They rallied for a new curriculum to include set theory, abstract algebra, calculus, and other university topics, to be introduced and taught from K-12.    However by the 1970's the New Math program was found ineffective.  It was difficult to implement because math teachers at the K-12 levels were not familiar with these topics themselves!

The third movement,  Math Wars over the NCTM Standards,  brings us to the 1990s and today.    In the 1980's standards were developed to support a more balanced, progressive curriculum.   Together with professional development, workshops to train new math teachers, these standards were generally well received.   Some wars erupted in the 1990's with a backlash against standards-based curriculum by traditionalists.   Other groups chimed in with various takes on the standard based teaching.   In 1996, studies showed that America's 8th grade students ranked 28th in the world,  well behind many Asian countries.    Hence, a move for deeper conceptual learning arose.    More research and math groups joined in the dialogue and today there are still on going media coverage of various cries for support for one side or the other.

Why would math education be a political battleground?

From this article,  I realise that Math education has deep implications.   Not only is Math is used to solve REAL LIFE problems,  the education of Math is linked to a nations scientific development, economic well-being and mental satisfaction.     At a time when the world is competing technologically and scientifically,  Math education is an underlying force behind a more competently educated population of students and scientists.    At a time when a nation is more interested in reform and inquiry,  our educational methods for Math also reform so we can better thinkers.    It is fascinating to see that Math Education is such a dynamic part of a society and nation.    How students are learning Math affects the future of the country.    I hope that those in power and authority will have the wisdom to find balance in math education methods, and focus on the learner, rather than our ego.

Student - Teacher Interviews

Assignment #1: Student / Teacher Interviews

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:

• Most students appreciate teachers who are patient, friendly, and care if you understand or not.  Being entertaining or funny was a bonus.
• Most students appreciate teachers who take the time to help OUTSIDE of class time. 
•  Most students (both the strong and weak ones) say they like Math, especially when they understand it, or when it is "fun". 
•  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.
•  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?
•  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?
6.  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.

Post micro-teaching thoughts...

It was certainly fun and enjoyable to learn about beading, stress relief tips, and even techniques for pressing acupuncture points when you are not feeling well!    We can certainly learn and teach alot in ten minutes.    Also with adequate preparation, and not being tiresome of bringing in the right supplies, it is possible to do some very exciting demonstrations within the four walls of a classroom!   At the same time I realise that it is also easy to misjudge our use of time even when we do our  lesson plan.    After I taught the three different things we can do with the chopsticks wrapper, we still had some time left.   So we ended up also talking about how to use the chopstick, and even how to make a toothpick out of a chopstick!     It is much easier to teach to a table of adults whom we already know.   They are supportive and helpful.   I wonder how teaching in high-school will be really like ...

Lesson Plan: What you can do with your chopsticks wrapper while you're waiting for your food to arrive.

	What?	How long?	Materials?
Bridges	Ask, "what do you do with them (chopsticks wrapper)?"  Don't waste them!	2 minutes	sushi restaurant menu
Learning Objectives	Students will learn how to make 3 fun things with their chopsticks wrapper.
Teaching Objectives	Engage and maintain group interest despite possibly noisy room.
Pretest	"Has anyone done this before?".  Ask each student to pick one of the 3 things they want to make this lesson, pass out one wrapper each.	1 minute
Participating Learning	show how to make: 1. chopsticks rest 2. party blower 3. learner chopsticks	5 minutes	wrappers, elastic bands, some chopsticks
Post-test	ask students to demo what they made.  Ask, "how can you modify this to make it different?".	2 minutes
Summary and Wrapup	"This is a great way to entertain others before the food comes,  and it's useful!  "	1 minute	menu, display of final products made by students

On Dave's Hewitt's teaching style

Dave Hewitt demonstrated unconventional ways of teaching mathematical concepts.    All his words, and actions, were well planned, and intentional.    Each tap on the board, and the location of that tap,  and timing of that tap, etc.  were done to lead the students to conjecture some pattern on the number line.    It forced student to concentrate with their ears, and think with their heads.    They were not presented with any numbers to stare at.    The teacher must carefully think through how sound and actions can guide the student's thinking,  and execute his plan accurately.   Even when delivered exactly as intended, there is still the possibility where one student's interpretation is different from another's.   Hence,  Hewitt also plans some assessment into his lesson.   He does this by repeating and returning to certain patterns,  by asking students what number we are at,  not giving out the correct answer, but rather, allowing the students as a group to participate, and self-correct their answers until all agree it is correct.   Overall, Hewitt's method is very clever,  somewhat intuitive,  while also counter-conventional,  takes arduous planning and thought.     However, it could instill a deep relational understanding in students that is unparalleled by conventional illustrations using pen on a board.

Questions for Math Teachers and 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. 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

I remember my grade 12 math teacher to be a calm not-too-exciting teacher who just systematically, day after day,  went through the textbook material.   He was predictable and while his classes were not extremely enjoyable, I did alright.

However,   he did do one thing that changed my life.   He encouraged me to write the Canada-wide Euclid Math Contest,  and provided me lots of old exams to practice with.     For the next few months,  another student and myself,  worked together at lunch hours to solve these problems.   The more we got into it, the more we loved it.    Later on, my dad started to work on these practice problems at home with me as well,  and we solved problems together.    The more we got into it, the more we loved it.   After doing many practice exams,  I ended up scoring in the top 10 percent in Canada, and won an award.  This really boosted my confidence and interest in Math.   It made me realise that  Math is a subject that I can master,  if I worked hard enough.     Previously I had assumed some common myths:   "if you're not a math person,  you won't get it",   "either you get it, or you don't",  or  "you have to be really smart to be good in math".    After doing the Euclid Math Contest,   I learnt that practice does make a difference.  Edison said that genius is 1% inspiration, and 99% perspiration.   I'd argue that once you do put in the "perspiration",  the "inspiration" will increase too!    I am thankful to my teacher for encouraging me.

Another math teacher I remember was my Math tutor in first year at UBC.  She was a 4th year student, and I really  looked  up to her.   I was completely lost in first year calculus,  despite being top of the class while back in highschool.    I just didn't get it.    My tutor was able to help me understand the concept, and show me how to solve the assignment problems.   It felt so good to finally get the hang of it.   I believe she used a combination of relational and instrumental strategies to help me understand, and I had the freedom to ask her questions when I didn't understand.    During lecture time, I was often afraid to raise questions in a big lecture hall of first-year engineering "keeners".

Example of Relational vs. Instrumental Approach in Math

The division of fractions is a good example how both relational and instrumental approach are useful.   A relational pictorial representation of the problems helps us understand the problem, and it's wording.   However in order to solve similar problems of a larger scale, having formula or fractions to manipulate makes the problem manageable and achievable.   When an answer is derived, one can return back to a relational mindset to ask oneself,  "does this answer make sense".   Hence, both methods complement one another.

Thoughts on "Relational Understanding and Instrumental Understanding" by Richard Skemp

Two words can mean different things for different people. And so it is also in mathematical teaching;  There is both an instrumental approach, and an relationship approach. It is not about who is right or who is wrong, but rather about how we can stand back to examine what we are actually trying to explain, and come to a mutual ground of understanding we can meet together.

The instrumental method is usually easier to understand, and quicker to answers to problems, and there is less information involved.  This is the more traditional method for math teachers.   The relational method is often more adaptable to new problems in the real world, and thus, more intuitive, perhaps easier to remember. Once a concept is understood relationally, the student is more likely to explore other problems which stem from the original one.  A student's understanding will grow beyond the scope of the original problem, to develop new problems,  or new ways of getting the answer.    Hence, achieving this basic understanding can be an effective goal in itself. However, with examinations, over-burdened syllabi, and other assessment difficulties, traditional teachers often shy away from relational understanding. This type of teaching and understand takes longer.