Citizenship and the Mathematics Classroom

This was a very interesting article about mathematics and the attempt to include citizenship in mathematics education.  I found it intriguing to note that the author follows a similar train of thought that we have been learning throughout the course.  That is, that we should not just teach optimal ways of finding ‘right’ answers to problems, but that we should provide them with a deeper understanding.  This deeper understanding will help student’s function better in society.  I had not ever thought of the fact that homework and having right and wrong answers may not be good for citizenship education.  I thought the author brought up an interesting point, that by doing this, we are judging a students’ thought process and not giving them the right to be freely creative in their thought process.  After reading this article I can only hope that I can take aspects of these points and use them in my own classroom.  From as far as I can remember, I have never really been given the opportunity to be entirely free and creative when doing math.  There has always been the underlying right and wrong answer that has affected the way I think about math.  I also liked the suggestion to “build community in the classroom” (Simmt, 5) by making sure students contribute in the classroom.  I feel this is a very important point because student participation will encourage other students to become active participants in mathematics.  I really appreciate how this article has tried to incorporate citizenship in the mathematics classroom, and provides the reader with ways to do this.  It is definitely an aspect I have never really thought much about before, but I will certainly consider from now on.

What-If-Not?

The What-If-Not method is a very interesting technique using a sequence of levels starting with choosing a topic/starting point, listing attributes of that topic, choosing an attribute and thinking What-If-Not, posing a problem and then analyzing that problem. I feel that this may be difficult to use for our microteaching lesson, mainly because we have created a very real-life application approach to our lesson plan. I think that to use the What-If-Not technique we will have to make our topic (algebra) more specific to begin the process.  If we did so, we could generate some characteristics about algebra before the lesson, or even have our students generate or add some characteristics they know about algebra during the lesson.   This would create a great base for understanding this topic. I think if we were able to create some What-If-Not statements it may also help to explore the topic, and allow students to get creative and work collectively as a group.

 Some strengths of this approach are:

1) It really allows students, or even instructors to start thinking of aspects or attributes of their topic they may not have thought of before.  This allows someone to be more creative within the math classroom, something that is often taken for granted.

2) This What-If-Not part of this approach is great because it creates a deeper understanding of the topic, and allows the instructor or whoever is using this method, to generate or pose many more problems than they normally would.

 Some weaknesses of this approach:

1)I think the main weakness of this approach is that it seems very extensive.  There are a lot of steps that involve a lot of time.  Although it points to towards a deeper understanding of the topic, I am not sure whether the amount of time spent generating these types of problems would be a benefit in the end. 

2)I also felt that delving into the What-If-Not’s might be confusing for some students.  Many people have a “math phobia” and I think that this could be potentially confusing for many students and put them off math even further.

 I think with lots of practice and more extensive knowledge of this approach, it could be useful in the classroom to spark interest and go into a deeper meaning of some topics.

10 Questions to the authors of 'The Art of Problem Posing'

1) I would like to know why the authors feel that after we solve a problem we don’t understand the significance of what we have done.  Could they provide an example of this?

 2) I would like to ask the authors how or what ideas they changed in their book for the third edition?  It would be interesting to see how their approach to problem posing has changed over the years.

 3) The Secular Talmund is something I have never heard of and I would be interested in finding out how they came to know this idea and have a little more information on what it is about.

 4) I found the fist chapter interesting from the point of view of the first question asked.  You could see the difference between the more simple to more complicated and relational thinking, making it more clear how we can begin to improve on problem posing.

 5) I would like to know why, when asking a question about isosceles triangles, those with ‘weak’ skills or knowledge come up with more robust questions?

 6) I was unclear about the discussion on ‘challenging’ the given’.  Why would you not accept something like a geoboard as is?  How would you change it?

 7) I liked how the authors used observations to create or pose new problems.  This seems like a useful tool, make observations about what you are looking at and only then begin to pose problems.

 8) Does the author find that by posing a problem that requires an approximate answer, the problem solver becomes more creative?

 9) I thought it was interesting that the authors suggested inquiring about the history of some of the problems.  This seems like a great way to become more familiar with the concepts and be able to have a better knowledge base for posing problems.

 10) I would like to ask the authors for more examples of internal vs. external exploration.  I was a little unclear how to go begin exploring this way in terms of mathematics concepts.

Letters from future students

Dear Miss Landon,

I wanted to write to you to tell you how much I enjoyed you as a math teacher, you were my favourite teacher throughout highschool!  Your classroom inspired me so much I have gone on to become a mathematician and I have been nominated for the nobel prize!

I really felt that you were so well organized that I could easily follow all of your lessons and this organization kept me in check.  I always knew when a test or quiz was, or what we had for homework.  I think this showed me that being organized is key for a future outside of school.

I appreciated that you were very approachable as well.  I never felt that I was going to be punished for giving a wrong answer, or that I couldn't come to you for help.  This has been so important for my career!  As a superior mathematician I can take risks and be creative, it doesn't matter if I don't end up with the right solution, I just go back and try again.

The thing that placed you apart from my other teachers was how you taught your lessons.  You didn't just sit at the front of the class and write down notes and algorithms and expect us to copy them down and do homework.  We were able to ask questions, talk to our neighbors about problems you gave us and even try learning things on our own.  

I especially liked the weekly problems you posted on the board.  I loved to go home and work through them on the weekends, no wonder I'm such a great mathematician! Those math books that you kept in the classroom were so interesting as well, no one else did that!  The time you read flatland to us was so much fun, it made the class go by so fast!

Well, I just thought I'd let you know how much of a difference you made in my life.  I promise to include you in my speech when I accept my award!

Sincerely

Peye Thagorus

Dear Miss Landon,

I needed to tell you that I hate math and that you were the worst math teacher I ever had!  You made tests too hard and all your lessons were so fluffy.  Why did we have to work with other students so much?  I just wanted to do the lesson and work on my homework so I could go home.  Your lack of notes caused me to fail tests and now I am in a prison trying to calculate how many more years I have to be here.  This is all your fault!  I hope you learn to teach math better someday.

signed

Jr. i

Hopes and worries.

I really hope that I will become a teacher like the one in my first letter.  I hope that I will be well organized and be able to create a balance between relational and instrumental understanding in my teaching.

I am worried that I won't be able  to create new ways to teach math other than just giving notes for students, or that I will fall into one category of teaching and not be able to combine methods.  I am also worried that I won't give good enough notes for my students to succeed in their studies.

Reflection of "Teaching the Marked Case" video

Teaching the marked case was a very interesting video.  For a while in many of our classes we have discussed engaging the students and using a relational understanding teaching method.  We have not however had the chance to see it in action.  I thought that seeing someone teach using a relational method was very helpful.  As a colleague mentioned in class, the instructor on the video was able cover more than one topic using his method.  This would be very helpful on a tight curricular schedule.  I thought that it also gave the students a great way of understanding how the math they were learning worked.  By the repetition and having students speak out loud I think that they would have a better chance of remembering the concepts they were being taught.  I did however have some concerns on the lack of notes.  If the students have nothing to look back on, how are they to remember what went on months ago?  Perhaps this teacher gave notes after his initial lesson.  As well, I feel that it would be important for the students to have a step-by-step method written down for them to practice which I did not see performed in the video.  I did, however, like how engaged the students in the video were and how many participated without fear of being wrong.  During the video it was also interesting to noticed how well managed the classroom was, so this is definitely a great aspect of this method.  I think that the marked case method has many positive features but could be used along with a more instrumental way of teaching to create and an even more effective lesson.

Summary and Comments on Battleground Schools

The view on mathematics education has varied between two politically linked ideas since the 1900’s.  The Progressive viewpoint focuses on a deeper level of understanding of mathematics by both the student and teacher.  It encourages teachers to take a lesser role and allow the students to take up more of the learning process.  The Conservative viewpoint focuses on teachers lecturing and students listening and copying notes rather than stimulating a deeper meaning of the content they are learning.

 The negative view of mathematics by parents, elementary teachers and teachers using the conservative method of teaching math have fueled three reform movements in mathematics: “Progressivist reform, the New Math and the Standards-based Math Wars”. (395)

 The Progressivist Reform in the early 1900’s focused on creating teachers who not only taught students how to achieve answers but also why the methods they used gave these answers.  It focused on involving the students in their learning rather than the common lecture method so as to create students who would become individual thinkers rather than followers.

 The New Math era of the 1960’s followed on the heels of the Cold war when competition with other nations became prevalent.  The New Math created by the SMSG, upon the anxieties of politicians, resulted in a curriculum stemming from university mathematics that many teachers and parents at the time knew nothing about.  The goal of this era was to create scientists and mathematicians without regard to those who may have not been interested.

 The Math Wars beginning in the 1990’s were battles between those we did and did not agree with the standards put in place in the school system by the NCTM.  When the TIMSS published their results comparing educational levels across different countries, opinions over who’s teaching methods worked better surfaced and the battle continues today.

 I was very interested to see that mathematics has been so political for so long.  It is interesting to see where our current curriculum stems from and where these ideas got their start.  I found that the Progressivist Reform is similar to what we as teacher candidates are experiencing now.  Our idea of lecturing and only teaching the how not the why is being challenged as we speak, similarly to the movement that occurred in the early to mid 1900’s.  I find it slightly upsetting that the New Math era really focused on creating students that would become adults that would rival other nations in scientific and mathematic knowledge. It seems that no thought went into the students well being and only what would benefit the politicians and make their country look good.  I really feel that there may be no end to the Math Wars that are continuing on today.  There are too many people with a vast array of opinions that have a say in the math curriculum today for anything to be resolved in a short amount of time.

Personal Response to Interview

When we fist began this assignment I really wanted to gain an understanding of what it is like to be a math teacher and how students feel about math in general.

 

Mr.Y gave us answers above and beyond what I was expecting hear.  I found his response about how to engage students with low motivation particularly interesting.  I have always felt that I as a teacher I would want to show students that math is fun and exciting but Mr. Y made me re-evaluate at which times I should use this enthusiasm.  Perhaps because there are so many students in our schools who have a math phobia I should be careful how I place this enthusiasm about math in the classroom.  Keeping an open mind and explaining to students who aren’t highly motivated that I understand that math is difficult and not always exciting may be the proper way to approach these situations. 

 

I found Mr. Y’s tip about making use of colleagues AND students to be highly interesting.  From this I realized that using brighter students to assist those who are having difficulties is not only a way to vary explanations but also a way to have students actively engaged in the classroom.  This is something we are constantly being asked to strive for in the teacher education program.

 

I highly appreciate the answers our student provided to our questions.  I found it interesting that the student we interviewed felt she would prefer to work alone rather than participate in group activities.  This was really the opposite answer I was expecting. 

I wish we had all had the chance to speak with Sam’s sister and some other students about our questions to get a more in depth look at how a student views math.

 

I enjoyed hearing some of the other questions that were asked to both students and teachers in class on Friday.  One group asked a question about whether students should have homework and how much, to both teachers and students.  I believe homework is important as and educator and I was surprised that most of the students interviewed in that group felt so as well.  I also found it interesting how the answer to this question varied so much in the teacher responses.  It really shows you how each person takes their own personality and ideas into the classroom.  Another group asked for tips to make the classroom more engaging and one teacher responded that we should act as a discussion leader and have the students to teach themselves.  I have been having some doubts about this method, which has been so frequently referred to by all of our instructors, so I found it encouraging knowing that there are teachers in the school system who are using it.  I really hope that I will be able to incorporate this style into my own teaching style in the future.