The Most Important Thing I Know About Mathematics Right Now

I have learned so much this semester about mathematics.  I began the semester apprehensive about my own skills in math as well as my ability to teach it.  As of right now, I feel confident that I have the ability to hone my own skills as well as make mathematics a more positive and empowering experience for my future students.

Although I have learned so much about different manipulatives and strategies to teach through problem solving, the concept that has stuck with me the most is the fact that the teacher is NOT the one who decides whether the solution to a problem is right or wrong.

wait... what?

It makes sense when you think about it.  Mathematics, more than any other subject, determines on its own whether it is "right" or "wrong".  I think that in every school subject the teacher can sometimes be viewed as the person who knows the answer to everything.  This assumption often closes the door to others who might have a different answer or explanation, and perpetuates the belief in the one correct answer.  Teaching students to own their problems by creating their own pathways to the solution as well as being able to check their answer against the problem is incredibly important - not only for cognitive growth but for self-esteem. 

On the subject of self-esteem, math seems to be where most students feel the most uncomfortable and unable.  By letting students have control of their own problems we can build in them a growth mindset - that they are in charge of their own abilities.  By telling the students who are faster at math what the "correct" answer is we leave behind the students who can get the answer with more time, and we create in those students a fixed mindset - they don't believe that they have the "natural" ability to do mathematics, and stagnate in their achievement.  

As a future teacher I think that this is one area that I will need to work on for myself.  I can imagine myself being tempted to direct students more than I should, and even give them the answer to some questions that they might be struggling with (although Jo Boaler says that struggle is incredibly important in learning mathematics).  I think that this concept is incredibly difficult for others in the class as well from what I observed over the last semester.  Hopefully we can overcome the urge to let others tell us what is right and wrong in mathematics!

This has been a very eye-opening semester for me in many ways, and although I know that I have a long way to go before I am ready to teach truly empowering mathematics, I have been given a good head-start.  Thanks to Mary for all of the great experiences and wonderful times that she has shared with us over the past three months!!

Resources in the Classroom

Looking at the resources that are available in today's Newfoundland and Labrador classrooms was a surprising experience for me.  I wasn't quite sure what to expect, besides maybe a lot of drill-work beginning in the elementary grades.  I was pleasantly surprised to see a lot of differentiation in questions, as well as a lot of upper-level taxonomies (justify, explore, analyze etc).  I think that having preexisting resources as an option and even as a part-time resource can be very comforting, especially as a way of brushing up on personal math skills.  Another thing that surprised me about the resources available is the large range of picture books that students can relate to and apply to mathematics that are available in the primary grades - but completely disappear in the elementary grades.  I think that it would be beneficial for students to continue to have resources that make mathematical concepts more realistic and tangible for them instead of just the textbook.

The classroom set of textbooks is a contentious issue, and I think that we can agree that it should not be the be-all end-all for teaching and assessing mathematics in the classroom.  I think that the extensiveness of the resources given (the amount of teacher guides, reproducible and practice questions) often give educators the wrong idea that the textbook is the main way to teach mathematical concepts to students.  In the classrooms where I've seen the the highest reliance on textbook resources I've also seen the greatest frustration in math - from those who find it too easy, those who find that the problems are too confusing and then those who don't believe that they can do it so they don't even begin.  I don't think these teachers think that they are doing anything wrong - why have all of this material available if we aren't going to use it?  But, as in our mathematics textbook Elementary and Middle School Mathematics - Teaching Developmentally says we must consider our class first and create lessons that will suit their interests and abilities, which is contrary to sole use of the textbooks.  The text also points out that we cannot drag our students through a worksheet that will assess their understanding of any particular concept and then assume that they have mastered it, which I think is a myth that is propagated by textbook use.

Although I am one of those who is comforted by the availability of preexisting resources, it is also interesting to think about the possibility of that money being spent better on creating your own set of resources.  I think that when we consider our own personal strengths and weaknesses as well as the abilities and interests of our students the money we put into textbooks could be better spent on creating a mathematics program that is for them.  I think that it would benefit not only the students but also the teacher because they would be more personally invested in the subject - rather than just passing on a curriculum that they didn't create with materials that they didn't choose. 

Today's exploration of resources definitely sparked greater consideration for how resources and money are doled out in the education system, and what the implications are for us as future educators.

This Hour Has 22 Minutes Presents: An Interesting Method of Calculating Probability

Canada did win the gold... does this make his reasoning sound?

A "Mathematial Revolution"

The organization YouCubed aims to revolutionize the way mathematics education is taught through free content and tips for teachers and parents.  My favourite part of this website was the focus on how we can change the way our students think and feel about mathematics.

Mathematical Empowerment > Mathematical Failure

The section that stood out for me the most was "Unlocking Children's Math Potential".  So many attitudes about math that I recognize in myself and my past classmates were highlighted.  Some of the concepts that really resonated with me are:

 1. Mathematics as a "gift"

•   It is so easy to believe that some have the natural ability to be successful in math while others do not and will not ever achieve what the "gifted" students do.  I experienced this in grades 5-8 when I attended a school for "gifted" students.  I could not calculate many of the same problems as fast as my peers, and for the longest time I believed that because I didn't have the gift of math that I was probably selected for that program accidentally.

2. Fixed vs. Growth mindset

• Mindset is the students' idea about their own abilities and potentials.  Many students with a "fixed" mindset believe that their abilities are static, and that if they are unsuccessful in a task the first time it isn't "meant to be" because they don't have the ability to complete it.  Students who persist and learn from their mistakes have a "growth" mindset.  Something that surprised me was that even high achievers (according to the article it is mainly high-achieving girls) have a fixed mindset.  I think this is because I usually relate the fixed mindset to students who don't attempt any task beyond what they are easily capable of (I used to be one of these students!).

3.  The relationship between speed and mathematical calculations

• In my background with math speedy calculating was always something that was valued, and it is one of the reasons why I disliked math.  Research now shows that expecting speed from students in mathematics is actually counter-productive and does not show the true potential of the student (in fact, stress from timed tests causes a block in the working memory where math facts are stored).  I wish they would have known this 15 years ago!

4.  The teacher's attitude is essential to how the students view themselves in mathematics

• This point is pretty obvious but I think that educators should be reminded about it often.  Young students are so impressionable and we can't allow any of our negative feelings about any subject affect our students.

After browsing this site I feel like I am not alone in my math experiences, and I definitely feel more empowered to teach mathematics in the way that is the most beneficial to my students.

What is Math?

When I try to summon my own definition of Mathematics, I tend to think of it in terms of the subjects it encompasses: addition, subtraction, geometry, algebra etc.  I knew it was much more than that so I  asked a friend what he thought the definition of what math might be to get a second opinion. After a long pause he said “Well, I guess it might be how numbers relate to one another”.  It is pretty clear to me that our definitions of math have been constrained by what we understood math to be in school, when in fact the scope of mathematics reaches far beyond that. 

So, to find out an all-encompassing definition of what Mathematics is, I turned to the internet.  I found a simple definition on Wikipedia:

Mathematics is the abstract study of topics such as quantity (numbers), structure, space, and change. (en.wikipedia.org/wiki/Mathematics)

Within this definition we find some words that we may not immediately connect to mathematics.  For me, these words are all of the ones that come after quantity.  When considering the definition it does make sense to include structure, space, and change as part of the study of mathematics, it is not something that many students are familiar with in the K-6 classroom.  

The internet was not so kind in my search of what defines "mathematical thinking".  There are courses one can take online to learn it, and books for sale on Amazon for purchase, but ne'er definition. So I'm gonna just go for a guess: Mathematical thinking is a set of skills that one is able to transfer to many disciplines.  Hopefully I'll be able to elaborate on this after Thursday's class!

P.S. If someone else finds a good definition, let me know.  Believe it or not I am not very good at googling.

How Schools Kill Creativity

One of the components of primary/elementary education that I am really interested in is the creation of school curricula. How do we decide what our children should be taught? How do we define "success" within this curricula? Sir Ken Robinson's talk "How Schools Kill Creativity" has made me consider these questions more deeply. 

The main point that resonated with me was the rigidity of our school systems.  We pass on a one-size-fits-all curriculum in the subjects that societal standards deem the most important, and our success is measured by how well we can conform to this curriculum. There is not often a place for risk-taking in the traditional methods of teaching, and as Robinson says we "... become frightened of being wrong". 

What does this have to do with mathematics?

Of the people that I know that enjoy math, most of them say that it is because math results in "one correct answer".  How do we reconcile risk-taking in math (or any subject) with this belief? Can we do it? I don't have an answer for that.... yet.

My Math Autobiography

My mathematics experience overall has been a series of ups and downs, with some years being more successful and some aspects being more difficult than others.  Something that stands out for me as a staple in the classroom from K-6 was long, double-sided drill sheets.  We would be given up to 30 problems on each side of the sheet to solve (with multiplication, subtraction and addition problems) where the speed of calculating answers was not explicitly, but rather implicitly indicated success.  

My best memory from mathematics happened in grade four when we learned our multiplication tables through song.  I still carry those songs with me to this day (although I still have a little bit of trouble multiplying numbers by 8, I was absent that day).  My worst memory would have to be in grade 5 when we were tracking the number of word problems that we completed on a community board.  Word problems were something that I wasn't very good at and I was in one of the last places on the chart, which was an embarrassing experience for me.  I was pretty neutral on my feelings toward math until that time, and continuous experiences like that in my competitive grade 5 classroom made me believe that I was bad at math until high school. I can't recall how my teachers felt towards the subject of math, but they may have facilitated the "competitive" environment that I always felt math was.

In high school I avoided math like it was the plague.  I didn't think that I would be capable of higher-level mathematics and only in my final year did I take the highest level required for entry into university.  As it turns out, the class that I so feared was the one where I felt the greatest success and where math became enjoyable.  The feeling of mastering a concept in math, when the process makes sense and answers become easier to find no matter how complicated, is very rewarding.  I think what made math work for me in that course was how the teacher explained WHY we did the things that we did.  This made the processes a lot easier to transfer to all situations.  A list of things that you must complete to come to the solution is no good when you come across an unfamiliar or slightly different problem.

Although I had a "math epiphany" I am still nervous about higher-level mathematics.  So far in university I have done only the two required courses for this program - Math 1090 and Math 1050.  Although I did well in these courses I was scared to challenge myself any further and ruin the good name I had given mathematics in my mind, and therefore did not do any electives.

I don't think I engage with math any more than the average person (calculating the cost of things etc.) and math isn't something that I feel strongly about either way, although I do appreciate its how it helps us to function in everyday situations.