Bonus Bits

Okay guys, so here is the thing: at the beginning of this semester I kind of anticipated having to write a few more blogs and so I collected a few more math quotes (and one picture) and I really can't let them go to waste. So here they are. I hope you enjoy.

If you think dogs can't count, 

try putting three dog biscuits in your pocket 

and then giving Fido only two of them. 

- Phil Pastoret

It is a mathematical fact that fifty percent of all doctors 
graduate in the bottom half of their class. 
- Author Unknown

How many times can you subtract 7 from 83, 

and what is left afterwards?  

You can subtract it as many times as you want, 

and it leaves 76 every time.  

- Author Unknown

A man has one hundred dollars

and you leave him with two dollars.

That's subtraction.

- Mae West

We do not learn by inference and deduction 
and the application of mathematics to philosophy 
but by direct intercourse and sympathy. 
- Richard M. Nixon

Pure mathematics is, in its way, 

the poetry of logical ideas. 

- Albert Einstein

One step back and Two steps forward

As this semester comes to an end and we (finally) reach the end of the course, I feel that my thoughts on teaching mathematics hasn't shifted a whole lot. That's not necessarily a bad thing, I just feel that I have already been exposed to many of the ideas brought forth in this course, therefore most of the enlightenment comes from a second glance at mathematics and a different set of glasses (as Dorothy would say). 

One idea I had that was challenged through this course was the fact that the teacher is not some all-knowing being for students to run up to and ask "is this right?" The onus really is on the student to determine if their response to an answer is "right" or not... or even if there is a "right" answer to the question. At least, it should be on the student. 

Also, through this course, we had the opportunity to participate in a math fair. If you had a chance to see me do all those activities (not to mention "practising" aka playing my game at menchies the night before) you would know how absolutely ecstatic that made me. I love games: all the games. By having this math fair we made math fun. I will definitely consider doing activities like this in my future classrooms. You just wait for the day when my students are the weirdos in high school being all like "math is fun!" and having everyone else think they're crazy for thinking that. 

Another idea I took from this course is the importance of making real life connections; making math practical. I think this is crucial if we want to create successful citizens of the world. Seeing my number sense (well, lack thereof) makes me feel like I have a duty to prepare my students better than I was prepared and to teach them practicality with math. Similarly, it is also important to have critical discussions about math. Not only will this be beneficial to their futures, it will also strengthen their connection to the topics and therefore deepen their learning experience. 

At the end of the day I think it is important to remember that math is all about problem solving and teaching math is all about creating a balance. Although all these "new" ideas are truly wonderful, we can't just throw out the idea of formalized testing and using worksheets altogether. We need to create authentic learning experiences that nurture our students' strengths and abilities, plays off of our strengths and passions as teachers and creates the best possible future leaders of the world. 

With this course in my pocket and a whole bunch of resources by my side, I not only feel very prepared to teach math, but more importantly, I feel very excited to get out there and start teaching math. So thank you Ms. (Dr.) Mary Mathematics and good luck with your future baby Einstein; you have been so positive, passionate and enlightening, it has truly been a pleasure. 

Not everything that can be counted counts, 

and not everything that counts can be counted. 

- Albert Einstein

Reviewing Resources

Although I have previously looked through various math curriculum guides along with some resources (specifically math makes sense) this past Tuesday was my first opportunity to compare and closely examine some of the mathematics resources from K-6 and it gave me a lot to consider. 

First of all, I loved the big books for kindergarten and grade one. They are cute and captivating and a great way to introduce math at an early stage. I was a little sad to see that there weren't any big books for grade two or three. 

The math resources that we looked at seemed to reflect the idea that students mature grade by grade. There is an evident jump in language found in books; this is especially evident when going from grade one to grade two. Then, in grade three, students start using text books and by grade five all the fun seems to be gone. There does also seem to be a reoccurring theme of pizza and candy when talking about math... So much for that new healthy lifestyle ideal that schools are supposed to be adopting.

Speaking about the text books... Although these can be a good tool when used properly and not too often, I found that the majority of the questions were close-ended. This type of question is good when ideas need to be practised and reinforced but I don't see it as a useful way to introduce new concepts. Students need to get thinking and relate ideas to real concrete situations, especially ones that can be found in the real-world. 

Also, from experience, I found that a few of the questions to be sort of "trick" questions. For example, today during my observation day, one question asked students to represent a number two different ways but both ways had to use decimals. Turns out they wanted one number to one decimal place (like 1.0) and the second number to two decimal places (as in 1.00). Even I had to ask the teacher about that one. 

Furthermore, I find that some concepts are laid out to be way more complicated than they need to. Like the idea of teaching four strategies to solve a problem and reinforcing the use of each strategy all the time when only one is needed and the others may make things harder for the child. That's may not be the best example, but I still stand by my point. 

I think the main point to take from all of this is to understand and know that there are plenty of resources out there for us future teachers to use but that we are not limited to just these resources. It is important that we use the curriculum simply as a guide and introduce other resources and materials as needed to suit our individual students' interests and learning needs.

The essence of mathematics is not to make simple things complicated,

but to make complicated things simple. 

- S. Gudder

Building Blocks of Math Education

Coined as a "new movement to revolutionize math teaching and learning," YouCubed is a new resource for teachers and parents alike who want to learn more about the best strategies to use when working with and teaching mathematics. Because this site is only now starting to get off the ground and won't be fully operational until a few months from now, it is hard to comment on the usefulness of the site. Nonetheless, here are some of my first impressions...

First of all, I think a site with the goal of "revolutioni[zing] math" has a lot of potential. I like the clean layout that is easy to follow. With that being said, I am not a huge fan of the fact that it is one continuous page. And, when you click on a link it opens in the same page so you have to constantly be clicking back and saving the documents you like elsewhere. That can be a pain in the butt, but thats only a minor functional detail of the site. 

When I did open up the available resources, I was interested and intrigued by what they had to say. One point that particularly stood out was in the document about Unlocking Children's Math Potential. It mentioned the importance of making mistakes. This was a point that I also noted in Sir Ken Robinson's TED Talk that I discussed a few posts ago. The thing that I liked about YouCubed's article is that it had evidence and research to support the claim that mistakes are important. 

I also noticed that a lot, if not all, of the articles were written by Jo Boaler, the driving force behind the site. I am hoping (and thinking) that as this site gets off the ground, there will be more contributors to make for a more diverse and reliable resource.

I liked that they had some ideas for math games for students but I would have liked it more if they were interactive, perhaps even designed for SmartBoards. Then again, there are already some sites like IXL that provide a great variety of interactive math games for students. (Plus IXL even directly relates to the different provincial curriculums, which is a huge bonus!)

Overall, I enjoyed this web-based resource. I like how they have an "In the News" section that keeps readers 'in-the-know' and refers to up-to-date information. I think YouCubed has excellent potential and an appealing design that will help make a mark on the mathematics education community. I look forward to checking back in a few months when the site is fully operational and ready to go. 

Somehow it's O.K. for people to chuckle about not being good at math, 

yet, if I said "I never learned to read," they'd say i was an illiterate dolt.

- Neil deGrasse Tyson

math·e·mat·ics

According to the Merriam-Webster online dictionary, mathematics is defined as "the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations." So it's basically anything and everything to do with numbers, right? But is that all mathematics really is?

When I asked my dad (who, by the way, is a high school math teacher) about what mathematics is, he quite simply told me that it is problem solving. When you do math, you solve problems. And when you think mathematically, it means you think logically; you plan and execute steps that will take you from point A to point B in a logical manner. 

Being my father's daughter, I would have to agree with him. Math isn't just about numbers, it's about problems and solutions. It's more of a way of life than a science or a school subject. Math is more than a text book of equations. 

In 2011, Get Real Math blogged about what it means to 'do mathematics.' They had some really great points which included a quote from Paul Lockhart, author of A Mathematician's Lament which states: 

“mathematics is about problems, and problems must be made the focus of a student’s mathematical life. Painfully and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process of doing mathematics — having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.”

In a more recent blog post (published in 2013), mathematician and programmer, Jeremy Kun, talked about the myth that math is solving for x. Here, he explains that "the primary skill [in regards to algebra] is in reasoning about complex problems in a principled way." Education.com also talks about what it means to do mathematics, showing how mathematics utilizes science verbs that indicate the process of "making sense" and "figuring out." 

So it seems to me that my dad is onto something... Mathematics is more than its textbook definition. Math is more than numbers. Math is problem solving. Math is life. Don't you agree? 

Go down deep enough into anything and you will find mathematics.

- Dean Schlicter

What is education really about, anyway?

Sir Ken Robinson's TED talk about How schools kill creativity:

If that is not something that gets you thinking about education, then I don't know what is.

Robinson discusses some issues in regards to our "current" education system (as of 2006 that is). The overall message is that we need to educate children's whole being rather than just their mind; we need to encourage, promote and support creativity and open our views to what intelligence really is rather than being compliant to the education system at hand. 

One idea that stands out to me is the importance of being wrong. If children are scolded too often for being wrong they will become (or maybe have already become) too scared to take risks, make mistakes and think creatively. I mean, have you talked to a child lately? Do you know how crazily imaginative those little buggers can be? One of the best things about children are their imaginations and, as a future teacher, I believe it is important to nourish and respect this creativity rather than educate them out of it. 

Robinson makes a point about how the purpose of education is to prepare students for jobs and, furthermore, to create university professors. Although I cannot completely disagree with this point, as it's no fluke that I have a French degree in a bilingual country and plan to continue my education even after I get a masters degree, I can still see where he is coming from and how this type of education system does not meet the needs of every student. In saying that, as much as I would like to tell students to follow their hearts and do what they enjoy, I don't want them living in their parents basement and working at McDonald's when they're 40 years old. There's gotta be a middle ground where education can nurture creativity while creating useful and productive members of society.  

Videos like this are great for getting me on the go. Although I often show indifference towards, well, most everything, education is an issue that I am passionate about. Watching this video in a class about teaching children mathematics just re-enforces the idea that education is not about the subjects you teach, but rather the ideas and concepts you re-enforce through teaching itself. Education is more about teaching children how to learn than it is teaching them how to count. If you don't teach children to think critically and solve problems then how do you expect them to become independent learners?

PS: Shakespeare had a dad? Tell me you never thought about that before, because I know I haven't. I suppose he had a mum too...

My History with Mathematics

When I look back at my past school experiences nothing really comes to mind. I feel like most of the memories that I have are configurations of stories that were continuously retold to me by my parents or grandparents. So, that's not such a great start...

What I can remember is that math was taught to us in a way that you would imagine school on a tv show. We, the students, were in our seats (often in small groups), and we learned from the teacher as she used the board at the front of the class to explain each concept. We had text books and homework and very little interactions and manipulatives. 

Also, when thinking about mathematics in the early grades one memory does come to mind... When I was in grade three we did those speed multiplication tests. I could never remember the solution to 8 x 8 so I always cheated off of my classmate, Stephanie. Funny thing is, if you test me now, that will probably be my quickest answer. The fact that this is the only thing I can remember about mathematics in my primary and elementary years shows how much of it really stuck with me and how big (well, little) of an impact those experiences had on my life. 

Thinking about my personal relation to mathematics, I would consider myself to be an average student. I recall getting mostly 4s on my report cards. I never remember doing an outstanding or an appalling mark on any tests, assignments or homework and that was about the only way I could judge my successes. Unfortunately, that still seems to be the case with my learning experiences, even now that I'm in university. Too much pressure is put on final grades that, often times, key concepts and outcomes of a course may decay into senseless notes that frequently become forgotten or dismissed. Useful knowledge becomes useless information. 

As I think my way through the grades and reach the high school level, I realised that I don't even know my high school math teachers off hand. They were nice and all, but not memorable. I can recall bits and pieces of experiences, all of which are fairly typical learning experience in that the teachers preached in front of the class to students seated in rows. I also remember a few in-class opportunities for small group work. (These experiences tended to result in one person already completing the problems assigned and the rest of the group just using that person's answers.)

In my first year of university, I took mathematics 1000 (intro to calculus). I had the marks and didn't want to waste two semesters working on math (to get the credits needed to apply for education). Up until that point, I thought I was decent in math; I never really had to try and yet I always got good marks. But with this course, things were different. I went to the math help centre and struggled with assignments. Needless to say, it was the last math course in my education career. 

When considering my experiences with math outside of school, I immediately thought about my summer job. Although most of my jobs have involved being a cashier, for the past couple years, I have also been using spreadsheets to balance accounts and record money. Each week I am in charge of reconciling sales and ensuring that the numbers balance out properly. I also ensure that my workplace has enough change on hand and make frequent runs to the bank. I remember one of my co-workers commenting on my work saying that they would have thought this to be a job for an accountant and I thought that was interesting. 

All things considered, I have nothing against mathematics. I see it as something practical and useful. It can be fun (especially since I like numbers and problem solving) but it can also be quite dreadful and confusing. I find I rely too much on calculators and other devices to "do the math" for me. I would like to feel more comfortable using mathematics in order to set the best example for my future students. 

P.S.: The answer is 64.

If people do not believe that mathematics is simple, 

it is only because they do not realize how complicated life is. 

- John Louis von Newmann