Why Learn Math?

I love love love this 3rd prompt from math blogger initiation.  Students are almost always surprised by how much I love answering this question.  I've been meaning to talk about this topic for a while now, and now I have even more excuse to do it right away.

I am sure you have encountered The Question in a mathematics classroom before.  "Why do we have to learn this?"  or "When are we every going to use this?"  or some variation of this.

Some classes seem to have it pop up more than others.  Some lessons seem to call for the questions.  If you quickly google these questions, almost always Math pops up as the subject in question.

If you find yourself faced with this question often - then I have a treat for you.  I am going to go through the many ways that I have answered this question, and the many reasons why we learn math. (of course, I don't claim it to be comprehensive).

1. Math improves your critical thinking skills

This is the answer I use the most often.  There's actually multiple ways of approaching this one - because follow up questions can always be asked: "why do we need critical thinking"  Well, now doesn't that seem like an easier question to answer?  It's almost like a straw man fallacy... except they made the straw man!

Okay, so this first response is the fact that understanding mathematical concepts and ideas are ways of training the mind to think in different ways, to different extents, and within different capacities.

Last year (also my first year teaching), we did push ups together in the first lesson (we did it together because I didn't want to just make them do it.  Plus... I kind of like push ups).  I preemptively asked them the question:

"When are we ever going to need this in real life?"

Do you see where I am coming from?  With some quick prompts, students realize that it is exercising their muscles, making them healthy, promoting better self confidence and image...etc.  But then I flip it around (if they haven't done it themselves already):  That's exactly the same with math.  We might not need the specific topics covered in mathematics, but they improve our thinking skills, make us healthier, and can promote better self confidence and image...etc.

If push ups are not your thing... then maybe this next one is:
"If you can dodge a wrench, you can dodge a ball."

If you can think through difficult logical/mathematical situations, then you are preparing yourself to think through other difficult life situations.

2. Math as your defense

This response also applies to the general idea of critical thinking.  Not to mention it can be considered as part of the first response.

What do the following have in common?

They are all protective suits of some kind.  There are lots of influences around the world now that is fighting for our beliefs, effort, time, money...etc.  There are lots of advertisement and ideas that uses mathematics, mathematical concepts (and if you want to go general and tap into other subjects - technical jargon, complicated terms...etc) to persuade us.  Having a solid understanding of the concepts and ideas are a way of creating our own protective suits against these influences.  Knowledge is power.

3. Math as your offense

Similarly in the same fashion, we can also think about increasing our ability to explain things in better ways.  We can supplement our ideas, knowledge, and beliefs with creative and convincing facts.  Facts that can come from mathematics (or other subjects can be associated with this response, once again)

4. Diversifying your portfolio/Because you don't know if you'll need it/enjoy it

There are 2 parts to this response, and is tied my second favourite way to respond.

Part A

"When do I ever need this stuff?"  That's a great question that maybe noone can answer.  But wait, not even you can answer this, and that's the point.

There has been a steep academic inflation over the past few decades -- and it's not done climbing.  Jobs are harder to find (even for us teachers), academic requirements overall are increased, and general day-to-day knowledge include ideas unimaginable by humans 100 years ago.  Mathematics is the basis for sciences.  Logical thinking, which is a large part of mathematics, is the basis for almost all disciplines.  So then the answer here should be a bit clearer.  By learning mathematics, we are extending opportunities and possibilities.  Opening doors instead of closing doors.

Part B

The question of why do mathematics usually comes with a sense of resentment.  It comes with a student's history of disliking math stemming from their struggles, lowered self esteem, "bad" teachers... etc.  But that doesn't mean they actually know, for sure, that they hate math.

Afterall, how can you hate something you don't fully know about or understand?  That's right, you can't.  It's like hating an unknown person that you've never met or heard about.  Mathematics is not just one thing.  It embodies infinite ideas, possibilities, concepts, and opportunities for learning.  There is an "ah-ha!" moment at the end of the struggling tunnel that you might find absolutely delicious.  You can't give up before you get there!

We don't know what our future holds.  Maybe we will find passion in something that requires mathematics.  Maybe we will find passion in some area of mathematics.

Don't let your past haunt your future.

p.s. there is a great poster that outlines what jobs need what aspect of mathematics.

5. Mathematics is a common language

There is a main idea from this video.  And that is to obtain a common ground in which to discuss things, organize things, and understand things.  Mathematics is one of those things.  It is naturally found everywhere. It is understood in almost the same way across our globe.  It is truly a universal language.  Understanding mathematics lets you open your eyes to the beauties of the world that you may not even be aware of!

~~

These 5 responses, and variations of them, are answers that I've come up with on my own (I think a lot... maybe too much), and have tried answering with.  Sometimes it comes as a planned prelude to a lesson (in which case it might come equipped with videos or pictures), sometimes it comes as an immediate response to a student, and sometimes it comes as a lengthy discussion with a friend (who may hate math).

Let me know what you think or if you have anything to add or if you want to discuss any of them (because I love talking about this topic)!!!

Singapore Math - Successful Meeting!

As I mentioned in these last 2 posts, one of my honeymoon destinations was Singapore.  I had wanted to contact their Ministry of Education to set up a meeting to talk about their math education.  My wish was granted!  There were some small hiccups along the way, but what's important was that I got some questions answered.

I had drafted some short questions to ask, and they were the following:

1. I read that you have made problem solving the main focus.  How do you achieve this?
2. How are your secondary mathematics curriculum structured?
3. What are the sample lessons like in a typical secondary math classroom?
4. How are students assessed and evaluated?
5. What do your textbooks look like?

Unfortunately I did not get to talk to a specialist on these topics.  Instead I had a meeting with someone who was knowledgeable about everything in general.  I won't put in any of my opinions in this post.  Instead, I will just try to summarize what I've learned.

Answers:

1. This question was not answered that well.  It seems that problem solving has been made the focus.  Most teachers still have their own individual touch to the lessons, so it was difficult to identify how it is achieved within each classroom.  It seems that the problem solving is primarily driven through textbooks and/or each teacher's approaches.

2. It is an interesting way of organizing.  The curriculum is separated by topics, but there are different levels of mathematics that students can take.  In Singapore they have standardized testing, and after elementary school, an exam score is collected and those are given to parents for consideration.  Each high school specializes in different areas.  So if your child is excellent in mathematics, but not humanities, then there would be a selection of schools recommended for your child.  The scores are ways of helping this.  The specialization is similar to the structure in Universities.  On top of this, students have 3 main streams that they can enter.

N(T) Level - Normal Technical Level, I was told.  These are for students who are doing courses that would be structured to be more application based.  This takes 5 years to complete

N(A) Level - Normal Academic Level.  The in-between level.  This takes 5 years to complete.

O Level - Ordinary Level.  These are for high academic achieving students.  This takes 4 years to complete.

Within each level, like O level, there would also be differences between each classes in terms of academic rigor.

3. I was not able to get a sample lesson since these differ from class to class (as I expected, I guess?) and depends on the teacher.

4. Students are primarily assessed through tests (I cringed a bit when I heard this, but that's ok.  There are still plenty of other things I can learn from this experience)  Students have 4 semesters in each school year.  At the end of each one there is either a test or national exam.

5. I was directed to a book store that pretty much specializes in selling textbooks.  When I got there, I was extremely surprised to see that 4 years (or 5) worth of material has been condensed into either 1 or 2 books.  They are not even that thick!  I opened them up and found it to mainly be summaries of concepts, followed by exercise questions.  While the questions are well made, I was a bit surprised to see that there are no activities within the textbooks being outlined.  I am guessing that these textbooks are meant to supplement, and that the teachers do their own activities?  I bought 3 of these textbooks for reference.  I have not had the chance to go through them all yet.  Something good caught my eyes in each of the textbooks that I bought (which is why I bought them...).  They are tremendously cheap!  Less than 20 dollars per book, compared to the 100+ dollars for the textbooks that we buy here.

All in all, it was a good experience.  We had some good conversations about their education system.  There were definitely things that I liked, and aspects that I did not like as much.  It was unfortunate that I could not set up something more substantial where I would be able to see a good lesson from a teacher (similar to the Japanese lesson studies that are out there), but maybe next time.  My wife was already a bit angry at me for taking time out during our honeymoon to go meet with the Ministry of Education for 3 hours.

[Activity] Angry Birds

Recently returned from my honeymoon -- so that means I will soon be writing up lots of blog posts!  I've discovered that there is so much that I want to blog about.  I just hope I will have time to blog about it all!

This probably won't come as something new for most math teachers out there, but I thought I'd throw in how I approached the topic.

When Angry Birds first came out late 2009 - 2010, it made a huge wave.  It made me a bit jealous because my phone at the time was a flip phone that barely performed the "phone" function.  When I finally got an iPhone (I don't remember exactly when that was...), Angry Birds was one of the first apps I downloaded.  I immediately went ahead to creating activities for Angry Birds.

What I did
I had 2 classes to try my activities on last year.

This picture was waiting students as they entered the room.  "What's the first question that pops into your head?"

I gave them some time to talk, get excited, argue, discuss, then I collected their responses.

We have already covered a lot of quadratic functions before this point, so it was relatively easy to lead them to exploring that aspect.  A lot of fun things also came out - gravity, angles,...etc.

*here are some more details in what we established in act 2*

We related to graphs, x-intercept, and basically had a lot to talk about when it came to these birds.

I then used the free Angry Birds Chrome to bring up the different birds.  It was a great experience because it is browser-based.  Since I had a smartboard in the class, I was able to put it right on there, and students can use the touch features to play around with the game itself.

I let the students play with them, and have them come up with "what's happening here."

Get Google Chrome if you are interested.  It comes as a free add-on, and you can play without paying a cent!

Here are the different birds I am talking about:

A lot of them generated very interesting discussions.  Even on the front of "what would the quadratic function look like", "how did the function change when you tap them".  The blue bird, for example, they talked about transformation of functions, what the functions would look like, and why did it split in the way that it did.

I don't completely remember what else we did in those classes (I have a memory of a goldfish when it comes to certain things, which my wife constantly reminds me about - which is why evernote has been a lifesaver), so I will stop talking about it there.  There were so many great conversations that happened during class...  which is why I may move to a different way of assessing students in the future.

In any case, I will move onto:

Potential future changes

• The general idea is a catapult.  What about building a real one with a slingshot for us to do outside?  Would that take too much time?
• Use Jing and VideoPad (or some alternative video editing software) to create a video version of act 1.
• Overlay the picture on Geogebra to give it a graph and more interactivity
• Maybe leave the "other birds" for a different lesson.
• I found this picture from a great photographer who does cool picture sessions with his kids.  His name is Jason Lee and here's his Flickr photostream.  I might use this picture in the future as well!  Maybe use Geogebra to separate the kids so it can be moved (or even with smartboard software this would be easy).  This would change the function every time I move the individual pictures.

• Create test questions based on this activity.
• Something that you guys suggest!

*I did more of an elaboration here*

Spiraling through the curriculum

Have you heard about Dan Meyer's 3 act approach to math education? The 3 act idea is not an exclusive idea to Dan.  There have been other three-part lesson concepts.  In Ontario, where I reside, the idea of a three-part lesson has been frequenting the professional learning community.  It has been backed by a lot of research.  I have continued to try this since I started teaching, and I have been finding a lot of success both on the front of understanding and engagement.  (Of course, the activities that I've tried are not perfect yet, but hopefully I'll continue to improve it!)

Two people in my school board had recently started structuring the activities differently (one of whom have been trying to blog occasionally).  They called it "spiraling through the curriculum.  The idea is that they would go through the curriculum entirely more than once.  Everything is activity driven.  Everything is student driven.  Mathematical concepts are not only natural consequences of the activities - the students want, need, and enjoy learning about them.  While there are similarities, note that this is not exactly the same as Jerome Bruner's concept of a "spiral curriculum."  Bruner's spiral curriculum involves continually returning to big ideas throughout the development of a student across the grade levels.  The type of spiraling that I am talking about is a bit different because topics, strands, and expectations are revisited repeatedly within the same course.  In a way, it's a tighter spiral.

In my mind, this type of spiraling is made possible due to the structure of our current curriculum where courses are driven by strands (like major topics) and overall expectations within each strand.  (Of course, I do not claim to know about how other countries in the world are structuring their curriculum.  Which is why I am eager to find out more about other countries like Singapore.)  Unfortunately not all teachers are teaching through expectations, big ideas, or activities just yet, but that's not a topic that I want to discuss at the moment.

Let me give you an example of spiraling.  Here are the strands and overall expectations in the Ontario (Canada) grade 10 applied curriculum:

The idea is that we would not introduce "topics."  Instead, we drive the classes through activities.  Activities would sometimes be short, and contained within a few classes.  Other times, the activities could span a week.  The concepts would come naturally out of the activities.  Every expectation is hit multiple times.  The first few activities may only scratch the surface (i.e. only the main concepts are introduced, and no formulas or specific calculation processes).  Then later on the activities become more extensive, because the students would have learned more about what they can do!

This upcoming year I plan on teaching classes completely through activities using the spiraling idea.  I imagine that the core of my classes are still the same (i.e. group-oriented, activity driven), but the difference would lie in how I structure the activities.  Assessment and evaluation would also be different, which would be an interesting task.  I have a lot of ideas for new assessment and evaluation methods which I am sure I will elaborate on in the future.

Note: The reason that I emphasized the differences between Bruner's spiral curriculum, and the spiraling that I plan on doing is because I do understand and recognize the debate around spiral curriculum and mastery curriculum.  The spiraling here addresses different aspects of what I want to achieve in the classroom.  In the future I will talk about portfolios for mathematical processes that I also plan on implementing this year.

Note 2: Another important distinction to make is that these 2 teachers are also promoting doing everything through these activities.  The chalk and talk portion is minimized, and exclusive to act 2 or consolidation.

$100 Tablet Project

Just a short post because I am exhausted from the travelling.

Came across this while browsing:

It seems that the goal has been reached already!  It's such a great idea. I am so very interested in the different ways that he and his students will be using these tablets in the classroom!  The opportunities are endless, and I would love to be able to provide this for my students as well.

[Inspired] Honeymoon - Thailand

I don't seem to be able to turn off my education-oriented mind, even when I'm travelling.

Edit: I am going to start a series called "Inspired" for things that inspire me about education, since my random mind seem to be inspired a whole lot...

As mentioned, our honeymoon will take us through Thailand, Malaysia, and Singapore.  Right now we're in Thailand, and I've finally obtained some reliable internet, and a tiny bit of free time from the tours.

I've been making a list of things that have inspired me on this trip.  Because of the way my mind works, the things that inspire me may be completely tangential.

Ok let's start with the first country -- and the one that I am still in:

Thailand

• Schedule flexibility

My wife and I booked a tour for Thailand.  When I arrived I was a bit surprised to hear that the itinerary has been modified due to circumstances.  When I stepped back and thought about it -- all the important tour features are still there.  The big ideas are the important stuff that we need to deliver to students.  This is on multiple levels.  The first level is considering the main focus of the entire curriculum.  What are the important aspects of the curriculum that a specific class need to know and learn about?  Lessons can be restructured around those ideas, but it should never take over and become a rigid structure that leaves no room for creativity.  The second level is considering the main focus of math education.  What are the important transferable skills that we want students to obtain from learning mathematics?  Clearly we don't expect the specifics -- like factoring, differentiating, or integrating -- to be "practical" for every student.  What we want is for students to be able to challenge themselves to think critically and effectively.  A schedule (lessons/activities) can be modified according to situations that arises in the country (classroom), as long as the important attractions (focus of curriculum/education) are shown.

• Listening beyond what has been said

Our tour guide speaks English, but it's a bit choppy and not completely fluent.  This made me think about how we interpret emotion from other people's conversations all the time.  Our tour guide interprets some of my questions as complaints about the tour (completely mistaken!  Although tired, we are loving this trip and this country), and offers solutions and adjustments as we go along.  This reminds me of what we do as teachers.  In a class, beyond just facilitating discussions and conceptual understanding, we are constantly re-evaluating how things are going.  We are always considering whether certain concepts need to be elaborated on, reinterpreted, or re-approached.  This is precisely why I've never done a traditional lecture styled class.  While I can still readjust with an entire class while lecturing, it is more difficult to differentiate without losing students to boredom (because they already get it) or confusion (because they still don't get it).  In a group discussion, activity driven classroom, it is easier for me to differentiate by circulating, constructing activities accessible from multiple levels, ...etc.  It is also easier to listen to students individually, identify what they need, and address it immediately (either to the whole class, or to the individuals),

• Numbers as familiar concepts

I don't read or speak a word of the Thai language.  But along with the foreign language, I am also recognizing a whole lot of numbers everywhere.  By relating numbers as a familiar concept, I found that I was able to make sense out of some of the things that I am seeing.  I guess I need some picture examples... maybe I can edit this post and upload the later...  But for example, I saw a giant 5.00 with a different letter beside it.  It was on top of an overpass, so I knew that it may be something that is similar to "meters."  I confirmed this idea when I saw the same symbol along with another symbol on the high way which indicated distances (the other letter must've stood for "kilo").  This is what I took away from this experience:

1. familiar ideas obtained from mathematics (or anywhere) is useful for reinterpreting new ideas
2. numbers are universal, and the ways which we use numbers are fairly similar as well.
3. a sense of familiarity is helpful for building links to new understandings!

In any case, I will cut this short and write more about being inspired later!

(btw, this post will probably serve as an example of how random my mind is) Edit: and thus sparking a new series!

Leaving for honeymoon!

As mentioned before, my wife and I are about to go on honey moon that is long overdue. And that departure date is tonight! We will be touring Thailand, Singapore, and Malaysia!

I got interested in seeing how Singapore approaches math education since it was mentioned to me recently by someone at a math conference. I explored for a few hours and recorded what I found out. I also contacted their ministry of education last week through e-mail to see if they would be interested in setting up a short meeting while I was in the country.  It would be nice to meet and chat with them, but it's definitely not the main goal of my trip -- I have to make sure to keep that in mind.  I've been having trouble truly letting go and enjoying the summer.  I am a math education addict.

I hope to continue blogging while we're on our trip, but alas it isn't guaranteed.

Understanding Fraction Division #2 - Part 1

From what I know, there are several ways of understanding fractions.  See the first one here.

I will probably try to compile all of these methods together in an organized fashion once they're all done.  For now I will just try to put them all down.

Without further ado, here's another way of thinking about fractions.  I will divide this into 2 parts with 1 example each, since it is a bit long, and I want to be a bit more thorough than last time.

Example 1:

We can extend our thinking with whole numbers, and now think about "how many groups of  1/3 are in 6/7?

First we have to make sure we’re talking about the same whole.  (I will probably elaborate on the definition of fractions later on and talk about the differences of different “wholes” and different fraction relationships) 

Note that both shapes have the same sized “wholes.”  In other words, their equivalent of “1” is exactly the same.  This is extremely important for establishing what we will do next.

We can see that it’s difficult to answer the question “how many groups of 1/3 are in 6/7“ right now since the sections are not arranged in a way that is easy to see!  So why don’t we separate them so that they contain “subsections” that are the same?

Let’s split each seventh in three equal sections

And each third into seven equal sections

This is what basically happened to our two fractions algebraically

 

And now we count how many groups of 1/3 are in 6/7

There are 2 complete groups of 1/3, and some leftovers that can be compared as “part of 1/3”

The leftover part has 4 equal subsections, and one group of 1/3 has 7 equal subsections.  So the leftover part that we have is actually 4/7.

Along with the 2 complete groups, we add on the 4/7, and we end up with

So we have:

Photo credit: ePractice.ca

Coming up in part 2:

• a different example where the second fraction is bigger than the first fraction in the division.
• generalization

Flipped Classroom - Pros, Cons, Solutions for cons?

Recently I was reminded of the idea of the flipped classroom when I was browsing some TedTalk videos.  My urge of jumping on tangents and clicking on related links to a topic lead me to Ted Ed.

Great idea, I thought.  But I recalled some concerns I had about it when I explored it a while ago.  It's about time I put my thoughts and intangible concerns into words, and I might as well do it here!

For those who aren't familiar, flipped classroom is the movement that teacher created videos drives lessons at home, and homework is done during class.

Things I like about flipped classroom

1. students going at their own pace
2. students can replay lessons if they want
3. leaves time in classrooms for discussions and problem solving -- the really meaty stuff

I can get behind these benefits for sure.  Students vary in their ability to pick up and understand new material.  Being able to go at their own pace, and replay if necessary, is a good way to address the issue.  I am also a big fan of having more time in class to do activities (not a big fan of class time just being quiet homework time... that's just lazy), and 1 on 1 help (maybe not even from me, but from other students in the class).  But... I still have some lingering concerns.

Spelling out some concerns

1. somehow still lecture based
2. it's not about the students ideas and their thinking!
3. missing opportunity to assess students and give them feedback on their thought process
4. content and understanding depends on the quality of the video
5. immediate application of the understanding
6. student access to videos at home?
7. time taken for video creation

Of course, there is always the worry that some teachers may just take the videos as an opportunity to do nothing the classroom - but let's toss those types of teachers aside for the sake of the discussion here.  My main concern revolves around the flipped classrooms being still lecture and memorization-based.  A very prominent resource for these types of videos that are pre-made is the ones at Khan Academy.  There are a lot of valid critiques of Khan Academy out there in the blogosphere, and I don't need to reiterate all of them. But as an example, over at Rational expressions, it was pointed out that Khan videos does promote precisely what Khan claims not to want to perpetuate - memorization.  Of course, memorization vs understanding in terms of their tangled relationships with each other is a difficult one.  I will likely talk about what I think about that topic in the future.  I was browsing Khan Academy, and was surprised to find Vi Hart being associated with Khan Academy.  I've seen some of Vi Hart's videos, and I definitely wouldn't jump to associating them together with Khan videos.  Dan over at Recursive Process had the same idea:

Vi compliments Sal Khan’s videos because the are done on the fly and you can see his thinking process, but then she scoffs videos that are all about the process of math (memorizing equations and processes).
I would argue that 99% of Khan Academy math videos are focused exactly on the process of math ignoring interest, motivation, and excitement; and hence are as dry and boring as a lecture can be.

Ways to avoid concerns
Okay, so here comes the main point of this post.  I am good at standing at multiple sides of the issue, especially when I haven't selected my own stance yet.  I think my position is relatively simpler than I first imagined.  I want to incorporate the use of videos, and I want to incorporate flipped lessons.  I have been playing around with Jing, and some video editing software to be confident enough about being able to create some of these types of lessons for next year.  However, it would be completely useless if I cannot address the concerns I had for the flipped lessons.  Here are some proposed methods of mediating the concerns:

• Create the videos myself

I certainly don't think I am the best teacher in the world, and thus my videos would be the best.  I just think that if I am creating the videos, then I would be aware of all the concerns that I have (for the video, current student dynamics, exercises to go with the lesson...etc).  In addition, I would be able to control the pace.

• Avoid lecturing through the video.  Incorporate pauses for students to try problems on their own.

This is my attempt to associate student ideas and thinking into this as well.  Instead of written homework, perhaps students can use these pauses to write down some responses (or use google docs or something??), and these written artifacts will be what I look at (more likely for me to do if it's done online than written pieces of paper flying everywhere).  This also addresses the idea of being able to assess student thinking, and being able to give feedback as well.

• Leave them hanging?

This is a weird one.  I know students need closure.  But maybe this closure can be given during class?  Maybe the videos can be set in 3 part videos?  1 for hook + interest, pause for student reflection and assessment, 2nd video for content and exploration (still seems to be difficult), perhaps with lots of pauses for students to work through, and 3rd video for "the answer" or consolidation (maybe this one is going through the methods/potential methods of doing problems??  In my mind, this would be the only video that would be similar to Khan videos in the sense that it would "teach" the students).  The 3rd video would be uploaded the next day maybe? or after the lesson has been done??

• Only do this a few times in a semester?

This is to mediate the fact that I may not have a library of videos ready for the move.  Also it addresses the other concerns of being able to provide thinking opportunities in other ways.  Hopefully given more time will also mean better quality of the videos and better quality of the questions.

Some remaining concerns:

1. If students don't have access to videos at home, then this would not be a viable option in the classroom.  Even if 1 student does not have access, it would defeat the purpose.
2. What is the best way of getting students to write up their thoughts and problem solving ideas?
3. What is the best way to get feedback back to them? (maybe this can occur during consolidation the next day?)

Well that's it for now.  I am still conflicted in terms of some of the concerns I have about flipped lessons, but it won't prevent me from trying it at some point!

[AnE] Assessment and Evaluation - Introduction - my interest

Assessment and evaluation in mathematics education has been an interest of mine for a while now.  I will be opening this series of topics that explores assessment and evaluation in various ways.  I will likely try to incorporate more to this once my graduate studies (on assessment and evaluation) begins in the fall.

First, let me introduce this topic by explaining why I am interested in it:

Why more so than other aspects of mathematics education?
I have tirelessly been watching, reading, and observing teaching practices.  I take a lot of it away with me.  In fact, I manage to take away at least one good thing from even the most horrifying approaches (not labelled by me, of course).  I find that there a lot of good ways of incorporating these together into a cohesive whole, and -- even if I don't incorporate them -- they are valuable teaching methods that I may use in the dynamic classroom that changes from year to year, class to class, and day to day.  But how do we evaluate?  I don't mean to portray "getting good grades" as the ultimate goal that the students should strive for.  That notion absolutely abhors me, actually, and I will likely touch on this topic later in the future.  Good assessment and evaluation should strive to accurately describe a student's performance, and be able to inform them in a way that is beneficial to their understanding and their learning.  "Accurately describe" is the heavy phrase that I've sneaked into the previous sentence.  How do we do this? What tools can we employ? What tools should we employ?  While I am becoming more confident that my students are critically thinking in class, I often find it challenging to evaluate and quantify the examples and anecdotal evidence in the classroom.

I will be looking more at assessment and evaluation processes in future [AnE] series (couldn't label "&" so it will remain as AnE as a label)

Singapore Mathematics

My wife and I got married last year.  We're finally about to go for our honeymoon next week (woot!), and one of our destinations is Singapore.

How is mathematics taught in Singapore?

I thought about this question suddenly last night, and couldn't sleep.  After some initial reading, it seems that they use textbooks that "have a consistent and strong emphasis on problem solving and model drawing, with a focus on in-depth understanding of the essential math skills recommended in the NCTM Curriculum Focal Points."

Wait, so it's a series of textbooks?  Not quite what I wanted, but the concept of problem solving is interesting to me, so I went on their website and read a bit about it.  It does sound appealing.  I haven't invested too much time reading the specifics of what they need to say.  I think I need to see a textbook for myself, in order to decide whether I like it.

But okay, let me step back.  I wanted to find out about how mathematics is taught in Singapore!  Damn my tendencies of jumping on interesting tangents.

I then found a secondary syllabus through the Singapore Ministry of Education.  And discovered lots to read over!  It certainly looks lengthy (maybe I will read this a bit later), but I was mainly interested in how they approached "problem solving."  There doesn't seem to be details here, so I moved on.

I found an exploratory study on ERIC, which contained a lot of interesting things to read as well.

Singapore has a centralized mathematics system, with detailed and consistent implementation procedures.

This little line on page x concerned me a little bit... but the skimming process continues...

Singapore’s textbooks build deep understanding of mathematical concepts through multistep problems and concrete illustrations that demonstrate how abstract mathematical concepts are used to solve problems from different perspectives. Traditional U.S. textbooks rarely get beyond definitions and formulas, developing only students’ mechanical ability to apply mathematical concepts.  

The main idea here I can definitely get behind.  Especially when they throw out a meaty concept like "deep understanding of mathematical concepts."

And this is where I stopped skimming.  There appears to be close to 200 pages, and that's not something I can handle in one sitting!

I think I will come back later and read that more carefully.  There are some lingering concerns of mine about this (like whether the delivery of the materials emphasizes exploratory learning, or whether it's still memorization), but I am an optimist when it comes to being able to extract something useful out of anything.

To be continued...

A roundabout journey to Mathematics Education

Never could have imagined my current footsteps when I was a kid.  Being Asian Canadian, I've certainly fought against the "positive" stereotype when I was a teenager, and well into my university years.  I acknowledged the fact that I was good at mathematics, but I definitely ignored the fact that I enjoyed it.  You know the stereotype.  You've probably heard about it in your school, or maybe even in your classroom.  "Asians are naturally better at mathematics."

I have a lot of concerns that immediately associates with this stereotype.  I am not even sure how I am going to organized my thoughts.  I will begin with the following study that examines the effect of positive stereotypes on Asian-American Women.

Although Asian Americans are often characterized as a "model minority" (Kao, 1995), this characterization may in itself be quite limiting (Lee, 1996)... Ho, Driscoll, and Loosbrock (1998) found that Asian-American students who performed poorly on a mathematical test were given substantially fewer points by graders than European-American students who performed identically...[When] one's poor performance may undermine the reputation of one's group,... Asian students may experience unique performance pressures when positive stereotypes of their ethnic group are activated. 

In addition to pressures that surrounds self-identity, a large number of these students who have immigrant families have increased expectations for their children.  To be honest, I cringe a little when I hear about "the Asian A."  Sure, there are factors that contributes to this notion, but generalizing the issue -- generalizing the people -- is the worse way of approaching society.

Here's another paper that examined stereotypes.

To make it one step worse, people are internalizing these stereotypes.  They believe it.

Last year I overheard a conversation where one non-white student said to a white student that "all white people are good at skating."  Class hadn't started.  I immediately interjected.  More students came in, and it became a good conversation about stereotypes.  A lot of students thought about these issues for the first time, which was great.  It ended up cutting a bit into the first activity of our class, but I thought it was definitely worth it.

Okay, back to my experience.  I ignored the fact that I enjoyed mathematics until I am in 4th year university, finishing up my honours in Biology.  It suddenly hit me that I was enjoying mathematics so much more than all of the other courses combined.  I decided to pursue a mathematics degree from that point forward.

A different roundabout happened with Education.  I enjoyed teaching others even when I was a teenager.  I initiated the peer tutoring program in my high school, and continued to tutor students well into university.  What happened?  Why didn't I pursue teaching as a profession?  Thinking back, I grew up in a family where teachers and educators are not well regarded as a viable occupation.  In addition to this, there was also the common society impression of "Those who can't do, teach," which were extremely hurtful, and speaks to the misconception surrounding teachers (sadly sometimes perpetuated by a small number of current teachers).

I am sure a lot of you can speak to this stereotype with your own experiences.  I will just let Taylor Mali speak for me:

I don't necessarily agree with the little examples he may endorse (for example I am a big fan of group work, and don't think I can stand not having group work happen in my classes), but it captures of spirit of the stereotype and our justified reasons for breaking it.

Flashforward to the present.  After my roundabout journey, I've landed in my current self.  Now I invested my time fully in mathematics education.  My past-time is basically reading about mathematics education, and thinking about education.  It doesn't feel like a chore because I enjoy it.  Recently with this blog, I am attempted to step out into the internet world, where I know more like-minded people roam.  I am fully immersing myself in mathematics education, and I love it.

Hopefully I will see some success and interact with others like me!

Understanding Fraction Division #1

Photo Credit: http://www.magazine.ucla.
edu/depts/quicktakes
/evolutionary-disgust.jpg

Fractions.  

Most students – and people in general – cringe at the sound of this word.  It seems to draw up the worst memories they had with mathematics.  I can probably spend a long time exploring the reasons behind this hate and disgust (and I probably will in the future), but I will do something else here instead.

I have always pushed and emphasized understanding of mathematics instead of memorization.  It is a much more enriching experience for the students, and much more interesting for me as well.  So I am going to spend a few blog posts exploring the different ways of understanding and exploring fraction.

The concept of “fractions” is a rich one.  There is so much to talk about.  E.g. what is a fraction? What are the different representations of fractions?  I will probably touch on the meaning of fractions in general in the future.  

I’ve recently attended the ministry funded CAMPPP (Collective Activities for MathGAINS Personalization, Precision,Professional learning).

At the conference, the theme was on fractions “with an emphasis on representing, ordering.”  While it was an excellent – one that I would come back to next year – it left me a bit unsatisfied in terms of exploring ways of explaining dividing fractions to students.  Namely: why does it make sense to “invert” the second fraction when we divide two fractions.

Enough context, let’s get to the first way I am going to attempt to explain the division of fractions.

Understanding Dividing Fractions #1

What does it mean to have “something” divide by 2?  It means we’re taking the “something” and dividing it into 2 equal groups.

So if the “something” was a number like 6, then the 2 groups would each contain 3 members.

In this approach, we are not only trying to divide the 6 into two equal groups, each containing 3.  We are specifically asking what is in one of those equal groups. In this example, it would be 3. Let’s set up this as our way of understanding division in general.

Therefore we can say: 

(Ok now I am going to take a bit of a leap, because the following is not immediately clear to me why this would make sense in an intuitive way)

But note that we can effectively write this as:

We can see that the “1” as a denominator means we are taking one of the groups from the initial set.

If you accept the above, then we can extend this idea to dividing fractions in general: If you have any set, dividing by a fraction x/y, you are essentially dividing it into x number of equal groups, and then keeping y number of those groups as what you end up with.

Ok let’s take a different example:

Applying the same logic as before, we have a set of 18 things (in this case, 18 blue circles on your computer screen).  First we divide it into 3 equal groups.

Note that each group has 6 members.  But now we look at the denominator, and it says 10.  Extending the same pattern and way of think, this means we need 10 groups of what we had before!  In which case, we are getting more than what we started out with (and if you rearranged the equation it should make sense that there are more).

And so we actually end up with 60.  There are some additional algebraic ways of looking at this as well.  We have effectively changed our operations to

Which is hopefully a less scary representation than the fractions in the beginning.

I suspect there are additional ways of playing with this idea that makes this clearer.  There are some lingering questions about intuitive approaches to this method… but I will leave that to another time.

Read on to Understanding Fraction Division #2