2+2 != Maths

October 30, 2011

Division as sharing helps with fractions.

Filed under: teaching — Numbat @ 15:50
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Hi All,

Thanks to a number of twitter references of late I have managed to really solidify my thinking on the concept of division.  I had never really given it much thought as division came a little too easily to my mind, however there are two very subtly different concepts of division.

The first, which is commonly referred to as “chunking”, involves breaking an amount up into groups of a certain size.  So, for example, 10 ÷ 5 becomes “break 10 into groups of 5”.  This is what I have found to be the more prevalent way of thinking about division amongst high school students.

The second method doesn’t appear to have a common name, and so I am going to call it the “sharing” method.  This involves sharing the amount between a given number of groups.  So, the same sum, 10 ÷ 5 becomes “share 10 items amongst 5 groups”.

The difference is subtle, and best represented graphically, as follows.

1) 10 ÷ 5 : Thought as 10 divided into groups of 5. How many groups of 5?

2) 10 ÷ 5 : Thought as 10 shared between 5 groups. How many does each group receive?

When teaching division as the opposite of multiplication, it becomes easy for students to concentrate on the “chunking” idea of division.  10 ÷ 5 = 2 because 2 x 5 = 10, or 2 groups of 5 equals 10. As stated, I have found this to be the more prevalent way of thinking amongst high school students.  However, I think the “sharing” method is probably the more “natural” way to think of division.

If I were to give 6 sweets to a group of three children, they would very easily calculate that they would each receive two sweets.  6 ÷ 3 = 2.  However, if I handed the same group of three an unknown quantity of sweets, they would still be capable of dividing them equally.

I very much doubt that would add up the sweets then divide by three. No, they would simply dole them out one at a time to each and end up with an equal share.

The two concepts of division become very important when it comes time to introduce fractions to students.  When presented with the fraction ¼, students who’s primary concept of division is chunking find the image of ¼ difficult to understand.  How can I split 1 into groups of 4?  It just doesn’t compute.

However, students who see division as “sharing” find it easier to compute that fraction.  “Share 1 pie among 4 students” is easier – simply cut the pie into 4 equal parts and dole them out.  This is what they’ve been doing as kids all along.



September 17, 2011

Update: iPad Gaming in Math and Science

Filed under: classroom — Numbat @ 17:45
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Hi All,

a few weeks ago I was really excited about this lesson I’d prepared for my Year 7 ICT class.

As the early indicators showed, the kids were even more excited about the class than I anticipated, perhaps too excited.  During the experiments they were far more interested in the “procedure” than the “recording” and while many of them obtained some excellent data, getting that data out of them in a usable format was quite difficult.

Still, the whole thing was a huge success both with the students and for myself and I am already planning on improving it for next year.  I can also see how I can expand on this lesson for stats class in Maths by introducing an analysis and graphs of the results.

The first thing I need to do is to give the students more time. If there’s anything I’ve learned this year it’s that I am underestimating the amount of time students take to do things.  Even simple things like fill in a short survey take them a lot longer that I expect.

With that in mind, I offer the following data to anyone who’d like to analyse some real world, student collected data.  I asked the students to fill in a short survey I’d created before they did their own experiment so that I could collect some data for myself.  It was a rushed job, a last minute thought and I needed to do better.  Next year I will plan it and present it to the students better, perhaps even get them to assist with the preparation, so that the data might be more comprehensive.

If you use this data and find some conclusions please come back and share them with us here in the comments.


The data file is in excel format and has notes for each of the columns on a separate sheet.  The data file can be downloaded here iPad Experimental Data

September 9, 2011

My first problem

Filed under: problem solving — Numbat @ 11:01
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Hi All,

I awoke a little “mathy” today.The first order of business most days for me is to check the blogs and twitter feeds that I follow.  It kickstarts the brain quite nicely.

I had to watch the following video twice.

The first time I watched, just as he got into the 2nd example (26 seconds into the video), my brain suddenly kicked into gear and screamed “No way!”.  As my early morning brain grappled with what I was seeing and James continued on with his explanation, it took a few minutes for me to realise the “proof” was staring me right in the face.

I immediately wondered if my students would see through these examples as quickly (or slowly as the case may be) as I did.  So there is the basis of my problem.  The first problem I have attempted to “create” I might add.  Please feel free to let me know what you think and make suggestions for improvement in the comments.

The problem:

Watch the first 45 seconds of the above video, then ask your students to either prove or disprove the method. You could make it a little harder (or perhaps easier) if you ask them to only use the 4 examples shown.


Thanks go to @jamestanton for the video and @ddmeyer for the inspiration.

September 6, 2011

SBG takes little steps

Filed under: teaching — Numbat @ 17:33
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A few weeks ago I stood up at the Maths staff meeting and made my pitch.  “This is SBG,” I told them, “I think it’s the way we should be heading.”  Overall I thought the session went as well as can be expected when someone they don’t usually associate with “change” gets up and tells them we all need to change the way we’re doing things.

Since then there has been some progress.  A few of the staff are starting to introduce smaller concept type tests.  While not truly SBG concept tests, they have trialled shorter more frequent tests with the same concepts being repeated.  And in almost every case they’ve found that the students are responding.  The students are engaging with these tests, they’re attempting more questions and are investing in their performance.  Some students are even staying back and making up time to improve their results.

While this may not be news to many who have implemented this previously, it is certainly a surprise that such a small shift toward SBG can produce such a significant change in attitude. Whether this is just a fad that the students will tire of or whether they will have the stamina to last a whole year only time will tell, but the initial signs are good.

What is also very interesting is that some of the teachers who have made the switch, and perhaps seen the biggest response, are those who I would not have considered the “strong” teachers.  This begs the question whether an SBG based approach may help some teachers more than others?

The last thing I’d like to comment on today is that we’ve now had two sessions working on creating the concepts list.  There have been many interesting and very useful discussions on how we generate our final list and how many concepts we need.  Some teachers have even remarked how we might reasonably need a large number of concepts to accurately identify a years worth of Math instruction….

I haven’t had the heart to point out the irony in those statements.  As it stands, our “concept list” is 10 items long.  It starts with  Chapter 4 Number Patterns; Chapter 3 Lines and Angles; and ends with Chapter 13 Maps, Coordinates and Directions.  If nothing else this work has caused the staff to think about exactly what concepts we’re teaching our students and to put those concepts into greater perspective than just Chapters 1, 5, 9.


August 22, 2011

Maths must be beautiful

Filed under: Musings — Numbat @ 20:06

Just wandering around the twitter-verse tonite and came across these in two separate tweets.

Coincidence?  Me thinks not!

“The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. … There is no permanent place in the world for ugly mathematics.

– G. H. Hardy

And then this..

August 18, 2011

iPad Gaming in Math and Science

Filed under: classroom — Numbat @ 17:23
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I cannot remember ever being as thrilled by both the prospects and the response to one of my lessons.

The aim was to get students to explore different settings to assist in extending their iPad battery life.  Rather than giving them a simple research assignment, I thought why not get them to actually prove or disprove these methods, let’s conduct a (semi) scientific experiment.

So I tasked them to design an experiment. They were required to research a number of different theories on conserving battery life and pick one to investigate.  They were required to write a hypothesis and a method to test their theories, to design a data collection method, to perform the experiment and then write up a conclusion.

My ultimate aim is to collect as much of this data that I can and do some statistical analysis on the results.  This would certainly be relevant for an older Math classroom although I doubt I’ll be getting these students to do this further analysis.  (These are Year 7 students, 12/13 years old).

The response has been phenomenal. I had a hunch that the students would respond positively, but I never envisaged the response I received. During the class discussion stage, virtually every student participated and we developed the framework for our experimental method collaboratively.  Even students who wouldn’t normally participate were right in it, and one particular student who normally participates with completely random and irrelevant remarks was on track and contributing thoughtful comments.

Two students in particular, who have hardly done any work all year, were almost the first to be ready.  They had done their research and planned their experiment in record time, and were even enthusiastic when I sent them back (repeatedly) to flesh out the process for their experiment.  I don’t recall ever having to write notes to parents asking them to “allow” their child to complete their homework, but I have written a dozen or so of those so far.

This started as a bit of a crazy idea that I thought I would run with, but I’m already seeing how I can expand and extend it for future years.  I can see how it could be useful in both Science and Math class, and the results are certainly relevant for my IT classroom.

If you give this a try in your class please report back on how it works out for you.


18 September:  there’s an update here which includes feedback and data from these sessions.

August 17, 2011

Just what is my job after all?

Filed under: teaching — Numbat @ 20:12
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I presented Standards Based Grading to my colleagues in the Maths department yesterday.  I was pleasantly surprised by the reception as I was expecting a very cold one but found quite a few of them nodding their heads in agreement at stages.

I was also surprised by the questions being asked.  Thoughtful questions which indicated that they’d accepted the premise and were asking about implementation.  That probably surprised me the most.

However, I was still struck by some of the firmly held beliefs of many of my colleagues.

In particular, I made a statement that I was no longer concerned by what a student had DONE, but rather I was very interested in what a student KNEW.  This differentiation appeared to be lost on most of the staff.  Having come from a background where the “yard of work” was the main focus of the entire department and the measuring stick we drummed into our students, I suppose I shouldn’t be too surprised.

In times past our “curriculum” consisted of a detailed list of questions that the students were required to complete. I suppose the theory was that if the students completed all that work then they must have learned something, right?  Unfortunately the answer was more than often, WRONG!

As I have explored implementing SBG I have come to realise that what a student DOES is perhaps irrelevant.  It’s what a student learns or knows that is relevant.  So now I need to unlearn all those skills I’ve developed that allow me to make a student DO something, and learn a whole new set of skills that will allow me to make a student LEARN something.


August 7, 2011

Balancing Questions with Content

Filed under: Musings — Numbat @ 13:11
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Teaching the same class to multiple groups has it’s challenges and rewards.  One of the best rewards is that it provides very quick feedback to what works and what doesn’t.  You don’t need three or four years to really figure out the dynamics of a particular lesson if you teach it multiple times in the same semester.

Recently I was engaged in some group activities where the classes were divided into groups of four and the students investigated certain websites and Apps.  The task itself worked quite well and most of the students were on task and actively engaged in learning.

When it came time to the group discussion, however, it was a different story.  I had two classes at the opposite ends of the spectrum.  In one, the students responded and worked well with each other in a very informal setting.  The questions and answers were orderly and stayed on topic.  Most of the students participated in some way, even it it was just sitting back and listening to their fellow student’s thoughts.  I came out of the lesson happy that we’d been very productive and covered the content.

The other group was the totally opposite.  The students did not respond and it was virtually impossible to keep them on track.  The questions and comments were mostly irrelevant and try as I might I was unable to get any momentum in the group discussion.  Two students in particular appeared to be actively sabotaging the discussion (although I don’t think this was a conscious decision on their part) although I got the feeling from the rest of the group that if those two were not present they would still have been totally disinterested.  I left this session feeling disappointed, knowing that we’d not covered even half of the content I’d hoped to.

So my question is how to balance the informal investigative/discussion/questioning style session with content delivery.  I understand that it’s all about the learning, not the delivery, but I still worry about covering all the things I’d like to cover.


July 21, 2011

Ironic behavior

Filed under: Irony — Numbat @ 19:47

Last Monday’s staff meeting was the same as usual, but what really struck me this time was the behavior of the staff.  A very good proportion of the staff were behaving in such a way that would cause them to have conniptions if their students were to do the same thing.

This got me to thinking about all the ways in which teachers ask their students to do things that most of them wouldn’t dream of doing themselves.  I watched with interest as most had either their mobile phones out (verboten for our students) or were either ebay-ing or facebook-ing on their laptops.

I know what 99% of those involved would say if they were asked.  They would say that they didn’t want to be there, that they were bored, that the subject matter was irrelevant to them and that they just couldn’t wait till the meeting was over and they were out of there.

I wonder how many of those same teachers would see the irony in either their behavior, their responses, or both?

July 2, 2011

Simple tasks can work.

Filed under: Musings — Numbat @ 08:18

I had an interesting session in IT classes this week. As a lead-in, I set the students a simple task in groups, to draw a picture which represented the Internet.  I must admit that going into the sessions I thought the task was a little weak and I was worried about the

Walking around the class while they were doing this, I was amazed at how they reacted.  There was genuine discussion within the groups, with students sharing their ideas about the Internet with each other.  Of the 40-odd groups so far, only one managed to stray off task.

What was even more interesting was that students were asking each other questions.  These are the same students who would never put up their hands and ask a question of me.

So, the question here is this…

How do I get this level of enthusiasm and engagement in my Math classes?

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