2+2 != Maths

April 22, 2012

Intro to Pythagoras

Filed under: classroom,teaching — Numbat @ 11:21
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I’ve been trying a lot of new things this year and I thought I would share my most recent experiment with you all.  It was time to introduce Pythagoras to my Year 8 (accelerated) class.  Here’s how I went about it.

Lessons 1 and 2

Firstly, I introduced the notion of doubling the square via the above video, which you can view on this page.  http://mathforlove.com/2012/02/pythagorean-theorem-part-1-video/

I stopped this video a few times to get student input. The first stoppage was at the 25 second mark, where I asked the students for ideas on how we can get a square twice the size of the first.  They were reluctant to offer suggestions, but at least none of them suggested a 2×2 square.  I was a little disappointed that I couldn’t get to discuss the error of that approach, but quietly happy that they were perhaps smart enough to know that wouldn’t work.

The second stoppage was at 1:10 (where asked) and I had the students look for ways to get half that square.  For this, I handed the students a piece of square paper (I have origami paper so I used that but any square paper would work).  One student was onto the solution straight away, but some of the others struggled a little till the solution was passed around.  I found this exercise quite useful and the students seemed to like it.

The third stoppage was around the 3:00 mark where I asked students to calculate the area of the square as shown above.  For this I handed out centimeter-grid paper which I created and printed from the following site.  http://www.math-aids.com/Graphing/Graph_Paper.html  I also set the students the task of calculating the area of squares for 1×3, 1×4, 1×5 and 3×4 diagonals.

I found this an excellent task.  The students really got involved and were genuinely engaged.  I was surprised at the number of students who had some difficulty drawing these squares on the grid paper, they didn’t understand the nature of the right-angle and had no process for ensuring that the diagonals were going to the right place. They could see that they didn’t get a square (they got parallelograms generally) but couldn’t figure out where they went wrong.

We built a table on the board of the side lengths and the areas and many of the students offered suggestions on patterns they thought were evident. I think next time I do this I will pick different side lengths so that one basic (but incorrect) pattern doesn’t appear. The students didn’t manage to come up with the pythagorean theorem themselves, but were easily led to the pattern and I sensed that they were happy to accept the pattern.  At this stage, we hadn’t even discussed triangles.

For the second part of this lesson I had the students construct their own right angled triangles (using the same grid paper from above to provide the right-angles) and measure the side lengths.  I then had them fill in the a, b, a2, b2, a2 + b2 + c2, c table and verify for themselves that it all worked (within the margin of error).  This part of the lesson didn’t go quite as well as I had hoped and I need to work on something here for next time.

The above took the better part of a double lesson and we resumed the next day.

Lesson 3

On the second day (third lesson) I had the students start with the above “puzzle”.  I like to start off with puzzles and the students jumped into this one with gusto.  I think the reason they took to it so well was that it was easy and there was little “thinking” involved.  The PDF file is available for download here : pythagorean triples PDF  The students got about 2/3 of the way through this file and started complaining about the repetition, and I suspect I need to cut the sheet down some for next time.

From here, I then proceeded to discuss Pythagoras Theorem and finding side lengths of triangles.

What I’ve found is that many students are more than happy to use the process from the above table to work out all their Pythagoras questions now.  They don’t want to set out the work in what I would call a “classic manner” (formula, substitution, manipulation, answer) but will happily write the numbers, square them and work out the missing values.

While I’m happy that these students actually understand the concepts involved, they’re complaining about the amount of time it takes to complete each question and I recognise that we need to do something about this at some point in time.

Overall, I’m happy with the way this intro to Pythagoras went. The students mostly enjoyed the activities and are more than comfortable with the “arithmetic” of Pythagoras. They are struggling with the “layout” and “rigor” I require but can do the Math which is more important at this stage I believe.

For next time, I think I need to work on the transition from the “calculate the area of the squares” to “measure the sides of the triangles”.  Perhaps I can get the students to create their own diagonals and work on those.  I also need to reduce the size of the “puzzle” I gave the students and work on the transition from that puzzle to the application to Pythagoras type questions.

I’m very happy that the students actually understand the concept of the squaring of sides of the triangles but I would be happier if the students were able to introduce a little more “rigor” into their working.  Perhaps that’s just a hold over from the old me?

Comments and Suggestions are welcome.



April 14, 2012

What do you mean I don’t need a formula?

Filed under: standards based grading,teaching — Numbat @ 14:16
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I have had a very hard time introducing concepts based assessment (aka standards based grading) to my Year 11 General Maths class this year. I have always tried to teach concepts and so assessing this way was a very good fit. I’ve also tried to emulate some of Dan Meyer’s “be less helpful” ideas. To say it’s been a tough task would be an understatement.

My students are all invested in “formula, formula, formula” and copying worked examples from “cheat sheets” onto test papers – these tasks they can do, but ask them to actually think about something, anything, and they run for the hills.

Imagine my pure joy, then, when during marking of my most recent concepts test I came across the following.  Here is a student who is complaining about having no notes, no formula and yet they were able to think the problem through and get the correct answer (albeit with the wrong units)

If only more of my students would actually engage their brains and take a chance to apply some of their knowledge to their work.

Contrast the above to this second student who has clearly found the right formula and put the numbers into the correct spot, but can’t use a calculator properly or recognise that the answer the calculator has given them is totally wrong.  This is a student with very little understanding of the question or of the concept.  A “formula” driven student who has probably experienced some success in the past but is totally out of sorts when asked something outside their comfort zone.

Clearly the first student has a better understanding of the concept and the processes involved, and I would like to think that if a few more students in this class could make the same effort then they would start to see Maths in a different light.

Unfortunately, many of my students would rather expend their efforts attempting to get me to “teach them real Maths”.


P.S.  I must admit to having a chuckle over this student’s comment about the photocopier. 🙂

March 31, 2012

Maths and my Lathe License

Filed under: standards based grading,teaching — Numbat @ 19:57
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When I was in Year 9 or 10 (I can’t remember which) I studied a year of Fitting and Machining. This was the “big leagues” at my school where we were ushered into a room with many grand looking machines and the smell of machine oil permeated everything.

Before we were allowed to even go near the lathes – the ultimate tool in that particular shed – we were all required to get our lathe license. This consisted of being able to name all 108 parts of the lathe as identified by our teacher. Our teacher handed out mimeographed sheets similar to the one shown here, although luckily the ones we received had each part numbered with all 108 names listed in order as well.

The “test” consisted of a sheet with a diagram of a lathe, with all 108 parts numbered, and we were required to write in the name of the part in the numbered lines below. A few of us got together and set about “memorising” the 108 names, in order. We managed to devise a pattern, almost a song, consisting of those 108 names. It wasn’t easy, but after quite a bit of “study”, we managed to memorise all 108 names. Come test time, we regurgitated those names, in order, in writing, and “passed” with flying colors.

Not a single one of us could identify any part of the lathe by sight or by function. All we had managed to do was to learn the words of the “lathe song”, very much like the words to our favorite songs of the time or of the national anthem. If our teacher had walked up to a lathe, pointed to a part and asked us to name it we would have been exposed as the frauds we were, but he didn’t, we had our lathe licenses and were allowed to use the machines.

The parallel between my lathe license and the way that many students are taught or learn Maths has become more and more evident to me this past year. Many students learn a “mantra” for each question. They follow a sequence of steps from start to finish never really understanding what they are doing. They reach a conclusion which they neither understand nor could repeat if the question were presented to them in a slightly different manner.

On twitter recently there was a bit of discussion about how students say they like Math, but what they really like is the fact that they can memorise a procedure and regurgitate that without having to think. I am beginning to realise that many of my students have not been asked to think in Math class too often.

This year in my concept tests I have disallowed worked examples. I am getting huge resistance from my Year 11 students in particular, many of whom have been bought up on being allowed to bring their notebooks into their tests. At parent teacher interviews the other evening I had one particular student announce to both myself and her parents that of course she knew the work and had learned it, but could only “do it” if she had a worked example in front of her. She was quite serious in demanding that I allow her to bring worked examples into tests. Luckily for me, I was able to convince her parents (and hopefully, eventually the student herself) that all she had managed to learn was how to translate a worked example into a test question by substituting the test values. In essence, she has learned her “lathe song” and will quite happily repeat it when requested.

This year it is my intention to actively prevent students from learning “lathe songs”. Hopefully I can get them to actually learn the concept and in doing so get them to engage their brains a little more frequently. It’s hard, time consuming and the students don’t appreciate the efforts yet. Hopefully one day they will.

What about others? Are your students learning “lathe songs” or are they learning Maths?

March 14, 2012

Pi Day

Filed under: teaching — Numbat @ 20:46
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this year was the first time I’ve partaken in Pi Day activities. I enjoyed it, and so did my students, even if they now think I’m even stranger than before. Many of them don’t understand how I can be passionate about numbers and Math but hopefully some of that will rub off on them.

For today’s activity I had my Year 7 students calculate Pi. We measured the circumference and diameter of a dozen different circular objects and then calculated out the Pi ratio.

I was happy enough with the results, and the students were definitely engaged in the activity. What continues to amaze me, however, is how little my students understand the actual process of measuring. We used dressmaking (or tailor’s) tapes and quite a few of the students had issues. Which end of the tape to measure, what the numbers actually mean, how to read the measurement off the tape. I will put that down to failure on my part – assuming that something so basic an action would be understood by all.

I was very happy with the way they went about the task, and many of them chose to use a spreadsheet to store the results rather than pen and paper – yet another example of giving the students choice providing a surprising result.

I still need to get much better at my follow-up and debriefing from these types of activities but the more I do the more I am becoming far more confident. I still worry the students won’t engage and will go haywire but that worry is reducing with every success. I suppose the inevitable fall is just around the corner but I shall face that when it arrives.


February 23, 2012

To the moon

Filed under: classroom,teaching — Numbat @ 20:06
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what to do in Maths class period 6 on a hot day when 1/3 of the class are off on excursion? I decided it was time to try something different.

The question is “how many times would I have to fold a piece of paper for it to reach the moon?”. Obviously, you have to assume it’s a pretty big piece of paper to begin with. I started with asking the students to estimate how thick a piece of paper was, or as one student corrected me, how “thin” was the piece of paper.

I then asked them to estimate the distance between the goal posts on the playing field outside the classroom (for the non Aussie’s amongst the readers – there is no standard length of an Aussie rules field).

I then asked them to estimate how many times they’d have to fold a piece of paper to fill the playing field. Then I let them loose. Most of them figured out you have to at least measure the playing field, and at least half the groups came up with a measurement that was reasonably accurate.

Only one group managed to make an estimate which was reasonably close – they calculated 10 times. The comment by the student was that it was a “surprisingly small” number. The actual number was closer to 17, but the lowest estimate by the students was 120. I will concede that 10 is surprisingly small against 120.

However, as a group they did not manage to convert that to the moon question. They all knew the rough distance to the moon (we’d used that number in a class a few weeks ago so they had a good idea), but they all still thought of a number in the thousands range.

Although I was disappointed in the quality of the estimates (even after the preparatory work) what I took away from this is that all the students engaged in the activity, and the discussions I overheard were very encouraging. “what’s the formula for ….”, “how do you work out ….”, “what should we do here…”

When I exposed the answer, a number of the students asked “how can that be”, and a couple of students attempted to explain it to them. I even heard one use the term “exponential”. I will count that part of the exercise as a success.

I was a bit rushed in this exercise as I wasn’t sure how long it would take and how long the students would need to measure the oval and make their calculations. I think I could prepare them a little better next time round. They were in no state after the exercise to debrief and that was a little disappointing. Next time I’d like to challenge the non-believers to prove me wrong.

Overall, however, I am very pleased with the outcome and I think many of the students saw a different side to Math than sitting in a classroom answering questions from a book. I’m hoping I can do many more of these sessions this year.


February 18, 2012

Did I win or lose?

Filed under: teaching — Numbat @ 16:18
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Last year I had one particular class who just wouldn’t take to activity based learning. The Year 7 ICT course comprised quite a few activities designed to get the students to interact with each other and to pickup ICT concepts along the way. It required them to think and to participate and this particular group flatly refused to do either. Having 8 other classes all doing the same course I think I can safely say that it wasn’t necessarily the activites at fault (although I won’t discount that entirely).

The last few weeks was supposed to be Scratch programming. In particular programming up a few games in Scratch. I just couldn’t contemplate this particular group and that activity. They had been particularly uncooperative for a few weeks so I thought I’d give them an opportunity to experience the alternative – boring old lectures and research tasks. I thought to myself that one week of doing that and they’d be ready to go back to what I (and most other groups) considered the “fun” stuff.

Well I couldn’t have been more wrong. They took to the “boring” old work extremely well. They were co-operative, their behaviour was much improved and they actually produced some finished work. I thought to myself “I’ve finally managed to win!”.

Over the summer break I was able to think about this some more, however, and have come to the realisation that in all likelihood I failed. The reason they took to the “research tasks” is that it was “easy”, it required little thought and even less participation. They could do a few quick searches on the Internet and copy out some words off the screen, words that they probably didn’t even understand. It’s mindless stuff for you and I, but for these students it was what they wanted – perhaps it’s very ease and “comfort” was what they were after all along.

It seems that in the battle of wits between this class and I, they finally wore me down and achieved what they were after all along, a class where they didn’t actually have to do any learning.

What do others think?


December 19, 2011

Our first attempt at SBG

Filed under: classroom,teaching — Numbat @ 16:01
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I make this post with some trepidation. Opening up our first attempts at converting to SBG is a little daunting. How will others react? Are we on the right track or have we completely gone bananas?

A few months ago I blogged about how I’d presented SBG to my Maths staff and how they’d started to think about and adopt the ideas in a small way. Since then we’ve been making some progress and have spent many hours going over our existing “work related” courses and coming up with a list of concepts.

We still have a ways to go but we have mostly finished the concepts list for our Year 7 course and would like comments from those of you who have been down this road previously. Please be gentle – this is our first attempt and I know that we have much to improve upon.

My own thoughts are that we have too many concepts, but I’m not really sure how to condense them any further. I think I’ll figure out some of that throughout next year but perhaps others can help.

So, without further ado, here is our list of Year 7 Concepts for 2012. Please make comments below.


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 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 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.


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