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 22, 2012

SBG from the Trenches

Filed under: standards based grading — Numbat @ 20:51
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So I’m full swing into my first year of SBG, it’s the third week of the year and I’ve managed two concept tests with two of my classes and have the other class scheduled for their second test shortly.

I am finding that the younger students (Year 7 and 8) seem ok with it (perhaps they haven’t been too heavily indoctrinated into the alternative) but my Year 11 class is very resistant. I am trying new things in the classroom as well to align my classroom practice with SBG and this is also causing me some grief. It seems that these students (Year 11 General Maths) are even less comfortable with change than some of the staff.

On the subject of the staff, overall I think that SBG has been received well. There are those who see it as a whole lot of work for no reason (thankfully in the minority) but mostly the staff see it as a positive and other staff are reporting their students are happier with the idea.

The biggest issue my Year 11’s have is that I’m not interested in the “yard of work”. They want me to tell them do Ex 3B Questions 1 to 7 left hand side. This is what they’re used to, they will then have something to aim for and they will have pages of worked questions (or hastily copied answers) to which they can point and say “Look, I’ve done my work” (or not as the case may be).

I am reaching for something more than squiggles on pages. I’m telling them “Look, do a couple of questions, if they’re easy and you understand them then skip to the harder stuff. If you’ve got this concept nailed, go back to those concepts you don’t understand and brush up on those.”.

Perhaps I’m aiming too high, but at the moment giving the students this freedom to work at their own pace and set their own goals is too far removed from the norm for them that they simply can’t cope.

Hopefully this will change soon.


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?


February 15, 2012

How Sorry Helped

Filed under: classroom — Numbat @ 18:10
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Yesterday I had a shocker in class. A problem arose that I was unprepared for. A problem which shouldn’t have happened. A problem caused by the kids being… well kids.

And I over reacted. Badly. I didn’t yell or scream (I tend not to do that) but I jumped on them hard. I issued ultimatums and outlined consequences which were, in hindsight, unreasonable. And the kids knew it. While most just bit their lips and took it, there were a few who tried to protest and I shut them down. I could tell by their body language that they were signing out.

The thing is that if these were adults, or even senior students, I think I would have been justified in my expectations. But these students are only Year 8 and my expectations were too high.

Last nite I spent considerable time thinking about what to do. Should I bunker down and ride it out – follow through on my unreasonable expectations and issue the consequences outlined? I knew I would have a number of students simply unable to comply with my timeline, was I prepared to make them suffer unreasonably? A few years ago there would have been no question, I’d be donning the protective gear and going full tilt.

This morning I walked into class and said “Sorry”. I admitted that I had over reacted and explained to them what they had done to upset me. We set new a deadline and put in place procedures to get us through to that deadline. I reiterated the consequences for failing to fulfill their part of the bargain and offered extra assistance for anyone who needed it. When I asked if this was acceptable they were enthusiastic in their response.

While it was certainly a hard thing to do, I will count it as a huge success. The students responded very well, including those who wouldn’t even talk to me yesterday. They worked, asked questions and were generally back on track.

I know that I won’t get away with the same thing again with this class for quite some time, but I feel like I’ve prevented a very dark few weeks. Hopefully I can build on that.


February 12, 2012

I used to think I was a good teacher

Filed under: Musings — Numbat @ 12:35
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I have a confession to make. I used to think I was a “good teacher” because I could do my “prep” on the way to class, or even better, I could do my prep once I’d got to class and opened the book to where we were yesterday.

“Oh Yes! Yesterday we did 3B so today it’s 3C” I’d think to myself, greet the students then start making notes off the top of my head. I’d be thinking about what examples to put on the board while I was creating my class notes on the spot and make up some numbers along the way. Once I’d managed to create a board full of notes, I’d quickly scan the questions in 3C to make sure I’d covered all the bases.

These days, I think I’m a better teacher because I spend quite a bit more time on my prep. I will think about a particular concept for ages, in the car on the way to work, during my morning walks and at other times when the brain is sitting mostly in idle.

When it comes time to actually sit down and “do my prep” I’ve actually thought about the concepts in some detail so that the actual prep time is shorter than it might otherwise be, but still a couple of orders of magnitude longer than when I was a “good teacher”.

I find that my notes (I am trying to do fewer and fewer of these by the way) are much more comprehensive and my examples cater to a number of different aspects of each concept. But perhaps one of the best outcomes from actually planning my lessons was completely unanticipated – the students respond a whole lot better. It seems that the students know when I walk into class and “wing it” and they know when I’ve put hours into the preparation and they respond accordingly.


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.


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