“The world as we have created it is a process of our thinking. It cannot be changed without changing our thinking.” ― Albert Einstein

I had a very interest conversation awhile ago and I have been thinking should I mention it here or not. After cogitating on it for sometime, I have decided it is worthy of discussion. One of the main issues in maths education is the desire to teach the way you were taught, which is fine, if you were taught well. While visiting a school, I spent time with a couple of teachers discussing 2-digit multiplication. D, an older teacher who openly declares she isn’t a ‘maths person’, wanted to know why we didn’t just teach the algorithm and procedure because the rest was too confusing and a waste of time. She described the way she learnt as,

“43 x 4; 4 times 3 is 12, write down the 2, put the 1 on the doorstep, 4 x 4 is 16, plus the 1 on the doorstep. The answers 172. Done.”

I ask what the students were doing. D showed me and explained as she went:

“4 x 3 is 12, and they write down 12 on the first line. On the second line, they put down a zero; don’t know why they do that. Then they go 4 x 4 is 16 and put it in front of the zero, then they add it up. Don’t know why they have to do it this way; it takes too long and it doesn’t make sense to me.”

I ask if the students understood what they were doing and she responded,

“Yes, but it takes too long, why don’t we just teach them the quick way like I was taught.”

Another teacher was trying to help me explain the need for conceptual as well as procedural knowledge, but D wasn’t having a bar of it; too long, not fast enough. When I asked if she knew why the zero was written down before the ‘16’, she wasn’t particularly interested because she didn’t need it her way. When I pointed out that the 4 next to the 3 was actually 40 and so it was 40 x 4, she was a bit taken aback and ‘yeah, yeah, I knew that.’

Someone came in at that point and ask me a question about the assessment they were doing, so I wandered off. When I came back, the other teacher was trying to explain the lattice model of multiplication to her. That went down like a lead balloon!

It is argued that teachers’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate from previous traditional learning experiences mainly during schooling. Once acquired, teachers’ beliefs are eventually reproduced in classroom instruction. It is also argued that, due to their conservative nature, educational environments foster and reinforce the development of traditional instructional beliefs. Although there is evidence that teachers’ beliefs influence their instructional behaviour, the nature of the relationship is complex and mediated by external factors.

Handal, B. (2003). Teachers’ mathematical beliefs: A review. The Mathematics Educator, 13(2).

If this is the case, how do you go about changing beliefs and attitudes? The last sentence perhaps gives hope. Beliefs can be changed but it takes time, energy, persistence, and a willingness to look at another way that may be difficult to master. This is not only relevant to older teachers, with many young teachers happy to teach the way they were taught, particularly if they are not overly confident when teaching maths.

Some of my most successful interactions have been with experienced teachers. Breaking down the initial barriers can be a challenge, (sitting arms folded, staring at me, thinking ‘what does she know that I don’t! Humph,’) but usually, when they realise, one: I’ve been where they are, and reasonably recently, and two: I do have practical, workable ideas that are something they can do, they warm to me.

A prefect example of this is teacher, J. J has been a teacher for as long as me and, when directed to meet with me at the beginning of this year, (all classroom teachers were meeting with me to discuss maths moderation tasks,) she came reluctantly. There was a definite, ‘what can you tell me that I don’t already know’ attitude. I will be honest; she made me work for my money. Trying to find out how I could support her met a brick wall and everything I offered the response was, “Yes, but…” Eventually, after what seemed like a very long 30 minutes, I finally made a break through, and “Oh, I hadn’t thought about like that.” 15 minutes later she left and I felt exhausted. The next meeting I had with J, she bounced in the door, full of stories about success and engage her students were with maths. Come the last meeting, J tells me how much she has enjoyed teaching maths this year and she never thought she’d say that. That only took a year and I really hope that J can continue to feel positive about teaching maths.

The problem is, permanent change takes longer than a year. Conventional wisdom suggest 3 – 5 years; for some it will take less, others more. Government and big business want change but they have to be prepared to go with the long haul and frequently they aren’t and to paraphrase Eddie Mercury and Queen:

They want it all and they want it now.

Back to some maths.

How could you teach 2-digit by 1-digit multiplication and 2-digit by 2-digit multiplication for both conceptual and procedural understanding? This is how I do it:

After using MAB blocks to demonstrate the concrete model of multiplication move on to grid paper. This is directly linking multiplication and area. There are a number of games that make this link when students are learning their multiplication facts,(just put multiplication area games into Google.)

3 rows of 15 is a bit difficult to answer unless I know that I double 15 (30) and add one more group of 15 (45), so how do can I make it clearer?

Using the distributive property, split 15 into 10 + 5 (this is how 15 is represented using MAB, which you have used during the concrete model)

Now we have 3 rows of 10 and 3 rows of 5.

Knowing the commutative property, (i.e. 3 x 10 = 10 x 3 and 3 x 5 = 5 x 3), we can see (10 + 5) x 3 = (10 x 3) + (5 x 3) = 30 + 15 = 45.

Using grid paper takes up time and space, so let’s just represent it as a region.

Moving onto 2-digit by 2-digit, again using the distributive property. Often when students are learning 2-digit by 2-digit multiplication, they do the 3fives are 15, write down the 5 in the ‘ones’ place, carry the one (pink above 1 in 15), 3 x 1 is 3 plus 1 makes 4. 1 x 5 is 5; 1 x 1 is 1. Write it down; add it up.

When we encourage students to only learn procedure, this is a common mistake, as they are reading 1 ten as 1 and not taking the true value into consideration. The area model is an excellent strategy to ensure that students do not ‘forget the zero’ as they can see quite clearly that they are multiplying by 10 not 1; yet 1 times 5 is often the language used with describing 10 times 5 in the algorithm.

Distributive property and the area model of multiplication are also vital for later on.

(This is just a side note, not something I’m teaching a three/four grade.)

Once students can see how the area model works, we need to move them on to using the algorithm. Again I would do this in steps, which again leads back to the students’ workings are the very beginning of this rant.

I know for some of you, this seems over kill, but I have found these steps provide a solid basis for both procedural and conceptual understanding; students can clearly see where each product belongs in the process and what it is representing so when they finally get to the ‘short’ method, they know what they are doing and why.

You will notice I have flipped 10 and 3 around in the final area model, simply because it makes it neater.

I am finishing with the final paragraph from the paper I included earlier:

Teachers’ mathematical beliefs are seen as self-perpetuating within the atmosphere of a system that promotes progressive teaching but in fact helps in maintaining traditional beliefs and practices. It was also argued that by the time an individual enters a teacher education program, these traditional conceptions are so solidified and entrenched in their personal philosophy that change to alternative beliefs is difficult although not impossible.

Handal, B. (2003). Teachers’ mathematical beliefs: A review. The Mathematics Educator, 13(2).

In case of D, very difficult! Okay, so this was a Maths overkill blog, but sometimes I really worry about the teaching, or lack of teaching, that occurs in some of our classrooms. Now getting down off my soapbox!

Last photo is one taken with my new toy, a drone. Unfortunately everything seems to be an angle. (Yes, I know there’s a maths activity in there.)

SO it begins….

I am going to try and do a post a week for the next 8 weeks as the count down to the end of the year and the completion of my role as an AMSI ORO (Outreach Officer) begins. Today, Sunday, I am up early and at the airport to catch a flight to Adelaide, South Australia. From there I pick up a car and drive 310 km to Port Augusta. I quite enjoy the drive as, while it can be very busy, it is a good road. The wonderful people at the Majestic Oasis Apartments look out for me and if I’m later than expected they check to make sure I’m okay. As a single traveler, frequently on the road, I greatly appreciate this as it’s nice to know someone is watching your back. I always get one of two rooms from where I have taken many photos of the Spencer Gulf.

Port_Augusta6 — View from the balcony, calm Spencer Gulf.

As I said I quite enjoy the drive. Sometimes on the way back to Adelaide, if the road is really busy, full of grey nomads and road trains, I will turn off and head for Wilmington and take the route through the Clare Valley, which takes longer but is quieter and prettier. If I have time I love wandering through the back roads of the Yorke Peninsula. The settler history of South Australia is fascinating and scattered all over the Flinders Ranges and the Yorke Peninsula. As I am moving to Adelaide in the new year, I will have the opportunity to capture digitally as many of these old places as I can. Maybe I’ll have an exhibition sometime.

I have got used to driving long distances over the past 4 years. I remember my first long drives in the Territory were a lot more difficult and Tim did most of the driving. The trick is to have a good book to listen to…Yay for audio books on mobile; no changing discs. Music tends to allow the brain to wander, which isn’t great when you are driving or it puts me to sleep (also not good.) A number of my colleagues recommend podcasts particularly true crimes but not really my cup of tea.

I have been very fortunate to have had the role of ORO for 4 years. I have learnt so much about the teaching and learning (T’n’L) of maths which has made me even more passionate about the importance of increasing engagement in maths and developing teacher confidence and capacity. Some of the most interesting aspect of T’n’L that I have learnt over the past 4 years have been not the actual content or pedagogy but the neuroscience, the mindset and the psychology of learning. I have thoroughly enjoyed developing my own depth of understanding around the T’n’L of fractions, decimals, ratios, rates and proportional reasoning. I always knew these were important components of mathematics but the research that demonstrates the direct links between competence in fractions and division in years 5 and 6 and general mathematical understanding and success at years 10 and 11 was a revelation.

playdoh7 — I’ll share this activity in a later post.

It was, however, the understanding of how the brain learns, the importance of mindset and adolescents’ beliefs when it comes to learning maths that have really challenged my way of thinking about T’n’L of mathematics. Students ‘switch on and switch off’ maths and often it takes only one teacher to do either. Maths is the poor cousin to English. As maths teachers, when asked what you do, are generally greeted with comments such as “I hate maths,” “I’m hopeless at maths,” “I had the worst maths teachers, that why I can’t do maths,” “I’m not a maths person,” or something of similar ilk. Maybe occasionally you might get a story about a great maths teacher and how they really help build confidence and self-perception of the speaker as a maths learner. If you announce that you’re an English teacher, you seldom get, “I had English” or “I’m hopeless at English.” We have to address the negativity around maths that exists in the wider community. And while parents’ perceptions and beliefs play a big role, we, as educators, must strive to make maths most relevant, useful and interesting in schools. And NO I am not talking about making it FUN. Professor Di Siemon, a well respected Australian mathematics educator, talks about success breeding engagement, not the other way around.

Bells and whistles will not guarantee success. Which is a problem when an excited teacher talks about how fantastic a lesson was, and a colleague asks for the lesson so they can ‘do it’ too. The first teacher may have delivered that lesson as the 3rd, 4th even 5th lesson in a sequence with the students experiencing success along the way, building their confidence and capacity. A one-off lesson, removed from its basis, is not going to have the same impact. David Ingham, a former mentor of mine, called these the Kaos Torturer’s theory of teaching. Max is strapped to the torture table while the Kaos torturer consults the Koas Book of Torture. He finally finds something and says, “That’s a goodie.” It is obvious he’s already tried something else which hasn’t worked even though it should but Max seems immuned. Too many lessons are presented as ‘goodies’ rather than considering what is really needed.

I am going leave it there for today as I have now arrived in Port Augusta. I want to talk more about Di Siemon’s work in the Big Ideas in Number, the work of Dr David A Sousa’s on how the brain learns and Professor Jo Boaler’s work on Mathematical Mindsets. Along with research into adolescents’ engagement and disengagement in maths and maths anxiety, these 3 have had a big impact on how I think about the T’n’L of mathematics in both primary and secondary schools.

I’m finishing with a sunset, taken in Port Augusta, but not today. I am planning to try out my drone tonight which I have recently bought. Hopefully some interesting photos to come.

MERGA in Perth

So where am I this week? Perth, Western Australia. Why? For the MERGA (Mathematics Education Research Group of Australasia) Conference. This is my first MERGA conference and already I have a few new ideas and/or elaborations for lessons.

Yesterday (Saturday) was teacher day so the workshops were very hands-on and focused on research that has very practical applications to the classroom. The keynote was delivered by Professor Doug Clarke, a name any maths teacher in Australia worth their salt, should know. He shared research on many things but perhaps the thing that hit home to me was about time, in particular wait-time after asking a question. Research from 35 years ago suggested that 1 second was the average time given to students to supply and answer before the teacher reframed, reasked or redirected the question. Waiting just 2 more seconds before doing so makes a huge difference. Interestingly, someone queried the validity of still refer to this research, because: ‘The teachers in that survey (35 years ago) were from a COMPLETELY different generation. I’m leery of basing change recommendations based on it.’ I visit classrooms on a regular basis and observe teachers teach, and my response is it is a valid now as then. Teachers often feel awkward in the silence and rush to fill it, even more so at secondary level than primary. And the problem at secondary level, the students quickly learn that if no-one responds the teacher will move on or supply the answer so very little thinking needs to be done.

I went to listen to Paul Swan first up. Paul is a well known WA Mathematics educator who always has good, practical ideas (https://drpaulswan.com.au/ ) In a Paul Swan session you know you will be playing with stuff as well as learning something. The workshop was titled “Problem Solving and Reasoning in The Early Years,” so we started with some subitising with dominoes. This quickly moved on into problem solving and using language to explain your thinking. Using dominoes, making a train (each domino is a carriage) that contains 10 dots. You have to abide by the rules of dominoes i.e. like to like. Now make the longest train with 10 dots. At a much higher level than Foundation (First year of school in Australia) use 5 dominoes to make a train with 20 dots. Lots of opportunities for older student to really think about addition of multiple single digit numbers. Then we moved onto attribute blocks as basic sorting equipment but again he extended in many ways including using all blocks to complete a circle where each subsequent block had two attributes different.

The second workshop I went to was with Jana Visnovska (The University of Queensland) and Pamela Vale (South Africa) called “On supporting students’ mathematical conversations:The case of fractions as measures. I found this session very interesting as teaching fractions is of particular interest to me. They talked about introducing fractions through measurement without direct reference to fractions. When we first start to teach measurement we use informal measures to encourage students to discover the need for formalised units of measure. They have written a wonderful picture book called “Mama Khanyi and the Pots” to introduce informal uniform measures and ‘smalls.’ The stick is the measure, but there are the 2 small, which fits along the stick exactly 2 times, the 3 small, which fits along the stick exactly 3 times etc.. What is being created is essentially a linear fraction wall, but you only have one 2 small, one 3 small, one 4 small etc., rather than 2 one half, 3 one thirds, 4 one quarters etc.. you do not refer to these in their fraction form but as smalls. I am going to explore this concept further as I think it has merit and may get students thinking.

$fraction wall 1$

Area Fraction Wall

Stick with a 3 small (exactly 3 fit the stick)

The last workshop was with Doug Clarke (keynote speaker) and Anne Roche (both from ACU). The research into teaching fractions, decimals and percentages these two, along with other colleagues, have done over the years is at the heart of the professional learning my colleagues and I deliver to teachers as part of our role as Schools Outreach Officers. It was a pleasant to sit and listen to them and an reaffirmation of the work we are doing in the area of fractions, decimals and percentages, a topic as I have mentioned already, I am passionate about. It was even better having the opportunity to share how I’d had extended a couple of aspects; e.g. The use of the part of a whole shape to scaffold parts of a collection and the importance of rewording the question if it is presented solely in abstract form.

$fractions2_1$

An interesting day over all.

Ticking one off the bucket list.

Recently I had the chance to tick one thing off my bucket list. Unlike people who consider climbing Mt Everest a must on the ‘to do’ list, even when you haven’t really got the necessary fitness levels but do have the money, (http://www.traveller.com.au/overcrowding-at-mount-everest-how-the-worlds-highest-mountain-became-a-tourist-attraction-h1f0gn), mine was a little safer and cheaper. I have wanted to see the Horizontal Falls in The Kimberley seen I first heard about them.

Hyperlinked to Australia’s North West Screenshot from Google Maps

I was back in Port Hedland for another round of school visits and had a long weekend so I drove up to Broome. If you have every done that trip, it really is 6 hours of boredom. There are some interesting spots along the way if you have the time for side trips; otherwise it’s pretty much flat with the occasional dump, river and floodplain. A new book that I’m keen to listen to, I find the best way to survive, particular travel by yourself.

The journey began next morning, getting picked up at my accommodation and taken to the airport where we boarded a single engine plane to fly to Cockatoo Island. The flight path takes you out over the Dampier Peninsula.

broometocockatoo_LI — Google maps (Accessed 15 June 2019)

The dry season had officially started and there tends to be a lot of burn off occurring so the atmosphere is pretty smoky. The Dampier Peninsula is mostly flat but once you fly out over King Sound and head for the Buccaneer Archipelago where Cockatoo Island is located, the landscape and geology changes.

The flat red/green landscape of the Dampier Peninsula

Smoke Haze

Cockatoo Island is, in itself, a fascinating place; a former mine and exclusive resort, only a handful of people now live there and only during the dry season. Alan Bond built it but like so many big dreams in the eighties, it went belly up. Someone is trying to revive it at the moment.

Empty houses on Cockatoo Island

Cockatoo Island

The plane, the plane

The boat

Off we go

After landing at the airstrip we headed for the beach and the boat. Cruising the old mining wharf was interesting. The iron ore is of such high quality that the company tried to follow the seam below sea level. It didn’t end well.

Wharf for iron ore

The trip out to the Horizontal Falls is fascinating, particularly if you are interested in geology. The rocks of the Kimberly were twisted and shoved up above the sea leaving amazing structures. My enthusiasm for the rocks can be directly blamed on my mum and the years of rock hound trips we were taken on as kids. I can’t go through a cutting with interesting geological structure without commenting on it.

The Horizontal Falls are actually two narrow gaps in the surrounding cliffs. The Kimberly experiences the some of the greatest tidal movements in the world, up to 12 meters. The amount of water (a great as one million litres of water per second) that attempts to rush through these gaps is greater than they can handle do water backs up creating a waterfall, whirlpools and an exhilarating boat ride. The boats can only pass through the larger gap as the narrowest gap is only 6 meters wide. The most spectacular view of the falls is from the other side of the narrowest gap as the water really backs up. People who take a tour that side don’t get to ride through the torrent as it is too dangerous. The tour I was on came in through the wider gap so we got to go through both ways of course about 3 times.

both gaps1 — Both gaps – the narrower one is in the distance.

Approaching the first gap.

Going throug the first gap

After that we headed back to Cockatoo Island, stopping for lunch and a walk on an island where the iron ore seam rising above sea level which creates a distinct division on one island, one side the beach is black sans, while the other is sand coloured sand.

Once back, we boarded the plane and headed out towards the Horizontal Falls for a fly over before heading back to Broome. Overall it was an excellent day and it was interesting to get a different view of Cable Beach.

from the air — Horizontal Falls from above

The following day, I drove to Derby, mainly because I had never been there; always interesting to visit a place you’ve heard about. Derby wharf has one of the great tides in Australia and while the water of the King Sound is usefully crystal clear, at Derby it is muddy, caused by the sheer volume of water that gets moved around. I also visited the Prison Boab, which has an interesting if cultural scarred history. Boab trees are only found in the Kimberley and western Victoria Region of the Northern Territory.

Prison Boab History

Prison Boab

Derby wharf

A quick trip but really interesting as the difference between the countryside of the Pilbara and the Kimberley. On the long boring trip back to Port Hedland stopped and watched a large cattle drive on the outskirts of Broome. Then somewhere between Pardoo and Port Hedland right by the side of the road I witnessed an amazing number of brolgas in a waterhole. I have never seen so many in one place; it is usual 2 or 4 in one spot. I couldn’t count how many were in this water hole.

On the road to somewhere……else.

Welcome to Port Augusta, South Australia. This is my second cluster area, working in 8 schools, primary and secondary. Unlike Port Hedland, my first visit as an outreach officer in February 2016, was not my first visit to Port Augusta. Everyone and anyone who travels by road or rail across this great country Australia has a very limited number of routes; National Highway 1, the longest national highway in the world apparently (although the section called the Savannah Way is pretty rough) or National Highway 2 (via Mount Isa and Tennant Creek.) There are other 4WD routes but if you want to travel from the Pacific Ocean to the Indian Ocean most traffic either takes NH2 or the southern part of NH1 and if you take NH1 you will go through Port Augusta.

And that’s what most people do; go through Port Augusta. You stop for petrol and amenities and that’s it. When Tim and I traveled from Darwin to visit his parents in Adelaide, we went through Port Augusta. It’s the last stop before Adelaide and the first on the way home. Its nickname is Porta Gutta because to be honest, what you see from the main road doesn’t do it any justice. The landscape is not green and welcoming with the fields on the way in from all sides, filled with grey salt bush, a few sheep, the occasional emu and road kill. Port Augusta used to have a coal fired power station as well, which probably contributed to the Gutta aspect.

Once you spend some time in Port Augusta, however, you discover a typical Australian regional town with all the pluses and minuses that comes with regional living. The closure of the power stations has created a real downturn in the local economy but there have been concerted efforts to create new jobs particularly in renewal energy with solar and wind. Unfortunately one of the solar projects hasn’t ‘got up’ due to funding.

The weather can be extreme, with the temperature hitting an all time high of 49.5⁰C on 24 Jan this year and weeks and weeks of no rain and high temperatures; even though it is on the water, the humidity can be very low. Other days it is so beautifully calm and the waters of the Spencer Gulf are like a millpond. When it does rain, it can absolutely bucket down

Port_Augusta3jpg — Port Augusta in the rain

and then, later in the year, you can be blown away in a major dust storm.

Port_Augusta4 — Port Augusta in a dust storm

Where I stay in Port Augusta, at the Majestic Oasis Apartments, is a lovely place with wonderfully friendly staff. (http://www.oasisportaugusta.com.au/ ) I essentially get the same room every visit which has a great view out over the gulf. Waking up and looking out over the water is a great way to start a day when you are away from home.

The big thing about Port Augusta is that it is on the way to somewhere. It is a great base to explore the Eyre Peninsula, the Flinders Ranges, the Claire Valley, the Yorke Peninsula, Gawler Ranges National Park and beyond. Spending many a weekend in Port Augusta, I have had the opportunity to visit all these places expect Gawler Ranges, which I hope to get this year, as well as a far north as Broken Hill and as far west as the Head of the Bight.

So next time you’re on your way to somewhere else, why not spend a couple of nights in Port Augusta, even if it’s only to visit somewhere else.

PA56 — Sunset over the Spencer Gulf from Port August

Back on the road again

Had my first real trip for the 2019 with a three week stint in Port Hedland, WA, which includes South Hedland. For those of you that may not know where Port Hedland is, it is in the Pilbara. Still none the wiser, check out their website: www.porthedland.wa.gov.au

I work in the 5 primary schools and the 1 secondary school in the immediate Hedland area as well as Marble Bar, which is 200+ km south-east of Port Hedland, into the outback. Marble Bar (www.aussietowns.com.au/town/marble-bar-wa) holds a world record but not a lot else, other than some amazing rock formation in the river.

20180824_094406 — Example of the rocks at Marble Bar.

20180824_093954 — Wide angle of the De Grey River which contains the rock formations that give Marble Bar its name.

While visiting the school, I popped down the local store , which by the way, sells fantastic homemade sausage rolls and other good Australian tucker, and when I got back to the school, I sat out on the porch outside the admin block. It was 45 C and there was silence; no insect sound, no bird sound, no human sound, just the gentle, hot breeze moving through the trees. It was just too damn hot and anything with sense was indoors in the air-conditioning.

I wasn’t planning on talking about the Pilbara, but if you have never been there, it is an incredible place, probably for all the wrong reasons. It is huge, the skies are vast, the land is red and everything is big. Sometimes it green, but mostly it’s red and brown.

PH3 — The dirt and the sky – Port Hedland, Western Australia

When I’m driving around this beautiful country I listen to audio books as you can’t always rely on the radio reception; my favourite in the Pilbara is Andy Weir’s ‘The Martian’ because I may as well be there, i.e. Mars.

The roads are pretty good because there are lots of mines around the place and huge road trains, 4 trailers long,

20170226_183858 — Road train heading to Port Hedland

work the smaller mines on the road out to Marble Bar or south towards Newman. The biggest hazard is probably live stock as cattle stations are so huge the paddocks aren’t fenced. Low flying flocks of small birds can be a bit of a problem too. I once took out about 4 wild budgerigars when a flock suddenly swooped in front of the car. I was devastated and stopped and ran back to check. While I’m standing on the side of the road looking at these poor little birds, a guy stopped and asked if I was okay. When I explained, he looked at me, shook his head and said, “These things happen,” then drove off. He was probably thinking a few other things as well.

But let’s backup a bit; to get to Port Hedland, I catch a 4 hour flight to Perth from Melbourne, generally sit round for a full working day (during eastern day light saving, there is a 3 hour time difference,) then fly 2 & half hours to Port Hedland. When I visit I stay for 3 working weeks because it is far too far to go for just 10 days as you loose 2 days just getting there and back.

When I first visited in February 2016 my overwhelming memories are of bigness; the roads are big, the trains are big, the trucks are big, the ships are big…. I’m sure you’re getting the picture. The trains can be between 90 and 330 wagons long, over 3 km in length. You don’t want to get to the rail crossing when one of the long ones goes past.

Distance is relative. No, I hear you cry, a kilometer is a 1000m wherever you are, but I would argue distance and length are two different things. Let me give you an example. Late last year, my very dear Uncle Rob passed away after a prolonged battle with cancer. Unfortunately I was in Port Hedland at the time of the funeral, which was in Auckland, New Zealand. I wrote down some of my memories of Rob and sent them to my brothers who would be attending the funeral. Afterwards I was feeling emotional and I needed to see as well as talk to a friend. Fortunately, Jac, my friend and colleague, was in Karratha, so I called her up and said, “Do you want to have a coffee?” As her answer was positive, I jumped in the car and drove to Karratha for coffee and then drove back again (a 500km round trip.) Telling this to locals, their response is, “Fair enough, done that myself more than once,” while responses from family and Melbourne friends was more likely, “You did what?” So back to my argument, distance is relative; when the nearest town is 250km down the road, a 500km round trip is a quick trip, while 500km in other places is at least an overnight journey.

20180311_134437 — The Claw, just outside Hedland, at the Marble Bar turnoff.

I love working in a place like Hedland as the people are generally so welcoming. Nearly everyone is from somewhere else, which means their immediate families are not around so people make their new families and support and care each other. Some people dropped in for a couple of days and are still there 20 years later; others last 6 months at the most. It is not an easy life, living in the heat, dust and isolation but if you are prepared to give it a go it can be very rewarding. I work with a number of young graduate teachers who have ventured out into the unknown, along way from home, friends and family, but the experiences they have, both professional and personal, will shape who they become. Some view it as the best, some as the worst. Part of my job is to hopefully give them positive and rewarding insights into teaching mathematics that will stay with them throughout their teaching careers.

A monster of a storm brewing over Hedland during the wet season.

If you ever get the chance, visit the Pilbara. You might be glad to leave but everything about the environment is overwhelming

At sunset, even the industrial landscape is beautiful.

I don’t get the point of decimals. I’m more partial to fractions.

While preparing a presentation/workshop on fractions for MAV, I came across some interesting research into the role conceptual understanding of fractions plays in later mathematical achievement.

As teachers, we all know that many students struggle to understand fractions; research has identified a number of probable reasons for this. I am not going to discuss them here but will at a later date because the teaching and learning of fractions and, by extension, decimals is an area of great interest to me.

half — Thank you to Bil and Jeff Keane for wrapping up one fraction misconception so succinctly. http://familycircus.com/comics/november-24-2012/

The research I want to talk about is the work of Robert S Siegler and colleagues. A psychologist and Professor of Psychology at Columbia University, Professor Siegler specialises in children’s cognitive development in problem solving and reasoning with much of his research focused “on how theoretical understanding of mathematical development can be applied to improving children’s math learning.” (http://www.psy.cmu.edu/~siegler/) One area of mathematical development he has collaborated with other researchers from around the world over the past 10+ years on is fractions: Why are they difficult to master for many? How important is a good conceptual understanding of fractions? And how do we teach fractions successfully to ensure and enable both procedural and conceptual understanding?

In an article published in 2012, the following was reported:

Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education.¹

In another paper² essentially the same findings were evident in Belgium, China and U.S.

So, coming back to Australia and the MAV presentation, I decided to delve a little deeper into the Australian curriculum and what it had to say about fractions. While I am just using the term ‘fractions’, I am including in that description decimals, percentages and ratios.

By the end of Year 6, students recognise the properties of prime, composite, square and triangular numbers. They describe the use of integers in everyday contexts. They solve problems involving all four operations with whole numbers. Students connect fractions, decimals and percentages as different representations of the same number. They solve problems involving the addition and subtraction of related fractions. Students make connections between the powers of 10 and the multiplication and division of decimals. They describe rules used in sequences involving whole numbers, fractions and decimals. Students connect decimal representations to the metric system and choose appropriate units of measurement to perform a calculation. They make connections between capacity and volume. They solve problems involving length and area. They interpret timetables. Students describe combinations of transformations. They solve problems using the properties of angles. Students compare observed and expected frequencies. They interpret and compare a variety of data displays including those displays for two categorical variables. They interpret secondary data displayed in the media.

Students locate fractions and integers on a number line. They calculate a simple fraction of a quantity. They add, subtract and multiply decimals and divide decimals where the result is rational. Students calculate common percentage discounts on sale items. They write correct number sentences using brackets and order of operations. Students locate an ordered pair in any one of the four quadrants on the Cartesian plane. They construct simple prisms and pyramids. Students describe probabilities using simple fractions, decimals and percentages.³

By the time a student is in Year 6, more than 50% of the mathematics achievement standards are ‘fractions’ related, yet my observations, when in schools, is that fractions are part of the explicit teaching and learning sequence for less than 25% of the time; that the content around measurement and decimals are taught separately and more time is spent on solving problems involving all four operations with whole numbers, than any other component.

Teachers argue that students don’t have the whole number place value concepts secure in their heads yet the place value patterns that exist in large numbers are inherent in decimals. The fact that a pattern exist seems, at times, to allude both teachers and students. Too many procedural short cuts are taught, like moving the decimal place, rather than the conceptual understanding around the resulting pattern when multiplying or dividing by multiples of 10.

base ten system relationships

http://www.cpalms.org/Public/PreviewResourceLesson/Preview/28707

I guess what I’m trying to get at is, conceptual and procedural understandings of fractions are important in developing overall mathematical understanding and achievement. If we, as teachers, are not allocating enough time to the teaching, learning and application of fractions, we are disadvantaging our students’ future mathematical success. Whole numbers are important, don’t get me wrong, but there comes a time when most real life maths is more likely to include fractions, decimals and percentages than large whole numbers.

If we want to see significant improvements in secondary students’ overall mathematical achievement, mastery of fractions (and division) is required. Building this mastery begins at Foundation level with sharing and continues throughout primary education, culminating with over 50% of the achievement standards in year 6 focused on fractions, decimals and percentages.

It is no longer acceptable that students leave school without the foundation knowledge, skills and dispositions they need to be able to function effectively in modern society. This includes the ability to read, interpret and act upon a much larger range of texts than those encountered by previous generations. In an analysis of commonly encountered texts, that is, texts that at least one member of a household might need to, want to, or have to deal with on a daily, weekly, monthly or annual basis, approximately 90% were identified as requiring some degree of quantitative and/or spatial reasoning. Of these texts, the mathematical knowledge most commonly required was some understanding of rational number and proportional reasoning, that is, fractions, decimals, percent, ratio and proportion. ⁴

Diane Siemon (2002)

Reference:

¹ Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., … & Chen, M. (2012). Early predictors of high school mathematics achievement. Psychological science, 23(7), 691-697.

²Torbeyns, J., Schneider, M., Xin, Z., & Siegler, R. S. (2015). Bridging the gap: Fraction understanding is central to mathematics achievement in students from three different continents. Learning and Instruction, 37, 5-13.

³ ACARA, (n.d.). Australian Curriculum – Mathematics. http://www.australiancurriculum.edu.au/f-10-curriculum/mathematics

⁴Siemon, D. (2002). Partitioning—The missing link in building fraction knowledge and confidence. Mathematics~ making waves, 411.

PHOTO: HSBooth Cable Beach, Broome, WA Australia

Sums, sets and sunsets

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Mathematics has beauty and romance. It’s not a boring place to be, the mathematical world. It’s an extraordinary place; it’s worth spending time there.

Marcus du Sautoy

Professor du Sautory words ring so true to me and is something I live and breathe every working day. It, however, is at times a struggle to persuade others that it is true. I have an incredible job; along with 7 colleagues, we travel the width and breadth of this amazingly beautiful and diverse country, Australia, working with teachers, school leadership, students, parents and the general community, trying to encourage and enable people to spend time in the wonderful world of mathematics. At the same time, we get to experience some stunning, and not so stunning, parts of the country. While we outreach officers are out on the road, there is also a talented team that manages marketing, media, research, careers awareness and mentoring around the country, as we all work towards raising the profile of mathematics across every aspect of the community.

I have had this job for 3 years now and, with 2019 the last year of the project, I decided to blog my way through the next 12 months, sharing stories, thoughts, discoveries, hopes, mathematics and sunsets.

The idea for this blog came from Chris, my brother-in-law, when he suggested I write a book called, ‘Sums, Sets and Sunsets,’ about my job and travel. Why sunsets? Well, it has become a standard practice among the team to share what we refer to as ABS (Another Bloody Sunset or Sunrise) and I frequently share my ABSs on my FB page so friends and family can see the beauty that this country can throw up.

I also thought it would be a great opportunity to share the amazing journey I have been on, both professionally and personally as, while I have 35 years plus experience as a teacher, the last 20 as a numeracy specialist, the last 10 as a teaching and learning coach, I have learnt so much over the past 3 years, I seriously wish I had another 35 – if only I knew then what I know now and what I will no doubt learn in the coming year.

I decided to go with a blog because there may be others out there in this wonderful global world that are interested in what I do, I think, I hope and I see. If there are, I hope you will join me on the journey.