Tag: fractions

“Now this is not the end.”

Now this is not the end. It is not even the beginning of the end. But it is, perhaps, the end of the beginning. Winston Churchill

First last days started this week; it’s sad, that after 4 years of visiting these school, their staff and students, there will be no, “I’ll see you next term,” as I walk out the door. Having said that, I am excited about the future so really, like Churchill’s sentiment, it is the end of the beginning. I have spent the last 4 years learning so I can confidently move to the next stage. I guess leaving the people I have met in Port Augusta isn’t too upsetting as I will be coming back in some role or another, and I will still be in contact and see many of them in and around Adelaide.

This week, I have had a number of discussion around planning for next year and what should be taught first. I have a number of views in this and most of them are not, as far as I’m aware, research based, but the result of years of teaching, talking with other teachers and the round-about of education. If you disagree, sorry and if you have research to demonstrate what I think is wrong, please point me in that direction.

I talk to teachers at all year levels about the importance of putting maths in context and my personal belief that starting with fractions, time, money and/or measurement, depending on the year level, are as good a place to begin the teaching and learning year as any.

Telling time using an analogue clock is a dying skill as most people, these days, use digital time. If you want students to learn to tell the time using an analogue clock then you have to give it a purpose. So teach it at the beginning of the year then, as the teacher, make the commitment to refer the students to the clock constantly. When they ask, “Can I go to the toilet/get a drink? What times lunch/recess/home time/PE? How much longer until…?” respond, according to level, “At quarter past/half past/quarter to/ 5 past/ 20 past etc., in 10 minutes, in five minutes….” Always look at the clock and refer the students to it, expect them to use the clock to tell the time and not view it as a room decoration. Time then becomes part of everyday life and not a 2 week unit. Once upon a time, (okay, when I was young, which was some time ago,) getting a wristwatch was considered a milestone, and it was an analogue watch, not digital. Most wristwatches are now digital; Fitbits or something similar.

Set up a classroom economy at the beginning of the year and have the students managing it by the beginning of term 2. Again this builds in money as an ongoing teaching and learning strategy rather than a ‘two week’ unit and it becomes part of everyday schoolroom life. There are lots of websites on the classroom economy as it can be adapted to work for all age levels. I have seen it working very well in a year 1/2 class, as well as in year 6 level. The sophistication of the economy is dependent on the year level, but, again, you as the teacher, must commit to it.

strip_fractions2

“Why my half bigger than your?” Reinforcing the concept that the size of the half is dependent on the size of the whole or collection.

strip_fractions3

Why fractions early on? In the national curriculum in Australia, half past and half are taught in Year 1; half, quarters and eighths of objects (whole) and collections and quarter to and quarter past are taught in Year 2; the concept of unit fractions and their multiples in year 3 along with telling time to the nearest minute, which, personal, I think is a huge jump so would focus on five minute intervals. Beyond Year 3, the curriculum begins to contain an increasing amount of content, with a lot of which is fractions, decimals and percentages focused. By the time students are in Year 6, more than 50% of the achievement standard has this focus, yet, in all honesty, these extremely important concepts are addressed as a separate unit of work maybe in term 3 for a few weeks and less than 25% of the teaching and learning time is spent on it.

Lastly, measurement. If you unpack the using units of measurement component of the Measurement and Geometry strand, the following action verbs reoccur; compare, order, measure, convert, all of which use number. Even in years 1 & 2 where uniform informal units are used there are opportunities to link measurements and number particularly with the extended subtraction concepts of missing addend and comparison.

My foot is 12 unifix long but I only have 8. How many more unifix do I need? or My height is 16 paddle pop sticks, Jane’s height is 10 paddle pop sticks. What’s the difference in paddle pop sticks?

Often strands are taught separately with students not seeing the connections between them, while teachers struggle with the ‘crowded curriculum.’ In Year 6 a content descriptor in the Measurement strand clearly states ‘connect decimal representations to the metric system’ while in the Number strand ‘multiple and divide decimals by powers of 10’ directly supports ‘convert between common metric units…’

Off back to Melbourne for 2 nights then back on the plane for my final trip to Port Hedland. Safe travels everyone.

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The stillness that settles at times over the Spencer Gulf, SA
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Same view at low tide and when it’s blowing a gale and dust is whipped up

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.

time

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.

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.

An interesting day over all.