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The evidence is in these sample evaluation comments from teachers in the Catholic Diocese of Canberra Goulburn. The teachers participated in our six day Working Like A Mathematician course (see Link List below) focussing on engineering 'aha' moments in number and pattern. (Emphasis in the extracts is the author's.)
Contact Doug Williams, 03 9726 8316, doug@blackdouglas.com.au to discuss 2014 professional development for your school, district or system.
How problems are solved and why we choose particular strategies is reflected upon and the use of multiple and varied ways to solve problems is encouraged. Students are encouraged to choose and use materials and strategies to investigate (often in groups).
Students are actively engaged in Maths and excited to build on previous lesson content. KIDS ARE GETTING IT - 'AHA' MOMENTS.
I have changed because I feel more confident in the way I can guide investigations in the classroom. Children deserve to feel success with Maths at their level and risk taking with their learning is encouraged and valued. I have changed because unless I change my way of thinking about how Maths is taught, the children won't change the way they think about how Maths is learnt. Changed Practice
Changed the way I think and feel about maths, how I would plan and program maths activities, what resources I would use that benefit my programming.
Why
Evidence
My planning and programming:
I have written evidence from my students to show their understanding and the continuous questioning provides oral feedback and understanding.
I have changed because I have more of an understanding of how to work mathematically and how to get students to work mathematically. I have fantastic resources to use to assist students to work mathematically.
I articulate my change in observing students being focussed and interested in the learning they are doing.
Thanks to Gina Galluzzo, Madonna Pianegonda, Fiona Pettit and the Diocese team for inviting us to work with their teachers. Every primary and secondary school in the Diocese has been represented in workshops over the past three years.
In 1984 Tim Finn suggested there's a 'fraction too much friction'. Before and since, many teachers have felt a fraction too much friction when teaching fractions, but this was not the case for the Canberra Goulburn Diocese teachers. See Link List below for stories from Year 0 to Year 6 confirming that much of the success described above occurred during planned units on fractions.
Teachers who receive Pot Boiler from Calculating Changes already know that a new video was added to Cube Tube at the beginning of the month. If you didn't know that, then perhaps now is the time to look. (See Link List below.) Jamie Kemp, St. Francis of Assisi, Calwell, took his Year 6 class and his camera into the school ground to explore Move Around.
This activity from Calculating Changes Free Tour easy to state, easy to start and involves heaps of mathematics related to number sense, sequencing on the number line, place value and decimals. It is an activity that can be used from Year 1 - Year 8, and beyond, depending on the range you choose on the number line. The children learnt lots about mathematics and Jamie's video makes it clear that he learnt lots about teaching and learning. The video includes a trial of the same activity with Year 3 and the two experiences, combined with what his school learnt, and continues to learn, from our Working Like a Mathematician course, has recreated how the school approaches the teaching and learning of mathematics.
Also, two of our Cube Tube videos are stored on the You Tube channel of the Association of Teachers of Mathematics (UK). One is built around Task 45, Eric The Sheep, and one around Task 154, 4 Arm Shapes. Both illustrate teaching craft likely to fascinate, captivate and absorb students in the process of Working Like a Mathematician. Links to the videos have been added to the cameos for these tasks. You can also access them through our Cube Tube link. See Link List below.
A surprising strategy that arose through playing with Arithmagon problems was the kids making a decision to 'share the difference' between the two bottom vertices of the triangle. What impressed me was their sense of the three equations having to be balanced. Even though they didn't quite get to the point of using those words, they implied it by "if we add on one to each of the bottom numbers, we have to take one from the top number." Using rich problems like Arithmagons reinforces the incidental 'informal' mathematical thinking that is being developed by students. Our challenge as teachers is to be sufficiently 'tuned in' to pick up the nuances!
When pressed, Matt explained further...
In the past week, I've really enjoyed my Year 7 students' reactions to working on Arithmagon problems. After a lesson of 'playing with the problem,' my students came up with the approach of picking any number for the top (say 5), and then finding the missing numbers for the bottom two vertices (13-5 and 8-5 in the example). They then turned their focus to what their total for the bottom edge was, and what the problem 'wanted the total to be.' They came up with the idea of 'sharing the difference' by halving between the two bottom vertices, which then gave them the solution immediately. Having a 'sense of power' over the problem, especially when many adults will struggle with the problem for a while, was a very powerful outcome for these students.
Ta-Daa!
An unplanned journey I went on with my students was when their initial guess for the first number required them to have a negative number in one or both of the bottom vertices. I was pleasantly surprised when they were able to deal with the directed numbers in their heads, without 'breaking stride.' The fact that the need to use directed numbers was embedded in the 'story-shell' of an Arithmagon problem meant it was a tool they just drew upon without really thinking about it. The drive to find a solution didn't get in the way of them not having met directed numbers too often before.
In thinking about the very common 'classroom chestnut' of how do we differentiate learning and cater for diversity, I thought of developing an example spreadsheet to show students. This can be used to challenge students to develop their own generalised Arithmagon spreadsheet that would solve any problem. Using speech or thought bubbles on the example can help scaffold teachers or students who have limited experience in using spreadsheets in this way.
Many students won't necessarily need to or want to pursue developing a spreadsheet solution, but it would be a significant challenge for many. An extension to this problem might be to write a computer program to achieve the same outcome using free Scratch software, available from http://scratch.mit.edu.
Please feel welcome to contact me via matt@skoss.org if you'd like to follow up on the spreadsheet idea or 'Write on-Wipe off' sleeves.
In this recent email, Dave Miller-Stinchcombe may have given you a reason, if you have Rotagrams, to talk with the art teacher(s) in your school.
Dear Ina,
I'm a maths teacher based in the UK. I have recently taken up drawing as a hobby, and have started using a Rotagram as a way of checking angles, particularly for perspective drawing and for portraiture. This works much better than a protractor, as I'm not cluttered up with numbers and can just use my eyes, plus the Rotagram has square sides so it can rest on a ledge, or be fixed to a clear surface and still used.I showed the art teacher at school what I was doing and she had never seen a Rotagram before, but loved the idea of using them to help students to draw in perspective. The Rotagram is superior to ...alternatives... for the same purpose (I feel), as well as having further applications in art. So a quick bit of googling, and I find you, who appear to be the only makers and retailers of Rotagrams I can find.
(I have included) a couple of pictures showing how the Rotagram can be used to help sight angles when making a drawing. Please forgive the drawing, I've only been learning for a few months. I hope these are useful to you.
Regards,
Dave
One of the very good reasons why Geoff Giles designed this tool to assist with learning about angles is, exactly as Dave has realised, because it isn't cluttered with numbers. And yes, Mathematics Centre is the world supplier of Rotagrams. See Link List below.
Keep smiling,
Doug.
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