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July 2022

In this edition of the News you will find:

Red Square  Meet A Mathematician

Red Square  You Go Where You Look

Red Square  Supporting Curriculum Shift

Red Square  Predict A Count

Red Square  Get to Know a Cameo
     ... Crossing The Desert
     ... McMahon's Triangles 2

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  • Meet A Mathematician

    Dr. Sophie Carr is director of Bays Consulting, a company she founded in 2009. The company's current team of 12 provides data science and mathematical modelling for a wide range of clients. Sophie also:
    • is Vice President for Education and Statistical Literacy at the Royal Statistical Society
    • serves on the General Council for the Institute of Mathematics and its Applications
    and was named the World's Most Interesting Mathematician of 2019.

    In an interview with Lynne McClure, Director, Cambridge Mathematics, published in Sip & Snack - Issue 40, July 2022, Sophie was asked:
    How would you change the school curriculum, if you had the chance? Why?
    The core of her answer is:

    If there was a way to encourage maths to be learnt through exploration, where pupils realise that mathematicians are wrong far more than they are right (and that this is nothing to be ashamed of) and which helps develop their curiosity, then I think this would be fantastic.
    Teachers who build their curriculum around Working Like a Mathematician, while making teaching craft choices intended to fascinate, captivate and absorb learners, know there already is such a way.

    Dr. Sophie Carr

    Photo: Leana Catherine ... Instagram
    Find Lynne's interview in the Link List below.
    It offers further insight into this mathematician's work and may encourage you to introduce Dr. Carr to your students. Pursuit of mathematics is genderless.

    Female mathematicians and statisticians are not 'exceptions' to the rule.
    Another quote from Sophie, this time in one of her own blogs (Hard Maths and empathy) also in Link List below.
    If you choose to find the few minutes needed to read these articles, consider doing it with the Working Mathematically page in hand (see Link List below) and ticking off the statements on it that are evidenced in the two texts.
    Special thanks to Lynne McClure for allowing Mathematics Centre to quote from her interview and to Sophie Carr for permission to use her photo.

  • You Go Where You Look

    It's a lesson I learnt the hard way during my motorcycle training course some decades ago.
    First time on the training ground tarmac, which seemed large enough to land a 747, rain, cones marking a practice course we learners needed to navigate. I finished flat on my back in the slippery grass and mud some metres off the blacktop with the bike between my legs.

    "You go where you look," my trainer advised as she helped me up in front of my class.

    She was right.

    While trying to steer around the nearest cone to the edge, I had been looking down at the closeness of my front wheel to greasy grass I didn't want to go to. Head up, I should have been looking around the cone to where I did want to go which would have helped me appropriately lean the bike into and up out of the curve.

    'You go where you look' is just as true for mathematics teaching.

    If you conceive of curriculum as learning to work like a mathematician, you will look for problems as your starting point and the teaching craft likely to encourage interest in them. A mathematician's work, a mathematician like Dr. Sophie Carr for instance, begins with an interesting problem. You will travel on valuing and refining questioning, reasoning, justifying and communication skills. Content is chosen to illustrate, develop, improve and assess student ability to work like a mathematician. An expanding skill toolbox is one of the consequences.

    If you conceive of curriculum as learning skills and processes you will look for a way to introduce the topic so learners get an idea of what's coming and its connection to what's been. You will travel on to sequenced worked examples so learners can see how to do it. Content is chosen to develop, improve and assess the number of exercises the learners get right.

    • It's easy to conceive of curriculum as learning to work like a mathematician and include plenty of skill and process learning.
    • It's hard to conceive of curriculum as skill and process learning and include plenty of questioning, reasoning, justifying and communication development.

    Describing only these two ways of looking at mathematics education does not imply presenting a dichotomy. They are two conceptions from the spectrum of views which exist among teachers, educators and education influencers. They are presented as an invitation to consider the consequences that flow from your own conception of mathematics education. You go where you look. Where do you, personally or as a faculty, want to shift your curriculum?

  • Supporting Curriculum Shift

    One reason why it's easy to start with a problem and include plenty of skill and process learning is that over decades teachers have helped us build our Content Finders and the teaching notes to which they lead. In Link List below you will find:

    • Task Cameo Content Finder
      - mostly for Years 2-10 and across all curriculum strands
    • Calculating Changes Content Finder
      - mostly Years K-6 and mostly the number strand

    Both Finders are an extensive alphabetical list of content items. Each item has its own list of available Mathematics Centre tasks or activities which involve that content. Each task or activity is linked to its teaching notes.

    Each Finder begins with simple instructions for using it, so if your curriculum tells you to teach algebra or addition or... next week, it will only take a few minutes to (a) find out if we have problems or activities to support you and (b) link to their detailed teaching notes.

    Note: Our Maths At Home division doesn't have its own Content Finder yet, but the content in each of the 74 activities is listed in its Activity Library. See Link List below.

  • Predict A Count

    The power and flexibility of this activity cannot be over-emphasised. Designed for Years 2 to 8 it has also been successfully adapted to many Year 1 classes and at least one Year 11 class.

    It's built around making and testing hypotheses, recording predictions, the neutrality of the calculator, self-correction, personal challenge and journal writing.

    Look for the activity in Link List below for rules and more details, then thread it. The given link is to the Maths At Home version of the activity, which is the one most recently written. From there it links to the Calculating Changes version for additional photos and ideas.

    Thread it - that is, use it for small amounts of time 3 or 4 times a week for at least three weeks - and watch the children's confidence with pattern and number grow. Here's a few testimonies from teachers to encourage you to look further:

    • Predict A Count has been a successful warm up activity for all students, particularly our strugglers.
    • [Evidence for change is the] ...way children approach and complete activities such as Predict A Count - enthusiastic, and getting more confident when explaining their findings and some are extending themselves.
    • My Year 2 children said it was: 'Fun / Awesome / Great / Excellent / Exciting / Wonderful / Interesting / Cool / Incredible / Okay / Puts a smile on your face / Brilliant / Amazing / Good / Getting confidence and learning / Magnificent / Makes you feel smart.
    • Thanks for an interesting and thought provoking two days. I have already tried out Predict A Count on my Yr 3/4 class with great success. One student ventured into negative numbers and even bridged from 3 to -2 subtracting by 5s. Impressive stuff ... and we called them 'underground numbers' as suggested in the PD. I've shared a few ideas with colleagues and they're excited about trying them too.
    • Staff have trialled Predict A Count and students are responding to this enthusiastically, especially with challenging their own abilities.
    Crossings out here are where learning is happening.

    Johnny has started out counting by tens from zero, but the written record suggests he has been 'parroting' expected answers, rather than recording predictions and checking with the calculator. The teacher comes along and notices.

    "Johnny it looks like you have been going very well. I want you to show me how you were doing them. The first few are easy so let's start at 100. Clear the calculator and write 100 on the screen for me."

    Johnny does.

    "Now press 'plus 10' but don't press equals. Write down what you predict will be on the screen when you press equals."
    "Okay let's see what the calculator says for 100 + 10. Press equals."
    "No problems. Cross out the 200 and write what the calculator says. Now we have a chain of correct answers from the top of the page down to 110. If we add another 10 do you think the answer will be 300?"
    "Uhh, no?"
    "Write your prediction, then check it with the calculator."

    Johnny has seen enough to realise that the answers up to 500 are wrong and the discussion continues. 130 was an incorrect hypothesis too so a fresh start has been made at the top right.
    Is there evidence that learning is happening from here? Yes.
    Is the learning stabilised? No.

    "Well done Johnny. That's enough for today. You can keep going tomorrow starting from 200."

    Predict A Count is a recommended activity in Working Mathematically with Infants.

  • Get to Know a Cameo

    Task 94, Crossing The Desert
    Crossing The Desert is content free in the sense of skills and processes, but content rich in the sense of the questioning, reasoning, justifying and communication skills which are the most important tools in a mathematician's work.

    The story shell is that a message has to be taken across a desert and the messenger must return with an answer. However one person can't carry enough food for the full journey and two people can't carry enough for both to do the return journey. Given they can bury food on the way, on which day should they bury food so that the message is delivered and both can return home having eaten their portion each day.

    Whether the task used in pairs as designed, in Mathematician Teams (a model explained in the Cameo) or with the whole class, learners will need to represent the problem with materials, and/or draw diagrams, discuss and refine what the problem means, recognise there are choices and explore choices consistent with conditions they have chosen. There is no 'the solution' to find. The focus is on justifying the solutions resulting from the choices the young mathematicians make.

    This process mirrors the work Sophie Carr and her team do daily.

    Several solutions are detailed in the Cameo, which also includes a link to a wonderful story of using the task in a Year 1/2 class at Clunes Primary School over several lessons.

    In the eTask Package this task is in the 'easy' set because it only needs 24 counters and two blocks, items usually available in any school storeroom.

    Task 107, McMahon's Triangles 2
    The equilateral triangles, which are the essential materials for this task, are divided into three parts. Picture just one of these triangles without its colours. For example, just one from the first diagram on the card. One question which could be asked is How many different ways can the three spaces be coloured using just four colours? - Red, Green, Blue, Yellow. That question is tackled in Task 148, McMahon's Triangles 1. Its answer is the source of the pieces for McMahon's Triangles 2.

    In this task the pieces are arranged to form what might be thought of as hexagonal 'place mats'. Question 1 gets things started with an activity anyone can achieve.

    Question 2 adds conditions about how the place mats display their colours. Things are getting tougher. The challenge however is a doozy. Use all 24 pieces to make a big hexagon with the same conditions as Question 2. This challenge is not likely to be completed in one lesson.

    And that's terrific because here we have another parallel with the work of mathematicians like Sophie and her team.

    McMahon's Triangles 2 is meant to be available to students over time. Perhaps it can be loaned out for home use. Perhaps it's best in a Maths Club. Experience has shown that if it is available there will be students (and teachers) who want to keep at it. Advise them to keep a record of partial solutions (and failures) along the way. A mobile phone might be a useful tool.

    In the eTask Package this task is in the 'more work' set because, apart from the task card, there is a page of the triangles to colour photocopy, laminate and cut into the 24 pieces.

Keep smiling,
Green Line

Link List

  • Did you miss the Previous News?
    If so you missed information about:
    1. Learning to Write a Maths Report
    2. Flash & Show
    3. Get to Know a Cameo
      ... Planets, Cube Numbers

Did You Know?

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