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April 2024

In this edition of the News you will find:

Red Square  New eTask School

Red Square  Negative Approach to Negative Numbers

Red Square  Using Computers Differently

Red Square  Making a Maths Mat with Matt

Red Square  Get to Know a Cameo
     ... Dice Differences
     ... Money Charts

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Green Line

  • New eTask School

    Welcome to the staff of Green Oasis School, Shenzhen, China. Kate and the team have just taken delivery of their eTask Pack (see Link List below). Perhaps later in the year we will hear from them about creating their Task Library guided by the package, which includes support for an in-house professional learning program. We'd love to see photos of the students working with tasks too.

  • Negative Approach to Negative Numbers

    Near the end of last term I was asked by a parent if I could help her Year 9 student with negative and positive numbers. The school uses Maths Pathways and something was said about having to get an exit ticket to finish a module and the questions didn't make any sense.

    I was in the area the next day so I dropped in to see what was going on.

    The ticket screen was well gone by then, but the parent had written it on a sticky note. Apparently you can try for an exit ticket as many times as you like, so parent and student worked together guessing and trying and returning to the screen and guessing again until they finally guessed well enough to win the ticket.

    They had been working like a mathematician using the strategy of guess, check and improve to solve the problem of how to pass the software. But they knew they had no idea how negative and positive numbers work.

    Not surprising really because the equations as written are mathematically incorrect and pedagogically unsound.

    Yes, I would have seen them written like this when I was a Year 9 student, but that was the 1960s. I thought maths education had progressed a little since then.

    Given these three equations were all I had to go on, I started by asking the student what they thought the first equation was telling them to do.

    Well, there's a bracket, so I have to do that first. So I have to subtract 9. But there's nothing to subtract it from so I can't start. I don't get it.
    Therein lies both the mathematical incorrectness of the software and its pedagogical weakness.

    • Subtraction, symbolised by the horizontal stroke at 'waist level', is a binary operation. By definition it is an operation that involves two numbers and the symbol is placed between the two numbers to indicate what to do.
    • Considering the first equation, the first time the symbol is used (reading from left to right as convention demands) it is used correctly (assuming for the moment that the next part of the equation is also a number).
    • The second time it is used, in (- 9), the symbol is, by definition, not understandable, as the student has correctly explained, because there is not another number preceding it within the bracket.

    Mathematically incorrect ... and since this exercise is an exit ticket from a module that supposedly has been focussed on learning the arithmetic of integers, one must assume that the whole module includes this incorrectness.

    Last century, as referred to above, when such equations appeared on blackboards and in textbooks, they were accompanied by four rules...

    I wasn't expected to understand integers, or their arithmetic. I was expected to memorize these rules, do what the teacher told me was the way to apply them and get right answers.

    • plus plus = plus
    • plus minus = minus
    • minus plus = minus
    • minus minus = plus

    This student's experience seems to be the modern day equivalent of post-second world war Australian mathematics teaching. Pedagogically unsound:

    • No concept of numbers having two types/states - positive or negative.
    • No consideration of the importance of the additive identity element - the number zero - as critical to the understanding of these states.
    • No concrete, kinaesthetic, visual or whole body involvement representations to help build a brain picture of how these numbers work.
    • No conversation with the teacher other than it being directed to learning what I have been told to learn.
    • No discussion with peers to help share the struggle of trying to 'figure this out' - just 'do your own work'.

    Since the 1970s, which was also last century, all of the items listed as missing have been available to mathematics teachers - and in more than one way. See Protons & Anti-Protons in the Link List below for one of these ways. I turned to this model to help this student because within it is a link to the Maths At Home activity of the same name, which the student and parent could continue together after I left.

    It's true that I have not seen the whole module that led to this exit ticket exercise. It may be full of wonderful diagrams and snazzy software representations designed to support learning. However, if that's true it has failed at least one student and it has presented at least one mathematically incorrect screen, which should have looked like this:

    +5 - -9 = +14
    ... positive 5 take away negative 9 = positive 14 ...
    -3 - -7 = +4
    ... negative 3 take away negative 7 = positive 4 ...
    -9 - -5 = -4
    ... negative 9 take away negative 5 = negative 4 ...

    I suspect the main reason why the screen didn't look like this is that there is no simple keystroke, for either the software designer or the responding learner, that will produce the positive and negative symbols. However, if the software is not up to the task:

    • do we keep the module to the detriment of the learner, or
    • drop the module for a pedagogical alternative that will benefit the learner?

  • Using Computers Differently

    Computers can be used in ways other than self-paced learning that isolates students from each other. They can be used to bring students together to learn, either in small groups or as the whole class. Picture Puzzles are one way to do this.

    Picture Puzzles are self-paced PDF slide shows built around

    one screen, two learners, concrete materials and a challenge.

    One learner cannot have a mathematical conversation with a screen. Two learners can have a mathematical conversation about a screen. Picture Puzzles generate that conversation through supported, achievable challenges that involve learners using their hands off the screen.

    36 investigations across 7 menus.
    Something for all year levels from 2 to 12.

    Some are designed as Investigation Guides for specific tasks, some develop key concepts and skills such as multiplication and fractions. All include an emphasis on visual learning, hence their name.

    From the Picture Puzzles starting page (see Link List below), you can:

    • Learn more about Picture Puzzles including the wide range of teaching craft features identified by teachers.
    • Read either the primary (Year 6) or secondary (Year 7) teacher article to get a feel for their first classroom experience with Picture Puzzles.
    • Investigate any of the 4 free Picture Puzzles with your class.
    • Explore the complete set of teaching notes for all puzzles - our recommendation is to at least explore notes for the Number & Computation C menu, which illustrates fraction concepts and operations using Cuisenaire Rods.

  • Making a Maths Mat with Matt

    For thirty years Matt Skoss has been developing Maths Mat activities, making giant plastic mats on which students can learn mathematics through whole body involvement and sharing his expertise and experience through maths education conferences around the world. Mathematics Centre records examples of the idea being used in UK, Sweden and Coonamble, NSW. (see Link List below). His most recent conference workshop was at the annual conference of the Mathematical Association of Victoria in December 2023. We uploaded a video from this session to Cube Tube as part of the December eNews.

    Teacher Eleni Pilafas was a participant in that session and followed up with Matt through Term 1 this year. The outcome was Matt, Eleni and Alex Cole getting together on a Saturday in the recent school holidays to make a Maths Mat for Croydon Hills Primary School. Above and beyond the call of duty, but a professional learning exercise in itself as you will discover when you watch the video of the event. See Link List below.

    Whole body involvement, also called physical involvement, is powerful teaching teaching craft at all levels. Muscle memory contributes to the creation of the brain picture related to the maths in the activity.

    Especially if you are a primary teacher, where space and time are used differently from secondary schools, there is a place for a Maths Mat in your school. If you are a secondary teacher, even if the mat may be less possible to use, physical involvement still has a place, for example, as described below in the Dice Differences cameo.

  • Get to Know a Cameo

    Task 34, Dice Differences
    The story shell is that six prisoners can place themselves in any of six cells, including having more than one person in each cell. Each morning the prison governor rolls two dice and their difference determines that one prisoner can go home from the cell with that difference.
    What is the best strategy for placing six prisoners in six cells numbered 0 to 5 so that all get home in the shortest time (least number of rolls).

    The outcomes of the experiment are not equally likely. There are, for example, far more chances of scoring a difference of 1 than there are of scoring a difference of 5. The task invites students to design an experiment, keep data and decide on statistics which help them make decisions about which placement strategies work better. There is also a good deal of arithmetic practice.

    In its whole class lesson life it cries out for acting out the story as a physical involvement activity. The cameo also includes a teacher's report on the success of the task in a remote Indigenous classroom and the extensions it spawned. In addition, for those who are Maths300 members the lesson plan offers software that extends the possibilities for calculating long run averages of placement strategies so they can be compared to determine 'the best'. There are examples of these calculations in the notes for the Menu Maths use of this task, which is linked in the cameo.

    In the eTask Package this task is in the 'easy' set because making it only requires dice and counters.

    Task 239, Money Charts
    The coins are essential to this hands-on task. Some may be able to symbolically reason through the several equation solving exercises underpinning the task, but having access to the coins allows so many more students to begin.

    The aim is to fill the empty spaces on the board. The spaces down the left and along the top are values that apply all the way across their rows or down their columns. The value of coins placed in other cells where a row and column intersect is the sum of that cell's leftmost and topmost values.

    Try the task yourself using coins. In what ways do you use the coins in your hand to help you think?

    The cameo explores the strategy for solving this problem; suggests how teachers can extend students' number sense with a mathematician's question; leads into tackling charts like this where the values are written rather than pictured; and highlights the value of students making their own chart puzzle as an extension.

    In the eTask Package this task is in the 'special' set because it requires the purchase of a set of play money. If that is already available and the necessary coins can be snaffled, then it's a bit 'more work' because as well as the card, there are two playing boards to print and laminate.

Keep smiling,
Green Line

Link List

  • Did you miss the Previous News?
    If so you missed information about:
    1. Number Slider
    2. More on Princess Catharina's Trinket Chest
    3. Get to Know a Cameo
      ... Keith's Kubes, Dice Footy

Did You Know?

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