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News

May 2024

In this edition of the News you will find:

Get to Know a Cameo
... Fold Up Houses
... Pascal's Triangle in Asia

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• Negative Approach to Negative Numbers (Part 2)

The previous eNews issue included an article critical of writing integer equations in this form: 5 - (-9) = 14.

The criticisms are both mathematical and pedagogical and it was clear that at least one Year 9 student was defeated by the mathematical incorrectness and the unsound teaching craft it brings with it. Please familiarise yourself with this article before reading on.

I am grateful to Elliot Tench, Sacré Coeur, Glen Iris, for taking the time to read and respond.

Dear Doug,

You are correct. The expressions are mathematically incorrect as the negative sign in the brackets has been written using a binary operation.
...
 I just spent some time looking through several well-known and widely used current textbooks and they all use the binary operation to represent negative numbers. This included the ICE-EM series which is recommended by AMSI. I also queried Wolfram Alpha which used the binary operation to represent negative numbers. The soon to be implemented Australian and Victorian Curriculum documentation also use the binary operation. On a positive note, NRICH, in their article on how to add and subtract negative numbers do 'write' their expressions correctly. Some students find adding and subtracting negative numbers confusing. I wonder whether sometime in the past some educators thought that by changing how expressions are written (incorrectly in this case) that it might make the concept less confusing. Unfortunately, this hasn't occurred but the new incorrect way of writing these expressions has stuck. Elliot also found this previous NAPLAN question.

At my school, we use counters to teach the addition and subtraction of positive and negative integers. The introductory learning activities we use are very similar to the approach you have taken with your Protons and Anti-Protons task. I find that using manipulatives helps students 'see' and understand what is actually happening when one subtracts (or adds) a negative number from another number.

Cheers and thanks,
Elliot.

The NRICH article presents several models for representing integer operations, one of which is very similar to Protons and Anti-Protons (see Link List below for the article and P & A). Although it isn't completely consistent in using + (positive) and - (negative) correctly, rather than the binary operation symbols + (add) and - (subtract), it does offer one place to find and review several teaching models and it does include these sentences:

In some ways, it doesn't matter which model you use, as long as you are pedantic about using the correct language to separate operations from directed numbers, and relate the model to the calculations. We would advise choosing one (or at most two) models at first, to avoid confusing students with lots of conflicting images.
If you follow this NRICH advice, and choose a mathematically correct approach to represent the calculations your chosen model affords, does the NAPLAN question give you any cause for concern?

Surely it brings us right back to where this discussion started ... with the words of a Year 9 student when referring to
5 - (-9) = 14:

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.
The NAPLAN question is mathematically incorrect, brings with it the implication of unsound pedagogy as previously discussed, and worse, because of the status of NAPLAN's data as a guide to what 'needs to be taught', is likely to encourage many educators to conclude, It's in NAPLAN so it must be right.. A step for many towards a blinkered view of how to create the learning environment.
An alternative response to the NAPLAN question would be to query the validity of this test item, if not the test itself, using the standard evaluation criterion: Does this question actually measure what it purports to measure?.

• Times Tables Learner

 Rows, Rectangles and Multiplication is a freely available Picture Puzzle developed as the core of a Maths At Home activity of the same name. It plots a pathway from a concrete, conceptual understanding of multiplication as an array, through using that brain picture to bridge into short multiplication using drawings of arrays, to representing long multiplication as sketches of rectangles, and developing a non-traditional, but effective and meaningful written record of the calculation. All of which could never be achieved in one 'lesson'. This is, and always has been, an activity you can return to several times on the multiplication learning journey. The activity has been revised and extended in the past month. The revision makes it clearer that this activity is intended for use over time and suggests a plan for breaking up the sequence. The extension is the inclusion of a personal Times Tables Learner early in the sequence to support parallel learning of automatic response times tables. It makes a direct link with the first section of the Picture Puzzle and includes challenges of its own.

Times Tables Learner (a 2 page PDF) supports students to learn times tables for themselves outside the classroom. It is embedded in the Picture Puzzle which is also designed to support self-directed learning. It can also be used in the classroom in Times Tables Teams of 3 or 4 students co-operating with and encouraging each other.

• If you have used Rows, Rectangles and Multiplication before, you might like to return to look at the extension in context.
• If you haven't, then as well as examining the possibilities for Times Tables Learner in your programme, you have the opportunity to discover how a Picture Puzzle works and to get a sense of the 'voice' in which Maths At Home activities are presented.

See Link List below for a link to the TT Learner and a link to the MAH activity.

• Get to Know a Cameo

 An imaginary company builds house frames using just squares and triangles so they can be transported flat pack to building sites. The pieces are then assembled on site. That's actually very close to the many houses are now built. The difference is the outcome. This company specialises in row houses and can build rows of any number of houses using the plans shown on the card, which drawn as nets. The challenge is If I tell you the number of houses in a row, can you tell me the number of squares and triangles to order?.

Here is a task that connects work in 3D space with pattern and algebra. The algebra can include generalisation in words or symbols, equivalent algebraic expressions, substitution, solution of equations, tables of values, ordered pairs, linear graphs (2 variables), simultaneous equations and probably more. This is a task that deserves far more than a brief visit, especially in secondary school.

In the eTask Package this task is in the 'special' set because there is no substitute for the Mini-Geofix on which it is based. The cameo provides a link to the manufacturer, who will be able to refer you to a supplier. As mentioned in the Whole Class Lesson section of the cameo, the larger Geofix or Geoshapes, also from GEO Australia are suitable, but the constructions are so much bigger. Therefore it is better to use them in groups of 3 or 4 students.

Task 144, Pascal's Triangle in Asia
 This wonderful task uses simple concrete materials to build a Pascal Triangle using instructions in Chinese. The card is the introduction. The Recording Sheet linked in the cameo contains the challenge. The cameo includes an explanation of the Chinese Rod system of numeration, which is base 100 and uses an empty space for zero. In the 'real world', chinese merchants carried around a bag of rods and a counting board to put the system into action, in the same way as we carry around a calculator today.

You will find decoding an alternative numeration system, an application of this to Pascal's Triangle, history of both Asian and European mathematics related to Pascal's Triangle, or is it Yang Hui's Triangle, comparison of base 10 numeration with base 100, aspects of probability, and more, including an Investigation Guide related to a Pascal's Pinball Machine provided by teacher Christine Lenghaus.

In the eTask Package this task is in the 'easy' set because it only requires match sticks which are easily acquired from a craft shop, and printing one extra sheet.

Keep smiling,
Doug.

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