Our Learning team gets together regularly to deep dive into the pedagogy behind a particular model, problem, or interesting question that has been presented to the team. This might come from new research, teacher feedback, or just out of interest! In these blogs we’ll be summarising the topics that come up and giving you an insight into how our learning team makes decisions and the kinds of problems they work with on a daily basis!

A little while ago our Learning team sat down for one of their regular pedagogy digs. Michaela Epstein, Mabel Chen, and Ammar Aldaoud were present. The topic this time around? The ‘charge model’ for presenting positive and negative integers, which is used across three modules in the ACMNA280 content descriptor (“Compare, order, add, and subtract integers”). This topic came from a few teachers who emailed in asking about the reasoning behind using this model specifically.

What is the charge model?

The charge model is one of a number of models used in Maths Pathway modules to teach concepts relating to integers. It is used alongside other models like number lines, thermometers, and MAB blocks.

Doing a bit of research, we found a paper by Judith B Kohn (“A physical model for operations with integers”), which was the earliest mention of this model we could see in the literature. Kohn puts forward this model as a solution to the fact that all operations can’t be represented with the alternative models mentioned above.

The charge model is based on using coloured circles to represent positive and negative ‘charges’. These charges can be added up to represent different numbers.

Before being introduced to the ‘charges’ —a more abstract concept— students are introduced to the concept of dealing with both positive and negative values using the idea of gift cards and bills. This is to give the students something more concrete and familiar to work with before introducing the model.

If you have a bill for ten dollars, you have ‘-\$10’. Whereas if you get a gift card for ten dollars you have ‘\$10’: Michaela (our Head of Learning) pointed out during this discussion that money is the first topic that has a reference to ‘negative’ numbers, as seen in accounting records from China dated to 200BCE!

Following this conceptual introduction using money, students are introduced to the idea of representing numbers as ‘charges’. This is similar in concept to ions in chemistry or similar.

For example, this represents a value of +5: And this would represent a value of -4: Following this, students are then introduced that a single positive charge and a single negative change cancel out to create a net charge of 0. This happens in the module ‘Balancing Positives and Negatives’, and is the main focus of this module.

Once students are comfortable with the idea that zero is made from positive and negative charges that ‘cancel out’, they can then use this system to model addition and subtraction with integers (supplemented with more ‘gift/bill’ examples): Subtraction

The subtraction situation is quite interesting. To be able to do these subtractions, more charges need to be drawn. As an example. for ⁻4-3, you start with 4 negative charges. You want to subtract 3 positive charges. Since there are no positive charges to start with you need to draw these into the picture. For every new positive charge that's drawn, a negative charge is also needed so that the overall value of the picture doesn't change. That is why we end up with more than 4 negative charges in the picture.

So why teach it this way?

Some of the feedback that sparked this pedagogy dig was teachers asking why we use this model in particular for teaching these concepts. The focus of the discussion was on what the pros and cons are.

The Learning team mentioned a number of benefits in their discussion of the model:

• Having multiple models for discussing any topic is valuable (integers are also taught with number lines, MAB blocks, etc)
• Having a model like this where it’s very visual can help students work things out in their head, specifically by activating their visual memory
• Separating the students from just using numerals forces them to make sure they they understand what’s really happening when they are performing these calculations, meaning that they don’t get too comfortable just applying a mental algorithm

Drawbacks the team pointed out:

• For teachers who have never encountered the model before, coming across it in the middle of the sequence of modules (i.e. after students have already been shown how the model works), might be a bit confusing.

Where to from here?

The team also discussed the idea of updating this module in the future to introduce students to the difference between an unary and binary operator. Because this model makes the difference between a negative value and a subtraction so explicit, it could be a good way to help students understand this difference more explicitly as well.

For the moment, the ‘Subtracting Integers’ does show this difference in the way that the minus symbol (-) and negative symbol (⁻) are formatted, but it’s not explicit. This change has been added to the Learning team’s backlog of content changes, for a future update.

What do you think?

Have you come across the charge model yourself in your practice?