Why be so negative?
When looking at mathematics curricula from around the world there seems to be 2 main schools of thought when it comes to scaffolding in the development of negative numbers for students. Before I get into this development though, what would be your answer to the following question?
Do you think the development of negative numbers be based on the development of positive numbers?
It turns out that most countries think that this is true and structure their development of the ideas surrounding integers on a similar model to how their curricula develop operational sense of natural numbers.
- Ideas surrounding what numbers are. Understanding size, names, patterns, place value.
- Ideas of addition and subtraction (having more or having less)
- Ideas of multiplication and division (having groups of, or sharing groups)
- Onto larger numbers, and then repeat from item 1.
This general development also applies to most countries curricula in the development of rational numbers as well (looking at fractions or decimals for example). Usually countries take this development over a few years of schooling as well. Extending on this, most curricula do exactly the same thing with negative numbers.
Get erin gallagher’s stories in your inbox
Join Medium for free to get updates from this writer.
Here is the development of negatives across 6 different countries. The colours indicate a matching of relative age groups of students.
What’s interesting here is how some countries like Singapore, leave all the negatives to one year whereas for the others in countries like the United States it’s over 2 years and Australia over 3. England and New Zealands curricula are defined over levels which is by default already multiple years of schooling. Also intersting is where negatives are first introduced. In Australia and England negatives are introduced before students start high school. In the United States it’s at the begining of middle school, and Canada, New Zealand and Singapore don’t introduce them till the beginning of high school.
My immediate thoughts on this is Woah!, it’s not a great idea to do as Singapore and the United states do and cover so much so quickly, in just 1 or 2 years of schooling. Surely it’s better to slow that train down. When I think of all the mathematics I have marked in classrooms, (particularly at the high school level), if there was a simple calculatory error it was typically either because it involved a fraction, or a negative. And so, if it is the source of many errors doesn’t it make sense to slow that development down and allow students the capacity to really understand what’s going on.
But then if you consider the International comparisons in mathematics between the US and Singapore you will get two very different stories. Clearly, the results on the international tests are not based on how countries develop the notion of negative numbers! So is there any real difference here or not?
Maybe not from a testing point of view, or not from an age group point of view, but I did think that generally, the scaffolded development and building of concepts over many years would seem to allow for a greater uptake of the concepts by students. It would also give capacity for some students who many have missed some key component in one year, the opportunity to make that connection in the following year. Seemed to make sense to me.
Let me digress a moment and tell you a story about an action research project I undertook as a relatively new high school teacher. Four of us all taught Year 7/8 mathematics. We were all about to start our unit on Area. The teachers all sat down and set out a plan, rectangles and squares for 1 week, then parallelograms for 1 week, then trapeziums the next week and rhombuses and kites for the last week. Under the advice and guidance of my mentor, he suggested that I alter that for my class. My class would do rectangles for three whole weeks. We would do rectangles in so many different ways, we would find rectangles in the environment, we would cover walls with newspapers, we would build them with blocks, we would cut them out, we would look at arrays and tables, we would take them apart and put them back together. And we did. We did rectangles until in that third week, all of my students were pleading with me to do something different. Then, in the final week we did all the other shapes. Parallelograms — just like a rectangle, but different; trapeziums — just like a rectangle, but different; rhombuses — just like a parallelogram, which is like a rectangle but different and kites — yep — just like a rectangle! My students smashed that common test like you wouldn’t believe. It seemed that spending a lot of time in the early days really laying the foundations meant that come the last week we could really fly. Connections could be made, my students didn’t have to learn four more formulas, they just related everything to the rectangle.
So, could the success Singapore has with doing all the negatives in one year, be related to stronger foundations students have developed in number and algebra already? Could the drop feeding approach taken by Australia and Canada actually benefit from being less scaffolded in terms of negative development, but more scaffolded by enhancing the time spent with algebra and fractions on positive calcualtions and manipulations?
It’s definitely given me food for thought, when should we delay the introduction of a new concept in preference for consolidating existing ones, and when should we take the approach to develop something slowly?
What are your thoughts?
