This weekend, I was tagged in a tweet from @KCost, asking what practical maths equipment people would recommend for primary schools, given a budget of just £100 per class.
Straight away the budget restriction means that some hard choices would have to be made, and this was the gist of my immediate tweeted response. But I did promise to write a more detailed response – so here goes…
I would not hesitate to put Cuisenaire rods and Numicon manuals and shapes at the top of the list under normal circumstances. But the reality is that £100 will not stretch far enough to make a meaningful difference.
A quick search on eBay revealed that £100 might be sufficient to buy enough second hand rods for a class, but it would take at least twice that to stock up with Numicon shapes, and that leaves no money for the manuals, meaning that much of the potential of these resources would remain untapped.
And of course, do we really want to blow our entire fictional budget on one resource? Probably not.
Clearly then, tough choices are required. Asking 20 teachers to produce a list would almost certainly give 20 different lists. Unsurprisingly we al have our favourites. So I asked myself whether there were any non-negotiables; any resources that EVERY teacher would agree on as essential on a £100 budget?
The short answer is ‘no’. After all, the full list of what’s available is overwhelming. Here is a quick brain-dump of possibilities:
Beadstrings, counters, dice, cards, Numicon, Cuisenaire, Abacuses (including Rekenreks), Tens Frames, Base 10 Dienes, Pegs, clocks, scales, 2d and 3d shapes, place value counters, part-whole mats, digit cards, digit counters, fraction cards, fraction cubes, money, dominoes, number lines, whiteboards, place value sliders, calculators, hundred squares, straws, spinners, counting sticks, number fans, flip stands.
Quite a range.
So while we must accept that there will never be an agreed ‘right’ answer, I think I may have a solution. I’ll call it the ‘Three Cs’: CUBES, CARDS, and COUNTERS. Here’s how I would proceed on a £100 per class budget.
Cubes (Linking and non-linking)
The humble cube is capable of so much. There are three main types we should consider.
The first type is commonly referred to be its trade name of Multilink, and these are cubes which can join together in 3 dimensions using Lego-style connectors. They are simply invaluable for demonstrating a whole range of concepts, such as area, volume, multiplication, division, fractions, perimeter, arrays, symmetry, and more. The ones linked to above are better than the old style, as they link and unkind more easily than the older ones, making them suitable for smaller fingers and better for children with less well-developed fine-motor skills. I also have a set of tiny ones, but mainly for practical reasons of transportation, something which does not apply in our scenario.
The second type is also referred to by its trade name of Unifix, which can also join together but only in a straight line. Interestingly, this limitation can also be an advantage in that it can hep children focus. In Key Stage 1 my preference is only to use cubes in a linear way – no chance of children becoming distracted trying to make a variety of animals or weapons, etc. so I tend to recommend Multilink stays in Key Stage 2 and Key Stage 1 children limit themselves to Unifix. I know, bah humbug. (The link above is currently offering 100 cubes for £7, and I’d say that 6 packs would be sufficient for a single class. In fact these are the exact ones I purchased myself to use when training teachers). I love using these to introduce Bar Modelling and fractions (much better than pizzas, trust me).
The third type of cube is named after the Hungarian mathematician Zoltan Dienes, and in Base 10 they are available as 1s, sticks of 10, square arrays of 100 and cubes of 1000, are a brilliant way to model the structure of place value. They are great for Y1 and Y2 as they can help children to learn the absolutely fundamental principle that ‘1’ does not always refer to 1 of the same thing (for example in the number 11, each 1 represents ONE OF SOMETHING DIFFERENT; a really tricky concept that takes time to develop – Dienes are perfect for helping children get to grips with this. I personally use this set, as I love the fact that ten of the ‘one’ cubes can be joined together to make a ‘ten’; this is a feature not available with the traditional sets.
In Years 2 to 4 they also offer the chance to model regrouping a ten into ten ones, of course, or ten tens into one hundred (and vice versa). This is very good for modelling the structure but obviously time-consuming, so there needs to be a progression.
In Years 5 and 6, I often return to the Dienes blocks to teach decimals; if we define the 100 square as ‘one’, then the stick can represent a tenth and the small cube a hundredth; such a powerful visualisation. Alternatively, the large cube is 1, we can show quantities as small as a thousandth.
Finally, and this is pretty cool, if you imagine the large cube is hollow, it actually has a capacity of exactly one litre – 10cm by 10cm by 10cm, and this is a real eye-opener; I often think about this when being ripped off at petrol stations…
Cards
Firstly, have you considered the humble playing card? Waddingtons are a brand of proven quality, and buying a set such as this one works out to be much better value than buying single packs. If I am buying packs that are likely to be damaged in some way or that don’t need to last as long, these ones at 2 packs for just £1 these are the cheapest I have found.
Playing cards are the ultimate CPA tool. Concrete, as they can be picked up, counted, shared, compared, etc. Pictorial because the spots are countable and in great patterns too, allowing children to practice partitioning visually, and Abstract, as they can simply refer to the numeral in the index corners.
Given time to think, I reckon I could teach almost the entire primary curriculum with playing cards! To celebrate turning 52, I created this book – a huge bank of ways to teach in the classroom using playing cards.
Secondly, digit cards – they’re probably exactly what you imagine them to be. Very cheap indeed, as you can quickly design, print and laminate them yourself, or buy inexpensive ones such as these from Autopress. Like playing cards, they offer you a lovely bridge between concrete and abstract but without the pictorial for partitioning element. Digit cards are great for modelling calculations and all manner of investigative work. Another benefit is the confidence they can give children in exploring ‘what if?’ questions; over the years I’ve found that for some children moving cards around feels less of a risk than writing things down.
Finally in the cards section, I find that the fabulous Place-value or ‘PV’ cards, are surprisingly under-used. In the world of CPA (Concrete, Pictorial, Abstract), it is easy to move too rapidly from the concrete (counting objects) to the abstract (equations etc). PV resources such as arrow cards and counters are a very effective bridge between the two. I’ve found nothing better than these for working with place value and bridging to formal multiplication and division structures The link above is to my downloadable ones; if you have a bit more money to spare, I thoroughly recommend these ones from Autopress Education; if you mention my name they will offer you a 10% discount on any order (disclaimer: they do NOT pay me for this, but I regularly use and recommend their resources on CPD as I think they’re great.)
Counters
There are two main types of counters worth considering. The first are differently coloured on each side. They are often referred to as ‘double-sided counters’, but the pedant in me was never happy with this name; after all, a one-sided counter would be an interesting thing to behold! I tend therefore to prefer ‘two-colour counters’. These are wonderful for showing the structures of addition and subtraction on tens frames, and for systematically showing how numbers can be partitioned in lots of ways. Not to mention arrays, equivalent fractions, and lots more.
The second type of counter are Place Value Counters. To avoid the effort of printing them yourself, this set are good value. As with PV Cards, these are a terrific bridge between the concrete and the abstract. Their advantage is that they are brilliant for using to model understanding of the formal multiplication and division algorithms, as well as regrouping for those and other purposes.
Place Value Counters Rock!
So there you have it – only an opinion, but hopefully some useful food for thought.
As a final thought, I thoroughly recommend the fabulous MathsBot website, specifically the virtual manipulatives section. You’ll find examples of everything I’ve mentioned here and a few more resources to boot. Free and easy to use, versatile, and a pretty good substitute if you don’t have the actual resources.