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A plea for ‘Slow Maths’ and having fun.
The greatest mistake, some say sin, in the teaching of mathematics a teacher can make is to move from the concrete to the abstract too soon in a child’s grasping of concepts. To rely heavily on counting and chanting and representing concepts with numerals and symbols will ensure that a number of your children will never be confident in maths, even developing a fear of failure, frustration and avoidance. ‘Slow Maths’ by its name is exactly that, ensuring hands on real items to manipulate, push and pull, pour and squeeze, slosh and slather. The slower one goes, with the emphasis on more talking and playing, arguing, suggesting, proposing and considering before asking and answering, the greater the chance of success for all. The more that maths can be seen and enjoyed as a game the greater will be the understanding, the building of a sound foundation from which to move forward with confidence. You as the teacher, or the parent, know all that but the rush for data and the pressure to cover the content in designated set time blocks is the curse our curriculum writers have thrown over our children.
The concrete objects do not have to be expensive coloured fanciful items supposedly to entice interest, like cartoon characters in the myriad computer program games that purport to teach maths in a fun way. Learning the secrets of the power and beauty of mathematics is far more enriching, satisfying and fun, without the gizmos of technology. They can wait. The first hurdle is to assure everyone that maths is a language that they can all learn and which will assist them for all of the lives, every day. Without the sound development of concepts through personal experience with concrete materials maths for many is just a mystery to be avoided where ever possible. The disaster of young learners going to a hand held calculator for answers has created a huge void in the understanding of mathematics. Getting an answer right is of little use if one can’t understand the concept, or give an approximate answer, within the range of possibility. They should be relegated to their rightful use, not the substitute for understanding. “I just got the decimal in the wrong place!” “$19.95 times eight?” “Where’s my calculator?” “Try about a hundred and sixty dollars, less 40 cents.” “I can’t find my calculator! Who’s got my calculator?”
A round one litre milk bottle weighs 42 grams with cap on so keep the cap on. When filled with water it weighs one kilogram if we ignore the 42gm and is a thousand millilitres in volume, or one litre. You can use 42gm less water if you need to be that accurate, maybe in upper primary classes. To make a very accurate and workable set of simple scales four milk bottles each filled with 250mls, 500mls, 750mls and a litre, each labelled with 250grams and a quarter of a kilogram, 500gm and a half of a kilogram, 750gm and three quarters of a kilogram and 1 kilogram will suffice for basic weighing experiments. A fulcrum and two pans on a beam, each with a plastic dish or pan completes the apparatus, all child made.
Three one litre milk bottles, each cap labelled A, B and C, one filled with water, the other two with heavier and lighter than a kilogram serve as a regular weigh in on arrival, putting them in order of weight, changing the caps daily and letting the children devise variations. Eventually do away with the water as it gives away the kilogram bottle. Add more milk bottles. Make all bottles except the kilogram bottles less than a kilogram and name them in order of heaviness. The aim is for everyone to get a pretty good idea of what a kilogram weight feels like.
Ten milk bottles with sand at the bottom for weight make a simple ten pin maths bowling alley, value of each bottle written on its cap. First to a hundred might be the aim, first bowl only counts. The variations with this game are endless when complex labels for caps are added to some: double your score, take eight off your score; you got it. The best way of bowling the different sized and weighted bowling balls, another variable to investigate, is to use a PVC pipe ramp, either whole or cut into a trough, that can have its height varied, another variable. A billiard ball, a golf ball, a bocce ball, a tennis ball a ??? Try lots of balls of varying densities and sizes when you are ready to add another variable into your investigations, into your game. Teach how to record outcomes. Simple graphing then has real meaning. A cardboard template to position the ten pins in exactly the same position every roll is essential.
Marbles of various colours and weights and sizes rolled onto a numbers map made from a thin piece of plastic, marble sized holes punched out on top of a number mat can make a very competitive game for three or four players. The mat with the numbers can be changed to add complexity. A surround barrier to contain the marbles is necessary. Given simple materials a group of children will make the whole apparatus as their activity. They can then be the supervisors when others play the game and do the score keeping. The numbers can be expanded to operations rather than single digits, four times four, thirteen minus five; you got it. It is important that for any maths game the number of players is limited as waiting and waiting for your turn is no fun. In a busy multiage classroom there are numerous options on hand at any one time, with Menus to manage the time aspect. Waiting is not an option.
A table set up with the old fashioned Bobs set, with cue and balls and the pockets numbered, is lots of fun and yours to vary by putting stickers on the arches over the numbers. If I have lost you there Google “game of Bobs”. You can make your own or a Dad or Mum, if asked, will make one for you. A very competitive maths game, with printed score cards, named with totals and left in a folder after each game, can provide the four top scorers of the week who might play an exhibition tournament for an audience, calling out the scores; you got it.
Thinking about measurement and metres and kilometres we might ask “How many steps do you take to walk a kilometre? How long would it take you to walk a kilometre? How long would you take to run a kilometre?” Your football field is a hundred metres long. If it’s not mark one out. You’ll need some stop watches as soon as ‘time’ comes into the activity. Every child should know their own statistics in regard to distances and time and the concept of time and distance. All easily done, lots of graphing and tables making. If you walk just the ten metre track and multiply that by ten do you get the same result as walking the hundred metres? Why are they different, if they are?
To make the keeping of data with statistics a reality your room has a height measuring stick and set of scales where everyone’s height and weight are recorded, as regularly as required, then graphed at the beginning of each term. Across a three year span and recording the data on the front of the Kid Tracking Envelopes maths has real meaning and purpose. Other measurements might also be added like foot size and so on. Obesity may be too sensitive a topic for you to contemplate such, and that could add another facet of learning. How much more does your class weigh at year’s end than at the beginning? Do we all weigh as much as a car? How can we find out how much a car weighs? How can that amount be visually shown? What would you use to show it?
If you have some sandy soil in your playground, at least five meters long, maybe your broad jumping pit, that you can wet down and smooth out, or you can get to a sandy beach, tracking can add an intriguing aspect to the above. With backs turned to the track, no peeping, the escapee makes passes along the smoothed sand, walking, walking backwards, limping badly, running, piggy backing a friend, using a walking stick, hopping, crawling; you got it. The ‘trackers’ have to identify how the escapee was travelling. Don’t use more than three passes at a time to analyse. Who are the world’s best trackers? Why did they have to be so good at their skill? Your escapee might even pretend to be a crow, a kangaroo, a dingo, a horse, a …., with the appropriate templates. You never know where this little activity may lead, but it heightens awareness, reinforces the concept of distance, and it’s fun. The five metre you are focused on relates to twenty in your football field. Ten football fields make a kilometre. How long is a marathon? How long would it take you to run a marathon supposing your time to run the hundred metres could be sustained? Try running the hundred metres at a pace you could keep up for an hour, two hours?? You got it.
There are two points that all children should have got a good grasp of in their first three years of primary school in relation to our nation adopting a decimilised or metric system in all aspects of money and measurement. We never went to hundred minute hours though, a step too far. The first is to regularly, every day that is, emphasise the advantages of our system, how easy it has made our maths, the same numbers always having the same proportionate value, no matter what measurement is being considered. This concept should be reinforced every time the numbers come up in Number Study and be highlighted all over ceilings and walls at every opportunity. I think you are ahead of me but I’ll still explain.
Five is always half, always. Twenty five is always a quarter, seventy five is always three quarters. Think the same with one eighth, three eighths, five eighths and seven eighths, 125, 375, 625 and 875. Twenty, forty, sixty and eighty goes up in fifths. However a third and two thirds have that three which can never be exactly represented in a tidy way when dividing ten or a hundred into thirds, but still gives a rough idea of three and a bit being the third, 3.3333’ goes on forever, as does 6.6666’. The only numbers that will not have been encountered in Number Study are the family of eighths so you might choose to leave them out till upper primary.
To simplify the concept which is one of the great shortcut strategies in maths, a chart can gradually grow. Headed at the top in columns is written: decimals, fractions, percentages, money $1, litres, kilograms, kilometres, hectares. For fun you can also throw in the hour column, the week column and the year column just to show that the limit of sixty minutes, seven days, twelve months not being metric, really is. Down the left side is written half, quarter, three quarters, one fifth, two fifths, three fifths, four fifths, one eighth, three eighths, five eighths, seven eighths if you choose, then one third and two thirds. You could also add the tenths column, gets a bit boring, and the sixths, though we tend not to use sixths very often. The sameness across the chart as the values are allotted reveals at a glance how easy they have just made their understanding of maths in everyday use. No matter where, a 5 always indicates a half of something and whenever they see 25 in any context a quarter comes immediately to mind, and so on.
The second point on the metric system, especially in relation to measurement, is that your children will continue to come across the old pre-metric terms for the rest of their lives for inches, feet and yards, miles, pounds and ounces will always have a place in literature and conversations. We’ll probably still be quoting the weight of newborn babies in pounds and ounces for another century. Another factor too is that the United States never converted to metric and most likely never will, so a confusion will continue. Even their gallon was different to our imperial gallon that we have abandoned. We’ll probably always talk of oil by the barrel, but don’t know how many litres that might be, nor care.
An easy reference point just needs to be available, though not a complicated conversion chart; yards are roughly a bit less than a metre, a mile is quite a lot further than a kilometre and so on. It seems a bit of an anomaly that we have continued to talk of boat speeds in knots but no one is ever sure of how many kilometres an hour that is, yet we manage. It’s also a bit hard to fathom out a fathom. And then there were ‘Twenty thousand leagues under the sea!” with Jules Verne. Enough! You probably have all that well in hand and are thankful that you don’t have to teach long division like two hundred and seventy five pounds, sixteen shillings and fourpence divided by seventeen as I was expected to learn, and later teach. That was when a calculator would have come in handy but they hadn’t been invented back then.
Graphs, tables, charts, tally marks, measuring tapes, rulers, stop watches, clocks, egg timers, set squares and compasses of all sizes, the bigger the better, scales and calibrated measuring vessels and more are all essential tools in developing the many concepts of measurement in maths; the capacity, value, amount, quantity, area, length, height, depth, width, weight, range, size, dimension, proportion, magnitude, mass, bulk, volume, capacity, extent, expanse and more. You also have a thesaurus.
Water, sand, play dough, blank cards, plastic or steel water troughs and tubs, disks, counters, marbles, buckets and bottles, jugs and cups and dippers, blocks of every shape and size, boxes, cylinders, you got it, are the basis for hands on concrete concept development tools. Why stop using them after Kindy, or Pre-school? Wrong, wrong and wrong. They can and should be used right throughout the primary school years of every child’s education. It is a great mistake, some say it’s a sin, to stop hands-on learning.
When the square number concepts are well understood one might then introduce the concept of cubic numbers, for volume can’t be represented in a two dimensional way. The Cuisinaire Blocks system was useful in this regard as the flats when stacked gave a clear view of volume, though I found the single cubic centimetre blocks too fiddly, preferring our home made three centimetre size for number study. Another popular system is Unifix but as they are interlocking and used in lines, not arrays of three, they reinforce counting which defeats the concept of the Barron Blocks system. There are numerous examples of structured maths materials available but whichever you should choose to use I warn against any system that would depend on counting, for it is not a visual representation that young minds can easily absorb. Counting has its place, but not in acquiring ready recall of number facts as described in Number Study.
Using any cubic centimetre type block system the squares of four when stacked four high become sixteen cubic centimetres, the squares of five when stacked five high become a hundred and twenty five cubic centimetres and the squares of ten when stacked ten high become that wonderful, that magnificent thousand cubic centimetres block. When we make an exact mould of the block we find it holds precisely one litre of water, our jug tells us so, and when we weigh that same litre of water it is exactly one kilogram, and the relationship is complete. Do all that in one fast lesson? I don’t think you have been listening. There are many discoveries to make if you go slowly enough. Five of the five centimetre flats make a hundred and twenty five cubic centimetre block and the ten by ten centimetre flats make a thousand cubic centimetre block. If you use the hundred and twenty five cubic centimetre cubes to build up to a thousand cubic centimetre cube you will need eight in all and each eighth will be 125cc, 250cc, 375cc, 500cc, 625cc, 750cc, 875cc and 1000cc. Those numbers seem to crop up everywhere and are well worth stressing. They can make what would seem to be a hard problem just so simple.
Year Five Mathematics Curriculum states this is the year for focus on learning about cubed numbers. A lot of rot if Number Study is being practised, as volume is integral to a child’s understanding with that litre bottle on display every day, which weighs that exact kilogram and that was discovered through mucking about with lots of cubes and cylinders and all sorts of containers. Slow Maths though doesn’t demand that complex calculations of volume, mass and area are able to be calculated. Slow Maths demands that a real understanding of the mathematical concepts are experienced in real life problem situations.
Of course all that is of little sense if it is just to show that the concept of volume is well understood. What uses can it be put to? What problems might your classroom be having where the concept of volume may be useful in solving them? I am of no help there as I don’t know your classroom and to make up silly examples as syllabuses usually do is to be avoided. Alright, I can think of a common problem related to healthy eating and drinking that is universal. “Is a 1.25 litre bottle of soft drink enough to keep you going for at least a fortnight?” The secret in making maths meaningful is to create real life learning situations where the learner can relate them to his/her own life. Just imagine where that bottle of soft drink may lead, across so many aspects of learning. I also wonder how much sugar is in that same bottle of soft drink.
For your own interest, and maybe as a challenge for some, the cubes increase in volume from 8cc to 27cc to 64cc to 125cc to 216cc to 343cc to 512 to 729cc to 1000cc. You might even call them cubic numbers as no one else does, just ‘cube’ or ‘cubed’. Can you find a pattern in that increment? If so what is the volume of the next cube, not using 11 to the power of three to find the answer? That’s not magic, just Mathematics.
I’ll leave you with one last cogitation: How can your children develop the concept of a Hectare? Let me know if you have a great strategy I can pass on.