Chanting Tables, a Poor Approach to Teaching Number Facts.
Since schools were schools, chanting and learning tables, multiplication number facts, have been universally accepted as one of the keystones of a good education. No one can deny that having ready number fact recall is essential to daily life. However there is a far more important aspect to learning essential maths. Many teachers missed out on this themselves so have little knowledge as to how best to teach it. Chanting times tables is a tried and tested strategy so who would want to challenge it?
There can be no excuse for any child by his/her tenth birthday not to have complete numerical dexterity in all number facts operations to one hundred. There are two huge flaws with learning and chanting tables, both of which are easily addressed when a number study approach is adopted. Apart from “knowing the tune but forgetting the words” the first flaw is why is a number fact for twenty-four so much more important than for twenty-three? Why can fifty-six only be a multiple of seven and eight and never be a multiple of six and nine? It can never be explained as a lapse of memory or forgetting the table. The second flaw is if number facts don’t ensure that children use shortcut operations their concept of mathematics is fatally flawed. Three lots of twenty-nine is ninety less three, that’s eighty seven. There is a whole raft of number strategies that are as important and more so than chanting tables. With a number study approach all of the times tables are thoroughly covered and learnt, then much more. Briefly the number study approach includes the following:
Children quickly learn to recognise numbers of objects or symbols when they are grouped in rows of three, without counting. Counting is counterproductive to learning number facts and combinations and should be discouraged. To be able to count is a required skill for many purposes but not operations with numbers. Example: When seven is presented as two rows of three and a single one, below, children quickly form a visual image of seven. To explore the properties of seven all permutations are explored and played with to saturation point. This is number study and forms an essential prerequisite to operations on numbers. Children move steadily from one to ten, simultaneously using the four operations of addition, multiplication, subtraction and division. Ten is represented by three rows of three with an extra one on the bottom. Story telling must accompany number study with real life stories to help the embedding process. eg Make six biscuits with your blocks (two rows of three). There are two children so how many biscuits will each one get? Gradually all children will master all operations and combinations up to ten. A chart slowly grows to illustrate these number facts and mental real life problems to accompany the mastery of these numbers and to put the visual symbols into memory. All the multiplication facts to ten are deeply embedded: two times, three times, four times and five times tables, but only to ten. The secret is in the grouping by threes. Children must not be allowed to count out numbers and progress should be as slow as the child’s real understanding. The proof of the uptake is when children feed back their stories about any given number to ten. Counting to whatever number teachers or parents may feel is adequate has no relevance to this number study. In a multiage grouping some children will skip ahead more quickly to the next grouping, eleven to twenty, whilst some will take longer to master the numbers to ten.
The same strategies apply from ten to twenty, all the time constructing the number facts charts to show the number relationships, plus the multiplication tables slowly expanding. Six, seven, eight, nine and ten times now get a run.. The study of seventeen though is as important as that of sixteen. It is two nines less one or it is four fours plus one. It is just three short of twenty and it is three fives and two more. Of course it is also a prime number and an odd number, and so much more. Later when thirty four arrives seventeen is exactly half. Amazing!
When satisfied, the learners who are ready begin the next phase from twenty to thirty. Every number must be thoroughly and carefully analysed and in as many media and situations as the class can discover. The process then continues in the same way up to one hundred, in groups of ten. This point is easily reached by the time a child is in year four, but some will be at least a year either side of that. If teachers emphasise a table chanting strategy so much time is wasted. There are only nine multiplication or tables chanting number facts from forty to a hundred excluding multiples of five, ten and eleven which are a waste of time if chanted as tables anyway. There are sixty numbers when a number study approach is used. You may make a case to include multiples of twelve if you choose to keep dozens as a number strategy.
A number study approach ensures that all numbers to one hundred are given more or less equal weight. Children build upon their multiplication table number facts with thorough understanding, plus the added advantage of using approximations as a real mathematical strategy. No child will have any problems giving the cost of seven donuts at $3.99 each or want to go looking for a calculator. Calculators should not be introduced before year five, and only when your 'Junior Mathematicians' have complete grasp of their number study to one hundred. There are many texts available which give the “short cut” strategies to ensure that teachers first know what they are to teach as a number strategy method. I call number study Slow Maths. Did you know that 67 is the closest number to two thirds? Fascinating isn't it?
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Three lads were applying for an after school job at their local hardware store. Bob went in first and returned with a big smile and thumbs up. “What did they want?” the other two eagerly asked. “You just have to say the four times table,” Bob replied. In went Gary and returned soon after, thumbs up and smiling, “Just the four times table.” Robert was last but came out downcast, shoulders slumped and in a glum voice to “What went wrong?” replied “I knew the tune but I just forgot the words.” If you listen closely you also can hear the tune in your head.
So will you go on chanting tables, singing tables, into the sunset or will you determine a time when you are satisfied your pupils are proficient in number study to one hundred? This is a strategy that works.
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Licence Requirements for the 20/20 Club
Multiplication Facts mastery.
The master copy has a place for the child’s name, the date and columns for answers, numbered one to twenty across the page. Each try for the Club will use two lots of twenty spaces for forty answers, thus the Twenty/Twenty Club. Generally children should be in their fourth or fifth year at school before being allowed to qualify.
By self assessment, or invitation from the teacher, children are able to try out for their licence. The teacher delivers five addition, five subtraction, five multiplication and five division number facts slowly and deliberately, but not quite slowly enough for children to count to get answers. After the first twenty, a further twenty is delivered as rapidly as possible giving children only enough time to write an answer before the next one is needed. Sheets are collected and marked at leisure. Children must not mark their own or each other’s. Award number fact licences and membership to the 20/20 club when a child consistently gets twenty slow and twenty fast number facts one hundred percent correct. That child is no longer required to take part in number study sessions, chanting tables or the testing program. They may voluntarily choose to have their licence endorsed again at a later date, by negotiation. They might also volunteer to be tutors to help less able children. Warning!! Do not assume that more able children will automatically become tutors, for they may choose to do other things with their personal time. For some it will be a delight to help those less able. "When do you know you really know something? When you can teach it to another."
Teachers will need to prepare at least six sets of number facts (6x40) in advance to ensure they have results to analyse. From weekly (or more frequent) results teacher focus on two aspects will be essential if the diagnostic value of the test is useful, ie the most poorly known number facts, and children who have low capacity for recall even in the first twenty slower set. These children will require number facts clinics in smaller groups and structured materials, groups of no more than six, for one needs to see and hear the strategies the child is using. Clinics are "Explicit Instruction" that works. Just chanting tables is a waste of time on everyone’s part and children will leave school still in wonder of why they can’t be sure of the right answer. “Seven times eight just might be fifty-four, or is it fifty-six? I know one of them is sure to be right, I think??
Teachers must bring to the fore as many times in the week the number facts they are targeting eg 56 is sure to be there. With concrete materials children must show, describe and prove, why only 7 x 8 or 8 x 7 is 56 and never 6 x 9 or 9 x 6. Children must be given more than just two digits on a board to be able to understand 54 and 56 and so on. The greatest cause of children failing to master mathematics is skipping quickly through the steps of Number Study I have described without concrete materials and real life examples, instead going to "sums on a board or in a text book of sheet of paper" so the data gatherers are happy. Children don't set out on their learning journey with the intention to fail.
A useful aide is the hundred grid found on most blackboards. Numbers one to one hundred should be entered in white chalk to begin. As the multiplication number facts are slowly developed and written over in red or yellow they clearly show how few multiplication number facts there really are to learn. Colour in all the multiples of ten, too easy, eleven, too easy and five up to sixty, too easy, all before beginning. The two times table is just doubling so poses few problems. When shown this children can regain their confidence and make it their responsibility to learn all of those that they are unsure of. Surprise surprise when children see that multiples of nine are the same as for ten, only dropping off an extra number as each multiple is increased, ie 7x9 is really 7x10, but 7 less. ie 63. It works just as well for 9x148 which equals 1480 minus 148. Amazing!
Chanting is more akin to a religious ceremony or ritual than understanding number facts. It has no legitimate place in the classroom for our Copycats, Stickybeaks and Scallywags. They may become disillusioned, even rebellious, and we wouldn’t want that, would we? Why does a chanting-tables based classroom put so much emphasis on 24, so little on 23?