Frequently Asked Questions
Both are equally important. The presence of instant recall, that is to say knowledge, is a pre-condition that allows higher level thinking to take place. So both are essential.
Observe whether the students are using a counting method. If this happens they are not using a part/whole strategy. If the student is not using a counting process then the answer may be by instant recall (knowledge) or by part/whole thinking. If this happens the teacher needs to ask a question that pushes the student towards a process typically by increasing the number size.
For example: Shiree says 5 + 6 = 11 very quickly and the teacher thinks she has instantly recalled this fact. The teacher now can ask 15 + 6 which pushes the answer beyond basic facts. If Shiree says something like '15 + 5 is 20, and so the answer is 21' the teacher can be sure the student has used part-whole reasoning.
At counting stages, which precede the emergence of part-whole thinking, the size of numbers is severely restricted, and there is normally only one way to solve problems. As such, counting represents a relatively low level of thinking. Whereas part-whole thinking opens up the world of large numbers and multiple strategies.
When under stress or just tiredness everyone, including adults, sometimes slip back backwards from their best way of thinking about a problem, and may appear to be between stages.
For example even a person who is very good at maths may find the number of days between 3rd July and 9th July by counting on their fingers rather than reasoning the answer is 9 - 3 then take away 1. Teachers should encourage the students to operate at their highest stage of thinking while acknowledging that occasional slippage is natural and not a concern.
Ten frames are an excellent piece of equipment and are extremely valuable to enable children to see groupings of five and ten. The long term objective in using tens frames, and other materials, is to assist students towards the instant recall of basic addition and subtraction facts.
The written form should not be introduced until students have a firm grasp of place value. Teachers need to recognise that students will not understand the number properties needed for the written form until part-whole thinking is available. Book One: The Number Framework, states that at stage six students will be performing column addition and subtraction with whole numbers of up to four digits (page 20).
Students can be taught the written forms with sufficient practice. But this does not require thinking or process. And, to be numerate as student needs to be able to check any written answer by mental checking. These require thinking rather than following a set procedure.
For example, if a student works out 48 x 52 is 246 that student should use estimation processes to see the answer is about 50 x50, and so the answer must be near 2500. So the thinking skills are not an optional extra even if the student is practised at using written methods.
Initially place-value is introduced at the Counting-all stage when children are reading and modelling numbers up to 20. Progressively larger numbers should be read and modelled by students. By the time students are ready for part-whole they should be able to model and write at least three-digit numbers. Teachers should note that a second and more difficult idea in place -value, namely for addition and multiplication ten units is swapped for one unit work ten times as much, cannot be understood until part-whole thinking occurs.
For addition and subtraction ideally the basic facts are instantly available before early additive part-whole reasoning emerges. If this has not occurred time needs to be spent on making the recall of facts automatic.
Example: Harry, who is a part-whole thinker works out 56 + 8 by counting 57, 58, 59, 60 then add 4 to get 64. Harry has mixed advanced counting and part-whole thinking because he does not know 6 + 4 = 10 automatically but he does know 8 is split into 4 + 4. Harry's lack of recall is inhibiting his ability to use part whole thinking efficiently. For multiplication and division, ideally the multiplication basic facts are instantly available before advanced multiplicative reasoning emerges.
Example: Belinda understands 6 x 68 is the same as 6 x 70 minus 6 x 2. But she has to work out 6 x 7 by a process like 5 x 7 + 1 x 7 = 35 + 7 = 42. By this time she has lost track of the original problem.
There is no point in trying to teach the basic facts before their meaning is understood.
For example, in the case of multiplication it is common to see children who knows 6 x 5 = 30 but cannot make up a problem that is answered by this fact. Provided the meaning of operations is understood teachers have many activities that will aid the learning basic facts. Often initially this is by process.
For example: recalling 5 x 5 = 25 and 2 x 5 = 10 a child may deduce 7 x 5 is 25 + 10 = 35 which is a part-whole process. This is fine for children who are not advanced multiplicative in their thinking, but teachers need to be aware that the inability to instantly recall will seriously effect advanced multiplicative thinkers ability to think quickly and accurately.
For example trying to share $377 among 5 people requires a student to 370 ÷ 5 = 7 tens with remainder 2 tens without having to work out the answers to the 5 times facts. Similarly early additive students need the instant recall of the basic addition/subtraction facts. Hopefully this occurs during the advanced counting stage of thinking.
Teachers need to be clear about the purposes of recording. From the student's point of view informal recording is mainly to lighten memory load.
For example a student may decide to work out a challenging mental problem like 3452 - 2645 by thinking 3452 - 3000 is 452, and 3000 - 2645 is 355, and record 452 and 355 to lighten the load. The student now can turn her attention to working out 452 + 355. From the teacher point of view she may want recording to help other children in a later discussion and/or enable the teacher to understand the students' thinking.
Refer to Book 3, Getting Started pages 4 to 8 and pages 12 to 16. Observing other teachers at work would be helpful here.
The relationship is close. The NZC Mathematics Standards for years 1-8 poster (PDF, 187 KB) illustrates the links between the levels of NZC, the Mathematics Standards and the Number Framework.
The mathematical processes skills are learned within the context of the subject strands and as such are not a separate strand. As the year level increases the amount of time spent on number drops. The algebra, measurement and geometry strands deserve roughly equal time with the statistics strand getting somewhat less time than these three.
To extend your able children remember that the vast majority of the activities in book 8 Number Sense and Algebraic Thinking are aimed at Advanced Multiplicative and Advanced Proportional thinkers. It is unlikely that even the most able year sixes would get through all this work. Other material includes Figure It Out levels 4, Books 2-6 especially 6; also the new Number Sense book. Books by Brian Bolt, Macmillan Maths Investigations and Problem Solving Lessons from the website . The Development Band Books are available. The Enrich-e-matics series are especially designed as enrichment/extension resources. The Number Sense books have excellent material. The Family Maths Books are very mathematically rich.
Initially children are put in like strategy groups based on data from the diagnostic interview, the Global Strategy Stage (Gloss) or strategy data from last year's class. Teachers should closely observe whether these stages are correct for an individual child. If it is either it is too high or too low he should be moved to a more appropriate group. Over time, as the children progress through a thinking stage at different rates, the teacher is faced with an organisational problem. Should more groups be created or should the more advanced children be added to an existing higher group? Teachers need to rely on their own experience and skills to make informed decisions that they are comfortable with.
Sometimes not. On average teachers of year one to three children are more familiar with group teaching than teachers of older children. They can be helpful here. It is part of the professional development culture of the school to discuss the desirability and the methods of organising groups for teachers unfamiliar with group teaching.
This is fine provided the purpose of doing knowledge like this is to provide the background needed for teaching thinking is understood and planned for.
Provided the work is purposeful and links clearly to the knowledge/strategy teaching that the teacher has planned, then activity sheets and textbooks are OK. The main danger here is that covering the work in a textbook ends up driving the teaching instead of the teaching driving the selection of the material.
In year one the large majority of maths time should be number. The amount of time drops year by year, so by year eight something like 35%-40% of maths time should be on number.
Knowledge activities that have already been taught, BSM activities, Figure It Out activities and activities from other mathematically known resources such as Think Mathematically and Think Maths, Nimble with Numbers and Number Sense.
No. Competence is indicated by the observing whether or not the students are using number properties. If a student can use materials or imaging but not number properties the teacher knows that they are in the process of constructing the desirable abstractions. Teachers can move everyone in the group on to a new activity and watch for the emergence of number property usage.
While students are using counting methods the majority of the time is needed to establish recall. As part-whole thinking emerges the balance moves towards more strategy teaching.
National data indicates that by the end of the project about one child in three is part-whole by the end of year three. This indicates that teaching part-whole thinking is a major challenge. But it is one no teacher can afford to give up on.
Strategy is thinking. Knowledge is those things that a student can instantly recall in order to think.
Example: Jan works out 9 + 4 by counting-on. The strategy Jan uses is that she must start counting at 10, (and not 9), she must say the next four words in the counting sequence, and she must know the answer to 9 + 4 is the last number she says. All this is strategy or thinking. The knowledge Jan must have is the counting forward starting from 10 without having to think what comes next.
Example: Jerry works out 89 + 89 by 90 + 90 minus 2 equals 178. The thinking that Jerry uses is to realise that 90 + 90 is 2 more than 89 + 89. The knowledge Jerry needs is 9 + 9 is 18. A process that he needs is 18 tens is a 180. A process now shows 180 - 2 is 178. These shows the close relationship between thinking and recall.