A place for valuing mathematical thinking
Please forgive the use of images for equations - it's the best I can do for now
Abstract
Place value numbers have been of great importance to mathematicians for millennia, precisely because they allow for the possibility of creating ‘automatic’ procedures in arithmetic, relieving mathematicians of heavy intellectual work when trying to perform mundane calculations. Today the decimal place value system has become the pre-eminent representation of number in human interaction. Place value and place value algorithms deserve due consideration in any mathematics curriculum taught in schools.
In structuring a curriculum for mathematics, which in Australia is taught as a compulsory subject from prep until Year 10, organising principles are necessary to provide coherence. Bloom’s taxonomy provides such principles, placing the acquisition of elementary facts as the starting point, and presents creativity and synthesis as the most demanding parts of curriculum. Place value numbers and place value algorithms are the synthesis of foundational ideas, such as the coordination of multiple operations and the field laws of arithmetic. Therefore, Bloom’s taxonomy suggests that place value, and the concomitant algorithms, are a demanding part of the school curriculum.
In contrast to Bloom’s, the Australian Curriculum encourages familiarity with place value at an early stage, and focuses on exploiting place value for use in a single arithmetic operation, typically limited to addition, subtraction, multiplication and division. Only after many years of learning arithmetic are students expected to coordinate multiple operations. Thus the curriculum order in Australia denies the student and teacher access to the tools that would permit satisfactory explanations of why place value algorithms work. Indeed this is so for many years after such algorithms have been introduced.
Beyond what may seem to be academic objections to the current curriculum order, there are some practical concerns. Research strongly suggests that where secondary students do not have an understanding of set procedures and algorithms, but rely instead upon rules, such students cannot robustly and reliably use such procedures over the long term, which leads to under achievement.
In order to understand why the current orthodoxy is so dominant we use the ideas presented by Skemp (1976) of Relational and Instrumental Understanding. While these ideas are useful in analysing the current situation, ultimately they present a false dichotomy when creating a coherent and rounded mathematics curriculum. Standard procedures are prized by mathematicians, and should be introduced at a stage when they can be used to promote mathematical reasoning, rather than hinder it.
Unpacking Place value
Zoltan Dienes was a 20th Century innovator in maths education and a creator of mathematics teaching materials, including the widely used Multi-base Arithmetic Blocks (MAB). He spent much effort creating materials to introduce the concept of place value to young children, as for him understanding place value was the foundation of arithmetic. Indeed Zoltan Dienes (1971 pg 46) talks about “the most fundamental principle of notation in arithmetic, that of place value”.
As Dienes’ goes on to write (ibid),
“When we write a large number like 24,579 what we really mean is:
”
It is clear from the above that ‘large’ multi-digit numbers, in a place value system, are the coordination of multiple operations. The notation of the decimal place value system is very dense, acting as shorthand for particular form of number sentence, or mathematical expression.
With respect to the Australian Curriculum [1] a child in Year 4 should be able to understand the number given by Dienes. However it is not until much later that the child will understand what the number means, at least in the sense of place value. For example:
Year 4 - Recognise, represent and order numbers to at least tens of thousands (ACMNA072)
Year 6 - Explore the use of brackets and order of operations to write number sentences (ACMNA134)
If a single ‘large’ number is shorthand for a number sentence, or expression, containing multiple operations, then coordinating two ‘large’ numbers in performing a calculation is a most demanding activity. Place value algorithms for performing a single operation with multi-digit numbers exploit the field laws of arithmetic and tacitly coordinate multiple operations.
In order to demonstrate how dense and involved such calculations are, an example is provided in Appendix A.
All this may sound unnecessary. Indeed much of the current orthodoxy seems to bypass all the messy mathematical concepts that we need to interpret the place value system with anything like its proper operational meaning. However if students and teachers do not understand place value, as it is understood by mathematicians, then what exactly is it that they understand, and what is being taught in the classroom? (For suggested classroom materials look at appendix B)
Some issues regarding the order of the Australian Curriculum
Given that place value numerals are tacitly composed of multiple operations, when a Year 3 wants to “Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)” what is really meant by the calculation 8,762 - 4,121 is:
Even if we do not use powers to write out place value, the Year 3 student will be expected to understand the calculation as something like
This knowledge will be used to re-arrange the calculation into a series of sub-calculations such as:
Even if such a re-arrangement is not written down by a Year 3 student, it is nonetheless invoked. Of course, there are other ways of re-arranging the whole calculation into sub-calculations. However, in accepting the veracity of any such a re-arrangement, the field laws of arithmetic must be invoked, even if these laws are not named.
The purpose of partitioning, rearranging or regrouping is to create a set of sub calculations. In order to coordinate these sub calculations, it is necessary to invoke the field laws of arithmetic. If the calculation is not re-arranged into a series of sub calculations, which necessarily invokes the field laws of arithmetic, it is hard to understand the purpose of partitioning. However, the associative, commutative, and distributive field laws are not addressed in the curriculum until Year 7 (ACMNA151).
The calculation presented here of 8,762 – 4121, when written in a way that exposes its mathematical elements, as above, shows the use of brackets, addition, subtraction, multiplication (even if we do not include powers). I present just four possible curriculum descriptors from the Australian Curriculum.
Year 3 - Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)
Year 6 - Explore the use of brackets and order of operations to write number sentences (ACMNA134)
Year 6 - Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers (ACMNA123)
Year 7 - Apply the associative, commutative and distributive laws to aid mental and written computation (ACMNA151)
The curriculum does not follow the principles set out in Bloom’s Taxonomy, as the Year 3 descriptor is a synthesis of Year 6 and Year 7 descriptors.
The issue is compounded, as the Year 6 and Year 7 descriptors, cited here, are also in the wrong order.
For even a simple number sentence, it is not possible to ‘explore’ the situation, as required in Year 6 (ACMNA134, as cited above), without knowledge of the “associative, commutative and distributive laws”, which is only addressed in Year 7 (ACMNA151).
Without a knowledge of the field laws, all that a student can do is follow more ‘rules’, usually reduced to the mnemonic BODMAS or BIDMAS.
For the Year 6 student the only way forward is by a BODMAS approach, as in the example below:
However, for a student armed with the field laws of arithmetic there are choices to explore. This student might choose to use the distributive law:
So far, an observer might wonder why this matters: a student learning BODMAS has the capacity to calculate the value of any given number sentence, even if there is no choice of approach. However as we have seen, in a place value system, numerals are a shorthand for a particular form of number sentence. BODMAS fails to recognise this, or that differing number sentences are operationally equivalent. BODMAS is focused on getting answers, not on exploring operations and relationships.
BODMAS is antithetical to a coherent teaching of arithmetic in a place value system. Yet BODMAS is forced on students and teachers, as number sentences are explored before the coordination of the field laws of arithmetic.
The adherents to BODMAS rules will not think “when we write a large number like 24,579 what we really mean is
”, but rather believe that when we write
we can calculate the answer, 24,579.
The descriptor “Apply place value to partition, rearrange and regroup numbers to at least 10,000 to assist calculations and solve problems” ACMNA053 from Year 3, ACMNA072 from Year 4 are proscribed by BODMAS.
Using the earlier example, in the calculation 8,762-4,121, knowledge of place value means this calculation can be represented, either in concrete materials or in writing as:
which, for the purpose of calculation, can be re-arranged into something like:
The various place value algorithms and ‘mental’ strategies are dependent upon the tacit use of such re-arrangements. This is explored further for a simple calculation in Appendix A.
BODMAS does not permit such re-arranging, but rather in BODMAS
from which we can only get an answer via a standard procedure such as a place value algorithm.
Here we can see that the failure to place the curriculum items in a coherent order directly prevents the mathematics curriculum from becoming an integrated whole in the mind of the learner. The algorithm, when written out using in-line arithmetic, is nonsensical, to the student who has been trained to follow BODMAS.
Rather than creating a coherent body of knowledge in the mind of the learner, the current curriculum order forces students to simultaneously hold divergent ideas: partitioning for assisting in calculations; and BODMAS for exploring number sentences. What the thinking student is to make of this can only be guessed at. These divergent ideas necessarily require some additional rules about when a given set of rules should be used, further adding to the burden of learning mathematics.
Those children who do not understand how partitioning numbers assists in calculation will not have the situation explained to them clearly for another four years, if at all. Such children must either put the development of their own mathematical understanding to one side in deference to the authority of their teacher, if they wish to get pages full of ticks, or appear to be remedial. Just because some children seem to ‘get it’ it does not mean the child can justify their actions, indeed it would appear this is not possible until Year 7, if the schools follow the curriculum.
Further, place value makes implicit use of powers. Students learn ‘rules’ for multiplying and dividing numbers that are derived from elementary ideas about powers, before those ideas have been met in the curriculum. I present five relevant content descriptors.
Year 4 - Recognise, represent and order numbers to at least tens of thousands (ACMNA072)
Year 5 - Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies (ACMNA100)
Year 6 - Multiply and divide decimals by powers of 10 (ACMNA130)
Year 8 - Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182)
Year 9 - Apply index laws to numerical expressions with integer indices (ACMNA209)
As before, these content descriptors are not in a coherent order. The place value system, which is heavily in use in Year 4 and 5, is dependent upon understanding the effect of multiplying by powers of 10, which is not met until Year 6. One cannot understand the meaning of multiplying by powers of 10, from Year 6 to a number represented using the decimal place value system without first knowing about the laws of positive integral indices, which first happens in Year 8. Further, we cannot know about dividing by powers of ten (from Year 6) without knowing the laws for negative integral indices, which is addressed in Year 9. So the Year 4 curriculum is dependent upon concepts not explicitly addressed until Years 6 and 8, and the Year 6 curriculum relies upon concepts that are first explicitly addressed in Years 8 and 9.
Thinking and questioning children in Years 3, 4, 5 and 6 may be dismissed as backward or slow, as their intellectual objections about arithmetic cannot be addressed, and they struggle to engage with the work.
Evidence of the detrimental effects caused by too early a focus on place value and standard procedures
The objections to the current orthodoxy are not purely academic. One of the conjectures of this article is that the current curriculum order comes at a real cost. This cost is in progress through the prescribed curriculum for some students, and in promoting mathematical thinking for all students.
In looking for evidence of how individuals are negatively impacted by the curriculum orthodoxy we present published data, in the form of interviews, from 1980s Brazil and 1980s England. While such students were not studying the Australian Curriculum, it is clear that the same orthodoxy, about the location of place value arithmetic, was present in the curriculum enacted in their classrooms.
Interview Data
Example from Mathematics in the streets and in schools”
The paper “Mathematics in the streets and in schools” (Carraher et al, 1985) gave a number of examples of students who could perform mental arithmetic well in the context of a market, and yet were incompetent in a formal, school like setting when answering what appeared to be the same questions. Presented here is the first interview example from the paper.
Informal test
Customer: I'm going to take four coconuts. How much is that?
Child: Three will be 105, plus 30, that's 135 . . . one coconut is 35 . . . that is . . . 140!
Formal test
Child resolves the item 35 x 4 explaining out loud:
4 times 5 is 20, carry the 2; 2 plus 3 is 5, times 4 is 20.
Answer written: 200.
This example is typical of the others presented in the paper. The child has all the necessary mental faculties to accurately perform the calculation given in the classroom. However, rather than seeing any possible context for the formal calculation, such as market trading, the student feels compelled to use a place value algorithm, when presented with the ‘classroom’ calculation. This algorithm is not linked to any sense making the child has, and the rules for using the algorithm are misremembered.
In this paper, of the five children interviewed, four appeared faultless when performing “street maths” (the last child got 17 out of 19 calculations correct). Of these four, only one could answer the same questions when translated into a formal question with a reliability of greater than 50%. One child, with a perfect score in the market trading, scored only 10% on the formal test. Despite their clear capacities, one suspects that such children would be seen as remedial by the classroom teacher.
Examples from Children's Mathematical Frameworks: Part Three Subtraction. (Hart 1987).
This study follows a teaching sequence that may be familiar to many current primary school teachers. The goal of the teachers was explicitly to teach students to use a recognised place value algorithm in the calculation. To help create an internal framework (or schema) in order to understand the algorithm, students were first exposed to concrete materials, especially to Dienes’ blocks (commonly called MAB) and Unifix cubes. The concrete materials were used by the students for sense making, but not in the way the teachers had planned.
“Ann (aged 8) had plastic Unifix cubes, which could be attached to each other to form a column of ten. When faced with "65 - 29" she put down six columns of ten cubes and five separate cubes, the second number "29" was used simply as an instruction. To comply with "take away 29" she removed two columns from the array and then a third from which she broke off a cube which was returned to the table. Finally she counted what was left on the table, providing the correct answer.
George (aged 12) was using Dienes' multibase bricks (MAB) which cannot be broken apart. His method was very similar to that used by Ann however. In his case the situation was confused by the teacher's part in the proceedings. The teacher said (for 62 - 19) "I want you to take away from that supply you've got there, one ten, well that's easy enough to do, isn't it? Just take that one ten away, get that off your sheet, now we've got to take away nine units as well. Let us see if we can do it." George said "Put it out of the tens and get one unit and put it in the units column." Both these children had practical methods of doing subtraction which matched the practical materials they were using; however neither method was connected to the supposed synthesis of practical experiences which was the algorithm.”
Hart and her colleagues made follow up interviews. Here is an extract of the first interview.
“four of the 8-9 year olds replied as follows:- (I: Interviewer; A: Adam; Al: Alan; An: Angela; N: Noreen)
“I: What number is now on the top line?
A: Two
I: You're quite right, absolutely on the top line and if you look at the top line of the sum, as opposed to the top line in the writing down, what number?
A: 29
I: 29 ... and is that what you started with?
A: No
I: Does it make any difference?
A: ... (pause) ...
I: What did you start with?
A: 305
I: I see, what number do you have now?
A: 29
I: 29. So what's happened? Has it changed?
A: Yes
I: Is it still 305 take away 97?
Al: No
I: Well, what's happened?
Al: I’ve changed it
I: Well, what's the answer to ... 208's the answer to which sum?
Al: 211 take away 97
I: I see. I didn't ask you that did I? I asked you 305 take away 97
Al: Shall I do it again?
The paper continues with similar examples. The students were able to perform the subtraction algorithm with complete success. However the rearranging of digits on the page does not appear to be linked to any sense making. So although the students may get good marks, they have no mathematical understanding of what they are doing: and this is after the teachers have introduced the algorithm using concrete materials.
Again place value and place value algorithms are too intellectually demanding for students who can otherwise manage similar calculations using their own strategies. The effects on students and their relationship to mathematics, of being asked to carry out procedures that do not make sense to them is a matter for speculation.
Examples from “Rules without Reason” (Brown 1982)
In the late 1970s and the 1980s a team at Chelsea (later King’s College London) carried out an extensive survey of children’s thinking at secondary school level.
This interview shows the result of a ‘weak’ student in Year 10, being fed a diet of procedures and algorithms during her time at secondary school, precisely because she has been identified as a ‘weak’ student.
“Margaret Brown: What would you use decimals for? Can you think of a situation in which you had to use decimals?
Susan: What, add?
MB: Yes, can you think of a situation except for a maths lesson, when you have to add decimals?
S: What, you mean you have to put the point there?
MB: Yes, not how you do it but why you might have to do it. Can you think of any real situation in which you might have to add decimals, or is it just in maths lessons?
S: Long multiplication you have to do, don't you?
MB: Yes, you would have to, but can you think of why you might be doing it, outside—when you leave school for instance?
S: Yeah, they're bigger than money aren't they?
MB: OK, in money you would. Anything else where you might use decimals apart from money?
S: Take away:
MB: Yes. Just in money, that's the only time you'd use them?
S: Yeah.
The latter conversation, with a girl at the end of year 10 shows quite clearly the effects of putting a priority on routine skills. Not only does she appear to have little idea of how or when to apply such skills, but the rest of the interview illustrated that she was rarely in fact able to recall and perform the algorithms reliably, presumably because she had so little conceptual understanding to which to relate them.”
“To summarize, many teachers say that they are attempting to compensate for children's lack of understanding by 'teaching by rote'. The evidence above casts considerable doubt on the wisdom of this policy. For a start, the lack of understanding means that children have to face an enormous memory-load which they cannot relate to their existing knowledge.”
Here is an interview with a boy in Year 7. David was in the top 6% of the ability range, and was answering a test item
David: Ah. I remember something. Shall I do it again?
I added a nought on each one 'cos you couldn't divide by that (indicates the 0.3). I added a nought so it'd make it 3.
MB: Do you think the answer's right?
D: I think you have to put it back again.
MB: You have to put it back again, do you?
D: Yeah (changes '200' to '20')
MB: Would you like to look at that one?
D: Oh yeah, I could have easily done it that way—that should be, e r . . . no, it should be bigger shouldn't it? It should be 200, I think.
MB: It should be 200?
D: Well (indicates 0.3) that's less than one—you'd be dividing it by less than one. Divide it by one; it's sixty, so dividing it by less than one it'd get more. So I think 200.
Margaret Brown further states:
“Even quite bright children sometimes found it difficult to remember routine procedures”.
“In fact none of the children unable to justify the technique were in the top half of the attainment range”
Discussion
The current Australian Curriculum for mathematics is very similar, in both content and ordering, to other mathematics curricula around the world today. Why this orthodoxy has arisen, and become so dominant is not clear. Why so much time in the early curriculum is devoted to working with place value, and so little time devoted to developing an understanding of place value is a matter for speculation. We will spend some time considering these matters now.
In understanding the current orthodoxy, rather than looking at broad socio-political matters, it is perhaps most useful to examine the possible relationship between individuals and the subject of mathematics itself. Skemp (1976) presented two different ways of engaging in mathematics, that he termed ‘Relational Understanding’ and ‘Instrumental Understanding’. These ideas are broadly similar to the ‘acquisition of techniques’ and ‘understanding of ideas’ presented by Dienes (pg 3, Dienes, 1971). Instrumental Understanding is dominated by rules, while Relational Understanding is focused on connecting ideas. Instrumental Understanding dominated both the teaching and texts in the 1970s, when Skemp and Dienes were writing. Little has changed, in this respect at least, and consequently for many, the Instrumental Understanding of mathematics is school mathematics.
Skemp, while seeing value in both approaches, considered that these two ideas were sufficiently divergent to be considered as two distinct subjects, both called mathematics. Here seems to be the central difficulty with the debates about the mathematics education: are we talking about mathematics as Instrumental Understanding or Relational Understanding?
For teachers and curriculum designers with an Instrumental Understanding it will seem entirely reasonable to believe that ‘automaticity’ with algorithms and procedures, and the necessary rote learning to perform these algorithms, form the elements that sit at the bottom of Bloom’s taxonomy.
The current curriculum ordering makes sense from the perspective of Instrumental Understanding. It is possible to spot sequences in the curriculum that build up procedural skills for place value, rather than focusing on building, and synthesising elementary concepts. For example here are two short sequences:
Year 2 - Group, partition and rearrange collections up to 1000 in hundreds, tens and ones to facilitate more efficient counting (ACMNA028)
Year 3 - Apply place value to partition, rearrange and regroup numbers to at least 10 000 to assist calculations and solve problems (ACMNA053)
Year 4 - Apply place value to partition, rearrange and regroup numbers to at least tens of thousands to assist calculations and solve problems (ACMNA073)
Or the following sequence:
Year 3 - Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)
Year 4 - Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)
Year 4 - Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)
Year 5 - Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)
Year 6 - Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies ACMNA129)
These two sequences follow an Instrumental view of mathematics, of increasing procedural skill, with the necessary rote learning presented as a pre-requisite.
However, those who favour Instrumental Understanding, as the core focus of school arithmetic, need to be aware of how fragile such knowledge is, as demonstrated in the interview data we saw earlier. Margaret Brown, in writing about curriculum design (Ch13 Leslie and Mendick, 2013) gave a summary of some pertinent research findings, from which we present just four points.
1. Children who can accurately recite number names in the correct order can still often obtain the wrong answer to counting problems.
2. Many Children have problems remembering mathematical facts and procedures for written sums like subtraction of three-digit numbers, and can only reconstruct the steps they have forgotten if they have sufficient understanding of the ideas underpinning the process.
3. Children who found the rules for doing written calculations difficult to remember at school could sometimes perfectly well perform the same calculation in everyday life using their own mental methods.
4. When doing written calculations, many children and adults chose to use methods different to those they were taught at school
As noted by Brown (1982), the memory of weaker students is often blamed for their failure. However there is no evidence that underachieving maths students actually have poor memories. Rather it seems that they have no framework within which to place the ideas they learn in class. In Brown’s interview with Susan, which we saw earlier, the secondary education Susan was receiving seems to have enforced her failure by denying her opportunities to develop her own sense making with number. By contrast we saw that the successful student, David, was not successful because of a superior memory, or greater automaticity, but precisely because he was able to think about what was written to see if it made sense.
It seems reasonable to state that neither a knowledge of “the facts” nor automatic skills is enough to ensure a robust and enduring ability to perform calculations reliably. Enduring success with routine procedures is almost certainly down to the ability of the student to link such procedures to their internal framework. As such it is important that the curriculum order supports the creation of this internal framework (or schema), by teaching the elements of place value arithmetic before combining these elements in the teaching of place value itself.
Despite the problems with an instrumental approach being used as a ‘foundational’ approach in the teaching of mathematics, instrumental techniques are an essential part of being able to operate mathematically. As Skemp (1976) notes, mathematicians regularly use instrumental techniques. Indeed for many mathematicians, time is often spent using relational understanding to develop instrumental techniques. Instrumental techniques act as a solution, not to one problem, but to a whole class of problems. For the current conversation an apt example of this is Napier’s bones [3]. Instrumental Understanding and Relational Understanding are important ideas in identifying the problems with the curriculum order, and help to identify divergent pedagogies. However these two ideas are ultimately a false dichotomy when looking at mathematics as a whole. Instrumental techniques need to be appropriately located within the P-10 curriculum structure, so that they can also be integrated into the student’s internal framework of understanding, and assist with the development of the student as a rational, thinking, reasoning person.
Being able to hone one’s thinking to such a degree that it can be codified into an automatic procedure is high goal for a mathematician. It is also an essential step for anyone wanting to write a computer program. In the modern world, in order to fully exploit the power of computers it is necessary to learn to codify one’s thinking into a satisfactory automatic process. Standard procedures in scholastic mathematics offer an opportunity to explore the process of automation, both in pencil and paper operations and in machine coding.
Standard procedures, including place value algorithms, transform operations with numbers, into the manipulation of numerals and digits. It is hard to think of a more formal process that is experienced in school. Given the formalism of standard procedures in arithmetic it only seems reasonable that they should be coordinated with the teaching of algebra, mathematical proofs, and computer programming.
There are clear mechanisms, by which the current curriculum order forces some students to put aside the development of their understanding for years, with a subsequent deleterious effect upon their learning of mathematics. However for those who believe in an instrumental approach to foundational mathematics, which appears to be in the majority, it is likely that some large scale, statistically sound, empirical data would be required before considering changes to the curriculum order. Dienes was hoping for such data when writing nearly 50 years ago (pg 15, Dienes, 1971). In contemporary Australia the socio-political situation would appear to make the necessary longitudinal studies less likely than a generation or two ago. The same would appear to be true in much of the world. Without such data it is hard to imagine the Australian Curriculum being appropriately restructured.
Despite difficulties of implementing alternate approaches to the curriculum, due to a national curriculum and regular standardised tests, curriculum changes can be implemented by individual schools, and individual teachers. For example, in the UK, Caroline Ainsworth [2] has used the work of Gattegno and Goutard to inform her primary school teaching, with the necessary curriculum changes.
It is beyond the scope of this discussion to set out in detail how such reforms can be undertaken by a school or teacher. However some personally preferred references to curriculum design, and to possible supporting teaching materials, including those used by Ainsworth, are set out in Appendix B, for those teachers and schools who wish to begin an inquiry into making changes.
In interview Ainsworth [2] cites a page from Madeleine Goutard’s book “Mathematics and Children” (1963), shown below. This is the work of a six year old boy in a regular school, after only six months of schooling, as the foot note shows.
Even if we dismiss the claim by Goutard that this is the work of an ordinary student, most teachers, schools, and curriculum designers in Australia would be delighted if this was the output of an ordinary Year 4 student. Note that this student is showing a capacity to deal with all the elements of place value numbers, while not yet explicitly dealing with the place value structure. When this child is formally introduced to place value there can be little doubt that he will be able to understand it in the way that a mathematician does.
The student creating this writing has experienced a curriculum focused on developing an internal framework; as such that curriculum can be described as Constructivist. However the term Constructivist covers a very broad spectrum of educational beliefs. For many, including Gardiner (2004), Constructivism is a pejorative term, suggesting free interaction with materials in the hope of students discovering mathematical ‘truths’. The skill displayed by the work sample above did not arise by accident, it occurred through detailed curriculum planning, as well as skillful teaching.
In conclusion, the Australian P-10 Mathematics Curriculum, in the round, prescribes a reasonable content for arithmetic. However there are both theoretical and practical concerns with the order in which that curriculum is addressed. Where educators consider that alternative formulations of the curriculum will better support students, then such possibilities should be investigated. Perhaps the most obvious case for such consideration is when a child has clearly been damaged by the mathematics schooling that has already been delivered, and has chosen to disengage.
References
Benson, I. (2011). The primary mathematics: lessons from the Gattegno School.
Brown, M. (1982). Rules without reasons?: Some evidence relating to the teaching of routine skills to low attainers in mathematics. International Journal of Mathematical Education in Science and Technology, 13(4), 449-461.
Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British journal of developmental psychology, 3(1), 21-29.
Dienes, Z.P. (1971) Building up Mathematics, Hutchinson Educational
Hart, K. (1987). Children's Mathematical Frameworks: Part Three Subtraction. Mathematics in School, 16(5), 30-33.
Gardiner, A. (2004). What is mathematical literacy. ICME-10, Copenhagen, Danmark.
Goutard, M. (1964). Mathematics and children: a reappraisal of our attitude. Educational Explorers.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics teaching, 77(1), 20-26.
Leslie, D., & Mendick, H. (Eds.). (2013). Debates in mathematics education. Routledge.
[1] https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/
[2]Caroline Ainsworth https://www.youtube.com/watch?v=ism0SJP5W2g&list=PLQqF8sn28L9wwVPLLO2VPOz3kL5A7CMlr&index=2
[3] Napier’s bones - https://en.wikipedia.org/wiki/Napier's_bones
Appendix A
(This needs some more work to transcribe across to this blog, but you'll get the idea)
The following demonstration is not intended to act as an explanation of the elementary ideas used a place value algorithm. Rather it is intended to highlight just how dense place value algorithms are, and to show the ideas that need to be explored, examined and understood before a proper justification for the steps taken in the algorithm can be given. Each step of the place value algorithm for subtraction is narrated three different ways: a ‘standard’ school model, a heuristic model, and something approaching a formal model, presented in English and in-line arithmetic, justifying the steps in the algorithm. Whilst there are different algorithms that can be used, and different presentations of the algorithm, I choose to present a ‘standard’ method which can readily be found in schools today.
The example chosen was presented by Hart (1987) as part of an inquiry into the teaching of subtraction using place value, including using Dienes’ base 10 blocks. The calculation we will consider is 305-97.
Step 1
Standard instruction
Write 305 in clear columns of units, tens and hundreds: the 5 in the 1’s column, the 0 in the 10’s column and the 3 in the 100’s column. Beneath this line, write 97, with the 7 in the 1’s column and the 9 in the 10’s column. Draw a line underneath the 97. Leave space for the answer and then draw another ruled line.
| H | T | U |
| 3 | 0 | 5 |
| - | 9 | 7 |
Heuristic/Vernacular explanation
We are not going to consider 305 as a single number, nor 97. Rather we are going to break them down into Hundreds, tens and ones, so that we can perform a series of separate calculations, the results of which can be combined as to have the same effect as if we had been able to do the single calculation of 305-97.
Formal explanation
Now we have partitioned up the numbers and operations we need to re-order the calculation – note we are simply re-arranging the calculation, there is no ‘doing’ yet.
Now we want to focus on operation with the digits, rather than the numbers themselves.
So the presentation of the algorithm – the setting out, achieves the recasting:
In order to understand even the presentation of the algorithm, as a mathematician does, has required four operations (addition, subtraction, multiplication and powers), the use of the distributive law twice, and extensive use of the associative and commutative laws of addition.
Step 2
Standard Instruction
Now starting in the units’ column: 5 -7 can’t be done (even if we know about negative numbers?). So we need to borrow from the other columns. We can’t borrow from the ‘zero’. So we borrow from the 3. We turn it into 2, and put a 1 in the tens column to make a 10. But we still haven’t added anything to the units yet. So we turn the 10 tens into 9 tens, and borrow one ten into the units to make 15.
| H | T | U |
| 15 | ||
| - | 9 | 7 |
Heuristic/Vernacular explanation of instruction
We start by taking 7 units away from 5. We know the overall answer to 305-97 will be positive, so we would like each column to have a positive answer. So we choose to partition 305 into 2 Hundreds, 9 Tens and 15 Units. 200 + 90 +15 is the same as 300 +5, just re-written. However, now every column has a larger number on top than on bottom. So the answer to each separate column calculation will also be positive, making it easier to interpret the answer at the end.
Formal explanation of instruction
We have already justified re-arranging the calculation of two numbers into a set of sub calculations involving the digits.
However we want to focus on the co-efficient of 10^0 for now. We could re-write the calculation, following the convention of written English to make things clearer. This is only possible using the commutative and associative laws of addition.
Indeed, even without re-writing the calculation, in choosing which part of the calculation to focus on requires us to be comfortable with the field laws of addition and the precedence rules of arithmetic that creates ‘terms’. If such knowledge has not been established, the student has no choices, and cannot see sense, or even ‘truth’, in the order of operations chosen by the algorithm.
We note that as the calculation is currently written, the co-efficient of 10^0 and 10^1 will be negative. So we choose to decompose and partition 3x10^2. For the sake of clarity we will do this before re-introducing the whole calculation.
Now we have the co-efficient of as greater than, or equal to, 9, which is needed for a positive final answer. But we need to create a co-efficient of which is bigger than 7. Therefore we continue.
So the final decomposition of 300, or , for the purpose of this calculation, is:
We put this back into the original calculation:
Using the field laws of addition and the distributive law we can write:
In order to compare this presentation to the compact place value algorithm, we tidy up the positive terms in each co-efficient. Now we write:
Step 3
Standard Instructions
Now looking at the Units column, 15 take away 7 is 8, put the answer on the answer line, and in the units column. Next look at the Tens column, 9 take away 9 is 0, put the answer in the Tens column. Lastly, in the Hundreds, there is nothing to take away from the 2, so we simply put 2 in the answer line in the Hundreds column. The answer is 208.
| H | T | U |
| 15 | ||
| - | 9 | 7 |
| 2 | 0 | 8 |
Heuristic/Vernacular explanation of instructions
No further explanation of the algorithm is needed, than that given by the instructions.
Formal explanation of instructions
So far we have re-written the calculation, using knowledge of place value, the field laws of arithmetic and the meaning of powers to arrive at:
We now calculate the co-efficient of each of the powers of 10.
Now given the definition of the number in a decimal place value system we can write an answer:
Appendix B
In agitating for change I feel some responsibility to assist those who seek to create change. However it must be understood that this appendix is not an argument for any particular form of curriculum, but rather is a personal list of places I would look at in considering curriculum change, and as such it is a bit of a mind dump.
In thinking about curriculum there are four broad layers: there are the guiding principles, used to create a meaningful and coherent program that may cover many years; there are the general pedagogic approaches of classroom interactions, that in themselves are very useful to consider when creating sequences of lessons over the short to medium term; thirdly there are the specific curriculum items and materials used in the classroom, which match our broad goals; and lastly there is the deliberate assessment of students’ progress through the curriculum (hopefully for the purpose of assisting them with further progress). No effort has been made to deal with these layers separately.
Mathematics in Color was the program devised by Caleb Gattegno, and followed by Caroline Ainsworth. The materials relating to this program are so extensive and divergent that they will be partially discussed and listed at the end of this appendix.
On the specifics of assessment materials that focus on concepts in arithmetic the CSMS tests are hard to beat.
The CSMS materials may help teachers identify what problems exist in the classroom, though they will do little, of themselves, to help resolve these matters.
On the specifics of classroom materials and related pedagogy, in Australia there are teaching materials available in alternative approaches to the curriculum, such as Maths 300 and the Mathematics Task Centre that may provide some assistance. Despite these projects predating the Australian Curriculum, as a pragmatic measure the projects seem to be adapted to the Australian Curriculum, and so may not be so helpful when looking at broad structuring.
https://mathematicscentre.com/
Margaret Brown set out a number of principles for choosing the content and order of a mathematics curriculum, as a summary of her experience of working in the field over many years in England (Ch13 Leslie and Mendick 2013).
Available at https://www.taylorfrancis.com/books/e/9780203762585
This book as a whole is worthy of consideration.
Brown was responsible for an assessment program called GAIM, which ordered the assessed mathematics curriculum by general cognitive abilities in ‘reasoning, problem solving and investigating’. GAIM materials are available on-line
https://www.stem.org.uk/resources/collection/2780/graded-assessment-mathematics-gaim
GAIM not only provides a broad structure within which various aspects of a curriculum can be located it provided materials that can be used for both summative and formative assessment with respect to these broad descriptors. These activities were useful enough, without the curriculum structure, that a number of them were re-worked into the Nuffield AMP
https://www.nuffieldfoundation.org/AMP
Related to GAIM was the London based SMILE program, which provided a pedagogical approach and classroom materials. These materials can also be found at the STEM e-library in York. There are so many SMILE materials, but most adaptable are the SMILE cards.
https://www.stem.org.uk/resources/collection/2765/smile-cards
Hillary Povey, who was intimately involved with SMILE, has written about the pedagogy of mixed ability classrooms, which was seen as central to the success of SMILE.
https://www.shu.ac.uk/about-us/our-people/staff-profiles/hilary-povey
These materials and approaches seem to have informed the research of Jo Boaler. There is a collection of publicly available materials from Prof. Boaler’s team at
The youcubed site presents a wealth of research, discussing the efficacy of different pedagogic approaches, as well as providing teaching materials for those wishing to emulate the findings. Boaler’s work has become linked with that of psychologist Carol Dweck, who developed the idea of Growth and Fixed Mindsets.
https://profiles.stanford.edu/carol-dweck
Another site with a cornucopia of materials, covering all aspects of the curriculum, is the Shell project website
Similar in spirit to GAIM assessment materials, in assessing general cognitive abilities, related to mathematics is Balanced Assessment program.
https://hgse.balancedassessment.org/
Zoltan Dienes is worthy of mention for many reasons (for one he spent a long time in Adeleide). The book I have, Dienes, Z.P. (1971) Building up Mathematics, Hutchinson Educational, is a great text thinking about general issues for curriculum design. Sadly it is out of print, but Abebooks might be able to help. Dienes believed in the development of conceptualisation years before formalism and mathematical writing. Attempting to develop concepts years before those concepts are given a literary form is very Piagetian. This is perhaps the most disputed part of his approach (oh for some serious longitudinal studies). For me, it seems that even if the abstract maths concept is not actually being taught, by using concrete materials without developing the related mathematical writing, then at least the mental capacity to handle the concept is being developed through the use of manipulable materials. Such issues aside, Dienes’ principles and Dienes’ stages of learning are valuable when planning a teaching sequence that will extend from a couple of weeks to a term in length.
https://subs.emis.de/journals/ZDM/zdm053a18.pdf paper discussing Dienes’ principles
https://www.zoltandienes.com/academic-articles/zoltan-dienes-six-stage-theory-of-learning-mathematics/ Zoltan Dienes’ posthumous website, with details of the ‘Six Stages’.
In Victoria the MAV is useful https://www.mav.vic.edu.au/Home
In the UK NRICH has a mountain of teaching materials – https://nrich.maths.org/
Gattegno, Goutard and the Cui (Cuisenaire) Program
Gattegno was a former research mathematician, and research psychologist, who worked with Piaget in creating the first English translation of Piaget’s work. He later rejected Piaget’s notions on concept building. Meticulous care and attention, exploiting his own high level of mathematical understanding was used to create a pedagogically sound curriculum ordering, focusing on ‘basic’ arithmetic. All this work is available as an ‘off the shelf’ solution. A little work is required to adjust to the thinking (but well worth it in my view). Further, this is not a complete maths program; it is an algebra and arithmetic program.
To get a flavour of the program there are some youtube videos
There are a series of videos by Caroline Ainsworth, who was inspired to try the program, starting with https://youtu.be/LUdL8GFZp2k
To see Gattegno in action, nearly 60 years ago start at https://youtu.be/ae0McT5WYa8 or if you prefer watching him teach in French https://youtu.be/Kw94gmzRrOY
Ainsworth was in part supported by the work of Ian Benson, who created a text, reviewing and updating the program. The book illustrations are in black and white – which is a nuisance – otherwise great book (but not available in soft or on-line versions).
Benson has a page devoted to the issue of Cuisenaire and Gattegno (great for making UK connections, but otherwise probably best left alone until you’ve got a handle on the ideas)
http://tizard.stanford.edu/users/ianbenson/
The general theory of designing an arithmetic curriculum is given by Gattegno in “The Common Sense of Teaching Mathematics”. This book can be seen for free as well as purchased in hard or soft copy at:
Gattegno’s classroom curriculum starts with “Text Book 1” – again free to read online.
http://www.educationalsolutions.com/books-books-for-teachers/gattegno-mathematics-textbook-1
There are two on-line shops that sell Gattegno’s materials:
Educational Solutions in Canada http://www.educationalsolutions.com/
The Cuisenaire Company in Wales https://www.cuisenaire.co.uk/index.php
Educational Solutions are awesome as all their texts are available to read online for free – so you can really try before you buy.
The Cuisenaire Company is the only place I know of where one can buy Goutard’s texts, which are focused on assisting teachers know how to get the most from the program, and avoid certain pedagogical pitfalls (as well as inspiring Ainsworth), here is perhaps the most useful one
http://shop.cuisenaire.co.uk/mathematics-and-children/
The Cuisenaire company also sells some materials developed the Association of Teachers of Mathematics (ATM). The ATM is a professional body of mathematics teachers, and was originally formed by Gattegno. One download from the ATM that is worth reading, if considering this approach is:
https://www.atm.org.uk/write/MediaUploads/Resources/Cuisenaire_Rods_and_Why_book.pdf