If modelling thinking is a key element of teaching, why do we appear to remove concrete apparatus when the mathematical concepts start to get more difficult?
Based on an earlier blog: https://chrischiversthinks.weebly.com/blog-thinking-aloud/talking-maths
It would be interesting to know when the concrete apparatus is withdrawn from the teacher explanation repertoire, as this can be seen as only useful for SEN children, yet, used effectively, can enable even more able learners to make connections through very clear visual manipulation. This can be demonstrated through the use of Dienes base 10 material to explore place value and four rules with decimals to three places, using the 1000 block to represent 1.
How do you know what a child is thinking unless you ask them directly to explain something?
We have become used to Talk for Writing, so why not Talk for Maths? If teachers and children engage in learning dialogue, the teacher can get a better view of how the children are thinking and the learners might become more secure in their willingness to have a go, especially when facing novel situations. We also talk of it being ok to make mistakes, especially in the context of Growth Mindset thinking. I would suggest that an openness to dialogue underpins GM, in that a child should be able to share insecurities and to be able to talk through a resolution. Learning to think and talk is an important stage in being able to do so internally, from the scaffolds developed through discussion and manipulation.
The early days of my teaching career were in a school that focused its approach on the work of two key figures in mathematics; it helped that the head was a County adviser for maths, so we also benefitted from regular visits by his colleagues in the inspectorate.
There were two key elements highlighted, logic and modelling mathematical thinking supported by continuous use of structured materials. The work of Zoltan Dienes was central, embodied in the structured approach created by Harold Fletcher, whose workbooks were the spine for mathematical activities by the children. For both, we were given the key background texts to read and understand. In this way, we avoided falling into the trap of just doing the activity booklets, as both the teacher guides and the senior staff accentuated the central place of concrete apparatus. This, in itself, was accentuated through staff training as a group or 1:1 coaching to need.
From https://www.stem.org.uk/elibrary/resource/30000 an extract of Fletcher’s background.
Harold Fletcher was seen as an outstandingly gifted teacher and educationalist. While he was always a firm believer in children being able to calculate accurately, he found from his own teaching that they could achieve remarkable results in other aspects of mathematics. Harold Fletcher considered the mathematics he wanted children to learn under six strands:
Number Pattern Shape Pictorial Representation Measurement Algebraic Relations.
With the help of a team of experienced teachers and educationalists, Harold Fletcher wove these strands into a teaching sequence which was called Mathematics for Schools. Examples of classroom activities are used to describe the mathematics, complete with teacher dialogue, diagrams and outcomes from recording.
Each element of number, addition, subtraction, division and multiplication along with place value was developed showing the use of concrete materials and styles of notation (many of which would be seen later in the Framework for Teaching Mathematics (NNS; National Numeracy Strategy).
Based on an earlier blog: https://chrischiversthinks.weebly.com/blog-thinking-aloud/talking-maths
It would be interesting to know when the concrete apparatus is withdrawn from the teacher explanation repertoire, as this can be seen as only useful for SEN children, yet, used effectively, can enable even more able learners to make connections through very clear visual manipulation. This can be demonstrated through the use of Dienes base 10 material to explore place value and four rules with decimals to three places, using the 1000 block to represent 1.
How do you know what a child is thinking unless you ask them directly to explain something?
We have become used to Talk for Writing, so why not Talk for Maths? If teachers and children engage in learning dialogue, the teacher can get a better view of how the children are thinking and the learners might become more secure in their willingness to have a go, especially when facing novel situations. We also talk of it being ok to make mistakes, especially in the context of Growth Mindset thinking. I would suggest that an openness to dialogue underpins GM, in that a child should be able to share insecurities and to be able to talk through a resolution. Learning to think and talk is an important stage in being able to do so internally, from the scaffolds developed through discussion and manipulation.
The early days of my teaching career were in a school that focused its approach on the work of two key figures in mathematics; it helped that the head was a County adviser for maths, so we also benefitted from regular visits by his colleagues in the inspectorate.
There were two key elements highlighted, logic and modelling mathematical thinking supported by continuous use of structured materials. The work of Zoltan Dienes was central, embodied in the structured approach created by Harold Fletcher, whose workbooks were the spine for mathematical activities by the children. For both, we were given the key background texts to read and understand. In this way, we avoided falling into the trap of just doing the activity booklets, as both the teacher guides and the senior staff accentuated the central place of concrete apparatus. This, in itself, was accentuated through staff training as a group or 1:1 coaching to need.
From https://www.stem.org.uk/elibrary/resource/30000 an extract of Fletcher’s background.
Harold Fletcher was seen as an outstandingly gifted teacher and educationalist. While he was always a firm believer in children being able to calculate accurately, he found from his own teaching that they could achieve remarkable results in other aspects of mathematics. Harold Fletcher considered the mathematics he wanted children to learn under six strands:
Number Pattern Shape Pictorial Representation Measurement Algebraic Relations.
With the help of a team of experienced teachers and educationalists, Harold Fletcher wove these strands into a teaching sequence which was called Mathematics for Schools. Examples of classroom activities are used to describe the mathematics, complete with teacher dialogue, diagrams and outcomes from recording.
Each element of number, addition, subtraction, division and multiplication along with place value was developed showing the use of concrete materials and styles of notation (many of which would be seen later in the Framework for Teaching Mathematics (NNS; National Numeracy Strategy).
Shape begins with an introduction to solid shapes before bringing in 2 dimensional or plane shapes. This is followed with measuring, area, capacity and volume before concluding with symmetry and tessellations. As with all aspects of the series it was stressed that concrete materials should still be used.
Pictorial Representation focused on students, from an early age being able to collect information, record it in pictures and most of all, think about it and use it for further number practice. The foundations for graphs were introduced before dealing with them further in Algebraic relationships. A final section on “How can I help my child?” contained some do’s and don’ts. A pdf of a parent guide is available from the STEM site above.
The second key character in my mathematical education as a teacher was Zoltan Pal Dienes (Pal anglicized to Paul). Looking up some detail, I came across his relatively recent obituary.
Pictorial Representation focused on students, from an early age being able to collect information, record it in pictures and most of all, think about it and use it for further number practice. The foundations for graphs were introduced before dealing with them further in Algebraic relationships. A final section on “How can I help my child?” contained some do’s and don’ts. A pdf of a parent guide is available from the STEM site above.
The second key character in my mathematical education as a teacher was Zoltan Pal Dienes (Pal anglicized to Paul). Looking up some detail, I came across his relatively recent obituary.
 |  |
DIENES, Dr. Zoltan Paul Obituary from http://www.zoltandienes.com/obituary/
Age 97, of Wolfville, Nova Scotia, passed away peacefully on January 11, 2014. Zoltan Dienes, internationally renowned mathematician and educator, was both a public figure and a much loved family man.
Zoltan was born in the Austro-Hungarian Empire in 1916, son of Paul and Valeria Dienes. His early years were spent in Hungary, Austria, Italy and France. He always had a fascination with mathematics, even hiding behind a curtain to hear his older brother’s maths lesson, for which he was deemed too young!
At 15 he moved to England. He received his Ph.D. from the University of London in 1939. Zoltan understood the art and aesthetics of mathematics and his passion was to share this with teachers and children alike.
He was fascinated by the difficulties many people had in learning mathematics and wanted others to see the beauty of it as he did. Consequently, he completed an additional degree in psychology in order to better understand thinking processes. He became known for his work in the psychology of mathematics education from which he created the new field of psychomathematics.
Referred to as a “maverick mathematician”, Zoltan introduced revolutionary ideas of learning complex mathematical concepts in fun ways such as games and dance, so that children were often unaware that they were learning mathematics – they were having a wonderful, exciting, creative time and longing for more. He invented the Dienes Multibase Arithmetic Blocks and many other games and materials that embodied mathematical concepts.
According to a Montana Mathematics Enthusiast monograph from 2007, “The name of Zoltan P. Dienes stands with those of Jean Piaget and Jerome Bruner as a legendary figure whose theories of learning have left a lasting impression on the field of mathematics education…
Dienes’ notion of embodied knowledge presaged other cognitive scientists who eventually came to recognize the importance of embodied knowledge and situated cognition – where knowledge and abilities are organized around experience as much as they are organized around abstractions. Dienes was an early pioneer in what was later to be called sociocultural perspectives and democratization of learning.”
I had a wry smile when I realised that his initials are ZPD, which is immortalised in the work of Vygotsky as the Zone of Proximal Development, something, I am sure that Dienes would have appreciated.
Conservation of number became a shot topic of conversation on social media during the week. It is, without a doubt an underlying concept in the learning of mathematics, akin to chunking of information to make subsequent thinking and manipulation easier.
Definition
Conservation of numbers means that a person is able to understand that the number of objects remains the same even when rearranged.
What is conservation of number?
Some of the materials being use for modelling may be less helpful, in that they might encourage children to start counting from one, for security.
Multilink or Unifix blocks are common in early classrooms. Where the mathematics takes children beyond tenness, breaking the chain into ten rods can be useful to accentuate that concept. It is heartbreaking to watch a child count, then have to restart the count because they have been interrupted. Making rods of ten would allow for interruptions and a means of continuity.
For this reason, I still have a preference for Dienes base ten equipment, as it allows early access to models of exchange, creating tens then hundreds. It accentuates place value and, using the function machine conceptualisation allows all four rules of number to be modelled effectively. With a visualiser attached to a class IWB, the modelling can be done on a large scale, enabling more to access. I recognise that in the days before such technology, there were visual limitations to modelling to large groups.
From Maths No Problem, the following accent on concrete apparatus seems to fit with this approach.
Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the Singapore method of teaching maths for mastery.
Dienes’ notion of embodied knowledge presaged other cognitive scientists who eventually came to recognize the importance of embodied knowledge and situated cognition – where knowledge and abilities are organized around experience as much as they are organized around abstractions. Dienes was an early pioneer in what was later to be called sociocultural perspectives and democratization of learning.”
I had a wry smile when I realised that his initials are ZPD, which is immortalised in the work of Vygotsky as the Zone of Proximal Development, something, I am sure that Dienes would have appreciated.
Conservation of number became a shot topic of conversation on social media during the week. It is, without a doubt an underlying concept in the learning of mathematics, akin to chunking of information to make subsequent thinking and manipulation easier.
Definition
Conservation of numbers means that a person is able to understand that the number of objects remains the same even when rearranged.
What is conservation of number?
- Conservation of number - the logical thinking ability to recognise that the numerical value of an object remains invariant with physical rearrangement - is a fundamental "cognitive milestone" during children's development (Crawford, 2008 p. 1).
- The concept of conservation was developed by Jean Piaget during the mid-1900s, who claimed it as "concrete operational" and, therefore, "unattainable" until children are of 7 or 8 years old (Halford & Boyle, 1985, p. 165).
Some of the materials being use for modelling may be less helpful, in that they might encourage children to start counting from one, for security.
Multilink or Unifix blocks are common in early classrooms. Where the mathematics takes children beyond tenness, breaking the chain into ten rods can be useful to accentuate that concept. It is heartbreaking to watch a child count, then have to restart the count because they have been interrupted. Making rods of ten would allow for interruptions and a means of continuity.
For this reason, I still have a preference for Dienes base ten equipment, as it allows early access to models of exchange, creating tens then hundreds. It accentuates place value and, using the function machine conceptualisation allows all four rules of number to be modelled effectively. With a visualiser attached to a class IWB, the modelling can be done on a large scale, enabling more to access. I recognise that in the days before such technology, there were visual limitations to modelling to large groups.
From Maths No Problem, the following accent on concrete apparatus seems to fit with this approach.
Concrete, pictorial, abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of maths in pupils. Often referred to as the concrete, representational, abstract framework, CPA was developed by American psychologist Jerome Bruner. It is an essential technique within the Singapore method of teaching maths for mastery.
Recently working with ITE trainees, in the conversation the idea of conceptualisation was raised. Using the simple example of 2+2=4, the trainees were challenged to explore the underlying necessary concepts to have a full grasp of the challenge. Twoness, fourness, addition (including synonyms) and equality, or balance, linked to balanced equations, eg 2+2=3+1. It was an eye opener to some.
Race to the flat
A very simple activity that can be very effective in supporting rapid calculation could be called race to or from the flat.
As long as you have Dienes base 10 materials and dice, this can be developed to cater for a variety of needs.
The rules of each game are simply described.
· Decide whether it’s a race to or from the flat (100 square). Decide whether, when the dice are thrown, the numbers are added together (any number of dice) or multiplied (two or three dice?).
· Dienes materials available to each group, plus dice appropriate to the needs of the group.
· Each child takes turns to throw the dice and calculate the sum or product.
· This amount is then taken from the general pile and placed in front of the child. The calculation can be recorded eg 3+4=7. This can provide a second layer of checking.
· If playing race from the flat, the child starts with ten ten rods, then takes an appropriate amount from these.
· Subsequent rounds see pieces added to the child’s collection; recorded as needed, eg round 2, 5+2=7 (7+7=14; the teacher should see one ten and four ones)
· The first child to or from the flat is the winner.
Altering the number of dice alters the challenge.
An extension could be a race to the block (1000 cube), or from the block, each child starts with ten 100 squares. If multi-sided dice, or different numbers of dice are available, the challenge alters yet again.
I came across some notes from some years ago, where I sought to put together examples from Dienes to enable colleagues to utilise the system to support their maths teaching.
Race to the flat
A very simple activity that can be very effective in supporting rapid calculation could be called race to or from the flat.
As long as you have Dienes base 10 materials and dice, this can be developed to cater for a variety of needs.
The rules of each game are simply described.
· Decide whether it’s a race to or from the flat (100 square). Decide whether, when the dice are thrown, the numbers are added together (any number of dice) or multiplied (two or three dice?).
· Dienes materials available to each group, plus dice appropriate to the needs of the group.
· Each child takes turns to throw the dice and calculate the sum or product.
· This amount is then taken from the general pile and placed in front of the child. The calculation can be recorded eg 3+4=7. This can provide a second layer of checking.
· If playing race from the flat, the child starts with ten ten rods, then takes an appropriate amount from these.
· Subsequent rounds see pieces added to the child’s collection; recorded as needed, eg round 2, 5+2=7 (7+7=14; the teacher should see one ten and four ones)
· The first child to or from the flat is the winner.
Altering the number of dice alters the challenge.
An extension could be a race to the block (1000 cube), or from the block, each child starts with ten 100 squares. If multi-sided dice, or different numbers of dice are available, the challenge alters yet again.
I came across some notes from some years ago, where I sought to put together examples from Dienes to enable colleagues to utilise the system to support their maths teaching.
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