You can find here detailed curriculum documents for classes 1-5 which use a combination of the NCETM Spines and White Rose Schemes of Learning:
I built this curriculum for my school which is a Steiner Academy, ie: a state-funded Steiner school. there are only 4 in the country, although there are many more private Steiner schools which may also find these useful.
The rest of the post below is the dissertation for my West of England Steiner Teacher Training (WESTT) course which accompanied the curriculum
Building a Steiner Academy Lower School Maths Curriculum
1 – Introduction
2 – Jamie York
3 – NCETM
4 – White Rose
5 – Rudolf Steiner and the mathematics curriculum
6 – The Concrete – Pictorial – Abstract model and the importance of language
7 – Growth Mindset
8 – Homogeneity
9 – Manipulatives
10 – The Judicious use of Technology
11 – Visual Representations
12 – Cognitive Load Theory
13 – Retrieval Practice
14 – Formative Assessment
15 – Deliberate Practice
16 – Conclusion and Next Steps
17 – Bibliography
18 – Online Sources and Resources
1 – Introduction
This project came about as a solution to a specific problem in a specific school although I hope that the products of these endeavours are useful more widely. The problem was that lower school teachers did not have clear enough indications of what mathematics to teach, how to teach it or when to teach it. In many cases they lacked sufficient subject knowledge to place learning in a wider context or to provide the rich tapestry of subject specific language necessary to allow the children to clearly grasp concepts. Teachers also lacked the time needed to research the what, the how and the why of mathematics and the result was that a child’s mathematical education became something of a lottery. Topics about which a teacher felt confident were covered in some depth but other topics may have been skimmed over or skipped altogether. This meant that children from different classes were receiving very different diets of mathematics and diets of vastly differing quality.
Part of our unique position as a Steiner Academy is that we are doing our best to deliver a Steiner education within the boundaries of state-funded education. Squaring this circle has not been easy and indeed the very future of our academy is currently being weighed by the gods as I write this. The show must go on in September however whilst we await the outcome of machinations beyond our control.
In our wonderful, strange, young, hybrid school the children sit SATS at the end of class 5 (year 6); we are currently exempt from KS1 SATS. Our KS2 SATS results are notoriously poor. With an average of 1 child per year group achieving age related expectations (ARE) in all three of reading, writing and maths. If we were to match the average primary school in the UK we would be getting approximately 18 children meeting ARE. If we consider maths alone, it is a little higher, around 3 per class, the national average being approximately 21 per class (based on 28 students in a class).
If one were to speak to my counterparts who run the literacy strand of the curriculum you would find many strong arguments for why the test does not match the curriculum and therefore we are very unlikely to have ‘adequate’ outcomes at KS2.
I’m afraid I cannot offer the same arguments for maths. Apart from percentages which we bring in Class 6 and algebra which we bring in class 7 the curriculum does fairly well match the test. Looking at the latest set of KS2 maths tests, percentages and algebra contributeonly 7 marks over all three papers. This is no reason why our children are not succeeding at maths. I would posit that they are not succeeding in maths for two reasons:
1: lack of time given to maths in the curriculum
2: lack of quality in the Teaching of maths.
If we were teaching the maths that is on our curriculum and if we were teaching it well then we would not have to answer awkward questions about maths progress in lower school.
But we cannot just blame teachers for the poor progress of students. School leaders must bear the responsibility for the progress of their students and the quality of their teachers. If the students are not learning effectively there are many questions we must ask ourselves. We must consider the quality of teaching and if it is not high enough we must provide support and CPD for our teachers. We must consider behaviour management and ask ourselves if we have supported our staff enough or given them enough training. We must consider all the systems and routines of the school and see where they are wanting and improve them. But underpinning all of these things is the curriculum. We cannot expect teachers to teach maths well without a full and detailed curriculum to follow.
One facet of the problem is “The Yellow Book” [The Tasks and Content of the Steiner-Waldorf Curriculum, Ed. Avison, K. and Rawson, M., Floris Books,secondi edition 2014 ]. A total of 20 pages outline the maths curriculum for classes 1 through 12 with just 5 bullet points for class 1. Maths is a subject which is to occupy up to approximately 200 hours of curriculum time in class 1. It represents exactly 50% of the main lesson blocks in classes 1 and 2. As a teacher, especially as a new teacher or a young teacher or a teacher for whom maths is not one’s major passion, I would be very nervous about launching a boat with such meagre rations.
The curriculum is our foundation upon which we can build a high-quality teaching and learning environment. Without it we are relying on the content knowledge that individual teachers bring. Some will be highly experienced and some will not. Some will be highly effective and some will not and the result will be inconsistencies which let children down.
So it became my mission to provide a bespoke and detailed maths curriculum for the lower school teachers to follow. This would form the foundations upon which we could build the rapid improvement in teaching and learning and student outcomes that we are required to produce.
2 – Jamie York
My first port of call, after The Yellow Book, was Jamie York [ https://www.jamieyorkpress.com/ ]. Now here one finds some more detailed guidance and some good advice. Notably that to review a topic, concept or skill means to actually practice that topic, concept or skill.
Mr York also offers some useful teaching tools to support our teachers in their efforts and highlights important pedagogical tensions such as the ‘tightrope’ walked between overwhelming one half of the class and underwhelming the other:
[Making Math Meaningful: A Source Book for Teaching Maths in Grades one Through Five, Fabrie, N., Gottenbos, W. and York, J., Jamie York Press, 2016 ]
“Two ends of the spectrum
Too much too soon. Class teachers can be pressured by parents of colleagues to prove that their own class is strong at math, doing only advanced material, and ahead (or at least not behind) other classes… the teacher is then inclined to move quickly through the material … all of this will ikey lose a good portion of the class, leaving them math traumatised and thinking that they will never become good at math. Such a class often enters middle school with a huge disparity between those who are good at math (the “fast students”) and those who aren’t good at maths (the “slow students”), with few students in between.
Math deprivation. Don’t mistake what has been said above as an argument to so as little math as possible. If a teacher avoids math because he or the students find it unpleasant, or the teacher keeps the math too simple, then the class won’t progress enough, and the students sense of number and ability to “think mathematically” wil be underdeveloped. Such students will likely have difficulties in later years when they have a teacher who has higher expectations.”
Mr York also throws in a few controversial statements which I support such as “we should be testing every day” and develops the concept of review in line with the concepts of ‘retrieval practice’:
“The second meaning of the word “review” relates to the word “practice”; the children should practice material from the current block … previous blocks, and previous grades… for a student to learn something permanently it should be reviewed the next day, the next week, the next month and the next year”
Jamie York is referring here, consciously or not, to the ‘forgetting curve’ which we will look at later in the section on retrieval practice.
Mr York also highlights another problem that I have found:
“The teacher needs adequate time to prepare for the math lessons. For many teachers, this is the most difficult issue. With everything that is demanded of the class teacher, there often isn’t enough time left to prepare adequately for the math lessons.”
This is exactly one reason why I felt it was so important to give the lower school teachers as much guidance and support as possible. We need to think about the cognitive load of our teachers as much as teachers do of their children. More about his in later sections.
Jamie York also provides much more detail than The Yellow Book and has attempted to map out each year’s maths curriculum block by block and we should be grateful for his contributions.
I do believe, however, that we can go even further, especially when i comes to the use of visual representations. The visual nature of mathematics cannot be overstated and the application of multiple representations is key to demonstrating the underlying structures of maths as well as empowering students to construct the meaning in the maths for themselves.
3 – NCETM
My quest led me next to the NCETM (National Centre for Excellence in the Teaching of Mathematics) [ https://www.ncetm.org.uk/ ] who have produced detailed curriculum guidance for teachers in three strands which they call spines.
Spine 1: addition and subtraction
Spine 2: multiplication and division
Spine 3: fractions
Spine 1 is complete from year 1 to year 6, spine 2 is complete up to part way through year 4 and spine 3, fractions, starts in year 3 and so far that is the only year complete in that spine, but eventually all three spines will be complete up to year 6.
The NCETM is clearly a ‘mainstream’ organisation and for that reason I think it has been overlooked by many in the Steiner movement. This has been to the detriment of our movement and not visa versa.
We must seek allies wherever they may be found and, in fact, many folk in the early years sector in mainstream and in primary are more closely aligned to our way of thinking than we might expect: we need to embrace them and learn from them. Maths, particularly, is one of those areas where there are many teachers and other professionals really trying hard to counteract the damage done by decades of poor teaching to which I’m sure the vast majority of adults in the UK can add their testimony.
Amongst its aims, the NCETM includes “to increase appreciation of the power and wonder of maths” [ https://www.ncetm.org.uk/ncetm/about].
Personally, that is definitely something I can get behind and the quality of the depth of thinking behind the NCETM spines is second to none.
Each spine is split into sections, atomising the concepts needed at each stage. Spine 1 has been broken down into 31 sections from 1.1: Comparison of quantities and measures in year 1 all the way through to 1.31: Problems with two unknowns in year 6. Every single section includes a PDF of the Teacher guide for that section which runs to around 20 pages. This will include the teaching points for that section of which there will usually be between 3 and 10. The teaching guidance will include many different visual representations which are recommended for teachers to utilise when introducing or developing a concept with the children. For some of the earlier sections the NCETM have begun including short videos which summarise the teaching points as well very nicely.
|An example of the teaching points from section 1.2: Introducing ‘whole’ and ‘parts: part-part-whole:|
Teaching Point 1: A ‘whole’ can be represented by one object; if some of the whole object is missing, it is not the ‘whole’.
Teaching Point 2: A whole object can be split into two or more parts in many different ways. The parts might look different; each part will be smaller than the whole, and the parts can be combined to make the whole.
Teaching point 3: A ‘whole’ can be represented by a group of discrete objects. If some of the objects in the group are missing, it is not the whole group – it is part of the whole group.
Teaching point 4: A whole group of objects can be composed of two or more parts and this can be represented using a part-part-whole ‘cherry’ diagram. The group can be split in many different ways. The parts might look different; each part will be maller that the whole group and the parts can be combined to make the whole group.
[ https://www.ncetm.org.uk/resources/50719 ]
Each section’s PDF will begin with an ‘overview of learning’ which is an in depth introduction to the main concepts introduced in that section. There is a strong emphasis on the language used by the teacher in order that concepts are communicated clearly and to avoid ambiguity. Sentence starters and frameworks are given which the teacher is encouraged to use with the children and also encourage the children to use so that they are using the correct mathematical language.
Each teaching point is then broken down further into subsections and detailed guidance given on the language around the ideas and some of the activities that a teacher might use. These notes are accompanied by pictures of various representations which the teacher may find useful. There are example questions that the teacher might ask to elicit the desired thinking on the part of the child and at every step the teacher is asked to consider different representations so that the children are exposed to every variation of a concept including non-examples.
An example of some visual representations with suggested phrases and questions:
The spines that the NCETM have produced come under the umbrella term of ‘teaching for mastery’. As with all ideas there is nuance and debate regarding what ‘teaching for mastery’ actually is or means but I cannot see how any definition of it conflicts with what we are trying to achieve in Steiner education. So much so that my advice to any Steiner school leaders would be to learn the language of mastery and apply it to the curriculum and the teaching practices that they already have and they will find much crossover between the Mastery approach and the Steiner approach.
Wonderful as these NCETM resources are, being limited to just the four operations and fractions means they do not cover the whole curriculum. So I was led to carry on searching for resources that our lower school teachers could use to support them to deliver the rest of the curriculum.
4 – White Rose
The search led me to a splinter project of the NCETM called Maths Hubs whose strapline is “helping schools and colleges lead improvement in mathematics education” [ http://www.mathshubs.org.uk/] . Although not without their critics, maths hubs are, to my mind, an amazing resource and are full of teachers dedicated to furthering their subject and pedagogical knowledge for the benefit of their children’s maths education.
The most notable of the Maths Hubs is White Rose [ https://whiterosemaths.com/ ] who have produced a large quantity of resources built on their ‘mastery’ schemes of learning. These schemes of learning are not as detailed as those from the NCETM and they lack the depth of guidance that the NCETM resources have but they do offer some guidance and examples of questions which a teacher might pose and also suggestions for challenging the confident mathematicians in your class.
Where there is overlap between the NCETM and White Rose I have ditched White Rose and gone for the NCETM due to the much greater level of detail in the notes and the wider variety of visual representations offered. White Rose has filled all the gaps in the curriculum including shape, measure, position and direction, money, time, geometry and statistics.
Dissecting both schemes of learning (NCETM and White Rose) and reassembling them to follow the indications of the Steiner curriculum has not been an easy task and the resulting product may be akin to Frankenstein’s monster or could be a thing of beauty and wonder but only time will tell. I believe in the power of collaboration and I am hoping that the lower school teachers in my school will be able to provide real-time feedback to me as next term unfolds in terms of what works and what doesn’t work so that the curriculum can be tweaked, prodded and pruned where necessary so that we really do end up with a worthy and practical tool which is a boon to the practice of the Steiner class teacher
5 – Rudolf Steiner and the mathematics curriculum
I have taken for granted most of the curriculum indications from the Yellow Book. I strongly support the later introduction of formal learning which our kindergarten offers and indeed is mirrored by many countries around the world and for which there is a wide research base bolstering the tradition.
There are three major references to the teaching of mathematics in Discussions with Teachers [https://www.rsarchive.org/Download/Discussions_with_Teachers-Rudolf_Steiner-295.pdf ]. In the 2nd lecture on the Curriculum he states:
“You must arouse in the children the powerful idea that 5 equals 3 plus 2, but that it also equals 4 plus 1, and so on. Thus, addition is the second step after separating the sum into parts, and subtraction is the second step after asking “What must I take away from a minuend to leave a specific difference?” and so on.”
I like this quote for two reasons. The first is that he is explicit in his use of the technical language associated with arithmetic which is supported by the NCETM documents. The second is that the ‘cherry diagram’ which is a common representation from the ‘mastery approach’ perfectly encapsulates this flexibility with addition and subtraction in relation to the parts and the whole to which Steiner is referring here.
He goes on to say the following about times tables:
“The point is that children should memorize their times tables and addition facts as soon
as possible after you have explained to them in principle what these actually mean — after you have explained this in principle using simple multiplication that you approach in the way we have discussed. That is, as soon as you’ve managed to teach the children the concept of multiplication, you can also expect them to learn the times tables by heart.”
So although we leave formal learning until the age of 6, there is no reason to go slowly once we begin the journey.
The final quote from this lecture surprised me:
“In the fourth grade we continue with what was done in the earlier grades, but we must now also make the transition to fractions and especially to decimal fractions.”
This is surprising in that it is apparently traditional (at least in my school) to leave decimals until class 5. But here Steiner is clearly stating that decimal fractions should be introduced alongside fractions. I am glad to see this quote here as it makes sense to me that this be the case. You will find that my Lower School Maths Curriculum mirrors this indication, introducing decimal fractions in the third main lesson block of Class 4.
I will also reiterate the quote which appears on the title page of this paper:
“The Aim of Waldorf education is to arrange all of the teaching so that within the shortest possible time the maximum amount of material can be presented to students by the simplest means possible.”
- Rudolf Steiner. [ https://www.rsarchive.org/Download/Soul_Economy-Rudolf_Steiner-303.pdf]
I hope that this quote is enough to do away with any anti-academic and anti-intellectual undercurrents that may be prevalent in some Steiner environments. Let us take the imparting of knowledge seriously and really highly educate our students. If we are to be better than the mainstream then our children should be more knowledgeable and more skilled in all the arts and the sciences, not ‘protected’ from the perceived dangers of learning and academia.
Interestingly, Gnomes do not get a mention, nor do princes and suchlike. According to Steve Sagarin [https://www.waldorftoday.com/2017/09/playing-steiner-says-twenty-two-myths-about-waldorf-education/], co-founder of the Berkshire Waldorf High School, the maths gnomes actually work against the grain of learning and against Steiner’s indications of making the mathematics real. According to Steiner, maths is a spiritual language by its very nature, we do not need to imbue it with fairy tales, quite the opposite: our job is to bring it down into concrete manifestations from the spiritual plane.
6 – The Concrete – Pictorial – Abstract model and the importance of language
Any literature on the Mastery approach, and indeed, any literature on quality maths pedagogy will reference the Concrete-Pictorial-Abstract (CPA) heuristic in some form.
I first met CPA through working with Bruno Reddy [http://www.mrreddy.com/] (of Time Tables Rock Stars fame [https://play.ttrockstars.com/]) circa 2014 but the idea has been around for a long time. CPA is based upon Bruner’s 1966 conception of enactive-iconic-symbolic modes of representation [https://pdfs.semanticscholar.org/7092/c5c0ad8019a8c5955af441d6af6fe15c812e.pdf?_ga=2.214222103.1206735086.1564241451-1919937537.1564241451]. I like these original headings as, although ‘iconic’ and ‘symbolic’ could be perceived as synonyms for ‘pictorial’ and ‘abstract’, there is a different feeling with ‘enactive’ as opposed to ‘concrete’. Although ‘concrete’ indicates a physical dimension to the learning of maths, ‘enactive’ requires us to consider how the pupils will interact with the concrete representations. The pupils need to physically ‘enact’ the maths in some way, be that with their bodies or with ‘concrete’ maths resources (which from here on in we will call ‘manipulatives’).
CPA became a back-bone of the Singaporean pedagogical toolbox in the 1980s and is mentioned in governmental documents regarding the teaching of maths:
“This [activity-based] approach is about learning by doing. It is particularly effective for teaching mathematical concepts and skills at primary and lower secondary levels, but is also effective at higher levels. Students engage in activities to explore and learn mathematical concepts and skills … . They could use manipulatives or other resources to construct meanings and understandings. From concrete manipulatives and experiences, students are guided to uncover abstract mathematical concepts or results … During the activity, students communicate and share their understanding using concrete and pictorial representations.”
- (Ministry of Education (Singapore), 2012, p. 23)
Much has been written about the CPA method which I cannot cover in great depth here, except to highlight a few main talking points:
The first is that although there is a natural progression from concrete to pictorial to abstract, there are benefits to be found in going back and forth between the modes: it is not a linear progression solely. There has been a fair amount of research on how the CPA approach can help those learners who are struggling but there is also evidence to suggest that the CPA approach is an effective way of stretching the most able in your class without just giving them larger numbers or accelerating them through the curriculum. The CPA approach will require students who have grasped a concept quickly to explain it in multiple ways using a variety of representations, pictorially and concretely. This will allow those students to deepen their understanding of the concepts at hand rather than superficially skimming over the top and ruching onto the next topic. Anecdotally, when the CPA method is brought into schools for the first time, the low-prior-attainers (LPAs) benefit hugely from the new approach while the high-prior-attainers (HPAs) initially struggle as they are asked to slow down and really show their understanding. As the NCETM states: “Mastery is achieved if children can use all modes of representations going back and forth between the representations” [https://www.ncetm.org.uk/resources/40534]
Another interesting development of the CPA model is the recent introduction of another letter in the acronym. Mark McCourt who is described as the UK’s leading authority on teaching for mastery now refers to CPAL with the addition of ‘L’ for language [Teaching for Mastery, Mark McCourt, 2019]. This is to highlight the importance of subject specific language being used correctly with the learners in order to support the learning of the key mathematical concepts. We have already seen how the NCETM and Rudolf Steiner himself have placed the technical language of maths at the heart of their approaches.
7 – Growth Mindset
The growth mindset theories from Carol Dweck [https://www.mindsetworks.com/science/] and Jo Boaler [Ability and Mathematics: the mindset revolution that is reshaping education JO BOALER: http://www.youcubed.org/wp-content/uploads/14_Boaler_FORUM_55_1_web.pdf] assert that a fixed mindset is a negative thing and that this can manifest in high-prior-attainers and low-prior-attainers equally. LPAs can get it in their head that they just ‘don’t have a maths brain’ or other negative thoughts associated with maths whilst the HPAs can acquire a fixed mindset about their own maths identity where mistakes are not permitted which reduces risk taking and the exploring of maths.
So the CPA approach can level the playing field somewhat in that the LPAs are more likely to gain the understanding that the teacher is hoping to convey and the HPAs are able to behave more like mathematicians and not just race through the material looking for the next big red tick on their worksheet. What is more is that the LPAs can often teach the HPAs a thing or two about the structure of mathematics through the concrete representations which again levels the playing field, or in other words increases homogeneity in the classroom.
8 – Homogeneity
When we talk of homogeneity in the mathematics classroom we are talking about the attainment distribution between the HPAs and the LPAs. For the last two years I have taught mixed ability groupings where the attainment distribution is as wide as it can possibly be for the age group. I have children in Class 8 (year 9) for whom counting is not yet a reliable strategy and I also have children in the same class who are happy exploring the content of further maths GCSE and are metaphorically chomping at the bit to do harder maths. This is a picture which has been observed by Mark McCourt [https://twitter.com/emathsuk/status/825249943271129088] . Where teaching for mastery works it is implemented at the start of schooling and teachers are careful to avoid the attainments gaps forming in the first place. So the middle school in which I teach demonstrates that a mastery approach has not been in place. What is very exciting is that we now have an opportunity to get this right from class 1 upwards. It will take strong leadership and plenty of staff training but luckily we have signed up to Mark McCourts Complete Mathematics platform which gives us access to plenty of free CPD on teaching for mastery [ https://completemaths.com/ ].
The UK Government is now taking on aspects of the Mastery approach from Singapore and now includes statements such as the following from the 2014 National Curriculum:
“The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace… Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content.”
This is only possible through the homogeneity of classes as a product of quality Mastery teaching from Class 1 upwards.
9 – Manipulatives
The CPA approach would not be complete without the expert use of manipulatives in the classroom. The curriculum that I have built has the use of manipulatives at the very centre and each block begins with a list of the manipulatives that the teacher is expected to use with guidance from the NCETM and the White Rose Maths Hub. These are the manipulatives which should be purchased by a maths department committed to a mastery and CPAL approach:
Counting sticks, Bead strings, Dienes blocks, ‘2D’ shapes, 3D shapes, Fake money, Dice, Metre sticks, Place Value Counters, Geo-boards, Clocks, Maths games (twister, snakes and ladders, City of Zombies, Shut the box etc), Double sided counters, Planks of wood, String, Balance scales, Measuring jugs, Cuisinairre rods, Multi-link cubes, Numicon, Algebra tiles, Protractors, Compasses, Rulers, 5-frames, 10-frames.
I won’t expand on all of the above, but will highlight some of the most important ones and ones which not everyone will be familiar with.
[I have place the image credits for this section at the very end of the post]
Counting sticks are a very versatile manipulative. Uses include to help with counting, times tables [ Jill Mansergh – Tables with a Number Stick https://www.youtube.com/watch?v=yXdHGBfoqfw&t=111s and 17 Times Table – Part 2 https://www.youtube.com/watch?v=G94gW8cnu9s], fractions and decimals. This one pictured above is from Amazon and has 4 different representations on each of its sides. It has quarters on one side, tenths on another, centimetres on another and a blank side for estimating or other uses
Bead strings come in a variety of sizes, the most common being 10, 20 or 100 beads. They are coloured in such a way as to highlight 5s or 10s. Good for ‘one more’ or ‘one less’, counting on, subitising, visualisation (hiding a certain number of beads), composition of number, subtraction as difference, place value, number bonds, partitioning, subtraction as taking away.
Dienes blocks (sometimes called base 10 blocks) are a fantastic resource for place value. They can be used to represent 1s, 10s, 100s, and 1000s or 1s, 0.1s, 0.01s and 0.001s which is a lovely way of taking a resource that a pupil has explored in class 2 and breathing new life into it in class 4.
‘2D’ and 3D shapes
The reason I have put ‘2D’ in inverted commas is because these are actually just representations of 2D shapes; they are in fact 3D. Great for shape recognition and introducing the technical language of vertices, sides, area, perimeter, volume, faces, edges etc. Also good for categorising.
Place Value Counters
Place value counters are another great way of representing place value alongside Dienes blocks. They are a step between the concrete and pictorial.
Geoboards are good for shape, position and direction, reflections and symmetry and patterns.
Cuisenaire rods are very versatile and can be used from reception right up to algebra in middle school and beyond.
Again, a very versatile manipulative, useful for counting, arithmetic, area, volume, sequences, more than, less than, estimation.
Numicon has taken the maths world a little by storm over the last few years. The number tiles are the most common manipulative that The Oxford University Press (OUP) produce under the title ‘Numicon’ although they include many different manipulatives in their Numicon packages. Although primarily aimed at supporting learning in the early years and primary, the OUP also produce intervention packages for KS3.
The Numicon Firm Foundations is aimed at Early years and includes peg boards and pegs for supporting the early acquisition of number, pattern and counting. Numicon Big Ideas for primary includes cuisenaire rods, spinners, number lines and a number track.
Although probably for your older students, algebra tiles are an ideal resource for introducing algebra. Interestingly, your students are more likely to appreciate the algebra tiles in middle school if they have grown up on a diet of physical manipulatives. If the first time your students see a manipulative is in class 7 then they may take some time to get used to it.
Some of these manipulatives have obvious uses, some not so much. The guidance from the NCETM and White Rose is quite specific about how these manipulatives might be best employed for each relevant topic but it is probably worth taking a look at a few of the most important manipulatives.
When adding two or more numbers there are a variety of strategies that one may use, these include:
Plus 1 (and plus 2, plus 3 and extensions)
Using near doubles
Now what is interesting is that some of the manipulatives are better at demonstrating the structure of different strategies so the teacher must select the optimal manipulative for each strategy.
For example, plus 1, 2 and 3 facts are the first step after counting. If I know that I have 4 beans and you give me one more then I have one more than 4 which is 5 and so on. Now, this can be demonstrated with a wide range of manipulatives including the bead strings, multi-link cubes, counters etc. But Numicon is not well suited to this.
Numicon is much better suited to the demonstration of odd and even and number bonds to 10, whilst 10-frames are better suited to decomposition or transformation.
So there is an art to the demonstration of mathematical concepts through the use of manipulatives. Most advocates suggest that the manipulatives are always available for everyone in the class. It is certainly bad practice to group your students by ability and give the ‘weaker’ students the manipulatives. This sends the wrong message about the purpose of manipulatives. They are not just there for the students who struggle. They are three as a resource for everybody to explore the structure and relationships inherent in the mathematics. As Peter Mattock in his recent book, Visible Maths, “The importance of having different ways to view even the most simple mathematics, in order for pupils to build up to more complicated ideas, cannot be overstated.” [Mattock, P. Visible Maths: Using representations and sturcture to enhance mathematics teaching in schools, Crown House Publishing limited, 2019
10 – The Judicious use of Technology
Steiner schools are traditionally low tech. As an antidote to the pervasiveness of screen culture I think low-tech is an important stance to take. I would, however, defend my use of the phrase low-tech over no-tech. What Steiner may or may not have said on the matter is inconsequential as it boils down to an extrapolation of his ideas as he died before personal computers were invented. I certainly wouldn’t want to see interactive whiteboards used in our school as much as they are in most mainstream schools. In fact, I am not a fan of interactive whiteboards at all. I do like projectors though. Projectors allow us to do a few things which I consider very important in the execution of maths teaching.
The first is modelling. When modelling a procedure or skill in mathematics, the closer the model is to the manner in which the pupils have to reproduce it the better. This is why I use a visualiser (document camera). A visualiser is basically a modern day over-head projector (OHP); it is a camera on a stick pointing at the table. The purpose of it is to project an image of the page onto the wall. The reason this is so valuable is that the pupils can see me writing with a pen, just like theirs, on a sheet of paper, just like theirs.
There has been much written about supporting dyslexic children in the classroom over the years and translating text written on a black-board in chalk into legible maths with a pencil on paper is an extra barrier that the visualiser removes. I use this every lesson with my middle and upper school children. I can see how in the primary classroom you might not use it every lesson but I do think it should be a common-place technique for modelling maths.
I use the visualiser in conjunction with a teaching technique called ‘standardise the format’ from Doug Lemov’s Teach Like a Champion 2.0 [Lemov, 2015] . This technique involves my producing a worksheet or booklet which all the pupils have and which I have a copy of too. When I model on the board I complete the model in a pre-designed space in the booklet so that the pupils can copy it down and follow along in an identical manner in their own booklets. Standardising the format of the worksheets in this way also allows me to check very quickly where the pupils are at as I circulate the room whilst they are working independently.
Some teachers write in an exercise book just as the students would do. This is a technique called ‘Board = paper’ also from Lemov where teachers literally model the skill of note-taking or other written work in maths or any other subject.
The visualiser also allows me to share pupils’ work instantly with the whole class. This is a fantastic way to give feedback as one can ‘live mark’ a piece of work, celebrate excellent work and encourage peer feedback. It is important to build the right culture around this as without setting the right tone in the classroom it can be a cause of stress. It is important to build a collaborative learning culture where pupils are able to offer and receive constructive feedback on their work.
Visualisers also allow a teacher to demonstrate the use of manipulatives to the whole class without everyone having to crowd around a table. Images can be frozen on the board and manipulations can even be recorded and played back on a loop.
The other judicious use of a projector in a classroom is the use of digital manipulatives. Recently there has been an explosion in the growth of online digital manipulatives. The website mathsbot.com has a whole host of digital manipulatives including algebra discs, algebra tiles, bar-modelling (more on that later), coins, counters and many many more.
These digital manipulatives are often more versatile than their concrete counterparts and are a useful step in the process from concrete to pictorial.
11 – Visual Representations
Within the ‘manipulatives’ section of each block I have included not just the physical manipulatives teachers will need but also some of the pictorial representations which will be useful for that section of the learning. The main visual representations cited which readers may not be fully aware are bar models, cherry diagrams and the gattegno chart
These have been around for some time and are very much en vogue in primary schools in the UK at the moment. They are another tool in the Shanghai Mastery approach and are a versatile tool for modelling all sorts of maths problems from basic arithmetic to ratio and simultaneous equations.
They are favoured by maths educators as a neat method of revealing the structure of the maths, especially in word problems. As with fostering homogeneity, the introduction of bar models should happen as early as possible so that pupils ‘grow up’ with this representation so that by the time they are in class 4 or 5 bar modelling is a natural way of representing maths problems. It is useful as it leads nicely into algebra without having to learn a new representation.
Cherry diagrams are a lovely visual representation of the part-part-whole nature of addition and subtraction; in fact, sometimes they are referred to as ‘part-part-whole’ diagrams.
They are similar to bar models but show the relationships in a slightly different way. They can be combined with the use of counters or multi-link cubes in a concrete fashion and different parts of the model can be covered up to reveal different relationships.
The Gattegno chart is a tool for place value:
The idea is that one can build up any number by choosing a number from each row. For example, if one wanted to build the number 43,182 one would point to 40,000, 3,000, 100, 80 and 2 in that order.
12 – Cognitive Load Theory
Cognitive Load Theory (CLT) was first researched by the educational psychologist John Sweller in the 1980s [https://impact.chartered.college/article/shibli-cognitive-load-theory-classroom/]. Cognitive load is used to describe how much capacity something takes up in working memory. Working memory is where thinking actually happens. It has a very finite capacity; it can only hold and process about four different items at a time. If it receives too much it fails – this is called cognitive overload and occurs when too many demands are placed on working memory at once.
Working memory is very limited, but our long-term memory has huge – almost infinite – capacity. It is here that we store our knowledge of facts and procedures. The goal is to stock our long-term memories with knowledge in a well organised, easily retrievable way and make recall of key aspects automatic. This frees up the working memory for new information.
One key finding of CLT is the ‘split attention effect’. This occurs when our phonological loop is overloaded. Our phonological loop is the part of the brain that deals with speaking, listening and reading. As a single part of the brain deals with all three of these formats, it is hard to attend to more than one at a time. If pupils have to listen to someone talking at the same time as reading something on the board, then their attention is split – the spoken and the written information compete for attention.
How many teacher talk and write on the board at the same time? I know I used to do this all the time before I learnt about CLT. Since learning about CLT I have introduced a technique which experienced maths teacher, Craig Barton, calls the ‘silent teacher’ method [How I wish I’d Taught Maths, Barton, C, 2018]. This basically involves the teacher silently writing on the board, it could be modelling a technique, instead of talking and writing at the same time. The purpose is to eliminate the split-attention effect. Students are to concentrate solely on the written model being provided and once the process is complete the teacher can then narrate the steps or invite comment or discussion from the pupils.
I personally have found this a powerful teaching technique which empowers learners to make sense of a technique in their own mind before discussing it with myself or peers. It is so ubiquitous in my classroom now that my students will pull me up on it if I forget and try to talk and write at the same time!
13 – Retrieval Practice
When we think about learning, we typically focus on getting information into students’ heads. What if, instead, we focus on getting information out of students’ heads?
Retrieval Practice Is Key In The Classroom
Retrieval practice is a strategy teachers can use to give pupils opportunities to have to try and remember things they have learnt previously; things they have begun to forget. Retrieval practice is quite simply giving children tasks where they have to try and retrieve an answer from their long-term memory. Each time pupils try and do this, that memory will become a bit stronger and a bit easier to find next time.
What is important for teachers to understand is that for the memory strengthening to happen, pupils must try to remember without any priming or reteaching from us. For retrieval practice to work, there has to be an element of struggle; it has to be at the very least, a bit hard to remember.
It works best when memories have been forgotten!
This ties into the Waldorf idea of “putting a topic to sleep” but is focussed on the “re-awakening” process which seems to be the important bit in terms of long term retention of content.
We saw in Section 2 that Jamie York is aware of the importance of retrieval practice even if he doesn’t use that phrase. And Steiner too was aware of the importance of practice:
“In intellectual life we tend to emphasize that the better the child understands something, the better our teaching is. We lay particular value upon immediate understanding, upon immediate retention. However, what the child immediately understands and retains does not act upon the feeling and will. Only what the child does repeatedly and sees as the proper thing to do under the given circumstances acts upon feeling and will…Thus, to give each child some task to do every day, possibly for the whole school year, is something that strongly acts to develop the will. It creates contact between the children, strengthens the authority of the teacher and brings the children to a repetitive activity that strongly affects the will.”
[Foundations of Human Experience, Lecture 4, Steiner, R, Translated by Lathe, R.F. and Whittaker, N.P, Anthroposophic Press, 1996 ]
I believe that what Steiner is referring to here could be correlated with what modern teachers call the difference between performance and learning [Learning Versus Performance, Soderstrom, N.C. and Bjork, R.A., Oxford University Press]. Performance is what can be observed and measured during instruction, ie: If I teach a class about pythagoras’ theorem and during the lesson the pupils appear to be able to apply the formula to a few problems then I can say they are performing well. Whether they have actually learnt it I won’t be able to tell until a future date when I test them on the skill or require them to utilise the skills again. The purpose then of retrieval practice is give the students ample opportunities to revisit the key concepts so that the skills and understanding move from their working memory into their long-term memory and we can be more confident that actual learning has occurred.
Becoming conscious of the power of Retrieval Practice (RP) as a learning strategy means we can make the best use of the brain’s natural recall functions; RP solidifies knowledge in long-term memory and also increases understanding.
Because students have a better understanding of classroom material by having practiced using this information, students can adapt their knowledge to new situations, novel questions, and related contexts.
RP also helps us to identify gaps in learning. In other words, not only does retrieval improve learning and help us figure out what students do know – more importantly, it helps us figure out what they don’t know. Teachers can adjust lesson plans to ensure common misconceptions are addressed and common gaps in knowledge are revisited if necessary.
14 – Formative Assessment
Formative assessment is what Jamie York was referring to when he claimed that “we should be “testing” every day”. In the same section he also states that “The major purpose of “testing” should be that the teacher is informed of the students’ progress” [Page 17, Making Math Meaningful: A Source Book for Teaching Maths in Grades one Through Five, Fabrie, N., Gottenbos, W. and York, J., Jamie York Press, 2016]. York hits the nail on the head here and this is what we mean when we talk of ‘formative assessment’.
Edglossary.org have a good summary of formative assessment:
“Formative assessment refers to a wide variety of methods that teachers use to conduct in-process evaluations of student comprehension, learning needs, and academic progress during a lesson, unit, or course. Formative assessments help teachers identify concepts that students are struggling to understand, skills they are having difficulty acquiring, or learning standards they have not yet achieved so that adjustments can be made to lessons, instructional techniques, and academic support.”
[ https://www.edglossary.org/formative-assessment/ ]
Some of the formative assessment tools available to the teacher include, Low- (or no-) stakes quizzes, Multiple choice questions, Exit Tickets, Mini-whiteboards, targeted and expert questioning of individuals, groups or whole classes, self-assessments and peer assessments.
Some of the benefits of formative assessment fall under the banner of ‘metacognition’ which is about helping students to learn about the learning process. Formative assessment brings the attention away from grades and onto the learning process. It highlights to the students the things that they do well and the things they need to improve so that they can take responsibility for their own educational growth.
For further reading about formative assessment I would direct readers to the voluminous work of Dylan Wiliam [https://www.dylanwiliam.org/Dylan_Wiliams_website/Welcome.html].
15 – Deliberate Practice
Deliberate Practice is the act of breaking down a complex process such as adding fractions into separate, individual skills and then deliberately practising those until they are easy. In this way cognitive overload is avoided.
Deliberate practice is in part designed to counteract the ‘curse of knowledge’ as explained by Wiemann in 2007 [https://www.aps.org/publications/apsnews/200711/backpage.cfm]. He described this as the difficulty one has in remembering what is was like to not know how to do something which you can do easily now. One example might be adding fractions. It may be that you can add fractions quite easily without having to think too much about the whys and wherefores about the processes involved. This is an issue if we are teaching novices. Novices do not have the detailed cognitive schemas and experience that ‘experts’ have. In order to teach adding fractions to novices we need to break it down into all of its component parts and teach each step separately and deliberately practice each step, building fluency and confidence on the way, and then finally at the culmination of the process we can put all the skills together and add the fractions.
There are 5 stages to deliberate practice which are:
- Isolate the skills
- Develop each skill individually
- Assess each skill individually
- Put the skills together in one task
- Practice through Retrieval Practice
If we take the skill of adding fractions then we can ‘atomise’ it into many sub skills, each of which must be mastered before we can add fractions. One way of atomising the skills might look like this
- Decide if fractions are in a form ready to be added (i.e. do they have the same denominator or not?)
- If they don’t, decide what a suitable common denominator might be
- Then transform both fractions into the appropriate equivalent form
- Then add the numerators
- Then decide if the answer needs simplifying or not
- If necessary, simplify the answer
So there are at least 6 stages to adding fractions which need to deliberately practiced. This again is going to reduce the cognitive load of our students and make it more likely that they will be successful in the long term.
Stage 3 of Deliberate practice is “assess the skills”. This is where our formative assessment strategies come in.
One effective way of assessing the skills as we go along the learning journey that I have been using recently is whole-class ‘diagnostic questions’. DiagnosticQuestions.com is a website created by experienced maths teacher Craig Barton. It is a website which contains over 40,000 maths questions all of which are in a multiple-choice format. The power of this format is that the three wrong answers are not randomly chosen but have been carfeully considered so as to draw out specific misconceptions that the children may have about the topic in question. For example:
This question is testing order of operations. The correct answer is 3.
Each wrong answer demonstrates a different mistake that children are likely to make.
To get 42, the children must have subtracted the 3 from 13 first.
To get 60, the children have subtracted the three and then added the two and then multiplied.
To get -5, the children have completed the addition before the multiplication.
As a teacher I can be reasonably confident that the most likely wrong answer is 42 as the children will have just complete the operation left to right, but by asking and getting whole-class feedback (using mini-white boards or a show of hands/fingers) I can quickly assess how many children are performing on this topic or concept and how much time I have to spend going over the order of operations again.
16 – Conclusion and Next Steps
I don’t feel right writing a conclusion for this project as I am only half way through a much longer process so ‘next steps’ feels like a more suitable title for this section.
Having completed the curriculum for classes 1-5 I now have a perhaps more challenging responsibility of ensuring its effective implementation. Our unique circumstances mean that out of eight class teachers in lower school next year, five are brand new to both the school and to steiner education. They are all qualified teachers, however, and between them bring several decades of experience in a variety of settings. I am hoping that the curriculum looks familiar enough to them with the White Rose resources that some of them will have met in mainstream but also looks new and exciting.
Initial feedback from various members of the team is that the curriculum documents do look useful so I am optimistic that teachers will find them helpful. I am also realistic and know that they are not final, polished products and I look forward to tweaking them based on the ongoing feedback I get from teachers.
I feel confident now that our lower school teachers have a strong and solid (if not perfect) base upon which to embark on their mathematical journey with their classes. But the job is not complete. There are still large gaps in the sections headed ‘resources’. This is my next big task, to begin to populate the scheme of learning with high quality resources to further aid the teachers and reduce their cognitive load. Over time I will be adding in worksheets, youtube clips and other links to games and activities which may be useful. I am also welcoming input from the teachers themselves and encouraging them to populate the scheme with resources that they have already created or found or that they build as they go through the year.
Alongside resourcing the curriculum, I need to tackle the big question of assessment. The question of how we assess the progress of students is a thorny one. Standardised assessments are OK to some extent but there is an argument that hey don’t match our curriculum so their efficacy is reduced, but if we create our own assessments then they are not standardized. My school has just signed up to Complete Maths from La Salle Education, founded by Mark McCourt which has built-into it an assessment tool which I will be exploring this year [https://completemaths.com/]. My concerns are two-fold. Firstly it is an expensive platform so may not be financially viable in the long-run and secondly, the tests are best administered online which will come into conflict with the low-tech disposition of our lower school.
Long-term, if the four Steiner academies retain their kindergarten phase and their later start to formal learning I would love to see a set of assessments which are common to all four academies. This would bring a certain level of rigour and commonality which we don’t currently have.
As well as resources and assessment, I also have to build the curriculum for classes 6-8 before September as we have a new teacher taking over in middle school as well so that is another big job ahead of me.
The successful implementation of this curriculum will require teamwork, collaboration and listening. Together we will need to embed the principles behind the curriculum, namely those of growth mindset, teaching for mastery, visual representations, cognitive load theory, retrieval practice, formative assessment and deliberate practice. It is no easy task but I am excited to be embarking on this journey.
I recently began a National Professional Qualification for Senior Leaders and its focus is the impact of the maths curriculum that I have just built. So in a year’s time I will be able to report back on how successful the implementation of the curriculum has been.
Until then, happy teaching and learning!
Avison, K. and Rawson, M. (eds), The Tasks and Content of the Steiner-Waldorf Curriculum, Floris Books, 2014
Barton, C. How I wish I’d Taught Maths: Lessons learned from research, conversations with experts, and 12 years of mistakes, John Catt, 2018
BOALER, J. Ability and Mathematics: the mindset revolution that is reshaping education, available from: http://www.youcubed.org/wp-content/uploads/14_Boaler_FORUM_55_1_web.pdf
Lemov, D. Teach Like a Champion 2.0, Jossey Bass; 2nd ed. 2015
Mattock, P. Visible Maths: Using representations and sturcture to enhance mathematics teaching in schools, Crown House Publishing limited, 2019
McCourt, M. Teaching for Mastery, John Catt, 2019
Soderstrom, N.C. and Bjork, R.A. Learning Versus Performance, Oxford University Press
Steiner, R. Discussions with Teachers, Anthroposophic Press, 1997
Steiner, R. Foundations of Human Experience, Lecture 4, Translated by Lathe, R.F. and Whittaker, N.P, Anthroposophic Press, 1996
Steiner, R. Soul Economy: Body, soul and Spirit in Waldorf Education, Anthroposophic Press, 2003
York et. al. Making Math Meaningful Book for Teaching Math in Grades One through Five, Jamie York Press, 2016
Online Sources and Resources
Bruno Reddy’s maths blog: http://www.mrreddy.com/
Carol Dweck on growth mindset: https://www.mindsetworks.com
Clare Sealy on Deliberate Practice: https://thirdspacelearning.com/blog/how-teach-primary-maths-deliberate-practice-in-education/
Craig Barton’s Diagnostic Questions: https://diagnosticquestions.com/
Dylan Wiliam on Formative Assessment: https://www.dylanwiliam.org
Jamie York Press: https://www.jamieyorkpress.com/
Mark McCourt’s Complete Maths: https://completemaths.com/
Mathsbot online manipulatives: https://mathsbot.com/#Manipulatives
Maths Hubs: http://www.mathshubs.org.uk/
National Centre for Excellence in the Teaching of Mathematic (NCETM): https://www.ncetm.org.uk/
Times table Rock Stars: https://ttrockstars.com/
Waldorf Today blog: https://www.waldorftoday.com/
White Rose Maths Hub: https://whiterosemaths.com/
Image sources for section 9 – Manipulatives