Whilst thinking about writing this blog, I went back to Richard Skemp’s book, ‘Mathematics in the Primary School’, published back in 1989 and promptly disappeared down a rabbit hole. Several hours later, this quote stuck with me:

“If the processes of multiplying numbers of any size are learnt as mechanical rules, little understanding and no creativity are involved. Children are simply being taught to function mechanically like a calculator, and for getting the answer quickly and accurately, it is much better for them to use calculators.” Pg 80

Relational vs instructional teaching

Skemp is a great advocate of teaching relational maths rather than instructional maths. He describes relational understanding as knowing what to do and why and instructional maths as rules without reasons.

In the first chapter, he provides arguments for both, and refers to the effect that assessments, tests, and exams, have on teachers’ decisions on how to teach. The vast majority of tests focus on getting the right answer and have a pass mark, so how the answer is achieved is not considered. This means that if the answer or process can be correctly recalled, the correct answer can be achieved with little or no understanding.

Skemp also lists ‘time’ and ‘an over-burdened curriculum’ as other reasons why teaching for relational understanding is a challenge.

Nearly 35 years later, have things changed?

I would argue yes, and no.

The need for relational understanding

I think we have a much better understanding of the need for relational understanding. Schemes of work that support teachers to deliver the curriculum are designed to support learning for understanding, but the tests children sit in maths are still about right and wrong responses, and how those answers are achieved is less considered – mainly because this is very difficult to do in a written test.

That means teachers are sometimes torn - I know I was.

I regularly asked myself the question, ‘Do I teach this properly or do I teach this in a way that the children will remember the rules and achieve the mark in the test?’. For some children, this doesn’t impact their progress but for others, especially those that struggle with maths, it does.

Without understanding what has been taught previously, it is very hard to progress, make connections, and learn new content.

Another challenge listed by Skemp was time. Teachers are still time-poor, and we still have a very full curriculum. So how do we support children to understand maths better, in manageable ways, to help create links and include creativity?

Finding a balance

In this blog, I can’t provide a solution for the whole of the maths curriculum, but I can offer a simple idea that will help when teaching and learning times tables. The Year 4 Multiplication Tables Check is one of those ‘hoops’ that our children must jump through and is purely about regurgitating a right answer. Many teachers are not fans of this assessment, and I think Skemp’s quote sums up why.

I do think it is very important for children to learn their times tables so that they can use and apply their learning. I meet a lot of children who can give me the answer to a multiplication fact but then can’t make connections to other learning (when trying to simplify a fraction, for example). I also meet a lot of children who, due to not having quick recall of times tables facts, feel they can’t do any maths.

Preparing for the Multiplication Tables Check (MTC)

In Rachel Brown’s blog Put your school in the driving seat for the Year 4 Multiplication Tables Check | HFL Education, she provides great advice on how to teach times tables in preparation for the MTC. I am going to drill down further and specifically look at those tricky facts. Many of the times table online practice sites, can provide a ‘heat map’ of children’s scores for the different facts. This means that you can pinpoint which facts are less secure. But to be honest, I am sure you could list the specific facts you know children find hardest to learn.

Once we know which facts need extra rehearsal, practice can be introduced.

Practice and rehearsal to make meaningful connections

The idea of a practice scaffold is to rehearse a multiplication fact in multiple ways, to make connections between different models and language, and to build relational understanding. They also provide opportunities for overlearning. Once the learning rehearsed within the scaffolds has been taught, the children should be able to access them independently.

A practice scaffold example:

The starting point provided by the teacher could be the calculation. However, it doesn’t need to be. In the example above, ‘Describe it…’, is the starting point.

The rest of the scaffold is then completed by the pupils, who must make connections to various representations of the calculation. This draws upon the CPA approach with the option to include a ‘build it’ element.

Variation in representation on the scaffold

Repeated addition

Understanding multiplication as repeatedly adding equal groups is essential. ‘Timesing’ is a common shorthand for this.

Array

The array, drawn or built, is a visual area model. When creating the array, you are showing the space it covers. This is a link missed by many. The array also clearly shows the commutative nature of multiplication.   

Count it…

The linear counting stick model shows that an equal group is being added each time, alongside repeated addition and the array model. Using a dual-coded counting stick enables a clear link to be made to multiples, for example, “the 6th multiple of 5 is 30”.

Commute it…

Knowing that when you learn one times table fact, you actually gain two, is a lovely ‘secret’ I enjoy exploring with children when commutativity is first introduced. The array clearly shows that the product stays the same whether you multiply the number of rows by the number in each row (6 groups of 5 in the example) or multiply the number of columns by the number in each column (5 groups of 6 in the example). Practically building the array with multilink cube enables you to physically turn the array to show this.

Describe it…

Interpreting the language of mathematics is often a stumbling point for children. In this box, vocabulary or sentence stems can be provided and if the scaffold is used repeatedly, children should be encouraged to vary how they describe the calculation.

Story it…

This box allows children to do something we very rarely do, which is require them to contextualise the calculation. Children regularly solve contextualised problems but here, they have to create the context themselves. It could be in the form of a question with a solution (as in the example), a question to be solved or a statement. Many children find this tricky but once they get into the swing of writing them, it is a great opportunity for them to be creative.

Adaptations for even deeper thinking

Practice scaffolds can be altered, depending on the age and confidence level of the children. You might want to include inverse facts, for example, so a box labelled ‘inverse’ or ‘fact family’ could be added. Space could also be included for building a model.

For older students, the language of factors could be included. This could be further extended through scaling and understanding of related facts such as 60 x 500.

Another box I sometimes add is titled, ‘Tell me something else…’ This allows freedom for children to be as creative as they wish.

Repetition is key

The practice scaffolds are designed as an overlearning task so need to be done repeatedly. Once the structure of how to complete them has been modelled and taught, they can be completed independently. I have seen them being used as an early morning task and as part of homework.

The same practice scaffold can be used multiple times to either practice the same calculation – for this, vary the starting point – or for different calculations that you know are the tricky ones. Every child in the class can have the same scaffold, but you might personalise the calculations to practise.

Some schools I have shared these with have them in a plastic wallet and the children complete the scaffold using a white board pen to reduce printing and make repetitive use simpler.

Maths as a tool for life

As Skemp said, a calculator is the best tool for mechanically solving a calculation, and nearly all of us carry one around in our pockets – despite my teachers telling me otherwise – so we don’t need our children to robotically recall facts.

We want them to know their times tables but in a way that encourages them to make and see connections, so they can use and apply the learning across their maths curriculum, not just to be able to jump though one of the many hoops put in front of them.

Is the teaching and learning of times tables a key focus in your school?  

The HFL Education Primary Maths team can work with you in school to develop the teaching and learning of multiplication facts through The Multiplication Package.

Find out more about the HFL Education Curriculum Impact Packages:

• Do we need to teach MRS GREN?
• Do pupils really need to memorise kingdom, order, phylum…?
• Do I need to teach the rock cycle? 
• Do pupils need to use the terms igneous, metamorphic and sedimentary? 

Are you relieved to find out that the answer to all these questions is, no? 

Throughout my ten years of supporting schools to reflect on and develop their primary science curriculums, I have had countless discussions about content that was being taught but was not actually in the primary science curriculum. 

It is not uncommon for schools to be teaching content that goes beyond the primary science national curriculum requirements, without realising it. This could be because the national curriculum requirements are not well understood. Sometimes it is because a scheme or resource has gone beyond the requirements and subject leaders and teachers are not aware of this.

In The 10 key issues with children’s learning in primary science in England report published in 2021, one of the issues identified was that children’s learning in science can be superficial and lack depth. An observation made in relation to this was that sometimes there was an ‘overload of inappropriately selected science.’ This was described as when, ‘Planning fails to consider the national curriculum requirements relevant to the age group resulting in teaching including content that is beyond the expectations.’ 

Ofsted also mentioned this in its Finding the Optimum report: 

‘In a few schools, pupils were being expected to learn content that was too technical. This was because they had not secured prerequisite knowledge first. For example, Year 6 pupils in one school were learning about genetics without having learned relevant prior knowledge. This was not surprising, given that in the national curriculum genetics is not introduced until the secondary phase.’

There are many potential consequences of teaching content that goes beyond the national curriculum requirements including:

• Misconceptions -Teaching some content too early can lead to superficial understanding or the creation of misconceptions. For example, respiration is sometimes discussed as being similar to breathing which is a misconception that can be hard to correct and is best avoided.  
• Less time available - Teaching additional content will mean there is less curriculum time available to secure the fundamental knowledge and understanding. This includes the disciplinary knowledge needed to understand and work scientifically and the time to practise working scientifically skills. 
• Too technical - Content which is covered within the KS3 and KS4 programmes of study, may be too technical, or pupils may not have the required prerequisite knowledge to understand it. This may result in pupils not understanding and reaching the conclusion that they are not very good at science and switch them off to science learning.  

The aim of any school should be to have an ambitious science curriculum that supports all pupils to understand the world around them. New learning should be connected to pupils’ experiences and prior learning to help them build rich and detailed schema. Jumping ahead in content can result in possible connections being missed.

Content that is sometimes taught in ‘rocks and soil’ in year 3 demonstrates this well. Although the topic title, ‘rocks and soil’, only appears once in the primary national curriculum it is part of learning about materials and should be connected to what pupils have previously learnt in KS1 about materials. This can be done by focusing on the content in the box below: 

Pupils can be supported to learn this content through a range of practical experiences that will develop both this substantive knowledge and also their working scientifically skills. This might include observing and testing rocks, observing uses of rocks (for example in buildings and graveyards) exploring how they have changed over time, researching fossils and exploring and investigating soil. (All of this is suggested in the notes and guidance section of the national curriculum for this topic). 

Unfortunately, what sometimes happens when pupils learn about rocks in Y3, is they miss the learning described above and are instead taught about the rock cycle and expected to learn and use terms such as sedimentary, igneous and metamorphic, which is the content outlined in the KS3 programme of study. By jumping straight ahead to this content, the opportunity to ground, connect and link new learning to what pupils already know may be missed. Some teachers also find that teaching the KS3 content has to be more teacher led and provides less opportunity for the practical work that develops working scientifically skills as well as substantive knowledge.

‘But what about challenge?’

One potential argument for the inclusion of additional content is that it has been included to challenge pupils. However, there is plenty of opportunity to challenge pupils’ thinking and get them to think deeply while focusing on content that is in the primary national curriculum.

For example, when comparing living and non-living things we can spend time correcting the misconception that plants do not move. This can be done through time lapse videos on Explorify or on the BBC Green Planet and through looking at examples pupils can see such as daisies and dandelions closing overnight or when it is very overcast. We can also introduce questions such as: ‘Is a seed alive?’ This aims to provoke thought, discussion and encourage children to justify their ideas.

Another good example comes from the ‘states of matter’ topic in Y4. We can challenge pupils to really think about the properties of solids, liquids and gases by looking at a range of examples. Through using many examples of new vocabulary or a concept, more is learnt about the concept being studied as different examples will allow different aspects to be made clear. In the Frayer model image below three examples of solids are given. The rock displays many of the properties pupils typically think about when thinking of a solid it is rigid and hard. However, the fabric is flexible, can change shape, and can possibly be torn which are not properties typically associated with a solid. Granular solids, like sugar, are interesting examples to discuss as they take the shape of the container and can be poured. This is because they are a collection of grains which are solid. Pupils can be guided to look at grains individually to realise individually they do have properties of a solid. Pupils could also explore interesting mixtures of solids, liquids and gases such as shaving foam, toothpaste or my favourite: cornflour gloop sometimes referred to as ‘Oobleck’ which acts like a solid when under stress but will flow like a liquid at other times. These examples will challenge pupils’ thinking and preconceptions and ensure they have stronger understanding of the properties of solids, liquids and gases which is a great foundation for the next stage of learning in this area. 

We could also challenge pupils by giving more time for pupils to take greater responsibility in planning, doing and reviewing enquiries which support their understanding of substantive knowledge but focus on developing the full range of challenging working scientifically skills. This is particularly relevant in UKS2 which lists a range of working scientifically skills that are not easy.

For example, in year 6, rather than spending lots of time trying to memorise Carl Linnaeus’s classification system (including kingdom, phylum, class…), pupils could plan their own enquiry to investigate variables that affect the volume of gas produced by yeast when learning about microorganisms. 

Including additional content

There may be occasions where a school chooses to include content that goes beyond what is outlined in the national curriculum and extend learning for good reasons, such as it reflects their school community or pupils have asked questions that the teachers decide to answer.

For example, one school I worked with decided to spend more time in KS1 ‘materials’ discussing the impact of plastics, particularly plastics in the ocean, as they had recently switched to using pasta straws in the school canteen and the children were curious about why this decision had been made.

Another school explored sensory impairment when learning about senses in KS1 as they had members of the school community with sensory impairment.

Extending learning through additional content, should only be done if it can be explained at an age-appropriate level and pupils are secure with the fundamental learning outlined in the national curriculum and have had opportunity for reinforcement and practice of this.

Where can you go to for support in reviewing choice of content in your science curriculum?

We have created a document (understanding the national curriculum requirements for primary science) listing some of the most common content that is taught but is not in the primary curriculum. The aim of this document is to develop understanding of content in the national curriculum and facilitate discussion about what is in a school’s curriculum to support leaders and teachers to make conscious choices about what they choose to teach and why. If your school subscribes to Primary PA plus, then this document is free to download when you are logged into the HFL website. If you are not yet subscribed to PA Plus, then the same document can be accessed for a small charge on our HFL shop.

For further support on reviewing content in your science curriculum, the first stop is to go back to the national curriculum paying particular attention to the notes and guidance section. PLAN assessment also has free knowledge matrices which outline key learning in each topic alongside prior and future learning. If you are interested in a wider debate about the purpose and content of a primary science curriculum you may also be interested in Primary Curriculum Advisory Group.

We also offer bespoke in school science support to help evaluate, develop and implement a science curriculum that is logically sequenced to enable all pupils to gain a strong foundation. If you would like to discuss this, please do get in touch: charlotte.jackson@hfleducation.org.