Think about this question: ‘Express 75 as a product of its prime factors’.

Many students are unable to answer questions like this in a test situation, despite having successfully performed similar tasks many times in class and often having revisited the topic in several different years. There are many reasons for this, but the language of the question itself is often the main barrier. Interestingly, it is not just the highly mathematical words like ‘product’, ‘prime’ and ‘factors’ that cause problems. A significant number of students struggle with the command word ‘express’ and get no further.

Tiers of vocabulary

In their 2017 book ‘Bringing words to life’, Isabel Beck et al. talk about three tiers of vocabulary:

Tier 1 – basic words with a single meaning that we use in everyday talk such as girl, book etc.

Tier 2 – high frequency words that occur in many areas, but are less common in everyday talk such as apply, explain, verify etc. These words may also have more than one meaning.

Tier 3 – less common subject-specific words such as numerator, equilateral, decagon etc.

It is important for us, as teachers of mathematics, to make sure our students gain familiarity with the Tier 2 words as well as the Tier 3 (that we often list as the ‘key words’ in our lessons). Alex Quigley, a Senior Associate at the EEF, talks about helping students ‘breaking the academic code’ to help them access the curriculum, citing the ‘vocabulary gap’ as a major barrier particularly for disadvantaged and EAL students. He also notes that students need to meet a word between four and ten times before they internalise it. This should make us reflect on how we deal with vocabulary – we might think we spend considerable time defining and working on a word when we first introduce it, but if we just assume it’s ‘sunk in’ thereafter we might be doing our students a disservice.

This is especially important in mathematics as there are so many words that have meanings in maths, but may be already familiar to students from their meanings outside the subject. Here are just a few examples:

Talk, talk, talk
Many (if not all) of the tasks we suggest at WRM are designed to be the basis of classroom discussion. The more often students get to engage in mathematical discourse, the quicker their understanding of Tier 2 and 3 words will grow. Take this question from our Year 7 scheme:

The above is an example task from our Year 7 schemes that encourages discussion.

Then listen
Listening to students’ discussion is a powerful assessment for learning tool, enabling teachers to really understand what students know and don’t know about a topic.

We also recommend the specific discussion of vocabulary through models such as the Frayer model:

Rather like goal-free problems, when students work in pairs or groups to find their own definitions etc., it’s not the answer they produce that is important, but the discussion round it that deepens their understanding of the word or concept involved. In particular being able to distinguish between examples and non-examples – things that are close in meaning but not quite the same – really enhances understanding. The more opportunities we create for mathematical talk, the more chances there are for students to get to grips with mathematical language concepts, and the better their learning will be.

Selected references:
Beck I, McKeown M & Kucan L (2002) Bringing words into life
Quigley, A (2018) Closing the vocabulary gap
Hattie J, Fisher D & Frey N (2017) Visible learning in mathematics

As one of the four number operations, subtraction is a concept that is familiar – and indeed possibly routine – to every primary school teacher. Yet, it is perhaps surprisingly complicated. Subtraction has three separate structures. Many pupils’ conceptual misunderstandings of subtraction originate in their lack of awareness of these structures, resulting in pupils at the end of primary school struggling to answer seemingly simple questions such as, “The population of Cambridge is 125,000 and the population of Bedford is 89,000. How many more people live in Cambridge than Bedford?”. Many pupils respond, Pavlovian-esque, to the word ‘more’ and add, calculating the answer as 214,000.

The first subtraction structure is the reduction model, and this is the model that is often mistaken for being the only model of subtraction, leading to such confusions as pupils (and many adults) referring to the subtraction sign as ‘take away’; this is only accurate when the subtraction follows the reduction structure.

Reduction is the most easily understood of the structures, and the one that children encounter most often in everyday life. Reduction concerns starting with a quantity and then this quantity getting smaller (for example, ‘Katy starts with 7 sweets and eats 3, how many sweets does she have left?’). The First….then….now story model works well when teaching young children the reduction model: ‘First Katy had 7 sweets, then she ate 3, now she has how many sweets?’.

These reduction stories need to be carefully considered to ensure that reduction which is being explored. Reduction involving small quantities should be represented concretely and pictorially, with children physically moving objects away and crossing out drawings, before using a single bar model with the subtrahend (the number being taken away) crossed out.

The second model of subtraction is partitioning. As pupils have become comfortable with subtraction being regarded as ‘taking away’, partitioning questions can prove challenging initially; pupils come to conclude that subtraction is only used when a quantity is being taken away. Partitioning questions demonstrate that this is not the case. When a subtraction question requires partitioning, the first number (minuend) is being split into two parts (the subtrahend and the difference). Crucially, nothing is being lost when partitioning occurs, yet subtraction is still necessary to calculate the answer. For example, ‘Katy has 7 sweets, some are marshmallows and the rest are lollipops. If 3 are lollipops, how many are marshmallows?’. No sweets are being eaten, thus 7 is not being reduced in magnitude. Rather, 7 is being split into two groups – lollipops and marshmallows. Concrete resources such as cubes (labelled as lollipops and marshmallows) support pupils’ understanding of this structure, before linking the creation of two groups of manipulatives to a part-whole model.

Thirdly, there is the comparative difference model of subtraction. This is distinct from both reduction and partitioning because with difference problems the ‘whole’ (the whole number of apples, or the whole number of pupils in the class) is not relevant to the situation (indeed, calculating the ‘whole’ is very often unhelpful when answering difference questions). This subtraction structure is therefore considerably removed from the reduction model (where the reduction of the magnitude of the whole is of central importance). A pictorial representation using a comparison bar model is helpful for pupils to use here, to avoid them relying on frequently misleading linguistic ‘clues’ in a difference question (the use of ‘more’ in the population example given above being such a clue). To consider the population question we discussed earlier, being able to represent this as a bar model makes the necessary operation explicit:

Maths Subtraction Blog for Primary School Teachers

The bar model shows clearly that the word ‘more’ in the question refers to the gap or difference between the bars, which each represent the population of a locality. Note that the gap is not shown as a bar but rather with an arrow: when bar modelling, bars are used to represent quantities which exist (so number of apples, pens, or in this case people). Frequent practice of difference questions (and use of the bar model to show what difference is being calculated) is crucial.

In conclusion, as teachers we need to be conscious of the three distinct subtraction structures, and ensure that our pupils have deep experience of exploring all three during their primary Maths education. Reduction is generally taught first, and as pupils progress through primary school, they encounter the partitioning and difference structures. Yet, it is crucial that pupils who are nearing the end of their primary career continue to read, understand and answer reduction questions. As with all topics within Maths, the mathematical language we use (and which we insist on pupils using) in the classroom when teaching subtraction is of critical importance if pupils are to avoid all too common misconceptions; specifically, the mathematical symbol – means ‘subtract’, not ‘take-away’.

Can you match these questions to their subtraction structure (some questions may include more than one structure)?
In a box are 5 cars. Two are Jane’s and the rest are Ian’s. How many are Ian’s?

I have 4 bananas. How many more bananas do I need to buy so that I have 10 in total?

Mount Everest is 8,848 metres high. Ben Nevis is 1,344 metres high. How much higher is Mount Everest than Ben Nevis?

There are 191 pupils in a school. 84 are girls. 23 boys leave the school. How many boys are left in the school?

Sarah has £985.00 in her bank account. She spends £110.00 on Monday. On Tuesday she spends half of what she spent on Monday. On Wednesday she spent £248.00. How much money does she have left?

Beckmann S (2004), Solving Algebra and Other Story Problems with Simple Diagrams: a Method Demonstrated in Grade 4–6 Texts Used in Singapore, The Mathematics Educator, 14 (1), pp42 – 46
Harris A (2000), Addition and subtraction, St Martin’s College
NCETM (2018), Additive structures: introduction to augmentation and reduction

How can Maths leaders effectively implement new approaches?

The idea of implementing a whole school change can be one that is often exciting and essential but at times daunting and over-whelming.

As leaders, we need to consider several aspects of school life to have the necessary intended impact such as timing, resources, delivering a clear and consistent message etc. In February 2018, The EEF produced the ‘Putting Evidence to Work: A School’s Guide to Implementation’ which states that:

“One of the characteristics that distinguishes effective and less-effective schools, in addition to what they implement, is how they put those new approaches into practice.”

The EEF suggest 6 foundations for establishing good improvement, point 3 being, “define the problem you want to solve and identify appropriate programmes or practices to implement”, and point 4, “create a clear implementation plan, judge the readiness of the school to deliver the plan, then prepare staff and resources.” This article intends to share some of the key things to consider when planning for effective change.

What is the problem that needs addressing?
As a leader, you may already be aware of the problem that you want to solve and the outcome that you want to achieve. We tend to have a sixth sense for noticing when something doesn’t feel right, this could be from conversations with pupils and/or staff, observations during a learning walk or a book look, perhaps it comes from a data trail. However, ensuring that your thoughts are valid and credible is very important; imagine spending your time and energy planning for change to find out that your initial thoughts are not as accurate as you hoped. Here’s where an audit could be used to give a precise picture of current practice and outcomes in your school.

Things to consider during an audit as well as current action plans:

Things to consider during an audit

Please note this this is not an exhaustive list, simply ideas to think about when defining the problem.

How are you going to address the problem?
Once you have a clear understanding of the problem and how it is impacting on the current practices of the school it is important to develop a vision for change. What will it look like? Who will drive it? How will you know it is having an impact? When will change start and when will it be embedded? Creating a clear implementation plan, or action plan, will help to focus on answering these questions and it will help to make the changes more manageable.
Paul Bambrick-Santoyo, in his book ‘Leverage Leadership’ states that:

“Action planning supports meaningful change.”

These meaningful changes often start happening in September, a new academic year to start a new approach, however when does the planning process for these new changes begin? June? July? Getting the action planning stage right sooner rather than later will help ensure the foundations for impact in September are firmly laid and ultimately ease the over-whelming feeling for leaders and teachers.

Here at White Rose Maths, we deliver a range of CPD sessions aimed at developing world-class maths teachers. Mastery pedagogy is at the heart of these sessions and fully embedding mastery approaches such as CPA or reasoning and problem solving can seem an enormous task that can be thought about and researched over the Summer and then introduced to staff during a staff meeting or inset day before the children come back in September. Let White Rose Maths help you make the best use of the time leading up to the new academic year.

How can White Rose Maths help?
White Rose Maths have developed a unique new training opportunity – Jigsaw Plus. This series of training sessions will give you all of the tools that you need to effectively audit Maths within your school, identify areas for improvement, and to develop a robust, personalised plan of action to address the areas for improvement. You will receive bespoke support from a White Rose Maths Primary Specialist and understand how to measure the impact of the actions put in place, and explore how to use White Rose Maths schemes effectively. The CPD sessions include:

Jigsaw Plus sessions

Delegates need to commit to attending on all three days of training. Click on the links below for more information about Jigsaw Plus.

Monday 20 May 2019
Monday 17 June 2019
Friday 5 July 2019

Thursday 23 May 2019
Friday 21 June 2019
Thursday 11 July 2019