Mindset Maths

Activity 1

Welcome to Secondary School

Across many Scottish local authorities, testing of first year pupils in maths is commonplace. For some this test is carried out in primary 7, prior to pupils leaving primary school, then marked by secondary maths teachers. For others, this test takes place within the first few weeks in secondary school. However, for any assessment the question of “What am I assessing?” and “Why am I assessing?” should always be considered.

Good assessment feeds into learning and teaching. In mathematics, learners are frequently tested, awarded marks and records of achievement are produced (a summative assessment of learning), in order to summarise the progress made, but this does little to promote learning. Effective assessment practices, however, enable teachers to build on learners’ prior knowledge, and match their teaching to the needs of each learner.  A teacher’s assessment practice is also influential in developing a learner’s sense of self (Boaler, William, Zevenbergen. 2000). Swan (2005) asks us whether students are being given low-level assessment tasks which are mechanistic and can be completed by imitating a routine or procedure without any depth of thought.

“Assessment approaches need to promote learner engagement and ensure appropriate support so that all learners can achieve their aspirational goals and maximise their potential.” Building the Curriculum 5: A framework for assessment

One of the most important factors in a young person’s learning is known as their Opportunity to Learn (OTL). It comes as no surprise that research has shown (Wang, 1998; Elmore & Fuhrman, 1995) that when pupils are given access to higher level content, they attain to better levels.

Given this, placing young people within specific learning sets in accordance with their test results can often be incorrect and damaging.

In August 2017, the Scottish Government introduced standardised testing through the Scottish National Standardised Assessments (SNSA) as a means to measure a child’s progress, provide diagnostic information and support teachers’ professional judgement. They were set out as part of the National Improvement Framework for Scottish Education, in line with Curriculum for Excellence (CfE), to assist teachers to help improve outcomes for all children.

Across Scotland, throughout primary 7, children are tested in reading, writing and numeracy (not mathematics). Although these assessments can be carried out early in the school year in order to provide diagnostic information, more often they are administered at the end of the school year to inform teacher judgements of whether a child has achieved a specific CfE level. No matter when they are administered, the results should always be shared with the pupil’s next teacher.

Click on the circles below to explore how we might assess for learning, and gather a rich range of evidence to document learning and encourage learning rather than the process of generating answers.

Activity 2

Setting or mixed ability?

There are several ways that classes or departments group may their learners: by ability grouping or 'setting', flexible grouping or mixed ability grouping*. Many have researched and written extensively about the benefits and pitfalls of each of these approaches, but can the structure of groupings have an impact developing a growth mindset culture?

Learning is a social practice, through which we come to know who we are and develop a sense of identity (Lave and Wegner, 2014). Through participating in a community of practice (in this case mathematics) learners come to not only learn mathematics, but also acquire a sense of who they are as learners within the social practice of mathematics (Lave and Wegner, 2014). Perpetuated by the evidence base that setting and streaming perpetuate social class divisions (Boaler & William, 2001) perhaps rather than attainment being the driver for pedagogic change we should consider social justice and mindsets in mathematics being the driver, and mixed ability being a vehicle through which we can affect equitable change? Furthermore, what messages are we sending to learners by putting them in set classes, based on a perceived ability?

Prof. Hattie and Jo Boaler explore these ideas in the following videos.

Although there is an argument that mixed ability groupings (or mixed attainment groupings) can promote positive mindsets in learners, the thought of a move towards this mode of delivery is often a contentious subject for teachers of mathematics. There are very valid concerns around the level of planning required to effectively attend to all learners, the spread of support required to ensure needs are met (whether that be through content, process or product/outcome or form of assessment) and the lack of equivocal evidence on the impact on attainment when compared to setting, amongst many others.

However, this is not to say it can’t be done, and with great success. Read the following to learn about some of the challenges faced by mathematics departments across Scotland when they moved to a mixed ability structure, and how they overcame them.

What barriers could you see if you were to implement mixed ability teaching?

How could you overcome them?

When consulting secondary mathematics departments across Scotland on the challenges they faced when moving to a mixed ability approach to teaching, common themes emerged. Click on the buttons to see what challenges they faced and action they took to begin to overcome these.

Activity 3

Using concrete, pictorial and abstract teaching styles

In order to become mathematical thinkers, learners must become not just procedurally fluent but also conceptually fluent. However, this can be tricky for some as the abstract nature of maths is often a mystery to learners. How do we overcome this?

Concrete-Pictorial-Abstract (CPA) is an approach to learning and teaching based on the works of several theorists; Jerome Bruner’s Theory of Constructivism where learning is an active process from which learners construct new ideas; Zoltan Dienes’ Variation Theory that children see, approach and understand the world differently and Richard Skemp’s intentional and relational thinking - not just understanding the how but also the why.

With CPA, learners experience and explore new learning:

• physically (concrete phase)

• they encode this learning into visual representations (pictorial phase)

• before relational and abstract understanding is developed, meaning they know what to do and why (abstract phase).

This example is taken from Bruner (1973):

“The concept of prime numbers appears to be more readily grasped when the child, through construction, discovers that certain handfuls of beans cannot be laid out in completed rows and columns. Such quantities have either to be laid out in a single file or in an incomplete row-column design in which there is always one extra or one too few to fill the pattern. These patterns, the child learns, happen to be called prime. It is easy for the child to go from this step to the recognition that a multiple table, so called, is a record sheet of quantities in completed multiple rows and columns. Here is factoring, multiplication and primes in a construction that can be visualized.”

Developed in Singapore, which consistently ranks high in PISA scores for maths and science, this three-step CPA methodology for teaching mathematics is an essential element of a maths mastery approach commonly known as Singapore Maths.

There is also a body of evidence that shows that when learners only practice and memorise procedures without developing a conceptual understanding, bringing meaning to their learning and applying it to different situations and contexts is difficult. What lies at the heart of a CPA approach is moving from the concrete to the pictorial representation, in order to make sense of the abstract, and equipping learners to make sense of the maths in front of them.

Watch the video below.

Further reading on this is available in the ‘additional reading’ folder below

Activity 4

Maths anxiety

As highlighted in the Making Maths Count report, 30% of Scottish learners reported that they feel very tense and nervous when doing maths work and more than 50% worry that maths will be difficult, outlining the OECD recommendation that “the issue of maths anxiety in Scotland warrants close attention”, and that the promotion of “a growth mind-set approach, and other methods of improving learner confidence such as positive learning dispositions, are essential to addressing maths anxiety and raising attainment.”

This was further explored by Ashcraft (2002) who found that those expressing a negative opinion of maths perceived the subject as “difficult and demanding”, “linked to feelings of anxiety or inadequacy”, “highly judgemental” whilst invoking a “fear of failure” and “reluctance to engage further to protect self-confidence”.

Further research was carried out with three focus groups. Young people in P4 – P7 were positive about maths but wanted to understand how it would be useful to them later in life. They were also highly attuned to their family members’ feelings towards maths, positive and negative. S1 – S6 pupils wanted to see the relevance of the topics taught (e.g. algebra). Parents and adult learners wished they'd known earlier  how much maths would impact their daily lives, especially without consciously being aware of 'doing' maths.

In addition, research from Winning Scotland (in partnership with Stanford University) involving over 11,000 Scottish pupils (4870 from secondary schools) found that not only does enjoyment of maths decrease with age, 58% of secondary students agreed that ‘Sometimes maths makes me feel anxious’.

Girls were significantly more likely than boys to agree that they feel anxious/nervous in maths sometimes. Of the non-neutral responses, 61% of girls of primary age agreed, rising to 80% of secondary girls. You can download the full research report here .

Watch the following videos on the subject of maths anxiety:

Activity 5

Homework

Studies have shown that, whilst the impact of homework on younger children is debatable, it's more effective for older children. The Education Endowment Fund (2018) summarise an increase in attainment that can vary between two to eight months of progress, depending on the homework. For example, routine practice tasks or learning vocabulary can have a two month progress increase, whereas a short and focused intervention style piece of work can show up to eight months of progress. Progress is also affected by how well the homework relates to class learning. Evidence suggests that levels of engagement with homework and its impact are correlated, with opportunities for learners to make connections with out-of-school life having been noted as increasing engagement.

Additionally, homework has been found to have far wider impact on learners rather than just progressing their learning or attainment, for example keeping parents abreast of what's happening in school and developing wider skills such as time-keeping, prioritising workloads and working to a deadline (See 10 Ps of Homework). Position and flow of homework should also be considered; does it continue the work in a preceding lesson or is it preparatory? If it’s preparatory is there a risk to participation in the lesson if not competed?

Cathy Vatterott (2010) identified five fundamental characteristics of good homework: purpose, efficiency, ownership, competence, and aesthetic appeal.

• Purpose: all homework assignments are meaningful and students must also understand the purpose of the assignment and why it's important in the context of their academic experience (Xu, 2011).

• Efficiency: homework should not take an inordinate amount of time and should require some hard thinking.

• Ownership: students who feel connected to the content and assignment learn more and are more motivated. Providing students with choice in their assignments is one way to create ownership.

• Competence: students should feel competent in completing homework. In order to achieve this, it’s beneficial to abandon the one-size-fits-all model. Homework that students can’t do without help is not good homework.

• Inspiring: A well-considered and clearly designed resource and task impacts positively upon student motivation. 

Read the following articles to explore further:

• Education Endowment Foundation
  

• Homework: What Does the Evidence Say? 
  

While there is a lot of debate associated with homework, the outcome should not be harmful to a pupil's learning. There is a wealth of information about the benefits of home work on the Stanford University website - here

Further reading is available in the additional reading folder.

Activity 6

Disconnect from BGE to Senior Phase

Curriculum for Excellence was envisaged to be a seamless, integrated curriculum from ages 3–18. As outlined in CfE Briefing Paper 6 , Progression from the Broad General Education to the Senior Phase, Part 1: The S3 Experience:

“New National Qualifications have been designed to build on the Es and Os of the BGE. As such, learning can progress seamlessly through S3 into the Senior Phase when, at an appropriate point, the learning will be measured within the qualifications framework.”

In its review of the BGE , the OECD states that “CfE represents an ambitious and important departure that has sought to develop a coherent 3-18 curriculum around capacities and learning, rather than school subjects, taking a different approach to assessment and national prescription from what was in place before”(OECD, 2015, p 37).

However, Building the Curriculum 3 states that “Subjects are an essential feature of the curriculum, particularly in secondary school. They provide an important and familiar structure for knowledge, offering a context for specialists to inspire, stretch and motivate. Throughout a young person’s learning there will be increasing specialisation and greater depth, which will lead to subjects increasingly being the principal means of structuring learning and delivering outcomes.”

Whilst the Es and Os outlining the 3–15 BGE were designed by Education Scotland, the design of the 16–18 curriculum was the responsibility of the SQA. Courses studied for qualification are designed around a nominal 160 hours of study with schools free to design the curriculum which best suits their local needs. Scott (2018) found that, in practice, this has led to a fragmentation in the late stages of BGE and into the Senior Phase, with schools offering between five and eight subjects in S4 (with a handful leaving exam presentation until S5). The Scottish Government has recently announced a review into the Senior Phase.

Activity 7

Utilising technology to teach maths

Are schools making the most of technology? Is pupils' use of technology in school representative of what they will face in the workplace?

What's wrong with today's maths?

World renowned researcher Conrad Wolfram argues that it’s an 80% different subject from what is required. Why?

Because computers have mechanised computation beyond previous imagination and do calculating really well. Today's maths education spends 80% of the curriculum time gaining expertise in hand–calculation methods and algebraic manipulation. The curriculum is ordered by the difficulty of the skills necessary to complete the calculation, rather than the difficulty of understanding the complexity of the topic.

“School mathematics is very disconnected from mathematics used to solve problems in the real world"

Now watch this video to explore this topic area further.

Similarly, Leonard Mlodinow in his book Elastic (2017) Discusses three levels of thinking:

He describes elastic thinking as “more of a bottom up process” and “where ideas come from” He also asserts that the elastic thinking, when compared to analytical thinking, is increasingly desired by employers in the modern world.

It could however be argued that Wolfram’s position dichotomises pure v applied maths.

“A typical mathematician does not actively try to be useful. Individual mathematicians are motivated primarily by a subtle mixture of ambition and intellectual curiosity, and not by a wish to benefit society, nevertheless, mathematics as a whole does benefit society.”Sir Timothy Gowers

This is also summed up nicely by Ayliean Macdonald in the video below (online channel here )

Please complete the questionnaire below.