Reflective diary: Summer Institute Week 1

Reflection point 1: As you see it today, what do you think is the vision or ultimate purpose of secondary education?

Drawing together academic research, student voice and my own personal vision and values, I think the ultimate purpose of secondary education is to give young people control over their lives, and to inspire them to use that control to become more than they thought possible.

Secondary education isn’t solely about individuals, though. Locally, nationally and globally the 21st century will be a perilous time. Students in school today will make a vital contribution to a more democratic, more prosperous and fairer world, and secondary school needs to prepare them to do so.

Reflection point 2: Reflecting on your own schooling, what do you think are the key qualities that a good secondary school education will develop in a young person? What are the key components for pupils to gain from school to indicate success in adult lives?

Strong subject knowledge is undoubtedly vital. Allied to this, the unpredictable nature of the knowledge and skills required to get a job and to contribute to society in the future means that students must want and be able to continue to learn throughout their lives.

Making positive choices and a positive contribution to the world around them will also require students to critically assess the information they receive from others, ranging from the views and influence of their family and peers, through to political viewpoints regarding global issues such as climate change.

Reflection point 3: What do you think is meant by the terms ‘challenging circumstances’ and ‘educational disadvantage’? Why is this an issue?

A school in challenging circumstances is one in which the outside influences on students lives, whether from parents/carers, peers, poverty, or previous educational experiences, combine to mean that students are unable to arrive at the school gates each morning ready to maximise their learning.

Educational disadvantage is what results from these challenging circumstances. It is the attainment, access and aspiration gap between pupils from high and low socioeconomic backgrounds, and it is the evidence that this gap widens as pupils progress through school.

This disadvantage is an issue because there is an underlying assumption of equality in the purpose of secondary education. This core purpose discussed above must be within the reach of every young person in the country. If, as at present, it is not, inequality will be further entrenched in families and communities generation by generation. By contrast challenging educational disadvantage can reverse this inequality. In Bristol this means young people having access to opportunities, control over their lives and the ability to positively impact on the world around them, no matter which part of the city they grow up in.

Reflection point 4: Use the SI Objectives and Competencies to start reflecting on which are your strengths and development areas.

One strength area for me is around leadership. Through previous academic and practical experience in this area I am confident working independently, taking risks and seeing tasks through. I am keen to continue to develop my capacity to show leadership qualities consistently, even when under pressure.

Another relative strength for me is my presence in the classroom and my ability to empathise and build rapport with young people, parents and communities. My CELTA tutors previously picked this out as a strength of mine, and the teacher in Clevedon school also remarked on it.

The most important development area for me at present is relating to maths-specific teaching theory, specifically understanding progression in maths topics and finding multiple ways to model and explain the most basic mathematical concepts for lower ability students. I am seeing some progress here and have written separately about this, but want to keep this as a focus throughout the Summer Institute.

A further development area for me is in understanding how best to maintain my resilience. I want to ensure that I have the habits, strategies and relationships in place to take a constructive approach to addressing ongoing weaknesses throughout the year. This will mean ingraining habits around taking ownership of feedback, having strategies to reflect positively when things go wrong, and ensuring I have strong professional relationships with colleagues and fellow cohort members so that we can help each other to maintain a positive outlook.


  1. Continue to develop my maths pedagogical knowledge, including strategies to meet the needs of every learner and ways to break down core topics into their simplest parts, with clear progression.
    1. The main resources available to me here are: my placement school maths department, subject studies sessions, and my fellow cohort.
    2. I will seek to plan for progression in my teaching practice this week, and seek feedback on these plans.
    3. This will help me to progress towards Teaching Standards 2, 3 and 5.
  2. Continue to develop relationships with the SW cohort and the teachers and support staff at Oasis Academy John Williams, to strengthen our collective resilience in preparation for September.
    1. I will use the two days in our placement school this week to build positive professional relationships with staff in my department and the wider school, as well as my fellow cohort members at John Williams.
    2. I will aim to continue to get to know the breadth of the SW cohort, seeking out those who I have spent less time with, as well as strengthening relationships with those I have found particular common ground with so far.
    3. This wil help me to progress towards Teaching Standards 1 and 8 by ensuring I can always remain positive in school, and that I am developing professional relationships effectively.
  3. A final target relates to my enjoyment of teaching. I came to the Summer Institute both excited and nervous about teaching maths. Standing up in front of a class last week has reduced my nervousness and increased my excitement – I really enjoyed teaching. I want to hold onto that and build on that in my next teaching practice lesson.

Sunlight on a gremlin

Before starting the Teach First Summer Institute I’ve read quite a bit about maths teaching. One thing that I’ve really struggled to make sense of is how to map progression in key mathematical topics. In particular, being able to track back from what I see as the key learning outcome of a particular topic to the simplest building blocks of that topic is a real struggle for me.

This post, then, is a bit of a celebration. If progression in maths feels like a bit of a gremlin, at the end of the first week of the Summer Institute I feel as though I’ve found the string that’ll let me pull back the blind and expose the gremlin to the sunlight (excuse the awful analogy!)

Taking advantage of some excellent subject studies sessions this week, and using the example of linear equations, I am starting to grasp what progression means in practice, and most importantly how to plan for progression.

It is surprisingly simple in practice really. Taking examples of linear equations, I worked with fellow trainee maths teachers to write all the linear equations we could think of, and then number them according to our perception of their increasing complexity.

Focusing particularly on finding the simplest linear equation we could, this process allowed us to look at equations at several National Curriculum levels. Most valuably, it means that when we come to teach solving linear equations and try to judge the level of the class, we will know where to go if pupils find the content either too easy or too difficult.

Just as important, going through this process is helping me to formulate a strategy for planning for progression in other topics too. Brainstorming all aspects of the topic, and the assumptions that underpin each of those aspects, before ranking these in order of increasing complexity will help me to know where to go when students find a topic too easy of difficult. The result is that I will be better able to help all students learn in every lesson, which is the point of it all after all!

After a busy first week of the Summer Institute, this nugget of progress stands out as the most exciting for me.

Instrumental and relational understanding in maths

Fascinating exchange with @Kris_Boulton about his post on simplifying area. This post is in part a response to that post, in part me trying to make sense of the relational and instrumental understanding debate.

Kris pointed me towards Skemp’s original article on relational and instrumental understanding, as well as Willingham’s article on the value of what Willingham calls inflexible knowledge.

Starting with the caveat that I don’t yet have any experience of teaching instrumental or relational understanding in the maths classroom, the exchange has stimulated lots of thoughts. Pulling them all together, I’m left with a few conclusions:

1. Relational understanding, whilst the ideal, is rarer and more difficult than Skemp implies.

…even relational mathematicians often use instrumental thinking.

This quote from Skemp sums it up for me – however valuable relational understanding is (and I do think it is very valuable), it just isn’t realistic for mathematicians in many day-to-day applications, be that as simple as working out the area of a trapezium or as complex as factorising quadratics that don’t have integer solutions. The instrumental approaches offered by the respective formulae are vital for mathematicians operating at all levels.

2. BUT in the case of area, a drive for simplicity actually leads me towards starting with relational understanding not instrumental understanding

In the simplifying area post Kris highlights an interesting approach to reduce the complexity of the formulae for area of four basic 2D shapes to three words: ‘length times height’. Keeping in mind the caveat above, it seems to me the approach works really well, and if anything promotes relational understanding more effectively than the four formulae.

But the point I want to focus on is whether it would be even simpler to start with relational understanding, and then teach the instrumental approach.* Perhaps its the naivete of the beginning teacher, but my instinct is to take an approach where students (who are confident in how to calculate the area of a rectangle) mark length and height on card rectangles, then cut them up to make the other three basic 2D shapes.

This approach seems engaging, manageable and most importantly gets at the ‘why’ of both the formulae for area and Kris’ simpler ‘length times height’. Subsequently teaching ‘length times height’ offers a handy shorthand for the relational approach, and is hopefully more meaningful as a result of previously exploring the relationship between each shape and the rectangle. Of course, revisiting and revising both the relational approach and the instrumental approach will aid both recall and understanding.

3. Inflexible knowledge is in a different category to instrumental understanding

Willingham makes a great case for inflexible knowledge – it seems clear that inflexible knowledge is a necessary step on the road to relational understanding. But it seems to me that inflexible knowledge is quite distinct from the knowledge that underpins instrumental understanding. Inflexible knowledge is narrow but meaningful. Instrumental understanding can be useful, may be quite broad, but isn’t meaningful in the way I understand the term. An example of the former might be knowing why the area of a rectangle is ‘length times height’, but not being able to use that to investigate the area of a triangle because I can’t visualise the relationship between the two shapes. An example of the latter might be using the ‘length times height’ rule to work out the area of a whole range of shapes, without understanding why it works.

As a result Willingham’s article isn’t, for me, a case for promoting instrumental understanding. It’s a case for promoting a particular, narrow form of relational knowledge.

Putting all this together my current thinking is that relational understanding is often tricky to promote. In some specific cases a mathematics teacher is justified in retreating from this ideal and instead fostering a narrower form of knowledge, at least initially. However, wherever possible this narrower form ought to be inflexible knowledge rather than instrumental knowledge.

I’m just beginning my teaching career and feel as though I’m stretching the limits of my own knowledge with this post – any comments on anything I’ve written would be really valuable for my own learning!

*As an aside, I am aware that this is almost exactly the approach that Kris describes as ‘doomed to failure’. Eek!

Challenging students to think critically

Teach Like a Champion Ch 9: Challenging students to think critically

  • Questioning can serve at least five purposes:
    • To guide students towards understanding new material
    • To push students to do a greater share of the thinking
    • To remediate an error
    • To stretch students
    • To check for understanding
    • Rules of thumb for designing effective questions:
      • One at a time
      • Simple to complex
      • Verbatim (don’t bait and switch): don’t change the question as a student is about to answer.
      • Clear and concise:
        • start with a question word
        • limit them to two clauses
        • write them in advance when they matter
        • ask an actual question
        • assume the answer is known i.e. ‘who can tell me’ not ‘can anyone tell me…?’
  • Stock questions: have similar sequences of questions that you apply over and over again in different settings.
  • Hit rate: should be above two-thirds but rarely 100%

Improving your pacing

Teach Like a Champion Ch 8: Improving your pacing

  • Pacing is the skill of creating the perception that you are moving quickly, when necessary.
  • Six techniques:
    • Change the Pace: through different activities with the same objective, with nothing longer than 10 minutes.
    • Brighten lines: make activities begin and end clearly and crisply, by setting clear, unexpected time limits, and having a clear signal for the end e.g. clapping hands.
    • All hands: if an activity is extended, create markers by involving lots of different pupils for short periods.
    • Every minute matters: keep a series of short learning activities ready so you’re prepared when a 2-minute opportunity arises.
    • Look forward: create purpose by e.g. putting an agenda up with a  ‘mystery activity’ on it or by talking about what students will be able to do by the end of class.
    • Work the clock: use countdowns in class and set timings for activities.