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.

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%

Check for understanding

Teach Like a Champion Technique 18: Check for understanding

Gather data constantly and act on them immediately (hard to do but important)

  • Consider answers to questions as data sets, and ask enough times to get a reasonable sample, across known skill levels.
  • Respond to right answers with how and why follow-ups to ensure you spot false positives.
  • Don’t rely on self-report
  • Use monitoring to assess the number and type of errors being made by students.
  • When errors are identified, you can respond in several ways:
    • Reteach using a different approach
    • Identify and reteach the problem step
    • Identify and reteach problem terms
    • Slow the pace and reteach
    • Reteach in a different order
    • Identify and reteach to specific students

Stretch It

Teach Like a Champion Technique 3: Stretch It.

The sequence of learning does not end with a right answer; reward right answers with follow-up questions that extend knowledge and test for reliability of knowledge. This technique is especially important for differentiating instruction.

  • Ask how or why.
  • Ask for another way to answer.
  • Ask for a better word.
  • Ask for evidence.
  • Ask students to integrate a related skill.
  • Ask students to apply the same skill in a new setting.

Embedded Formative Assessment Ch 7: Activating students as owners of their own learning

  • Self-assessment depends on both meta-cognition and motivation
  • The most important thing for self-regulated learning is to encourage students to pursue personal growth not well-being. This can be encouraged by:
    • Sharing learning goals
    • Promoting a belief that ability is incremental
    • Make comparisons between students more difficult
    • Make feedback a recipe for future action
    • Use every opportunity to give learners control of their learning.
  • Practical techniques:
    • Traffic lights used to focus revision efforts.
    • Green/red disks on desks to indicate when a teacher is going too fast.
    • Or use red to ask a question of another student.
    • Choose three of a selection of reflection statements to respond to at the end of a lesson.

Embedded Formative Assessment Ch 6: Activating students as instructional resources for one another

  • Peer tutoring can be as effective as teacher tutoring because of the different power relationships.
  • Effective cooperative learning requires group goals and individual accountability.
    • Some schools achieve this through a ‘secret student’ approach whereby a single student’s behaviour is reviewed and if good the class gets a reward.
  • Practical techniques:
    • 3B4ME
    • Peer evaluation of homework
    • Two stars and a wish
    • Getting students to write down end-of-topic questions in groups
    • Classifying errors and sharing with a peer with strengths in that area.
    • Student reporter summarises the lesson in plenary and answers remaining questions.
    • Get a buddy to sign off on a checklist e.g. labelling a graph, with accountability for their peer’s work in this area.
    • Assign group reporter at random so everyone has to be prepared to do it.
    • Get students to review and present on a particular aspect of work.