Photograph of Simon Mitchell
The towns and villages to which I travel
Face-to-face tutorials make it easier to build a strong rapport from the outset. With younger pupils, hands-on learning with value counters, protractors and compasses is also more natural.
I am always happy to schedule face-to-face tutorials - without restriction or additional cost - within 15 minutes' travel of Graffham. That includes Cocking, Bepton, Heyshott, Hoyle, Amberham, East Lavington, Midhurst, Easebourne, Lodsworth, Selham, River, Tillington, Upperton, Petworth, Byworth, Duncton, West Lavington, and of course, Graffham - the towns and villages within the smaller crimson ellipse on the map below.
I also travel further afield: to Petersfield, Haslemere, Billingshurst, Pulborough, West Chiltington, Storrington, Arundel, Chichester, Emsworth, Fittleworth, Bignor, East Dean, Chithurst, ... indeed, throughout the area between the two crimson ellipses.
The minimum chargeable duration for these home visits is 2 hours.
Pedagogy of a maths tutor in 1000 words
Each pupil is different – true, but not enormously insightful. Physicists, economists, architects, and chefs all create simplified models to explain, predict, and inform policy or procedure. Educators are no different – we select from a myriad of tools and techniques, informed by simplified models.
Bloom's taxonomy
Bloom’s taxonomy describes learning as progressing through a hierarchy of cognitive attainment. First, we remember newly encountered knowledge.
we can state the double angle identity for sineThen, we understand – we grasp the meaning behind our learning.
we can explain how the graph of sine is symmetric about the origin, referencing standard trigonometric propertiesThen, we apply our knowledge in straightforward situations until we attain confidence.
we can solve 3 sin(θ) = 1, for θ in the interval 0 ≤ θ ≤ 2πThese lower-order skills underpin higher-order learning.
We analyse – we break down structure, identify patterns, and contrast methods.
to simplify sin(x) / {1 − cos(x)} we compare two methods: multiply top and bottom by {1 + cos(x)}; versus substitute cos(x) = 1 − 2 sin²(x/2), to find which is more efficientNext, we evaluate: not just the skill, but also how new learning connects with existing understanding; and how impactful is the new learning – is it internalised as an underpinning of an existing big-picture model?
evaluate Charlie’s method: 'to solve cos(2x) = sin(x), I rewrote cos(2x) as (1 – x²)'Lastly, Bloom suggests we create original work, innovative solutions, and new internal models.
Learning styles
Felder, Kolb, Martinez, Gordon-Bull, Keirsey, Myers-Briggs, Herrmann, Honey-Mumford, McCarthy, Gregorc, and no doubt others I have yet to encounter, have all proposed pigeon-holing the population – most often into four pigeonholes – each with their own preferred learning style. There is remarkable commonality between these models, many echoing Carl Jung, the founder of analytical psychology.
At this point, the teacher can pause and apply these insights. Develop a series of ten lessons that cycle through Bloom’s taxonomy. And in each lesson, provide different approaches to learning – e.g. theoretical, practical, reflective, pragmatic [Honey-Mumford] – to appeal to the diverse learning styles within the class.
The tutor and tutee work one-to-one. Let’s finesse ...
Myers-Briggs as a Bayesian prior
Myers-Briggs provides one vocabulary. Drawing on Jung, it describes four different ways in which we may preferentially perceive information: introverted sensing (Si), extraverted sensing (Se), introverted intuition (Ni), and extraverted intuition (Ne).
I do not regard these as immutable categories; rather as a Bayesian prior: an initial expectation against which to observe a pupil. Like a bottle of Chablis: the label sets the expectation; the tasting provides the evidence.
A tailored approach to each individual tutee
Around 45% of us are introverted sensors (Si)
Favour established knowledge, accumulated experience, detail, and orderly progression. Of the four patterns described here, this most naturally follows the familiar progression through Bloom.
The rest of us beat our own path through the taxonomy.
Around 25% of us are extraverted sensors (Se)
Defined by a need for immersion, action, and immediate feedback. They live in the physical-now. The traditional Bloom path (memorising theory before applying it) feels painfully slow and disconnected from reality. They do not want to read the manual; they want to press the buttons and see what happens. Their entry point into the taxonomy is usually right in the middle.
Phase 1: Apply → Analyse
- Skip the lecture: Do not start with, “Today we are learning about gradients.”
- Start with action: Start with a physical graph, or a real-world scenario.
- Trial and error: Let them guess. Let them mess up. Se users treat errors as feedback.
Phase 2: Understand
- Introduce concept: Once they have pointed to the steepest part of the graph, then explain the concept of ‘gradient’ and how to measure it.
- Visual logic: Use diagrams, colour-coding, and spatial relationships.
Phase 3: Remember
- Repetition: Se users have low boredom thresholds. They can drill (they are often natural athletes/musicians who understand practice), but it must feel like a challenge.
- Speed runs: “See if you can solve these three questions in under two minutes.”
Around 10% of us are introverted intuits (Ni)
Cognitive style is convergent, holistic, and future-focused. They are constantly looking for the Grand Unified Theory – the single best path, or the underlying essence of a concept. Their learning is conceptual. They struggle deeply with the bottom of Bloom's taxonomy: they cannot just memorise a formula; they need to know where it fits in the maths’ universe.
Phase 1: Understand → Analyse
- Start at the end: Don’t introduce integration with the power rule. Say, “Let’s calculate the area of a shape with curved edges.”
- Provide the framework: Give them the macro view. Show them how algebra connects to geometry, which connects to calculus.
Phase 2: Evaluate → Create
- Proofs and principles: Pythagoras' theorem is conventionally illustrated with three squares. Replace them with three similar pentagons. Does the relationship still hold? Why?
- Efficiency: “We could solve it this way, but is there a faster way?”
Phase 3: Remember → Apply
- The necessary evil: Only after they respect the concept will they tolerate the tedium of practice questions.
- Reframing practice: Frame practice as calibrating intuition to ensure accuracy.
Around 20% of us are extraverted intuits (Ne)
Those with strong extraverted intuition (Ne) naturally prefer an inverted or non-linear approach. They need to see the whole complex picture first, before they care about the details – the facts or formulae. They thrive on patterns, possibilities, big-picture connections, and innovation.
Phase 1: Create → Evaluate
- Start with the puzzle: Do not start with, “Here is the formula.” Start with a complex, open-ended problem or a paradox.
- Trigger Ne exploration: “How would you measure the height of that building without a ladder?” Let them brainstorm wild ideas.
Phase 2: Understand → Analyse
- Conceptual understanding: Once hooked, explain the concept visually or metaphorically. “Calculus is the maths of infinitely zooming in.”
- Forming connections: Show them how this new topic connects to something they already know. “This is just like the geometry we did, but now in three dimensions.”
Phase 3: Apply → Remember
- The Trojan Horse: Now that they understand the concept, introduce formulae as helpful tools.
- Systematise: Help them build the structure they naturally lack.