Teaching methods, problem solving and pedagogic content knowledge

What this heading covers

Three related things: how mathematics should be sequenced, the named methods a question will list as options, and what a mathematics teacher knows that a mathematician does not.

The Primary TET heading is unusually explicit about the last one — mathematical problems based on deep concepts, to judge teaching ability in applying pedagogical issues. That is the Board saying it will set mathematics questions whose point is pedagogy. The arithmetic will be easy; what is tested is whether you can see what a child would do with it.

The sequence: concrete → pictorial → abstract

1. Concrete. Real objects handled — counters, sticks, bottle caps, base-ten blocks, a number line drawn on the floor, money, folded paper.
2. Pictorial. Drawings and diagrams standing for the objects — a bar, a number line on paper, a picture of four groups of three.
3. Abstract. The symbols: 3 + 4 = 7; 1/2 + 1/3 = 5/6.

The symbol is the destination, not the route. Most difficulty in primary mathematics comes from starting at stage three, or from leaving stage one before it has done its work.

Two things a question tests here:

• The stages are not age bands. An eleven-year-old meeting negative numbers for the first time needs concrete work (a thermometer, a number line, debts) just as a six-year-old meeting addition does. A new concept restarts the sequence, whatever the child's age.
• Moving too slowly is also an error. Children left at the concrete stage with no push towards the symbol do not generalise, and generalisation is the subject.

Van Hiele's levels, for geometry

The same argument in a form the paper names directly. Children move through:

1. Visualisation — a shape is recognised by overall appearance. "It's a rectangle because it looks like a door."
2. Analysis — properties are listed. "It has four right angles and opposite sides equal."
3. Informal deduction — properties are related. "A square must be a rectangle, because it has everything a rectangle has."
4. Formal deduction — proof.

The teaching consequence: asking for a proof from a child at level one is asking for something the level does not contain, and instruction pitched more than one level above the child fails regardless of how well it is delivered. A question describing a child who insists a tilted square is "a diamond, not a square" is describing level one.

The named methods

The paper gives these as options, so each needs to be recognisable in one line and matched to where it fits.

Method	What it is	Fits
Inductive	Many examples → the child sees the pattern → the rule is stated	Introducing a rule or formula
Deductive	The rule is stated → applied to examples	Practice once the rule is understood
Analytic	Start from what is to be proved or found; work backwards to what is known	Problem solving, proof
Synthetic	Start from what is known; work forwards to the result	Presenting a finished solution
Heuristic	The child discovers by investigating; the teacher sets the situation and withholds the answer	Concept formation, investigation
Project	An extended real task using several skills at once	Integration, application, group work
Laboratory	Hands-on measurement and verification with materials	Mensuration, geometry, data
Problem-solving	A non-routine problem with no rehearsed method	Developing reasoning
Play-way	Games, puzzles, rhymes carrying the mathematics	Early primary, number facts, fluency

Two pairings that get confused, and that questions exploit:

• Inductive and deductive are opposite directions. Inductive is examples → rule, and is the one for introducing. A question asking which method suits teaching a formula for the first time wants inductive.
• Analytic and synthetic are also opposite directions, but about a single problem rather than a rule. Analytic works backwards from the goal — "to find the area I need the height; to find the height I need…". Synthetic works forwards. Analytic is how a problem is solved; synthetic is how the solution is written up, which is why textbooks look synthetic and thinking is not.

The syllabus's preference, across all of these, is for methods where the child does the mathematical work. Lecture and demonstration are not forbidden, but a question offering "the teacher explains the rule clearly and gives examples" against "the children investigate several cases and state what they notice" wants the second.

Problem solving

Routine against non-routine

A routine problem applies a method the child has just been taught. A non-routine problem does not announce its method — the child must decide what to do. Only the second develops problem solving; the first develops fluency, which is also needed and is not the same thing.

Polya's four steps

The framework the paper names:

1. Understand the problem. What is given, what is asked, what are the conditions. Restate it in your own words; draw it.
2. Devise a plan. Look for a related problem, a pattern, a simpler case; work backwards; guess and check; draw a diagram; make a table.
3. Carry out the plan, checking each step.
4. Look back. Is the answer reasonable? Does it satisfy the conditions? Is there another way? What does this method generalise to?

Step 4 is the one skipped and the one asked about. Looking back is where a method becomes transferable rather than a thing that happened once, and a question asking which step children most often omit wants it.

A worked example

A rectangular field is 24 m long. Its perimeter is 80 m. A fence costs ₹35 per metre. What does fencing it cost?

Understand. Given: length 24 m, perimeter 80 m, rate ₹35/m. Asked: total cost. The breadth is not given and is not needed — the perimeter already is the length of fence.

Plan. Cost = perimeter × rate. No need to find the breadth at all.

Carry out. 80 × 35 = ₹2,800.

Look back. Reasonable? 80 m at roughly ₹35 is roughly ₹2,800 — yes. And notice what the problem offered: the length of 24 m is extra information, there to tempt a child into computing the breadth first. A child who finds breadth = 16 m and then adds up four sides reaches the same answer by a longer route; a child who multiplies 24 × 35 has used the wrong length entirely.

That is the shape of a pedagogy-flavoured mathematics question: the arithmetic is one multiplication, and what is really being tested is whether you noticed that a given number was a distractor.

Pedagogic content knowledge

Shulman's term, and the Primary TET names it as a heading.

• Content knowledge — knowing the mathematics.
• Pedagogic knowledge — knowing how to teach, in general.
• Pedagogic content knowledge (PCK) — knowing how to teach this content: the representations that work for this idea, the misconceptions children bring to it, the order the sub-ideas have to come in, and the examples that make it visible.

PCK is what a teacher of mathematics has that a mathematician does not. Concretely, it is knowing:

• that children will add across when first adding fractions, and why;
• that a square shown tilted is often not recognised as a square;
• that "take away" language makes subtraction-as-difference hard to see later;
• that the area formula is best met after tiling and the perimeter after walking;
• which of three correct explanations a seven-year-old can actually use.

None of that is mathematics, and none of it is general pedagogy. It is the overlap, and the syllabus names it because teacher education that teaches only the two separately produces teachers who know the subject and cannot teach it.

The mathematics teacher

What the heading expects, briefly: a teacher who is confident in the content (a teacher who fears mathematics transmits it), who treats errors as information, who uses materials and the children's own methods, who asks rather than tells, and who believes every child can learn mathematics — which the syllabus states as a professional commitment rather than an opinion, because the belief changes what the teacher does when a child struggles.

A question of the kind the paper sets

A teacher wants children to discover the formula for the area of a rectangle. The most appropriate method is: (a) deductive (b) inductive (c) synthetic (d) lecture

(b) inductive. The children tile several rectangles, record length, breadth and the number of tiles, notice that the count is always length × breadth, and state the rule themselves.

The word that decides it is discover. Deductive would state the formula and apply it; lecture would tell it; synthetic is about presenting a solution, not forming a rule. Any question containing discover, investigate, find the pattern or arrive at the rule is pointing at inductive or heuristic.

Mistakes that cost marks here

• Swapping inductive and deductive. Examples → rule is inductive, and is the one for introducing.
• Swapping analytic and synthetic. Analytic works backwards from the goal.
• Treating concrete–pictorial–abstract as age stages. A new concept restarts the sequence at any age.
• Skipping Polya's fourth step when asked to list them, or when asked which is most neglected.
• Describing PCK as content plus pedagogy. It is specifically knowledge of how to teach that content, including its misconceptions.
• Choosing lecture or demonstration where an option has children doing the mathematical work.
• Asking for formal proof from children at Van Hiele's first or second level.
• Using every number given in a problem. Extra information is a deliberate feature of these questions.

In one line

Concrete before pictorial before abstract and restart it for every new concept; inductive to introduce and deductive to practise; analytic works backwards and synthetic forwards; Polya's fourth step is the one everybody drops; and pedagogic content knowledge is knowing the misconceptions that attach to this particular idea, which is what makes a teacher of mathematics rather than a knower of it.

In the classroom

The cheapest way to build pedagogic content knowledge is to collect it. A teacher who writes down, for each topic taught, the wrong answers that actually appeared and the representation that finally worked is assembling exactly the knowledge the heading describes — and it is specific to the children in front of them, which no textbook can be.

For problem solving, the single most valuable habit is protecting step four. A class that has found an answer wants to move on; the teacher asking "is that sensible?" and "could we have done it another way?" is where the method becomes something the children own rather than something that worked once.