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Developing Conditional Knowledge in Maths

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Gemma Scott

@directormaths

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"I get it when we do it in class but then I don't know what to do in tests...."

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The problem with a maths lesson…

Title

Worked examples

Practice

Problem solving/ exam style questions

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Norwegian anyone?

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A Curriculum Approach

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Three types of knowledge

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Declarative knowledge

FACTS

VOCABULARY

SYMBOLS

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Procedural knowledge

Adding fractions

1

Finding the mean

2

Completing the square

3

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Conditional knowledge

“Conditional knowledge is an understanding of when and how to use something we already know, for example, using different strategies in different situations” (Larkin 2009)

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"I get it when we do it in class but then I don't know what to do in tests...."

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“Successful problem-solving is (…) not a just an activity but an outcome of successful learning of the facts and methods, and their useful combination as strategies” (Ofsted, 2021)

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What makes a good problem solver?

  • Big knowledge base to draw from
  • Exposure to similar problems before
  • Resilient

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Big Knowledge Base to Draw From

  • Manage space in the working memory
  • Automate prior knowledge (declarative and procedural)
  • Increase retrieval strength

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Exposure to Similar Problems Before

  • “Expertise is largely domain-specific
  • Strategies for solving problems are topic-specific
  • Planned into the content sequence for that topic”

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  • ”Problem solving as a generic skill that pupils can transfer to multiple topics and sub-domains” (Ofsted, 2021)

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Resilient - “able to withstand or recover quickly from difficult conditions”�

Help

Help students recover from knocks

Create

Create success

Foster

Foster the right conditions

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"I get it when we do it in class but then I don't know what to do in tests...."

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A Curriculum Approach

Define the declarative and procedural knowledge

Secure high quality teaching and learning

Maintain retrieval strength

Equip students with generic skills

Provide time to practice

Develop a ”problem solving” curriculum

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Sequencing

  • Retrieval
  • Consolidation

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Lesson 1 – Activating prior knowledge

Diagnostic questions

Minimally different questions

”Not on autopilot”

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Work out the exterior angle

72

A

108

B

36

C

540

D

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Sometimes questions in exams aren’t as straightforward as the ones we give you in lessons so we want to make sure you thoroughly understand a topic. We’re going to ask you MINIMALLY DIFFERENT QUESTIONS to see if you what you really understand about this topic.

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These questions are designed to only be slightly different from one question to the next. You need to think about how these changes will effect the answer.

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We are going to use the following to guide us on this.

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You need to think :

REFLECT

EXPECT

CHECK

EXPLAIN

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Sometimes questions in exams aren’t as straightforward as the ones we give you in lessons so we want to make sure you thoroughly understand a topic. We’re going to ask you MINIMALLY DIFFERENT QUESTIONS to see if you what you really understand about this topic. You need to think REFLECT, EXPECT, CHECK, EXPLAIN

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Lesson 2 – Developing Generic Skills

Goal free problems

Chains of reasoning

Same surface, different deep

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Goal free problem

What can you work out using the information given?

Here is a regular 10 sided shape

In your exams you will see problems that you have never seen before, and we haven’t seen them either.

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We want you to be able to solve complex problems that require lots of steps.

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We’re going to start by giving you a GOAL-FREE PROBLEM

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(This is where we give you the basics of a question and try to imagine what kind of problems the examiner can create with that information)

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Goal free problem

Make up some questions about these and answer them.

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Same Surface Different Deep

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Lesson 3 – How far can we take it?

Increase the number of steps

01

Decrease the amount of scaffolding

02

Change the amount of autonomy

03

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Task

ABC is part of polygon. How many sides does it have?

 

A

B

C

D

Angle ABC and angle CBD are in the ratio 4:1.

CBA is part of a polygon. How many does it have?

1.

2.

3.

4.

If the size of the exterior angle is x and the polygon has y sides then how many sides would a polygon with exterior angle 2x have?

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Show that AB is parallel to CD

The angles in a triangle are in the ratio 1 : 2 : 3

Mr Hemsley says, ‘this is a right-angled triangle. Is he correct. Show your reasoning.

Work out the size of the smallest angle in the triangle.

 

5.

6.

7.

8.

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9.

10.

11.

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4mm

1mm

The polygons below are cogs in a watch mechanism. Approximately how many times will the small polygon revolve if the large polygon makes one complete revolution.

12.

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The problem solving curriculum

Skills develop over time

Provide scaffolding

Model using metacognitive processes

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The problem with a maths lesson…

Title

Worked examples

Practice

Problem solving/ exam style questions

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Reflection Questions

  • How do you support students to develop declarative knowledge?
  • How do you support students to develop conditional knowledge?
  • Can we “teach” problem solving?
  • What conditions are effective in facilitating problem solving?
  • Are my students effective problem solvers? Why?