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Increase responsibility for thinking and participation

Discussion --> Argument

Explainer --> Listener

Expectations placed on explainers and listeners on themselves and others

Think about why the approach we are taking facilitates the learning of students

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Green --> Typical classroom where teaching is transference

Move up through discussion contexts

What are students doing in explaining role, clarify, defend justify solutions

Students should be comparing and contrasting solutions, think about questions, ask them

Students should disagree and challenge

Rather than recall --> comparing and contrast - higher order thinking

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Effective Teaching Practices - NCTM (2014)

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Effective Teaching Practices - NCTM (2014)

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Use this to reflect on engagement to mathematical enquiry

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Development of solution

Problem posing and solving

Students formulate their own questions

Investigation --> multiple pathways

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Stillman, G. A., & Brown, J. P. (2019). Lines of inquiry in mathematical modelling research in education (p. 259). Springer Nature.

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A statement that appears reasonable but the truth of which has not been established

When we are investigation we begin with making conjectures

The Conjecturing Cycle - Mason, Burton and Stacy (2010, p.69)

Focusing on similarities = formulating a conjecture

Emphasis on communication of mathematical ideas using appropriate language

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Specialising: Looking at specific cases

Generalising: Moving from specific instances to making statements about a wide class of cases

DO NOT GENERALISE TOO EARLY

Reference at least three specific cases before a relationship between cases can be considered

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VCAA (2002) mathematical thinking identification:

1. Make conjectures

2. Test the truth of conjectures

3. Identifying counter examples

4. Generalising

Look at figures made by joining the midpoints of sides of various polygons

1. Investigate the nature (type of polygon)

Polygon with least number of sides = Triangle

Scalene (no sides equal)

Isosceles (two sides equal)

Equilateral (all sides equal)

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In all polygons, the midpoint figure is the same type of polygon (same number of sides as original polygon)

Therefore the midpoint figure created by joining the midpoints of sides of a convex polygon will have the same number of sides itself

Triangle ABC produces a midpoint figure DEF which is also a triangle with 3 sides

Quadrilateral ABCD produces a midpoint figure EGFH which is also a quadrilateral with four sides

Pentagon ABCDE produces FGHIJ a polygon with five sides

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1. CONJECTURE: Have you noted down observations and formulated an idea?

Note down observations

2. CHECKING: Does the conjecture hold for all known cases based on evidence?

Check the diagrams

3. TEST: Disprove the conjecture

Polygons are all convex --> midpoint figures for crossed and concave polygons

Make prediction on 6 7 8 side polygons

4. Modify conjecture to test of crossed or concave polygons

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Two line segments joining midpoint could join at a straight angle reducing the number of sides of the midpoint figure by one each time this occurred

One line segment rather than two, midpoint polygon may have fewer sides than predicted

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Single counter example is all that is needed to disprove a conjecture

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Single counter example is all that is needed to disprove a conjecture

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Every convex polygon has N sides

Each sides has one midpoint

Hence there are n midpoints

if N midpoints are joined in line segments, the convex polygon constructs a midpoint figure with n vertices

All segments joining must lie inside the convex polygon

No two of them can join at a straight angle

Therefore: The midpoint has n sides, which is the same number of sides as the original polygon

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MIDPOINT INVESTIGATION: MathsWorlds Yr 8 Texbook (pg. 292) 

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MIDPOINT INVESTIGATION: MathsWorlds Yr 8 Texbook (pg. 292) 