Inquiry and Communication in Maths and Stats: Training students to be self-starters who persevere and communicate their thinking. Presented by Jeremy Brocklehurst

Goals of workshop today: “Open-ended holistic task” : - What are they exactly? - What makes them good vs a waste of time? - Why ?? - Does this lead to bigger questions about what Mathematics is ? - How to help all students with them. Warning: I have some ideas but more questions than answers! This is where you come in!

Success criteria for this workshop!

Those open-ended problems everyone just loves !! .

How many coins could we fit on the floor of this classroom?

How many coins could we fit on the floor of this classroom? How could we get the greatest possible money value using a single layer of coins?

What’s the minimum length of fencing needed to enclose a paddock holding 50 cows if each cow requires 25m2 to graze?

The quadrilateral ABCD has vertices A (0, 0), B (1, 4), C (5, 3), and D (7, 1). P, Q, R, and S are the midpoints of the sides of the quadrilateral. If the points P, Q, R, and S are joined, a new quadrilateral is formed. What type of quadrilateral is PQRS? You must show the co-ordinate geometry methods you used to get your answer.

The following graphs are distributions of ages in two populations The following graphs are distributions of ages in two populations. The United States of America (based on US census data) and Facebook users: Describe features of the distributions comparatively. Aim to make at least three statements. Support your statements with statistical evidence.

Questions to discuss: Why give students open-ended ‘holistic’ tasks? Is there any research around this or is it just a “fad”? Finish by 10:12

Questions to discuss: What do students find most challenging about open-ended Maths or Stats tasks? Finish by 10:15

Questions to discuss: What is required for a person to become engaged and engrossed in them? Finish by 10:20

Questions to discuss: What about students who insist on being “spoon fed” ? Finish by 10:25

Questions to discuss: Other issues? Finish by 10:27

Writing rubrics can be useful

What the OSEM writing frame means: O stands for Obvious observations Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence Support each observation with numbers or calculations. M is for Meaning: So what? Write down what each of your points tells you about the original problem/situation. What else could we find out?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: So what? Write down what each of your points tells you about the original problem/situation. What else could we find out?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: So what? Write down what each of your points tells you about the original problem/situation. What else could we find out?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: So what? Write down what each of your points tells you about the original problem/situation. What else could we find out?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: Make sense of it. So what?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: Make sense of it. So what? Write down what each of your points tells you about the problem/situation.

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: Make sense of it. So what? Write down what each of your points tells you about the problem/situation. Why is this relevant?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: Make sense of it. So what? Write down what each of your points tells you about the problem/situation. Why is this relevant? What else could we find out?

What the OSEM writing frame means: O stands for Obvious observations: Write down one thing you notice, then leave a gap, and write down something else you notice. Try to get 3 obvious things. S is for Specific: Under each obvious observation, explain what you mean so that another person would get it. E is for Evidence: Support each observation with numbers or calculations. M is for Meaning: Make sense of it. So what? Write down what each of your points tells you about the problem/situation. Why is this relevant? What else could we find out? Towards Merit Towards Excellence

Examples of student work…

Observe (look for obvious) %

Specific with Evidence (numbers) %

% Meaning:

% Meaning: So what?

Meaning: So what? What does this tell us about the situation? % Meaning: So what? What does this tell us about the situation?

In the Level 1 Multivariate Data standard (1 In the Level 1 Multivariate Data standard (1.10), my students found OSEM a useful guide when writing their conclusions:

1.10 Multivariate – Conclusion Obvious

1.10 Multivariate – Conclusion Obvious (Make call: Yes/No)

1.10 Multivariate – Conclusion Obvious Specific

1.10 Multivariate – Conclusion Obvious Specific (about what? – popns / variable)

1.10 Multivariate – Conclusion Obvious Specific Evidence

1.10 Multivariate – Conclusion Obvious Specific Evidence (justify, support with stats) Meaning

1.10 Multivariate – Conclusion Obvious Specific Evidence (justify, support with stats) Meaning (so what?) Context (does it make sense etc.? Sampling variability

1.10 Multivariate – Conclusion Obvious Specific Evidence Meaning

1.10 Multivariate – Conclusion Obvious (Make call: Yes/No) Specific (about what? – popns / variable) Evidence (justify, support with stats) Meaning (so what?) Context (does it make sense etc.?) Sampling variability

1.10 Multivariate – Conclusion

1.10 Multivariate – Conclusion

And even in Level 2 Coordinate Geometry

And even in Level 2 Coordinate Geometry Obvious observations Specific (labelling) Evidence (calculations) Meaning (so what?) (Relate back to problem)

2013 Level 2 Probability external Compare and contrast the two distributions. You should discuss shape, centre and spread in relation to the context.

Finish by 10:40 Reference for SOLO Taxonomy symbols: Pam Hook: http://hooked-on-thinking.com  

Algebra in Years 9 and 10 Key ideas in teaching it: Start with meaningful problems that involve either: The students themselves and can be acted out. Something that might interest them (saving money for something, get to know common interests of class members). Students creating their own problems is a powerful tool to engage and motivate learning.

Important messages for students to get about Maths : Maths is like fitness or strength, not like height. The Maths classroom should be a place to try things . Mistakes are good! When you realise your current way of understanding something is wrong… How do you feel? This is when you learn the most! No pain, no gain!

Important messages for students to get about Maths : Good mathematicians Are not born good at Maths. They train and persevere! Look for patterns and enjoy discovering them Write down what they notice and any ideas. Represent them using words, pictures, graphs. Break problems into small steps and write these down.

Creation! The power of creating your own maths: Students creating their own problems is a powerful tool to engage and motivate learning.

Number Crunching Machine What’s my rule? Open eXcel document to play game. Also handouts that go with it.

Play “What’s my rule”. Play “What’s my rule”. Discuss strategies – organise in a table to look for patterns. Play again. Make up own. Represent on a graph.

Handout has tables and grids.

Reasoning! Convince a sceptic that your way works. Example of types of questions we want students to be able to answer with patterns:

Teaching proof – why does my rule work? Question: How do you know that your rule will work for any case? Which answer below is most convincing? Why?

Teaching proof – why does my rule work? Question: How do you know that your rule will work for any case? Which answer below is most convincing? Why?

Teaching proof – why does my rule work? Question: How do you know that your rule will work for any case? Which answer below is most convincing? Why?

Our scheme… Basic scheme: topics, essential content and assessment dates. Teacher’s guide: Detail including links to curriculum and to good resources and rich tasks to choose from.

1. Basic Scheme

2. Teacher’s Guide (bank of resources and links to NZ Curriculum)