SOLO in Mathematics Mitchell Howard Lincoln High School

Activity 1: Discuss for 2 min with the person next to you 1.Why did you come here tonight and 2.What do you hope to get from this talk?

Aims for my talk 1.A brief explanation of SOLO 2.Why SOLO in Mathematics? 3.SOLO Pedagogy 4.A look at LHS Maths dept journey

What this talk is not I’m not explicitly going to talk about the revised standards. You can see some excellent talks on these through the NZAMT site or by attending one of the CMA level 1 writing workshops.

Who is this Aussie? Mitchell Howard HOD at Lincoln High School B.Sc/B.Ed, M.Ed Lived in NZ for 3 years Previously taught in NSW, UK, ACT and Victoria.

Why am I giving this talk? Lincoln has a school wide SOLO focus Lincoln has been working with Pam Hook ‘Hooked on Thinking’ for the last 4 years. I’ve been in a group of 3 Maths teachers who have met with Pam roughly each term for the last 3 years.

Pam Hook sessions Former Science teacher now Education researcher and consultant. Responsible for the maps and symbols Doesn’t claim to have answers for Mathematics. Facilitates discussions. Puts our discussions in SOLO speak Maths dept not as progressed as other dept’s such as English, Science, Social Science

Thinking at Lincoln The focus is on ensuring students achieve deep learning outcomes and “learn how to learn”.

Activity 2: Describe map - SOLO Much like a spider diagram or brainstorm Students could do this on a template, or just sketch up in their books or on mini white boards or scrap paper

Activity 2: Describe map - SOLO Use the map to write what you know about SOLO? Then write a statement about what you think SOLO Taxonomy is SOLO Surname of HAN What this talk is supposed to be about The next passing FAD in education SOLO is the surname of a cool space guy from Star wars. It is also a word which is being a used a lot in education at the moment. Tonight I’m attending a course about it. We will come back to your statement soon

Everyday SOLO Language UNISTRUCTURAL MULTISTRUCTURAL RELATIONAL EXTENDED ABSTRACT PRESTRUCTURAL

Prestructural What does it mean? What do you know about SOLO? Err….. What??

Prestructural What does it mean? At the prestructural level of understanding, the student response shows they have missed the point of the new learning.

Unistructural What does it mean? What do you know about SOLO? Err….. It’s got some funny symbols!? !

Unistructural What does it mean? At the unistructural level, the learning outcome shows understanding of one aspect of the task, but this understanding is limited. For example, the student can label, name, define, identify, or follow a simple procedure.

Multistructural What does it mean? What do you know about SOLO? It’s a thinking taxonomy with funny symbols and a type of mark scheme.

Multistructural What does it mean? At the multistructural level, several aspects of the task are understood but their relationship to each other, and the whole is missed. For example, the student can list, define, describe, combine, match, or do algorithms.

Relational What does it mean? It’s a way of structuring your thinking. It follows on from having your ideas to being able to link your ideas together by explaining or comparing & contrasting them to show a greater understanding of a topic. Rubrics can be used to assess their level of achievement. What do you know about SOLO?

Relational What does it mean? At the relational level, the ideas are linked, and provide a coherent understanding of the whole. Student learning outcomes show evidence of comparison, causal thinking, classification, sequencing, analysis, part whole thinking, analogy, application and the formulation of questions.

Extended abstract What does it mean? It’s a way of structuring your thinking. It follows on from having your ideas to being able to link your ideas together by explaining or comparing & contrasting them to show a greater understanding of a topic. It then allows you to formulate your own prediction or generalisation, discussing the topic in question. I predict that if I use SOLO Taxonomy within my lessons, I will see an increase in Merits and Excellences as students learn to structure their answers better and they can transfer their knowledge to another context. What do you know about SOLO?

Extended abstract What does it mean? At the extended abstract level, understanding at the relational level is re-thought at a higher level of abstraction, it is transferred to another context. Student learning outcomes show prediction, generalisation, evaluation, theorizing, hypothesising, creation, and or reflection.

Self Assessment: So What Level do you think you were at with your initial statement about SOLO Taxonomy?

How do the symbols relate to NCEA? ACHIEVED MERIT EXCELLENCE

The Verbs

Activity 3: Apply the verbs to mathematical concepts With the people around you and using the verbs as a guide: –Think of some activities, concepts or topics in the junior curriculum (years 9 and 10) which might match up to the levels of thinking? –Discuss whether the verbs given are appropriate in Mathematics. –Can you suggest any other verbs that might work for Mathematics?

The Hattie and Brown Asttle example: Algebra patterns Given: How many sticks are needed for 3 houses? How many sticks are there for 5 houses? If 52 houses require 209 sticks, how many sticks do you need to be able to make 53 Houses? Make up a rule to count how many sticks are needed for any number of houses Houses123 Sticks59___

Why SOLO in Maths Problems worth solving require a much deeper level of thinking Gives students a framework to structure their thinking. Feedback: Where am I going? How am I going? Where to next? Students assess where their level of thinking is at and recognise what they have to do to progress their understanding. Can give structure to open ended tasks. Allows for differentiation. Key Competencies

Key competencies: Thinking: Makes explicit the structure of thinking Managing self: Self assessment see how to move from one level to the next. Using language images and text: Relating to others: peer assessment/tutoring Participating and contributing:

SOLO pedagogy Using the structure to plan units of work Using the terminology and referring to the symbols in class discussion. Using rubrics to differentiate for open ended tasks. Using SOLO maps Using SOLO maps with rubrics

Equivalent Fractions: Doing versus Understanding Pictures Algorithm –Double top and bottom –× or ÷ numerator & Denominator by same number Relate to +/- Fractions of different denominator Relate to ratio Algebraic Fractions

Fraction Tiles

I can’t find any combination of tiles that are equal to ½ I can find sets of tiles that have the same denominator (or are of the same colour) that add up to ½. I can find sets of tiles of different denominators that add to ½ I can explain how to find combinations of tiles (of different denominators) that add to ½ I can explain and relate fraction tiles to other representations of fractions such as Number sentences. I can write a general rule about the mathematics involved in this activity which makes it quicker and easier to do this kind of task with fractions that are not included on this set of fraction tiles.

The equation The Number pattern The Picture (graph) Using the terminology and referring to the symbols in class discussion: Level 1 - Understanding quadratic patterns and graphs The Context. (dot diagram or skateboard ramp etc)

The equation The Number pattern The Picture (graph) The Context. (dot diagram or skateboard ramp etc) Understanding quadratic patterns and graphs I have two answers for x when y=0 I have one positive and one negative answer The parabola cuts the x –axis twice (2 roots) I can only have a positive answer for the number of people Has an x squared Differences not the same Is a parabola Some of the dots form a square shape

The Picture (graph) The Numbers The Context A Guide for responses in Level 3 Statistics internals The Points on the graph are going down hill from left to right The gradient of my regression line is negative As one of my variables increases the other decreases My smoothed data looks non-linear But I have a high r squared smoothed data will tend to get increased r squared value

Gives structure to and differentiates for open ended tasks. Yr 9 Measurement project: Design back yard. Must have –A scale diagram –An area no greater than 0.4 hectare. –a pool or pond with a minimum capacity of 10 kilolitres –a raised garden bed which contains a minimum of 3 cubic metres of soil

NI have not shown how I arrived at my answer. My pools capacity is less than 10 kilolitres A My pool is a rectangular prism that holds at least 10 Kilolitres of water. MMy pool has sections which are of different depths OR I have used a more complicated shape* E The bottom of my pool has a section with a graduated depth. This means it is on a slope. E+I have made a generalised formula to calculate the capacity for shapes that I have created.

Graphic organisers designed to help students to structure their thinking prior to writing. They have associated rubrics which can be done as self, peer or teacher assessments. Some times they work in combination E.g. the ‘Compare contrast’ Map is done as two separate ‘Describe Maps’ put together. Probably more relevant for statistics than for much of pure Mathematics

We have used compare contrast for Statistics –Comparing two ways of organising data squares –Comparing two distributions Heart rate before and after exercise Recall before and after memory training Graphs –Two equations of graphs (see later) –Log v’s exponential –Step and leaf v’s boxplot –Parabola v’s Cubic

Data Squares

Observations Describe the shape –E.g. a straight line with some spikes or bumps which represented double ups Describe the spread –The tallest person is 20cm taller than the shortest –Most people were around the same height with a few students much taller than the rest Where is the middle? –There are two students who are in the middle and their height is 157cm

What I noticed 1.The heights didn’t always go up by the same amount. 2.The tallest person was a girl and the shortest person was a boy 3.The double ups (or clumps) of data seemed to occur towards the middle of the data What I wondered 1.What the graph would look like if I left gaps to show how spread the heights really were 2.What is the relationship between heights of year 9 girls and year 9 boys 3.How does the relations ship of heights of boys and girls change as they get to the end of school?

Analysing how parts contribute to the whole, and how each part functions PART WHOLE whole part if part missing then... part function is … PART WHOLE FILTERS 4 THINKING : SOLO : RELATIONAL © 2004 Hooked on Thinking

Deconstruction of the formula To gain an understanding of the turning point form of quadratic equations,. To develop skills on how to systematically play and test components of the formula to determine their purpose. So when students are faced with unfamiliar or complicated equations and graphs, they will have the confidence to have a play and make a start. To understand something it’s a fairly natural behaviour to pull it apart to see how it works. Hopefully we can put it back together with a better understanding. E.g. History of Medicine, car mechanics, toys, transistor radios etc.

Student AStudent BStudent CStudent D y=2(x-3) 2 +1y = -2(x+3) 2 -1y = ½(x+3) 2 -1y = -½(x-3) 2 +1 y=2(x-3) 2 y = -2(x+3) 2 y = ½(x+3) 2 y = -½(x-3) 2 y=(x-3) 2 +1y = -(x+3) 2 -1y = (x+3) 2 -1y = -(x-3) 2 +1 y=2(x) 2 +1y = -2(x) 2 -1y = ½(x) 2 -1y = -½(x) 2 +1 Task 1:Draw each of the 4 graphs and describe changes in a)The formula b)The graph

Task 2: Pair up with each of the other members of your group and compare/contrast your formulas and graphs

Task 3: Using your work from Task’s 1 and 2, try to draw the parabola’s for the following equations without the use of your graphics calculator or Graphmatica. Do your first predictions in one colour, and then use your calculator or Graphmatica to check your answer. Make any corrections in another colour.

Analysing how parts contribute to the whole, and how each part functions PART WHOLE whole part if part missing then... part function is … PART WHOLE FILTERS 4 THINKING : SOLO : RELATIONAL © 2004 Hooked on Thinking

Analysing by determining causes and predicting effects CAUSE and EFFECT possible cause possible cause possible cause event CAUSE and EFFECT possible cause possible effect possible effect possible effect possible effect FILTERS 4 THINKING : SOLO : RELATIONAL © 2004 Hooked on Thinking

Possible events I use a linear model for my Forecasting My ISE’s are increasing over time I use a exponential model for my Forecasting I use a polynomial model for my Forecasting I have an outlier in my data I have centred moving mean

Linear model All types of data (exp, polynomial etc) when viewed over a short time period will look linear. Steady (consistent) increase/decrease Piece wise Smoothed I start at a particular point on y axis Will have a reasonably high r 2 due to generation from smoothed data Will continue with a steady increase or decrease in the future Possibly several straight lines Cause and Effect

Summary - Basic SOLO is an tool for measuring levels of thinking. The Hot Maps are a graphic organiser to help structure thinking. The Maps are a precursor to writing. You basically have to write to demonstrate thinking. Rubrics are used to assess the writing, not the maps.

Lincoln Maths dept Journey 1.Classifying course content 2.Making rubrics 3.Planning units of work 4.Using our first maps 5.Writing rubrics for maps 6.Statistics – yr 11, 9 and 13 7.Fraction Walls 8.Problem solving

Thanks for listening