Teaching As Inquiry in Mathematics NZAMT 2011 Louise Addison laddison@mcauleyhigh.school.nz Ideas Background - My class My plan for teaching - Key Ideas in Maths vs Essence Statement Teaching as Inquiry - School goal “To promote student ownership of learning through the Teaching as Inquiry cycle” Focusing Inquiry - Development of Teaching as Inquiry goal Teaching Inquiry - It’s not what you do it’s the way that you do it - Three obligations - Cause, be and do - Teaching for tomorrow Learning Inquiry Resources Flip Video Key ideas in Maths Posters Plan for the year Appraisal system at McAuley High School Focusing Inquiry Sheet Teaching Inquiry Sheet Teaching considerations Observation 1 (Argumentation and Proof) - EGG technique - Number Proofs - Power Point Learning to learn - Think Board - Poster - Wait time - What do good learners do ‘Revision’ - Algebra assessment Making connections (Structure and Generalisation) - Relationships tasks - Geogebra files Observation form 2 - Quadratic modelling Work in progress Teaching as Inquiry 1 KEY IDEAS IN MATHEMATICS 2 1 5 7 2 4 6 3 5 4 1-2 Change and variation Students uncover stories in which variation is omnipresent. 6 SSG Structure and generalisation Students unlock stories using models, abstractions, and representations. Weeks Argumentation and proof Students tell stories using evidence and reasoning. Term Theme Statistical Reasoning Involves identifying problems that can be explored by the use of appropriate data, designing investigations, collecting data, exploring and using patterns and relationships in data, solving problems and communicating findings. Statistics also involves interpreting statistical information, evaluating databased arguments, and dealing with uncertainty and variation. Algebraic Reasoning Number involves calculating and estimating, using appropriate mental, written, or machine calculation methods in flexible ways. It also involves knowing when it is appropriate to use estimation and being able to discern whether results are reasonable. Algebra involves generalising and representing the patterns and relationships found in numbers, shapes, and measures. Geometrical Reasoning Geometry involves recognising and using the properties and symmetries of shapes and describing position and movement. Measurement involves quantifying the attributes of objects, using appropriate units and instruments. It also involves predicting and calculating rates of change. AOs Big Ideas Contexts Learning to learn Assessment - plan and conduct surveys and experiments using the statistical enquiry cycle determining appropriate variables, cleaning data, using multiple displays, and re-categorising data to find patterns, variations, in multivariate data sets, comparing sample distributions visually, using measures of centre, spread, and proportion, presenting a report of findings. - plan and conduct investigations using the statistical enquiry cycle justifying the variables used, identifying and communicating features in context (differences within and between distributions), using multiple displays, making informal inferences about populations from sample data, justifying findings, using displays and measures. PPDAC Variation Within and between distributions Sampling effects Variables and measures Informal inference Us and them Census @ School – finger length Birth weight data Baby names Titanic Kiwi Kapers The eyes have it Using data to verify Worry questions Use of representations Design process Justify findings 1.10 Investigate a given multivariate data set using the statistical enquiry cycle (4) - Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance. - Investigate situations that involve elements of chance: comparing discrete theoretical distributions and experimental distributions, appreciating the role of sample size, calculating probabilities in discrete situations. Distribution Sample size Compare expected and experimental Simulation Hospital Problem Games Birth numbers Baby names Questioning Using data Compare and contrast Theoretical reasoning 1.13 Investigate a situation involving elements of chance (3) - find optimal solutions, using numerical approaches - solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns - relate graphs, tables, and equations to linear, quadratic, and simple exponential relationships found in number and spatial patterns - relate rate of change to the gradient of a graph. Connecting representations Parameters Starting points Island Paradise Geogebra tasks Create the picture Growth and motion modelling Use of models Use of technology Visualisation Making connections Abstraction 1.3 Investigate relationships between tables, equations and graphs (4) Generalise properties of operations Algebraic identities Expressions and Equations EGG technique Always, sometimes, never Squares and derivations 26 x 24 Generalisation Turn it into something I can solve Using a concrete example Using models 1.2 Apply algebraic methods in solving problems (4) Vitruvian Man Ohakune Carrot Body measurements Accuracy Using prior knowledge 1.5 Apply measurement in solving problems (3) (1.1 Apply numeric reasoning in solving problems (4)) 3-D Triangles Proofs Deduction Proof Use of formula (2 out of 3) 1.6 Apply geometric reasoning in solving problems (4) - generalise the properties of operations with fractional numbers and integers - generalise the properties of operations with rational numbers including the properties of exponents - form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns. - convert between metric units, using decimals - deduce and use formulae to find the perimeters and areas of polygons, and volumes of prisms - find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders - apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures - calculate volumes, including prisms, pyramids, cones, and spheres, using formulae. - deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties - recognise when shapes are similar and use proportional reasoning to find an unknown length - use trigonometric ratios and Pythagoras’ Theorem in two dimensions - deduce and apply the angle properties related to circles. Level of precision Metric system Area / Volume Derivation of formula Optimisation Modelling Similarity Angle rules Geometric Proofs Loci Proportional reasoning Trig ratios and PT Revision 3 4 5 Focusing Inquiry What is important (and therefore worth spending time on), given where my students are at? The focus for this teaching as inquiry cycle is going to be developing students understanding of different algebraic representations and the links between them. This is traditionally a very challenging area for students in mathematics and I would like to explore how the use of technological and other visual representations can best support the development of this understanding. Baseline data: Curriculum Level Algebra Topic Test Graphs Topic Test Algebra Exam Graphs Exam 3 0 0 3 3 4 3 10 11 8 5 3 7 5 5 6 13 2 0 3 The above table is my class’ results from Year 10. From this it is clear most students are about levels 4/5 of the curriculum. They also struggled to replicate class test results in the exam – this is important as both the standards for this topic are externals. McAuley High School National results Standard Title N A M E N A M E 90147 Use straightforward algebraic methods and solve equations 19.6 45.7 34.8 0 33.9 31.5 28.3 6.2 90148 Sketch and interpret graphs 23.9 56.5 17.4 2.2 23.9 43.2 23.6 9.3 Highlighted areas show where McAuley students do better than the national average, however this is mainly because only our top third of students sit the exam whereas nationally about the top two thirds would be sitting. Students were also be given the practice examination at the start of the unit next term so that a baseline can be established – the results were Not Achieved for all students in both papers. The highest number of questions students had correct was 2 out of the 16 in the procedures paper and 3 out of the 18 questions in the graphs paper. The majority of the class (15 out of 18) had no questions correct in either paper. This shows students have already learned: From the Year 10 data it is clear that students have a good procedural understanding of basic Algebra skills, however they are not as proficient at demonstrating this understanding in an exam situation. The graphs understanding of the majority of the class is at Level 4 of the curriculum currently which suggests they have good understanding of linear relationships only. However this understanding was not demonstrated in the pre-test and thus the teaching of linear understandings will also be necessary within this topic. These students need to learn: A key understanding for the new standard in algebra is linking the three different representations of graphs, equations and tables. In order to move to Level 6 of the curriculum work on quadratic and exponential relationships and equations is needed. My goal for this inquiry cycle is: To investigate the use of teaching strategies that support the development of students’ understanding of quadratic and exponential algebraic representations, in order for them to meet the requirements of Merit and Excellence in the external examination. 6 Teaching Inquiry What strategies (evidence-based) are most likely to help my students learn this? In this teaching inquiry, the teacher uses evidence from research and from their own past practice and that of colleagues to plan teaching and learning opportunities aimed at achieving the outcomes prioritised in the focusing inquiry. Observation 1: I was very happy with this lesson in the way it engaged all students to work towards a conceptual understanding of algebra. This process was very new to the students who are more used to a textbook approach to algebra. Use of the EGG technique enabled all students to access the excellence requirements of this topic and enabled a future focus on what mathematicians actually do. This was quite a teacher directed task and whilst most students were very engaged, a couple did need reminding to participate fully. Over this topic I would like to develop student initiated reflection on ownership of their learning by using a greater variety of assessment for learning strategies. Key information / research about this topic: Mathematics BES E-learning research from at Auckland University Paper written for e-learning maths Masters level course “how does your technological knowledge effect how you use technology as a pedagogical tool” Te Kuaka – being the agent to cause learning Use of instrumentation and instrumentalisation theory Feedback article from PD Teaching and learning opportunities I intend to trial: Use of geogebra to build representational thinking Use of algebraic proofs of number properties Links to learning to learn for mathematics with student evaluations of their development in these Use of teaching model for linear relationships to build a structure for the quadratic and exponential understandings needed 7 1. Effective Teaching is characterised by a commitment to three obligations: - the obligation to cause successful learning - the obligation to cause greater interest in the subject, - the obligation to cause greater confidence, feelings of self-efficacy and intellectual direction in learners Wiggins, Grant. (2010). What’s my job? In Robert Marzano (Ed.). On Excellence in Teaching. Bloomington, IN: Solution Tree Press. p 11. 2. The Impact of Cause on Teacher Qualities and Actions Aitken, G. (2011). Excellence in Teaching in a Faculty of Education. IN: Te Kuaka. pp 4-6. 3. Teaching for tomorrow by Ted McCain 1. Resist the temptation to “tell” 2. Stop teaching decontextualised content 3. Stop giving students the final product of our thinking 4. Make a shift in our thinking – problems first, teaching second 5. Progressively withdraw from helping students (Be less helpful). 6. Re-evaluate evaluation 8 Lesson plan 1 – Algebraic generalisation This lesson explores the use of algebraic generalisation to generate algebraic identities. Key Competencies Number and Algebra Level 5 Generalise the properties of operations with fractional numbers and integers. Number and Algebra Level 6 Generalise the properties of operations with rational numbers, including the properties of exponents. To explore algebraic generalisations of number strategies Task Introducing Exploring Assessment Extending Working like a mathematician – Andrew Wiles example Use of example to explain EGG technique of explain, give other examples and generalise x + x + x + x = 4x Working in teams of 4 to explore different situations of Task One Summarising key ideas from Task One – complete Task Two for homework (link to consecutive numbers) Proof “Show that the sum of two consecutive numbers is always odd” “Show that the sum of three consecutive numbers is always divisible by three” Success Criteria Generalising and representing the patterns and relationships found in numbers. Aim Essence Statement Generalise ideas (GI) Use generalisations (WS) Use words and symbols to describe patterns and generalisations (WS) Use appropriate vocabulary to explain ideas (V) Compare and contrast ideas (CC) Critically reflect (CR) Participate actively in a collaborative team or community (CT) Curriculum Objectives Mathematics Level 7 Manipulate rational, exponential and logarithmic algebraic expressions. I can generalise from a number strategy I can explain why an algebraic identity is always true I can use identities to manipulate algebraic expressions I know key algebra vocabulary and recording conventions Pedagogy Making connections to prior learning and experience: Sometimes true / always true / never true Focus on why as well as how Accept all answers, lead academic discussion Learning Use of language to support explanation: variable, co-efficient, LHS, RHS Use of algebraic recording conventions: = sign, identity symbol , no times sign, coefficient first, letters of alphabet Learning to learn Use of examples and linking strategies Use of mathematical language Linking number and algebra Facilitating shared learning: Observation of groups to explore current understandings and look for opportunities to develop further Linking of factorised expression to expanded version. Development of ‘rules’ in mathematics Working in a group Encouraging reflective thought and action / Enhancing relevance of new learning: Supporting students to make links, sharing of light bulb moments How is the variable being used? Link to other representations – what would a graph of each ‘side’ look like? Nature of proof “Show” Use of key ideas The difference between proof and example Students can work independently to derive the common algebraic identities Use of correct vocabulary Simplifying, expanding and factorising tasks Link to 1.2 assessment, expanding, factorising and proofs 9