Tasks and learning mathematics Anne Watson University of Oxford DfE 2010

Nature of evidence Published research, usually comparative studies Review of research on pedagogy to Smith report Key Understandings in Mathematics Learning (Nuffield) 20 years of teaching and curriculum development with teachers, teacher educators worldwide Research into the practice of exceptional teachers working in the English context International knowledge about task design

Being careful with research findings Do evaluation strategies match curriculum aims? Innovation v. roll out Cultural context Research shows ‘what is good/possible’, not ‘what is best’ or ‘what will always work’

International Congress of Mathematics Education (ICME) task workshop Task types known to develop knowledge, understanding and application: –Draw on students’ perceptions and past experience, and offer extensions –Afford conjectures and feedback on effects of actions –Constrain choices to fit models, images & alternative representations of conventional mathematics –Exploit students’ search for familiarity, similarity/difference, and effect

Example £2500 is invested at 2% per annum. What is it worth after 2 years, 10 years, 7 years, 27 years?

Find the number mid-way between 28 and and and and and and and and 46

Example

You have a sheet of black card 150 cm. x 150 cm. You have to make a Hallowe’en witch hat. What possible heights can you make?

Concepts Puzzles, problems, situations which can be understood, but not resolved, using current knowledge Classifying mathematical objects - new classifications Interpreting multiple representations - new notations and new methods Evaluating mathematical statements - truth, usefulness, domain of applicability

Advantages Flexible, adaptable, knowledge Can construct and reconstruct meaning Misconceptions may arise for usual reasons, but are resolved through microtasks Mathematics with meaning Students do better in test questions that require adaptation than …

Shortcomings Takes time (coverage) Takes time to establish appropriate habits Memory maybe, and fluency Teacher knowledge is challenged (secondary)

Applying concepts problem to read and understand decide whether to use statistical, algebraic, logical or ad hoc methods identify and select variables coordinate mental, graphical, numerical, representations select facts, operations and functions to apply … and how to apply them apply appropriate knowledge of situations and operations to interpret

Advantages Realistic application of known mathematics Creates a need for new mathematical methods and knowledge Draws on everyday reasoning in mathematical contexts, so more students are included New opportunities with every new task - less chance for students to ‘lose track’ Habits of identifying variables and deciding methods Good preparation for workplace Students do better with unfamiliar test items and multistage problems than …

Shortcomings need to understand both context and maths need experience of making good and bad choices ad hoc, numerical or visual approaches dominate purpose can be confused: to understand the situation better, or the maths, or to learn new mathematical ideas?

Procedures for fluency: repetition and not much change for understanding: careful change of variables; reflect on outcomes for retention: layout, visual, representations, pattern and rhythm to challenge usual misconceptions/errors: special cases, comparing similar cases, focus on meaning, correction

Advantages Can anticipate answers and difficulties Can generalise and extend methods to more complex situations Can develop algorithmic understanding Automatisation of key procedures A page of ticks boosts confidence Can be automated (online worksheets with good quality feedback and adaptation) Understood by society (parents, outsiders etc.)

Shortcomings Misapplication of methods; repetition of errors; hard to adapt to unfamiliar situations; difficulty of multistage problems Hard to recognise when to apply methods Textbooks need research-based principles (China) Associated with dislike of subject and boredom It is what machines can do Need to reflect on answers in order to fully understand method Confusion between purpose: fluency or conceptual understanding or retained knowledge Does not prepare students for higher study Traditional use does not develop the potential advantages

Learning from effective teachers ( Microtasks more important than task-types: Exemplifying and specialising Completing, deleting, correcting Comparing, sorting, organising Varying, reversing Conjecturing, generalising Explaining, justifying, convincing, refuting, proving Representing Stop press GCSE results …

Examples of microtasks Is it always, sometimes, never true? What do you get if you change … to …? Make up three examples like this Make a connected chain from … What is the same and what is different about…? Provide the missing steps in … Can you swap the property and the definition and define the same objects? Of what is this a special case? Explain the role of … in … What is wrong with …? Verify that … means the same as … What cases does/doesn’t this work with? Provide missing steps in ‘if … then …because …’ arguments When is … a good notation for …? When is … a good method for …? What has to be included in … to make …? Find a relationship between … and …

Example