Deep Progress in Mathematics: making a difference Anne Watson Stirling, March 2007

The four squares problem

Self-awareness of progress What do you know now that you didn’t know before? What can you do now that you wouldn’t have done before?

Sara’s teaching input Personal organisation and emotional attachment Triggering recall Simplifying Beyond 3, 4 and 5 Beyond whole numbers This relates to … Recognising shifts of focus Expressing in shape, number, symbols Access ‘at their own level’ but outcome well beyond original ‘level’ Going on and on …

Improving attainment in mathematics project ten teachers who set out to make a difference very little obvious common practice common underlying principles

Improving Attainment in Mathematics Project Aim: to develop students’ habits of mind so they become better learners Deep progress means:  Learning more mathematics  Becoming better learners of mathematics  Feeling better about learning mathematics

Project beliefs That all can learn mathematics That some students ‘don’t’ rather than ‘can’t’ That focusing on mathematical thinking is an appropriate way to approach improvement

What is mathematical thinking? Thinking hard in mathematics lessons What else …… ?

Beliefs Right to access mathematics All students have the right to, and are capable of, full engagement with the subject. Development of reasoning and thinking All students are entitled to learn maths in ways which develop thinking and confidence in problem-solving Rights and responsibilities as citizens All students are entitled to have access to the maths necessary to function in society, beyond the minimal

…more beliefs Learners’ identity Learners can change their goal from ‘to finish’ or ‘to fit in’ to ‘to learn’ Taking account of reality Exam results are important; changing habits is hard Maths as a source of self-esteem Success in maths can be a source of self- esteem and empowerment

Similar principles; different enactment What do you do if a student does not bring a pencil to class? Do you allow students to choose methods? Do you repeat what students say? Do you give written work? How do you use/develop self-esteem of students?

Giving time to think and learn Timing for learning; not just for coverage  e.g. spending three weeks on differences

Visualising All students use a range of senses to engage with mathematics: visual, tactile and physical experiences, and visual imagination

2 3 ? e.g.

Giving choice Teachers expect as much self-direction from students with lower achievement as from any other group e.g. choice of order to do tasks; of examples to work through; of ways to express work

Dealing with complexity Students can learn rigorous mathematics if rigour is discussed explicitly  What is a good mathematical reason? Students can learn complex mathematics if there is discussion and design to help them sort out complexity  Careful use of variation

Use of pattern e.g. 17 – 9 = 27 – 9 = 37 – 9 = …

Developing proficiencies Looking for patterns is natural  so can I present concepts using patterns?  so can I control variables so the ideas are easy to see? Matching my ideas up to other people’s is natural  so can I use matching different perceptions in lessons? Creating own examples is a natural exploration method  so can learners’ own examples be incorporated into lessons? etc……

Establishing working habits It takes time, persistence and imaginative methods to alter behaviour. Disrupt old expectations (including the teacher’s).  boardroom discussions  multiplication worksheets

Interacting and responding verbally Students participate in the creation of mathematics in the classroom  e.g. choose numbers, letters, examples; start from what they already say about a topic

Shifts for teachers They can’t ….. They don’t ….. They don’t, so how can I give opportunities and support so that they do …..

Success Project students did at least as well as comparison groups in standard test questions, and significantly better in questions which required some adaptation of original thought. In addition they were more willing and able to engage effortfully with non-routine work and extended explorations.

Knowing that students have been learning more To what questions can 48 be an answer? Give me an example of a four-sided shape...

Rebecca’s story 2.40 p.m. Thursday in the final week of term There has been a fight at lunchtime This is a bottom set The TA is attached to one student only She is being observed by her tutor They ‘ should ’ be doing functional/everyday/relevant maths She has been told not to bother to teach them simultaneous equations They know it is not on their syllabus ….

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European Declaration of Human Rights Article 5: No one shall be deprived of his liberty save in the following cases … :  (d) the detention of a minor by lawful order for the purpose of educational supervision Article 9: (1) everyone has the right to freedom of thought Article 10: (1) everyone has the right to freedom of expression Article 14: … without discrimination on any ground such as sex, race, colour, language, religion, political or other opinion, national or social origin, association with a national minority, property, birth or other status

Pocket PAL: Building Learning in Mathematics Prestage, DeGeest and Watson (Continuum) Deep Progress in Mathematics Watson, De Geest & Prestage: ATM website (MT157) or my website: Raising Achievement in Secondary Mathematics Watson (Open University Press) Institute of Mathematics Pedagogy,July/Aug 2007: Institute of Mathematics Pedagogy,July/Aug 2007: