Mathematics Teaching and High Leverage Practices

Mathematics Teaching and High Leverage Practices Seán Delaney, PhD Marino Institute of Education Notes from Presentation 1. Mathematics education at primary and post-primary levels has traditionally been quite segregated with teachers at one level unsure about what happens at the other. 2.The bridging framework developed to support Project Maths is a helpful initiative that helps to "bridge" the gap that has existed between the two levels and it identifies how mathematical content at one level is linked to content at the other level. 3. Perhaps more focus could be put on the skills or mathematical practices that students are expected to develop at each level. Here is my attempt to match the practices at each level and to relate them to the mathematical practices outlined in the U.S. Common Core curriculum. Primary - Skills

Post-Primary - Objectives

Understanding and recalling Implementing

Conceptual understanding Procedural fluency

Applying and problem solving

Strategic competence

Reasoning

Adaptive reasoning

United States - Practices (Processes and proficiencies) Attend to precision Use appropriate tools strategically Look for and express regularity in repeated reasoning Make sense of problems and persevere in solving them Reason abstractly and quantitatively Look for and make use of structure

Productive disposition Integrating and connecting Communicating and expressing

Model with mathematics Construct viable arguments and critique the reasoning of others Attend to precision

4. An important question for teachers at both levels to answer is what teachers want students to be able to do mathematically. Is mathematics mainly about memorising mathematical facts or is it about thinking mathematically? Both matter, but traditionally the former has been emphasised in schools more than the latter. 5. Textbooks don't help - most have a heavy emphasis on routine, repetitive drill and predictable operations. Preferable would be more emphasis on problems that require students to think about how to solve them and which operations to use, that allow students to do a small or large amount of 1

the problem according to their knowledge of the topic, that provides interesting material for subsequent discussion, that tap into more than one mathematical topic, that focus on quality of student engagement with the mathematics rather than the quantity and that allow students to learn mathematics through the problems. Mathematics problems in this "replacement unit" have some of these features: (http://seandelaney.com/wp-content/uploads/2013/11/PDSTArea5Maths-Pupils1.pdf). Magdalene Lampert describes such an approach in her book Teaching Problems and the Problems of Teaching. 6. Mathematics teaching makes demands on teachers' mathematical knowledge over and above the complexity of the mathematics involved. Measures developed by Deborah Ball and her team (of which I was a member) - http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf - attempt to identify and tap into such mathematical knowledge for teaching. 7. Maths lessons within Ireland, and within other countries, tend to resemble each other more than they resemble lessons in other countries. According to Stigler and Hiebert (The Teaching Gap), this is because teaching is a cultural activity and most of us teach as we were taught. This is typically a stronger influence on teachers than initial teacher education. 8. In response to this and other problems with initial teacher education, high leverage practices have been identified in order to help beginning teachers gain mastery of teaching. A high leverage practice is a teaching practice “in which the proficient enactment by a teacher is likely to lead to comparatively large advances in student learning.” When it is done well it gives “teachers a lot of capacity in their work.” (Ball et al, 2009). 9. The criteria used for identifying high leverage practices are that they:        

Support work that is central to mathematics Help to improve the learning and achievement of all students Are done frequently when teaching mathematics Apply across different approaches to teaching mathematics Can be articulated and taught Are accessible to learners of teaching Can be revisited in increasingly sophisticated and integrated acts of teaching Are able to be practised by beginners in school placement settings (Ball et al, 2009).

10. Although the list is an evolving and fluid one, here are high leverage practices that have currently been identified. They are from the website www.teachingworks.com. The ones highlighted in yellow are the ones that are explicitly focused on in the Teaching and Learning module in Marino Institute of Education and were discussed in the workshop and in relation to the videos shown. 1. Making content explicit through explanation, modelling, representations, and examples 2. Leading a whole-class discussion 3. Eliciting and interpreting individual students’ thinking 4. Establishing norms and routines for classroom discourse central to the subject-matter domain 5. Recognizing particular common patterns of student thinking in a subject-matter domain 2

6. Identifying and implementing an instructional response to common patterns of student thinking 7. Teaching a lesson or segment of instruction 8. Implementing organizational routines, procedures, and strategies to support a learning environment 9. Setting up and managing small group work 10. Engaging in strategic relationship-building conversations with students 11. Setting long- and short-term learning goals for students referenced to external benchmarks 12. Appraising, choosing, and modifying tasks and texts for a specific learning goal 13. Designing a sequence of lessons toward a specific learning goal 14. Selecting and using particular methods to check understanding and monitor student learning 15. Composing, selecting, interpreting, and using information from methods of summative assessment 16. Providing oral and written feedback to students on their work 17. Communicating about a student with a parent or guardian 18. Analyzing instruction for the purpose of improving it 19. Communicating with other professionals

Thank you for attending the session and for your constructive participation in it. You can find out more about my work at www.seandelaney.com.

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