As we reach the last few classes of our course together (:( I will miss our discussions!), I would like you to write one more blog entry -- one where you take a look back over your personal blog and in some way reflect or sum up some of the most important things you take away from the work we have done together. Please finish your blog post with three burning questions about 'math education: beyond the early years' that you are left with at this point!
To reflect for myself what we have been discussing, questioning and learning over the past 11 weeks or so, I've done the following summary of many of the topics of our classes. I hope that this might be helpful to you as you think over your own questions and thoughts.
-Susan
EDCP 553-16 Math education: Beyond the Early Years
Topics covered in first 11 weeks
1) From Jo Boaler: student centred inquiry-based vs. teacher-centred lecture based classes. What elements in our pedagogy make mathematics seem beautiful, boring, open, closed, exciting, routine, etc. for learners? How do we teach learners what mathematics is and how they might connect with it?
2) The nature of mathematical understanding. How does this understanding relate to:
• language & symbolism
•executing algorithms
• body, gesture, physical models, imagination?
How to assess students’ mathematical understanding?
Is it possible to ever reach an ‘ultimate’ understanding, even of a very elementary concept (like, say, multiplication)?
3) What do math teachers need to know?
• Can teachers’ pedagogical content knowledge be tested by a written test? Would this be a good idea?
•Teachers’ abilities to compress and pry apart concepts
•How to design teacher education to help teachers gain deeper and broader PCK of ‘big-idea’ topics in math?
•Many meanings and representations of mathematical concepts: for example, division/ fractions, squares and square roots (which we explored in groups)
4) Considering high-stakes, standardized testing:
•What agendas do these serve? (at school, city, provincial, national and international levels)?
•How are these tests changing, and why?
•Do they/ should they drive curriculum and pedagogy (or vice-versa – or some other kind of mutual influencing)?
•What equity issues (locally, nationally, internationally) are raised by these kinds of tests?
5) A problem-solving curriculum: Is it possible/ desirable to connect mathematics learning with
-Actual problem solving in the world?
-A mastery of logical processes?
- General approaches to dealing with a problematic situation (persistence, ingenuity,collaboration)?
-Specific heuristics for working with a difficult (mathematical) problem? (reframing the question, thinking of simpler examples, etc.)
-Bringing in knowledge and approaches from other disciplines and practices?
- Is mathematical problem-solving only a matter of doing word problems after instruction, as an application of knowledge, or can it generate knowledge and reflective learning?
6) Word problems: What are they, and how do they relate to problem solving more generally? The strangeness of word problems as a genre.
7) Repurposing textbooks and other existing curricular materials: How to open up closed problems to promote problem-solving, problem-posing, and more holistic ways of understanding variance and invariance for students learning new mathematical concepts?
8) What significance does standardized testing have to different stakeholders in education (students, teachers, administrators, parents), explored through role play.
9) Experiencing non-standard problem solving and meaning-making by trying examples together.
10) How does curricular change take place in a jurisdiction (like the new BC provincial curriculum)? What forces in society and in schools promote and resist curricular change? What do we make of the proposed new BC K-12 math curriculum? What do we want students to learn in math, and why? Is there any way to satisfy everyone with any particular curriculum – and if not, why not?
11) What might it mean to teach mathematics in multisensory, multimodal, embodied ways – integrated with more abstract, conceptual and symbolic/ language-oriented approaches? Trying examples from the Graphs & Gestures research project.
12) Questions of equity in mathematics education:
• Do ‘real-life’ activities within a cultural community, etc. necessarily translate to equity in terms of math learning?
• How does multilingualism play into equity issues in math?
• Is it helpful (or not) to integrate adults from the community into school classes to promote intercultural equity?
• Does group work or cooperative learning necessarily solve any equity problems – or might it contribute to making them worse?
• Can we make progress toward more equitable math education? If so, how?
13) Examples, example spaces and mathematical generality and abstraction:
•Is it enough for teachers to ‘give’ examples, or should students be helping generate them? If so, how do they learn to do so for a new topic, without simply parroting textbook examples?
•What pedagogies support establishing better, more carefully designed spaces of examples, non-examples and counterexamples?
•Is the design of exemplification and mathematical tasks a key element in improving learners’ more holistic understanding of new mathematical topics and concepts? If so, how could new teachers learn to do this better?
•Since assessment drives curriculum, how could assessment be designed to accord with better, more flexible, more student-generated exemplification and task design?
•How could teacher-led and student-centred pedagogies interact in optimal ways to keep learners engaged and developing deeper, broader mathematical understanding?
To reflect for myself what we have been discussing, questioning and learning over the past 11 weeks or so, I've done the following summary of many of the topics of our classes. I hope that this might be helpful to you as you think over your own questions and thoughts.
-Susan
EDCP 553-16 Math education: Beyond the Early Years
Topics covered in first 11 weeks
1) From Jo Boaler: student centred inquiry-based vs. teacher-centred lecture based classes. What elements in our pedagogy make mathematics seem beautiful, boring, open, closed, exciting, routine, etc. for learners? How do we teach learners what mathematics is and how they might connect with it?
2) The nature of mathematical understanding. How does this understanding relate to:
• language & symbolism
•executing algorithms
• body, gesture, physical models, imagination?
How to assess students’ mathematical understanding?
Is it possible to ever reach an ‘ultimate’ understanding, even of a very elementary concept (like, say, multiplication)?
3) What do math teachers need to know?
• Can teachers’ pedagogical content knowledge be tested by a written test? Would this be a good idea?
•Teachers’ abilities to compress and pry apart concepts
•How to design teacher education to help teachers gain deeper and broader PCK of ‘big-idea’ topics in math?
•Many meanings and representations of mathematical concepts: for example, division/ fractions, squares and square roots (which we explored in groups)
4) Considering high-stakes, standardized testing:
•What agendas do these serve? (at school, city, provincial, national and international levels)?
•How are these tests changing, and why?
•Do they/ should they drive curriculum and pedagogy (or vice-versa – or some other kind of mutual influencing)?
•What equity issues (locally, nationally, internationally) are raised by these kinds of tests?
5) A problem-solving curriculum: Is it possible/ desirable to connect mathematics learning with
-Actual problem solving in the world?
-A mastery of logical processes?
- General approaches to dealing with a problematic situation (persistence, ingenuity,collaboration)?
-Specific heuristics for working with a difficult (mathematical) problem? (reframing the question, thinking of simpler examples, etc.)
-Bringing in knowledge and approaches from other disciplines and practices?
- Is mathematical problem-solving only a matter of doing word problems after instruction, as an application of knowledge, or can it generate knowledge and reflective learning?
6) Word problems: What are they, and how do they relate to problem solving more generally? The strangeness of word problems as a genre.
7) Repurposing textbooks and other existing curricular materials: How to open up closed problems to promote problem-solving, problem-posing, and more holistic ways of understanding variance and invariance for students learning new mathematical concepts?
8) What significance does standardized testing have to different stakeholders in education (students, teachers, administrators, parents), explored through role play.
9) Experiencing non-standard problem solving and meaning-making by trying examples together.
10) How does curricular change take place in a jurisdiction (like the new BC provincial curriculum)? What forces in society and in schools promote and resist curricular change? What do we make of the proposed new BC K-12 math curriculum? What do we want students to learn in math, and why? Is there any way to satisfy everyone with any particular curriculum – and if not, why not?
11) What might it mean to teach mathematics in multisensory, multimodal, embodied ways – integrated with more abstract, conceptual and symbolic/ language-oriented approaches? Trying examples from the Graphs & Gestures research project.
12) Questions of equity in mathematics education:
• Do ‘real-life’ activities within a cultural community, etc. necessarily translate to equity in terms of math learning?
• How does multilingualism play into equity issues in math?
• Is it helpful (or not) to integrate adults from the community into school classes to promote intercultural equity?
• Does group work or cooperative learning necessarily solve any equity problems – or might it contribute to making them worse?
• Can we make progress toward more equitable math education? If so, how?
13) Examples, example spaces and mathematical generality and abstraction:
•Is it enough for teachers to ‘give’ examples, or should students be helping generate them? If so, how do they learn to do so for a new topic, without simply parroting textbook examples?
•What pedagogies support establishing better, more carefully designed spaces of examples, non-examples and counterexamples?
•Is the design of exemplification and mathematical tasks a key element in improving learners’ more holistic understanding of new mathematical topics and concepts? If so, how could new teachers learn to do this better?
•Since assessment drives curriculum, how could assessment be designed to accord with better, more flexible, more student-generated exemplification and task design?
•How could teacher-led and student-centred pedagogies interact in optimal ways to keep learners engaged and developing deeper, broader mathematical understanding?
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