mathematics-for-teaching where are we with all of this? brent davis university of british columbia

Mathematics-for-TeachingWhere are we with all of this?

Brent DavisUniversity of British Columbia

Overview

• Topic 1: Overview of MfT

• Topic 2: Sites of possible collaboration

between Departments of Mathematicians and

Faculties of Education (research and

instruction)

What is MfT?

• the sorts of mathematical understandings that are needed to teach mathematics well;

• in particular, understandings that support abilities to• interconnect concepts,• interpret idiosyncratic responses,• prompt multiple interpretations,• trace misunderstandings,• structure appropriate learning experiences,• extend/elaborate learning activities.

Why should mathematicians care?

Teachers and children play a profound role in shaping

mathematics by, for example,

• selecting and emphasizing the subset of concepts that

constitute “math” for most people,

• emphasizing and de-emphasizing particular interpretations of

and interconnections among concepts, and

• gate-keeping -- deciding who will and who will not be

permitted to pursue further study in mathematics.

Some Context for MfT

• “Mathematics education” is a young field …

• … that has tended to move from one obsession to

the next every few decades:

• 50s/60s: Structure of mathematics (ESM)

• 70s/80s: Student learning (FLM; PME; CMESG)

• 90s/00s: Teacher knowledge (JMTE)

Some Context for MfT

• Studies of MfT have a checkered past:

• Begle (1979) noted no correlation between teachers’

disciplinary knowledge and their effectiveness.

• Monk (1994) noted a slight negative correlation after a threshold

of 4 or 5 courses.

• Conversely, the Conference Board of Mathematical Sciences

(2001) noted a relatively strong positive correlation between

courses in mathematics pedagogy and teacher effectiveness.

Issues with early MfT research

• coarse measures of “mathematics knowledge” (typically, counts

of stock university courses)

• similarly coarse measures of “teacher effectiveness” (typically,

student scores on standardized tests)

• short-term studies

• studies organized around uninterrogated beliefs about what

teachers should know (typically, more advanced or deeper

math)

Implications for MfT Research

• Needs for more nuanced, fine-grained tools.

• MfT is about knowing math differently, not merely about

knowing more math.

• In particular, MfT has something to do with being able to

decompress ideas.

• A major aspect of MfT has to do with the figurative substrate of

mathematical concepts.

• The nature of knowledge isn’t a critical issue in MfT, but the

nature of knowing is.

Trends in MfT Research

• Ball/Bass/Hill: • concept unpacking/concept analysis

• job analysis

• large-scale, longitudinal, comparative studies

• Adler: • semiotic analysis of resources and strategies

• role and nature of problem solving in MfT

• Davis/Simmt:• figurative substrates of mathematical concepts

• teachers’ enacted (tacit and explicit) knowledge

FRAMES

mathematics

socio-cultural theory

cognitive science

Trends in MfT Research

• Ball/Bass/Hill: • concept unpacking/concept analysis

• job analysis

• large-scale, longitudinal, comparative studies

• Adler: • semiotic analysis of resources and strategies

• role and nature of problem solving in MfT

• Davis/Simmt:• figurative substrates of mathematical concepts

• teachers’ enacted (tacit and explicit) knowledge

Foci

explicit mathematics

“appeals”

implicit mathematics

What is multiplication?

• repeated addition;

• grouping.

How is the topic of multiplication addressed in the K–12 curriculum?

DAVIS/SIMMT

K–12 multiplication involves …

• repeated addition;

• grouping;• sequential folding;• many-layering (literal meaning of ‘multiply’);• the basis of proportional reasoning;• grid-generating;• dimension-changing;• intermediary of adding and exponentiating;• opposite/inverse of division;• stretching or compressing of number line;• rotating a number line.

DAVIS/SIMMT

How do the teachers know?

• For the most part, the images, gestures, metaphors,

applications, analogies, etc. aren’t explicit,

• but readily enacted,

• that is, they’re available to the teacher, but not necessarily

available to the learner,

• perhaps hinting at why so many students seem to fall apart

in Grades 7–9.

DAVIS/SIMMT

… which isn’t really news …

from van de Walle & Folk, Elementary and Middle School Mathematics: Teaching Developmentally

• “All of these ideas should be part of students’ repertoire of models for multi-digit multiplication. Introduce … different representations as ways to explore multiplication …” (ch. 8)

However …

• “… exponent is simply shorthand for repeated multiplication of a number times itself, for example, 34 = 3 3 3 3. …”

• “… That is the only conceptual knowledge required.” (ch. 20)

• Yikes. What about negative or fractional exponents (let alone irrationals, matrices, etc.)?

• In fact, teachers invoke a broad range of images, including events (explosions, population growth, decay) and graph-based curves/gestures.

So …

• Where might we go?• And, in particular, how/where might

mathematicians and math educators might complement one another’s efforts.

Six Sites …

Site 1 - How is math structured?

Prevailing conceptions of mathematics shape its pedagogy.

FOR EXAMPLE:• RATIONALISM - logical argument, pointing to

sequential curriculum (core image: directed line)• FORMALISM - “structure of the discipline,” pointing

to emphasis on abstract principles (core image: pristine architecture)

• STRUCTURALISM - meaning arising in the interminglings of experiences and interpretations, pointing to emphases on physical exploration and active sense-making (core image: web)

• COMPLEXITY THINKING - nested and intersecting domains of emergent activity, pointing to recursively elaborative, hub-centred pedagogy (core image: decentralized network)


WHERE HELP IS NEEDED …• Might mathematics be productively productively interpreted as a

decentralized network?

• If so, what are the highly connected ideas (HUBS)?

• Some candidates:


WHERE HELP IS NEEDED …• Might mathematics be productively productively interpreted as a

decentralized network?

• If so, what are the highly connected ideas (HUBS)?

• Some candidates:


• boundary• shape• containment• quantity• equivalence• binary operations• variable

• equation• function• coordinate graphing• rate of growth• sequence/pattern• chance• proof

Discussions of mathematics tend to be framed the assumption that it’s logically derived.

Site 2 - How is math produced?

In fact, mathematical insight seems to be aboutthe logical and the analogical,

rigor and intuitiveness,the explicit and the implicit,

rational argument and empirical evidence,accident and intention.


Perhaps most importantly: Humans aren’t inherently logical creatures; logical thought rides atop our

capacities for making and elaborating associations.

A brief diversion (video set-up)• Deborah Ball’s “Shea” video

• Context: Grade 3 classroom

• Question: Is 6 an odd number, even number, or both?

• Task: identify some of the announced and/or enacted meanings for “even.”

• Challenge: There are at least six distinct images/definitions/ analogies/gestures indicated … some simultaneously



INSERTSHEAVIDEO

Some interpretations of “even”:

• Sets of objects that can be split into subsets of 2;• Sets of objects that can be split in half (“fairly”);• Sets of objects that are symmetric.• Whole numbers than can be divided by 2;• Symbols on a number line that can be identified by

alternating odd/even/odd/even etc.• Quantities produced by adding other quantities to

themselves.

Some interpretations of “even”:

• Sets of objects that can be split into subsets of 2;• Sets of objects that can be split in half (“fairly”);• Sets of objects that are symmetric.• Whole numbers than can be divided by 2;• Symbols on a number line that can be identified by

alternating odd/even/odd/even etc.• Quantities produced by adding other quantities to

themselves.


WHERE HELP IS NEEDED:

• What are some of the figurative (metaphoric, gestural,

analogical, etc.) underpinnings of major concepts?

• Which figurative underpinnings are most useful for which

concepts (e.g., for “function)? And which are best

avoided?

• Which underpinnings must become embodied/intuitive/

enacted/automatic/tacit? (NOTE: This question might be a

rephrasing of the HUBS question.)


Overwhelmingly, it is assumed that research mathematicians

are solely responsible.

Site 3 - Who defines mathematics?

There is an emergent realization that research mathematicians share responsibility

with teachers and children,business and industry,

the military,policy-makers, ….


WHERE WORK IS NEEDED:

• Clearly, mathematics teacher education has to be more of a

joint project of mathematicians and educators …

• … and perhaps research mathematicians and practicing

teachers should be taking the lead in MfT research,

• thereby legitimating it as a branch of mathematical inquiry

• and giving better access to NSERC and other funding.


Discussions of mathematics knowing tend to be framed by a

monadic (individualist) model of understanding/knowing.

Site 4 - Where is math?

… it would be perverse not to infer that for large stretches of time [mathematicians] are engaged in a process of

communicating with themselves and one another; an inference prompted by the constant presence of standardly presented formal written texts (notes, textbooks, blackboard lectures, articles, digests, reviews, and the like) being read,

written, and exchanged, and of all informal signifying activities that occur when they talk, gesticulate, expound, make

guesses, disagree, draw pictures, and so on. (Brian Rotman)


Clearly, mathematics knowing is distributed across humanity,


prompting a suggestion for a more collective approach to mathematics learning.

Where work is needed:


Recalling that one of the most potent influences in how we teach is how we were taught,

Can courses in mathematics (esp. MfT) become

courses in collective engagement?

Discussions of MfT tend to be focused on grade-specific expertise.

Site 5 - How is math distributed?

The evidence suggests that integrated longitudinal awarenesses of topics contribute to

more effective pedagogy

in large part because teachers find ways to leave definitions open for elaboration.


Where help is needed:

Can all-at-once MfT courses be developed?(Once again, this might be a variation of the HUBS question.)


Site 7 - What is math ability?

There’s a pervasive belief that mathematics ability is fixed and pre-given.


“[The] belief in the importance of innate talent, strongest perhaps among the experts

themselves …, is strangely lacking in hard evidence …. The preponderance of

psychological evidence indicates that experts are made, not born.” (Philip Ross, Sci Am)

The keys to cultivated genius seem to be:• early starts,• extensive, persistent engagement,• expert assistance,• effortful practice.


WHERE WORK NEEDS TO BE DONE:

How do we help teachers develop senses of “where to go next”? (Deep math helps.)


Can teaching/learning resources be structured to help?

Might public school teaching be more actively presented as a viable and respectable career

choice for mathematics majors?

We mimic (and re-experience) the affective/ emotional behaviors that we witness.

Site 8 - How is math experienced?

How are mathematics educators (including us) experiencing/manifesting:- pleasure in the mathematics;

- curiosity and surprise;- delight in difficulty;

- comfort with disagreement?

Where we might work together:

Site 8 - How is math experienced?

There are indications that emotioning is happening in some university-based

mathematics courses.

How can we work together to research the differences that it makes?

Trying to define, structure, and cultivate a branch of mathematics that is:

• overtly a decentralized network,• attentive to figurative substrates,• mathematically legitimate,• conscious of collective dimensions,• unfragmented,• lending itself to effortful study;• emotionally engaging.

So where might we be going?

• stronger sense of what teachers know,

• clear sense that enacted MfT is currently different

from formal preparations in math,

• no hard evidence of the relationships between MfT

and student learning,

• sparse resources,

• emerging, but still scanty support from the

mathematics community.

And where might we be?

mathematics-for-teaching where are we with all of this? brent davis university of british columbia

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