Putting the “T” in TCMProfessional Teaching Practice in Mathematics

Lew RomagnanoromagnaL@mscd.edu

NCSSM TCM ConferenceJanuary 2009

My Claims• When we talk to each other, we don’t talk enough

about teaching– By “we” I mean both the non-representative subset of

teachers in this room and the larger universe of mathematics teachers in the U.S.

– Talk about good problems is only the beginning

• The field knows a lot, but not enough, about good teaching

• What we teachers know about mathematics teaching–and we know a lot–is craft knowledge

• We can build professional knowledge of teaching, and we need to.

Without a professional discourse about classroom practice, education is in a weak position to improve itself.*

* Lampert, M. (2001). Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press (p. 30).

Some DefinitionsTeaching consists of classroom interactions among teachers and students around content directed toward facilitating students’ achievement of learning goals.*

Teaching is the practice of structuring activities of studying in relation to particular content and particular students.†

* Hiebert, J. & Grouws, D. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371 - 404). Charlotte, NC: Information Age Publishing. † Lampert (2001), p. 32

Some Definitions• Craft knowledge is private, idiosyncratic,

rooted in specific contexts,and it leaves with us when we go.

• Professional knowledge is public, shareable and shared, testable and tested.*

* Hiebert, J., Gallimore, R. & Stigler, J. W. (2002). A knowledge base for the teaching profession: What would it look like and how can we get one? Educational Researcher 31 (5), pp. 3 - 15.

The Fabergé Egg Problem

Fred donates 5 of his Fabergé eggs to a local museum. The curator decides to display 3 eggs at a time, side-by-side, and to change the display each week. How long before she’ll have to repeat a display?

Is this an example of teaching?It is not an example of the teacher standing at the front of the room and engaging with students in the teacher initiation - student response - teacher evaluation (IRE) discourse pattern* so common in U.S. classrooms. For many, classrooms like this one, in which the students seem to be doing all the work and the teacher seems absent, is one of the many things wrong with the reform movement of the last 20 years. (That is, since 1989 A.D., or 0 S.E.)

* Mehan, H. (1979). Learning Lessons. Cambridge MA: Harvard Univeristy Press.

Is this an example of teaching?I claim this is an example of teaching, because each moment of this segment is the result of many decisions and actions by the teacher, and interactions among teacher and students. For example,

– The teacher chose or designed the problem, – The teacher posed the problem to students, using an

overhead– The teacher had the students work on the problem, in

groups of 3 - 4, for some time– The teacher shifted to a whole-class discussion– The teacher had certain students present, in a particular

order

Thinking Through a Lesson*

• Selecting and setting up a mathematical task

• Supporting students’ exploration of the task

• Sharing and discussing the task

* Smith, M. S., Bill, V. & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. Mathematics Teaching in the Middle School 14 (3), 132 - 138.

Selecting and Setting up• What are your goals for the lesson? What do you

want students to know and understand as a result of the lesson?

• In what ways does the task build on students’ previous knowledge? What definitions, concepts, or ideas do students need to begin work on this task? What questions will you ask to help students access prior knowledge and experiences?

• What are the ways this task can be solved?– Which of these methods do you think students will use?– What misconceptions might students have? What errors

might they make?

Selecting and Setting up• What are your expectations for students as they

work on and complete the task?– What resources or tools will students have to use in

their work?– How will the students work (independently, in small

groups) to explore this task? For how long? What group size? How will groups be constituted?

– How will students record and report their work?

• How will you introduce students to the task so as to provide access to all students while maintaining the cognitive demands of the task?

Supporting Students’ Work• As students work on the task, what questions will

you ask to:– focus their thinking? What evidence will let you know

how they are thinking?– assess students’ understanding of key ideas, strategies,

or representations?– advance students’ mathematical understanding?– encourage students to share their own thinking with

others, and to assess their understanding of the thinking of others?

Supporting Students’ Work• How will you ensure that students remain engaged

in the task?– What will you do if a student (or group) does not know

how to begin, or quickly gets frustrated?– What will you do if a student finishes immediately? How

will you extend the task to provide additional challenge?– What will you do if students focus on non-mathematical

aspects of the task (such as the format of their work)?

Sharing and Discussing• How will you orchestrate the class discussion so

that you accomplish your mathematical goals?– Which solution paths do you want to have shared during

the class discussion?– In what ways will the order of the solution paths help

develop students’ understanding of the ideas that are the focus of the lesson?

– What specific questions will you ask so students will• Make sense of the ideas you want them to learn?• Expand on, debate and question the solutions being shared?• Make connections, look for patterns, make generalizations?

Sharing and Discussing• How will you ensure that, over time, each student

has the opportunity to share his or her thinking and reasoning with peers?

• What will you see and hear that lets you know that all students understand the ideas you intended for them to learn?

• What will you do tomorrow to build on this lesson?

Is this an example of teaching?• So, I claim that this is an example of teaching,

even though the teacher didn’t say much, at least in this short segment.

• Is it an example of “good” teaching? What is good mathematics teaching?– Is it using real-world problems?– Is it using small groups?– Is it using calculators and computers?– Is it the result of teachers who have taken a lot of

mathematics courses?– Is it all about concepts rather than procedures?

Effective Mathematics Teaching*• Teachers and students attend explicitly to

conceptsThey have coherent, structured, explicit and connected discussions of the key mathematical ideas.

• Students struggle with important mathematicsStudents expend effort to figure something out that is not readily apparent.

* Hiebert, J. & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 371 - 404). Charlotte, NC: Information Age Publishing.

More DefinitionsBy struggling … we mean the opposite of simply being presented with information to be memorized or being asked to practice what has been demonstrated. … the process begins, said Dewey (1910) with “some perplexity, confusion or doubt.” It continues as students try to fit things together to make sense of them, to work out methods for resolving the dilemma. Deep knowledge, said Dewey (1929), “is the fruit of the undertakings that transform a problematic situation into a resolved one.”*

* Hiebert & Grouws (2007), pp. 387 - 388.

More Definitions

I would like to revive a more general and positive interpretation of the term … “studying” to mean any practice engaged in by students in school to learn. … I use it to include activities like inquiring, discussing,

thinking, reading carefully and examining closely.*

* Lampert (2001), p. 32.

Sadly, the research also points to an almost total lack of these features in U.S. mathematics classrooms.

It is hard to argue with emphasizing big ideas and making mathematics problematic for students, and a growing body of research suggests that such teaching leads to increased student achievement.

Another Claim

Teaching is enormously complex, but it can be learned (and improved) if it is approached in a professional way. That is, we can use a common language and theories of teaching and learning to wrestle with the problems of teaching in a public, collaborative way, and thereby get better as a community.

A Model of Teaching Practice*

Teacher

Mathematics

Students

* Lampert, (2001)

A Model of Teaching PracticeTeacher

Mathematics

Students

Teacher

Mathematics

Students

Teacher

Mathematics

Students

Time

The day-to-day continuity of teaching practice

Teaching…*…to establish a

classroom culture;…while preparing a

lesson;…while students work

independently;…while leading a whole-

class discussion;…to connect content

across lessons;

…to cover the curriculum;…students to be people

who study in school;…the nature of

accomplishment;…the whole class;…closure.

* Lampert (2001)

A Model for Teaching* Dims of classrooms Core Features

The nature of classroom tasks Make mathematics problematicConnect with where students areLeave behind something of mathematical value

The role of the teacher Select tasks with goals in mindShare essential informationEstablish classroom culture

The social culture of the classroom

Ideas and methods are valuedStudents chose and share their methodsMistakes are learning sites for everyoneCorrectness resides in mathematical argument

Mathematical tools as learning supports

Meaning for tools must be constructed by each userUsed with purpose – to solve problemsUsed for recording, communicating, and thinking

Equity and accessibility Tasks are accessible to all studentsEvery student is heardEvery student contributes

* Hiebert, J. et al. (1997) Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.

Lesson Preparation Questions• What are the mathematical concepts about which we

want students to construct deeper meaning? (What are the mathematical learning goals for the lesson?)

• What are some tasks that could engage students in making sense of these concepts?

• What do we have to do to prepare and manage these tasks?

• How might these tasks, and our plans to manage them, influence students’ mathematical thinking?

• How will we determine who has learned what about these concepts?

Assessment Planning Questions• What are the mathematical ideas I want to

assess?• What are my expectations for student

understanding of these ideas at this time?• What evidence will meet my expectations?• What sources will I use to collect this evidence?• How will I document the evidence I collect?• How will I report the inferences I make from this

evidence?

Even if teachers use these frameworks to make their thinking explicit, why won’t this lead only to improved craft knowledge for each of the teachers?

How might we use these frameworks to develop professional knowledge of mathematics teaching? That is, how can knowledge of teaching become public, shareable and testable, so it can grow and improve over time?

Teachers could work together,• Treating the lesson as the unit of analysis “The individual lesson is a big enough unit of teaching to contain all of the

complex classroom interactions that influence the nature of learning opportunities for students. At the same time, the individual lesson is the smallest natural unit for teachers that retains such interactions.”*

• Treating lessons as experiments– “[Lesson] plans are local theories of teaching and learning with the

planned lesson serving as an example.”*– Planning includes:

• Making hypotheses about students’ learning trajectories• Testing hypotheses by gathering data on students’ thinking

– Analysis focuses on effects of teaching on student learning

* Hiebert, J., Morris, A. K. & Glass B. (2003) Learning to learn to teach: an “experiment” model for teaching and teacher preparation in mathematics. Journal of Mathematics Teacher Education, 6, 201 – 222.

An Example: Japanese Lesson Study*

Working as a team, teachers

• Identify a problem or an issue

• Set goals

• Plan the lesson

• Teach and observe it

• De-brief and revise

• Report publicly* Fernandez, C. & Yoshida, M. (2004). Lesson Study: A Japanese Approach to Improving Mathematics Teaching and Learning. Mahwah, NJ: Lawrence Erlbaum Associates.

A Student Teaching “Research Lesson”

Some context:• Research Lesson planned for 2 weeks by a team: 2 student

teachers, 2 mentor teachers, a college supervisor, an outside student teacher, and a math ed faculty member

• Taught by the 2 student teachers; observed and de-briefed by the team plus two additional math ed faculty members

• 8th grade algebra lesson– The lesson’s mathematical focus was on linear functions and their

representations (size of 15% tip as a function of check amount)– The team’s research focus was on learning more about selecting

and orchestrating a rich task

• We’ll see a brief excerpt from an hour-long de-brief

One Last ClaimResearch Lessons slow down the whirlwind of practice, allowing us to study it together:

– More detailed, public discussions of mathematics, teaching and learning, grounded in specific practice

– Explicit testing of conjectures about specific teaching decisions and actions and their effects on student learning

– A process that supports teachers’ learning how to think and talk about practice

What I’d like to see• Sessions at conferences like TCM that offer really

interesting ideas and problems, along with explicit discussions of how to build lessons around them, ways students will think about them, difficulties students might have with them, evidence of learning to look for when students struggle with them, ways to orchestrate sharing of ideas about them…

• Gatherings in schools where groups of mathematics teachers plan, design, observe and analyze–that is, study–the problems of teaching

romagnaL@mscd.edu

Putting more emphasis on the “T” can make some real differences in the “L” – students’ mathematical learning.