Learning to Teach Learning to Teach Purposefully: Purposefully:

Purposefully Learning to Purposefully Learning to TeachTeach

Doug JonesDoug Jones

Appalachian State UniversityAppalachian State University

Prepared for the 2007 Conference of the

North Carolina Association of Mathematics Teacher Educators,

Chapel Hill, April 20, 2007

* Boone

Radius = 4

Area = ?

Go ahead, work it out. We can wait...

Radius = 4

Area = ?

Hint: This arose in the context of regular polygons.

Radius = 4

Area = ?

Radius = 4

Area = ?

The problem above was posed to a 10th grade geometry class as an opener one day during their study of regular polygons. (I was there to observe a student teacher.) The teacher (Ms. H) asked Michael (not his real name) to work/ talk through the problem for the rest of the class. She provided no hints or other direction and allowed him to work out loud by himself. (This is a normal activity in her class.) The following exchange took place over the next 5 minutes.

Michael: Um, o.k. It’s equilateral, right?Ms. H: (no response)Michael: OK. Draw the altitude from the top to the bottom. Ms. H draws in the altitude. Michael: Now that line is 4.Ms. H: Which?Michael: The base. Half of it.Ms. H writes a 4 below the base. (students whisper)Michael: And that angle is 30. No, 60.Ms. H writes 60 for the measure of the base angle.Michael: So that makes the height 8, right?Ms. H writes 8 for the height. Students get restless. Ms. H shhh’s them.Michael: Is that right?Ms. H: You know I’m not going to answer you. Go on.

Michael: No, the side is 8. Now draw the line from the vertex to the height line.Ms. H draws this in.

4

60

Michael: No. That’s 4. So the top part also is 4 and the bottom part (of the altitude) is 2. I remember that.Ms. H writes “4” and “2” and erases the 4 below the base.Michael: So the height is 6 and the base is, um, what is it? The side is 8; no, it’s not. The side is the hypotenuse. OK. The height is 6 and the angle is 60. So that makes the … that part of the height is 2, right? So √3 times 2.Students are really restless now. Student teacher appears a little uncomfortable. Some students start to protest that Michael is wrong and that he is taking way too long. Ms. H pleasantly Shh’s them.Ms. H (to Michael): Where?Michael: There. On the base. So, OK, now I got it. The area is ½ times 6 times 2√3. Ms. H: Done?Michael: Yes. No. Double it.Ms. H is very pleased.Ms. H: Good job, Michael. (to class) OK? What did he do?

4

2

2√3

As a mathematics teacher, what are good questions to ask?

As a mathematics teacher educator, what are good questions to ask?

As a mathematics teacher educator, what are good questions to ask?

Well, it depends on why you use the scenario and what you intend it to be an example of.

If you are concerned with engaging students, you ask certain questions. If you are concerned with assessment, you ask others. If you are concerned with uncovering mathematical relationships, you ask still others of this scenario.

Here are a few questions we might ask of our teacher education students and of ourselves:

Based on what the student (Michael) did, • What have you learned about what he

knows about regular polygons?• What would you have learned if you (or

students) cut him off earlier?• What connections do you understand in

a new way about regular polygons, about students, or about teaching?

• How would you have handled the class?

In what way do you see purposeful teaching in this scenario?

In what way do you see purposeful teaching in this scenario?

What supports the teacher (Ms. H) in maintaining and manifesting that purpose?

Pedagogical Content Knowledge

• 1986: PCK characterized by Shulman and his colleagues as a result of the study “Knowledge Growth in Teaching”

• Concluded that the missing paradigm for growth was purposeful attention to the subject matter as it pertains to pedagogy, particularly as manifest in real cases.

Place in history of teacher education

• 2000 years ago: “What distinguishes the man who knows from the ignorant man is the ability to teach.” (Archimedes, from Metaphysics)

• 1000 years ago: All universities were normal schools. The highest form of understanding was measured by the candidate’s ability to teach. (Ong, Ramus, Method, and the Decay of Dialogue)

• 100 years ago: 950 of 1000 points on teacher licensure exams concerned content knowledge (1875 California State Board Examination for Elementary Teachers)

• 20 years ago: Emerging research on teaching shifts emphasis to general pedagogy (Houston, 1990, Handbook of Research on Teacher Education)

• Today: PRAXIS exams, particularly for secondary, emphasize both pedagogy & content (www.ets.org)

PCK comprises…

• The most useful forms of representation of the subject’s big ideas.

• The most powerful analogies, illustrations, examples, explanations.

• The ways of transforming the subject matter so that it is comprehensible to learners.

• An understanding of the students’ perspective, and what makes the subject matter easy or difficult

Significance

PCK represents a drawing together – of content and pedagogy, of teacher and student, of theory, purpose, and practice.

Remember Michael and Ms. H?

Ms. H intentionally made use of what she knew about mathematics and about her students. In speaking with her after the lesson, she remarked that she needed her class to review regular polygons by hearing someone talk through a problem. She knew that she could call on Michael to do that. She wanted the class to make better connections between central angles and side length and with sides of special angles. Further still, she knew that in order to learn the material well, Michael needed to talk through a problem.

“Michael and the equilateral triangle” is a case of using a student-centered approach to effect better understanding and connections by the class as a whole

Ms. H used what she had come to know about

• Particular mathematics (regular polygons and special angles)

• Particular representation (diagram with important information hidden)

• Particular pedagogy (student explanation)

• Particular students’ perspectives (who she could call on)

“Learning to teach purposefully”

is linked to

Pedagogical Content Knowledge.

Lesson Study

• Formal (Japanese) Lesson Study was introduced into US in the late 1990s as a result of video research related to TIMSS and the book, The Teaching Gap (Stigler & Hiebert)

• An extension of the culture of teaching in Japan it is characterized by a commitment to excellence and an intention to bridge theory, practice, and focused observation

Lesson study…

• Represents case knowledge, (remember Shulman’s call?) from which both theoretical and practical knowledge may be derived

Remember Michael and Ms. H?

We participated in a very abbreviated version of a segment of Lesson Study at the beginning of this presentation. We examined a portion of a class with the intention of raising questions for and about teaching.

What would make Lesson Study effective for people learning to teach?

Essential Elements of Lesson Study(per Mills College Lesson Study Group)

• Think carefully about goals of lesson, unit, subject• Study and improve best available lessons• Deepen subject-matter knowledge• Think deeply about long-term goals for students• Collaboratively plan lessons• Carefully study student learning/behavior• Develop powerful instructional knowledge• See our teaching through the eyes of students and

colleagues.

Significance

• Everyone who intends to improve,

will improve (with the help of colleagues).

• Lesson Study intends to effect and support a particular culture of teaching and learning, one in which purposefulness is highly valued and cultivated.

U.S. education suffers not from a lack of good programs, but from a lack of demand for them.

(Elmore, 1999-2000 writing about school leadership)

However,

• NCTM produced a video about lesson study, presumably in anticipation of demand.

• Teachers College and Mills College have active research groups on Lesson Study.

• Numerous studies and dissertations have been undertaken on Lesson Study in the past 10 years.

• Currently over 250 Lesson Study groups exist in schools in the U.S.

“Purposefully Learning to Teach”

Is linked to

Lesson Study

Reason-To-Be

• Remember Ms. H?

• Remember how she seemed determined to get Michael to think through the problem?

• Ms. H has a very clearly defined, and very compelling (to her) Reason To Be: “It’s not about me.”

Reason-To-Be

• Literally, “justification for existence”• What compels the person to become a

teacher– a passion• What the person intends to contribute –

a purpose• What the person wants to be

remembered for – a set of principles

Big Rocks

If you don’t get the big rocks in first, you won’t get them in at all.

A robust reason-to-be is purposefully used and pervades

the entire professional life.

Ms. H is a prime example.

So also was Georg Polya

“In teaching, as in several other things, it does not matter much what your philosophy is or is not. It matters more whether you have a philosophy or not. And it matters very much whether you try to live up to your philosophy or not.”

--Georg Polya

… and probably many people we collectively know.

Significance of paying attention to the RTB

• Potential to focus teacher on coherence.

• Potential to facilitate purposeful teaching in a broad sense.

• Potential to help teachers better draw theory from their teaching experiences

Another case: Bruce the beginning teacher

(a case of the centrality of mathematics and service)

Began college as math major and athlete• Explored teacher certification, took

introductory courses• Transferred to university• Kept math major• Worked as a substitute teacher for a

semester• Entered a master’s certification program

Bruce’s Reason-To-Be

“To serve the mentally, emotionally, and physically impoverished students at the secondary level through making mathematics meaningful.”

• Focus on Service• Focus on Mathematics, particularly critical

thinking

Mathematics per Bruce

• Reasoning

• Analytical thinking

• Purposeful, not rote

• Generalizing

• Problematic

Bruce and Planning

• Plan for student perspective

• Plan from a general approach, but accommodate to students’ needs

• Plan to model mathematical thinking

• Plan for authenticity and application

Bruce and Instruction

• Must serve students’ ability to think critically

• Model mathematical reasoning

• Link to methods of proof

• Purposeful, intentional

• Engage in pedagogical problem solving

Bruce and Assessment

• Must provide for valid inference

• Derives from learning to write proofs

• Abstract and intentional

Bruce and Collaboration

• Models organization and analysis

• Intentional

• Has as a goal to get beyond the individual’s boundaries

The Reason-To-Be is linked to all of it…

• As a light to guide teaching

• As a lens for analyzing lessons

• As a foundation for a purposeful career

What’s the Punch Line?

If you don’t know your “why,” the “what doesn’t matter and the “how” is irrelevant.

What do we do?

Blend these three constructs into an orientation to developing highly qualified teachers of mathematics to learners.

What do we do?

Blend these three constructs into an orientation to developing highly qualified teachers of mathematics to learners.

I cannot promise that it is easy;

but it is worth it.

The Field as I see it

T1 T2

Information ------> knowledge ------> wisdom

T1: Information is transformed into knowledge by study.

T2: Knowledge is transformed into wisdom by purposefully and humbly acting on what one knows.

Attention to Pedagogical Content Knowledge can help us teach more purposefully.

Purposefully analyzing our teaching can help us and our students better learn to teach.

A compelling Reason-To-Be can serve as a foundation for coherence.

When we reach for these, even if outside our grasp; our heaven will be a compelling life’s work.