Professional Development for Facilitators

Professional Development for Facilitators
1. Joys/Concerns/Questions facilitators have
2. History of the National Council of Teachers of Mathematics publication of the ‘Standards’
3. Reform Movements in Professional Development
4. Challenging Beliefs
5. Mathematical Proficiency
6. Constructivism
7. Dealing with Resistant Teachers
8. Professional Standards for Teaching Mathematics
9. Practice-based Instruction
10. Revisit Joys/Concerns/Questions
Here we provide a guide for the professional development of the teacher-facilitators for the Number and Computation K-1, 2-3, and 46 courses. This is also an accompanying powerpoint with the materials. We have not indicated specific breaks in the implementation
of these materials so that you can identify stopping places that fit the needs of your group. For example, one group using these
materials may choose to stop the first session after introducing the reform movements in professional development (Part 3) while
another group may choose to stop after the discussion pertaining to constructivism. Use the materials in ways that make sense to you.
We hope these materials help you think about ways to best prepare your facilitators to offer productive and empowering professional
development to your teachers!
Part 1: Joys/Concerns/Questions
Begin by providing opportunities for teacher-facilitators to identify their joys/questions/concerns about acting as facilitators for these
courses. Ask them to write their questions/concerns/joys on an index cards and to keep the cards throughout the initial facilitator
session. At the end of the initial session, give them an opportunity to reflect back on their initial joys/questions/concerns and modify
as they desire before turning in the index cards. Use the information on the index cards to address throughout these sessions the
specific joys/questions/concerns your teacher-facilitators have identified.
Remind the facilitators that the primary purpose of the Number and Computations courses is to build teachers’ number and
computation sense and then in turn, that of their students. Ask facilitators to define number sense. Accept all responses and facilitate a
discussion. At some point share some definitions of number sense such as the following:
“…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing
them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989).
Flexibility in thinking about numbers and their relationships.
“Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the
numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5).
Part 2: How did the current wave of mathematical reform begin?
Numerous reports have criticized the practices of mathematics teaching in the United States (National Commission on Excellence in
Education, 1983; National Research Council, 1989). These reports noted that the achievement of American students is far less than
that of their international counterparts in the area of mathematics. The publication Everybody Counts: A Report to the Nation on the
Future of Mathematics Education by the National Research Council was interpreted as a strong indicator of a need for the U.S. to reexamine mathematics education. It established a need for change in the way math had previously been taught. In response to calls for
change, several documents were written to characterize goals for reform in mathematics education by describing a vision of what it
would mean to be mathematically literate in today’s society, providing guidance to those involved in changing mathematics
curriculum and teaching. These materials offered a challenge for educational reform and provided a basis on which reform could build
as steps were taken towards improvement. The following describes the publication and intent of those documents
(http://www.nctm.org/standards/content).
•
Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989)
In 1986, the Board of Directors of the National Council of Teachers of Mathematics established the Commission on Standards for
School Mathematics as one means to help improve the quality of school mathematics. This document, which is the product of the
commission's efforts, contains a set of standards for mathematics curricula in North American schools (K-12) and for evaluating the
quality of both the curriculum and student achievement. As school staffs, school districts, states, provinces, and other groups propose
solutions to curricular problems and evaluation questions, these standards should be used as criteria against which their ideas can be
judged.
The Standards Documents were written by a cross section of math educators including classroom teachers, teacher educators,
supervisors, educational researchers, and university mathematicians.
The Standards is a document designed to establish a broad framework to guide reform in school mathematics in the next decade. In it
a vision is given of what the mathematics curriculum should include in terms of content priority and emphasis, The challenge we issue
to all interested in the quality of school mathematics is to work collaboratively to use these curriculum and evaluation standards as the
basis for change so that the teaching and learning of mathematics in our schools is improved. Here is what we should be teaching, but
how are we going to teach it?
•
Professional Standards for Teaching Mathematics (NCTM, 1991)
The Professional Standards for Teaching Mathematics is designed, along with the Curriculum and Evaluation Standards for School
Mathematics, to establish a broad framework to guide reform in school mathematics in the next decade. In particular, these standards
present a vision of what teaching should entail to support the changes in curriculum set out in the Curriculum and Evaluation
Standards. This document spells out what teachers need to know to teach toward new goals for mathematics education and how
teaching should be evaluated for the purpose of improvement. We challenge all who have responsibility for any part of the support and
development of mathematics teachers and teaching to use these standards as a basis for discussion and for making needed change so
that we can reach our goal of a quality mathematics education for every child. We have established what to teach and how to teach it,
now how are we going to assess it?
•
Assessment Standards for School Mathematics (NCTM, 1995)
The Assessment Standards reflect the values and goals associated with the type of assessment system that must be achieved if the
reforms envisioned in the teaching and learning of mathematics are to become a reality. We challenge teachers, school district
personnel, and state or provincial officials to read and reflect on these standards, critically examine their own current assessment
systems, and then work to develop new systems that are consistent with the reform vision.
These 3 documents represented an historically important first attempt by a professional organization to develop and articulate explicit
and extensive goals for math teachers and policy makers. It’s an ongoing process that needs to be periodically re-examined, evaluated
and revised to remain relevant.
•
Principles and Standards for School Mathematics (NCTM, 2000)
Principles and Standards for School Mathematics describes a future in which all students have access to rigorous, high-quality
mathematics instruction, including four years of high school mathematics. Knowledgeable teachers have adequate support and
ongoing access to professional development. The curriculum is mathematically rich, providing students with opportunities to learn
important mathematical concepts and procedures with understanding. Students have access to technologies that broaden and deepen
their understanding of mathematics. More students pursue educational paths that prepare them for lifelong work as mathematicians,
statisticians, engineers, and scientists.
Its recommendations are grounded in the belief that all students should learn important mathematical concepts and processes with
understanding and describes ways students can attain it.
PSSM has 4 major components (Handout 1). The first are Principles for School Mathematics which reflect basic perspectives on
which educators should base decisions that affect school mathematics. These Principles establish a foundation for school mathematics
programs by considering the broad issues of:
1. Equity – high expectations and strong support for all students
2. Curriculum – more than a collection of lessons; it must be coherent, focused on important mathematics, and well articulated
across the grades
3. Teaching – requires understanding what students know and need to learn and then challenging them to learn it well
4. Learning – students must learn math with understanding, actively building new knowledge from experience and prior
knowledge
5. Assessment – should support the learning of important mathematics and furnish useful information to both teachers and
students
6. Technology – essential in teaching mathematics; it influences the mathematics that is taught and enhances students’ learning
Second are the Content Standards for School Mathematics:
1.
2.
3.
4.
5.
Number and Operations
Algebra
Geometry
Measurement
Data Analysis and Probability
Third (Handout 2) are the Process Standards for School Mathematics:
1.
2.
3.
4.
5.
Problem Solving
Reasoning and Proof
Communication
Connections
Representation
Finally, is a component discussing the issues related to putting the Standards into action and outlines the roles played by various
groups and communities in realizing the vision of the Standards.
This vision of mathematics teaching and learning is not the reality in the majority of classrooms, schools, and districts. Today, many
students are not learning the mathematics they need. In some instances, students do not have the opportunity to learn significant
mathematics. In others, students lack commitment or are not engaged by existing curricula.
Major thrust of the reform movement in mathematics has been the effort to replace the current, obsolete, elementary mathematics-ascomputation curriculum with a mathematics curriculum that genuinely embraces conceptual understanding, reasoning, and problem
solving as the fundamental goals of instruction (Battista, 1994). The Standards documents offer teachers interpretations of
mathematics reform, but they are not an individual instructional program ready to be implemented.
These suggestions for mathematics reform offer a promise for implementation because they address improvements of the education
system as a whole.
The vision of the NCTM’s standards cannot be realized without the support of classroom teachers. Teachers must learn the skills and
perspectives assumed by new visions of practice and unlearn some of the practice and beliefs about students and instruction that have
dominated their professional lives to date. For this to happen, serious engagement in professional development is necessary.
Part 3: Professional Development Reform
Ask participants to share their professional development experiences (workshop? Inservice?)
The term ‘professional development’ of teachers in mathematics education means the opportunities offered to educators to develop
new knowledge, skills, approaches, and dispositions to improve their effectiveness in their classrooms. Professional development
traditionally aimed to enhance the knowledge and skills of teachers through presentations, course work and workshops. It concentrated
on conveying information and providing ideas. The focus was on bringing expertise to teachers in order to improve their classroom
work with students. ‘Experts’ distributed ideas, tips, and guidance, while participants collected handouts and ready-made activities.
The dominant ‘training’ model focused primarily on expanding an individual teacher’s repertoire of well-defined and skillful
classroom practice. This approach offers teachers an assortment of resources, but the teachers’ learning ended with the completion of
the program rather than continuing every day in their classrooms.
Professional development has evolved from a focus on individual growth to a more systemic integrated perspective on enhancement
across cohorts of teachers (Loucks-Horsley, et al., 2010). Teachers may appear to be enthusiastic about restructuring their classroom
practices during in-service workshops, but they may find the change to be very difficult to implement in their classroom.
There are several problems facing professional development in mathematics education. First, it is critical to try to develop an
understanding of the issues in mathematics education, even though teachers may have a primary desire to take ready-made materials
back to their classrooms. Short-term gains often limit long-term opportunities for growth. Second, the instructional approaches that are
called for when working with students (a wide array of learning opportunities that engage students in experiencing, creating and
solving real problems, using their own experiences) are often denied to teachers when they are the learners.
A Paradigm for Professional Development in Learner-Centered Schools
From Too Much
Focus on teacher needs
Focus on individual development
To More
Focus on student learning outcomes
Focus on individual and system development
Transmission of knowledge, skills, and strategies
“Pull-Out” training
Generic teaching skills
Fragmented, piecemeal, one-shot experiences
District direction and decision making
Professional development as trainers
Just some people’s job
Professional development for teachers
Professional development as a “frill”
Loucks-Horsley, et al, 1997
Inquiry into teaching and learning
Job-embedded learning
Content-specific teaching skills
Driven by a clear, coherent, long-term strategic plan
School direction and decision making
Professional development as facilitators, consultants, planners,
coaches, and trainers
Everyone’s job
Professional development for everyone
Professional development as essential
The Professional Teaching Standards focus on the essential components for the professional development of teachers of mathematics.
They comprise the threads that are woven as the fabric of successful mathematics teaching: personal experiences in contexts that
model and value good mathematics teaching; ongoing development of knowledge about mathematics, students, and teaching;
numerous and diverse opportunities to apply knowledge and experience through practice; and the gradual assumption of
responsibilities for professional growth and change. Ideally, the weave of the fabric will evolve and change, reflecting the numerous
stages to be explored in the career-long development of mathematics teachers (NCTM, 1991).
Standards for the Professional Development of Teachers of Mathematics (Handout 3):
Standard 1: Experiencing Good Mathematics Teaching
Notes: We want to redirect mathematics instruction from a focus on presenting content through lecture and demonstration to a focus
on active participation and involvement. We don’t want to ‘deliver’ content; rather we want to facilitate learners’ construction of their
own knowledge. You teach the way you were taught because that’s all you know. Consider the problem 156 ÷ 4. How would you
teach a student to solve this problem? In my PD sessions we talked about how you don’t want to just ‘teach’ students a procedure.
Long division should evolve from an understanding of what division means. We started with a simple problem (15 ÷ 3) and used
counters to model the problem. This lead to a discussion of the difference between sharing (partitive) and grouping (measurement)
types of division. We used Base 10 pieces to model larger quotients (72 ÷ 6) and related division to multiplication using rectangular
arrays. We talked about using problems as a context for helping students understand why they are performing the procedures. Share
the M&M problem (4-6 Module Session 2) and ask them to solve it. Share student work for how some 4th grade students solved the
problem. (Supporting Documents can be found in Handout 4.)
Mathematics teachers should model good mathematics teaching by• posing worthwhile mathematical tasks;
• engaging teachers in mathematical discourse;
• enhancing mathematical discourse through the use of a variety of tools, including calculators, computers, and physical and
pictorial models;
• creating learning environments that support and encourage mathematical reasoning and teachers' dispositions and abilities to
do mathematics;
• expecting and encouraging teachers to take intellectual risks in doing mathematics and to work independently and
collaboratively;
• representing mathematics as an ongoing human activity;
• affirming and supporting full participation and continued study of mathematics by all students.
Standard 2: Knowing Mathematics and School Mathematics
Notes: Segue from m&m problem to talk about what it means to know mathematics. Would you have ever thought students would
solve problems the way the students did? Why do we only teach the standard algorithm? (It’s the only way most teachers know). Do
you have the content knowledge to teach the math you are teaching? Teachers need opportunities to revisit school math topics in ways
that will allow them to develop deeper understandings of the subtle ideas and relationships between and among concepts. This is hard
to do when teachers already know a procedure for solving particular problems. Problems can seem so simple to them because they
already know how to do it. That’s why we investigated addition and subtraction using Base 5 pieces. The participants were taken from
the ‘known’ (standard Base 10 procedures) into the ‘unknown’ (Base 5). Because it was unknown, they had to use the materials that
were available to them. The purpose was to help them think about what a number system is and how it works. If I had given the
problem 17 + 35 and asked them to use the pieces to solve the problem, they all would have done the computation in their heads and
modeled the answer with their pieces. They didn’t know how to do the computation in their heads so they had to use the materials to
solve the problems. This helps them to experience ‘disequilibrium’ that their students often experience.
Communication (discourse) is critical. Using concrete materials or representations helps students explain their thinking. In a
traditional class, a student might explain how he solved the addition problem 17 + 35 by saying: 7 + 5 is 12. I put the 2 down and
carry the 1. One plus 1 plus 3 is 5. So it’s 52. Not only does this explanation only demonstrate a procedure, but it doesn’t do it very
well. Are you adding 1 + 1 + 1? No, you’re adding 10 + 10 + 30. Relate to the ‘1’ problem.
A child with experience using concrete materials might say:
‘One 10 plus 3 tens is 40. Seven + 5 is 12. 40 + 12 is 52.’
The education of teachers of mathematics should develop their knowledge of the content and discourse of mathematics;
including• mathematical concepts and procedures and the connections among them;
• multiple representations of mathematical concepts and procedures;
• ways to reason mathematically, solve problems, and communicate mathematics effectively at different levels of formality;
and, in addition, develop their perspectives on• the nature of mathematics, the contributions of different cultures toward the development of mathematics, and the role of
mathematics in culture and society;
• the changes in the nature of mathematics and the way we teach, learn, and do mathematics resulting from the availability of
technology, (maybe use microwave example?)
• school mathematics within the discipline of mathematics;
• the changing nature of school mathematics, its relationships to other school subjects, and its applications in society.
Standard 3: Knowing Students as Learners of Mathematics
Notes: Teachers need to examine children’s thinking about mathematics so they can select or create tasks that will help children build
more valid concepts of mathematics. Should be active participants and impose their own sense of investigation and structure. Teacher
expectations are founded on knowledge and beliefs about who their students are and what they can do (relate to my methods’ student
interviews and the ‘high’ and ‘average’ kids). Mathematics is learned when learners engage in their own invention and impose their
own sense of investigation and structure. How does mathematics appear to a 5-year old? 8-year-old? 12-year-old? Teachers must be
able to perceive mathematics through the minds of their students. We must use math in a context that makes sense to our students
(Share Spring Fest example). If we use a context that students can relate to we can get away from teaching things such as ‘key words’.
My method’s students interviewed 2 practicum students with about 25 different questions. It was amazing how many of them had no
idea of what to do. Several number picked and added on every single problem. If the problem meant something to them personally
(For example if a student was saving his money to buy a toy. If he knew how much the toy cost and how much he had), he would be
very interested in figuring out how much more money he needed.
The continuing education of teachers of mathematics should provide multiple perspectives on students as learners of
mathematics by developing teachers' knowledge of• research on how students learn mathematics;
• the effects of students' age, abilities, interests, and experience on learning mathematics;
• the influences of students' linguistic, ethnic, racial, and socioeconomic backgrounds and gender on learning mathematics;
• ways to affirm and support full participation and continued study of mathematics by all students.
Standard 4: Knowing Mathematical Pedagogy
Notes: Teachers learn how students think about math – should focus on students’ understanding of mathematical concepts and
procedures. Teachers need to employ strategies that will help them develop the participation essential to engaging students in
mathematics. This includes posing worthwhile tasks and knowledge of the ways of representing mathematical ideas. Modeling
mathematical ideas through the use of representations (concrete, pictorial, abstract). We tend to teach abstract first. If the students
don’t get how to use the procedure, we show them with manipulatives. Relate to building arrays to demonstrate 2-digit multiplication,
transferring this knowledge to Base 10 grid paper which will show the 4 partial products and then the algorithm. Relate the discourse
we use when teaching the traditional algorithm. We are not referring to what we are actually doing. Teachers tell me their students
can’t add fractions with unlike denominators so they tried to use manipulatives (Cuisenaire rods) and that only confused them more.
That’s because you’re moving backwards instead of forwards.
The continuing education of teachers of mathematics should develop teachers' knowledge of and ability to use and evaluate• instructional materials and resources, including technology;
• ways to represent mathematics concepts and procedures;
• instructional strategies and classroom organizational models;
• ways to promote discourse and foster a sense of mathematical community;
• means for assessing student understanding of mathematics.
Standard 5: Developing as a Teacher of Mathematics
Notes: You do NOT want to be that teacher a person is referring to when they talk about their bad experiences in math. You want to
be the trailblazer and provide direction to others on how to plan and teach math. Collaborate with others (why we brought you here).
Think about what traditional math lesson plans look like (model, practice, test). For new ways to teach we need new ways to plan. We
must start with the students.
The continuing education of teachers of mathematics should provide them with opportunities to• examine and revise their assumptions about the nature of mathematics, how it should be taught, and how students learn
mathematics;
• observe and analyze a range of approaches to mathematics teaching and learning, focusing on the tasks, discourse,
environment, and assessment;
• work with a diverse range of students individually, in small groups, and in large class settings with guidance from and in
collaboration with mathematics education professionals;
• analyze and evaluate the appropriateness and effectiveness of their teaching;
• develop dispositions toward teaching mathematics.
Standard 6: Teachers' Role in Professional Development
Notes: Beyond the classroom teachers must evolve as participants in a wider educational community – read, talk with colleagues,
initiative to press for changes, speak out on current issues. For change to occur, teachers need to:
-
Have a perturbation
Have a commitment to change
Construct a vision of what the classroom could be
Project themselves into that vision
Decide to make a change within a given context
Be a reflective practitioner by comparing their practice with their vision
(Shaw & Jakubowski, 1991, p.13)
Teachers can take an active role in their professional development through such activities as•
•
forming special-interest groups within their schools to investigate ways technology might better enhance their teaching;
participating in summer programs to learn new topics in mathematics such as statistics or discrete mathematics;
•
•
•
meeting with teachers from neighboring school districts to explore how they can jointly offer advanced mathematics courses
for their students via telecommunications;
working on curriculum renewal with other mathematics faculty to change the nature and kinds of courses that are being offered
and align their program with the Curriculum and Evaluation Standards;
joining local mathematics associations, attending meetings, making presentations, and assuming leadership roles.
Teachers of mathematics should take an active role in their own professional development by accepting responsibility for• experimenting thoughtfully with alternative approaches and strategies in the classroom; Look at student invented algorithms?
• reflecting on learning and teaching individually and with colleagues;
• participating in workshops, courses, and other educational opportunities specific to mathematics;
• participating actively in the professional community of mathematics educators;
• reading and discussing ideas presented in professional publications;
• discussing with colleagues issues in mathematics and mathematics teaching and learning;
• participating in proposing, designing, and evaluating programs for professional development specific to mathematics;
• participating in school, community, and political efforts to effect positive change in mathematics education.
• Schools and school districts must support and encourage teachers in accepting these responsibilities.
Part 4. Challenging Beliefs
There is a large body of mathematics education literature concerning teachers’ beliefs and their impact on practice (see Thompson,
1992, and Philipp, 2007 for syntheses). Many of these studies seek to explain this phenomenon by investigating teachers’ systems of
beliefs and the ways certain beliefs influence practice under particular circumstances (Leatham, 2006; Skott, 2001).
Essentially what we know from this research is that beliefs about mathematics and beliefs about teaching and learning mathematics
impact what a teacher plans, teaches, and assesses as well as how he/she interacts with students in a mathematics classroom.
Therefore, it is imperative that we attempt to get teachers to become more explicitly aware of their beliefs about mathematics and
about teaching and learning mathematics.
Two activities are used to help facilitators become more explicit about their beliefs: 1. Draw a mathematician at work and 2. If
mathematics were an animal, what would it be and why. (Handouts 5 and 6)
For the first activity, ask facilitators to think of a mathematician at work. Ask them to think about what this person is doing and where
this person is as well as the tools this person is using. Once they have had time to draw what they imagined, spend some time having
people share their drawings. In particular, look for drawings that show only one person – which could mean that the drawer thinks of a
mathematician as someone who works alone – and this is not necessarily true! Look for drawings of mathematicians that demonstrate
some kind of stereotype, like crazy hair, thick glasses held together with tape, or a pocket protector. Ask facilitators to speculate what
beliefs might underlie the choices made by the drawer. End by making the point that the intent of this activity is to try to draw out
some beliefs that may be held implicitly and to challenge those beliefs.
The second activity has the same purpose – to draw out beliefs that may be held implicitly. Make sure to have facilitators think about
why they drew the particular animal they drew because oftentimes we may assume the reasons and we may very likely be incorrect.
For example, when I used this activity with elementary children, one student drew a monkey. I made the assumption that his reason
was because he thought of math as mimicking what the teacher does or “monkey see monkey do”. Surprisingly his reason was because
he likes math and he likes monkeys because he thought both were “fun”.
At this point, ask the facilitators to identify 3-5 beliefs they have about mathematics and about teaching and learning mathematics.
Take some time to discuss their beliefs and why they think they hold those particular beliefs.
Share the following quotes from classroom teachers to demonstrate that teachers’ beliefs about mathematics and about teaching and
learning of mathematics can change:
Kathy Statz, Third-Grade Teacher: I got good grades in high school algebra. I learned the procedures that the teacher
demonstrated. I thought that was what mathematics was about; if you could memorize a procedure then you could do math. I
didn’t even know that I didn’t understand math because I didn’t know that understanding was part of math….I have become a
confident problem solver by working to understand my kids’ strategies (Thinking Mathematically, pp. 74-75).
Jim Brickweddle, First/Second-Grade Teacher: There are things [in math] that I don’t understand. I am a learner along with my
students. Most of us teachers don’t use the invented algorithms that the kids do. Teachers who don’t have a broad
understanding of math might end up restricting kids who are thinking outside the box. I saw a teacher ask a child to solve 20 x
64. The child said, “It will be easier if 20 is 4 times 5, then I can find what 5 64s are then add that 4 times.” The teacher wasn’t
sure if this would work. I tell teachers, you need to be comfortable with feeling uncomfortable (Thinking Mathematically, pp.
111-112).
How teachers interact with their students can also provide a window into understanding their beliefs. We start with the following task:
How would you respond to this student who answered the following task as shown?
Which of the following helps you with 12 – 7 = ?
a. 12 + 7 = 19
b. 2 + 5 = 7
c. 5 + 7 = 12
d. 4 + 5 = 9
Have facilitators share their possible responses and reasons for their responses. Share the following possible responses and ask what
kinds of beliefs are underlying these responses:
• No. You should have chosen (c) because it is part of the fact family.
• No. Choose another answer.
• What is 12 – 7? How did you get your answer?
• Explain how 2 + 5 = 7 helps you find the answer.
• Can you show me your reasoning using materials?
You want them to understand that how you respond gives a glimpse into
• how you are listening,
• what you are listening for,
• what you ignore, and
• your beliefs about mathematics, mathematics teaching and learning.
It also sends a message to students about what is important in your classroom.
Share with the facilitators the following transcript between the second grader (C) and teacher (T) so that they understand why the child
chose answer b.
T: What is 12-7?
C: 5
T: How did you get that?
C: Well, I know 12 is 2 away from 10, so I broke 7 into a 2 and a 5. Then I took away 2 from both 12 and 7 so that I had a 10
and a 5. I know what 10 - 5 is. It’s 5.
T: Why did you choose (b)?
C: Because it’s 2 + 5 = 7 and I used those numbers to find the answer.
A framework that can help teachers think about how they are interacting with their students is from Davis (1996) and described
different ways of listening (Handout 7). Below is a brief overview of the framework along with the different beliefs that can be
aligned with each kind of listening.
Different Ways of Listening
Evaluative
• Response seeking
• Listening for particular responses
• Set learning trajectory
Possible Corresponding Beliefs
Evaluative
• Math is about getting answers
• Strategies to get answers are decided by the teacher
Interpretive
• Information seeking
• Making sense of students’ sense-making
• Listening for particular responses
• Set learning trajectory
Interpretative
• Math is about making sense
• Reasoning is big part of mathematics learning and
assessment
• Strategies to get answers could be decided by the student
Hermeneutic
• Moving with the students
• Mathematical ideas are locations for exploration
• Student contributions essentially direct the learning
trajectory of the class
Hermeneutic
• Math is about exploring ideas and making sense
• Teaching math is about capitalizing on students’ ideas
• Strategies to get answers are decided by the student
We have embedded various activities and tasks in the Numbers and Computations courses to help teachers become more aware of
their beliefs (e.g., class activities, reading Young Mathematics at Work, responding to blogs, observing videos, interviewing children).
Facilitators need to look for opportunities to engage teachers in discussions about their beliefs in a supportive, not accusatory, way.
Part 5: Mathematical Proficiency
The National Council of Teachers of Mathematics in Principles and Standards of
School Mathematics (2000) identified five process standards for doing
mathematics: problem solving, reasoning and proof, communication, connections,
and representations. One of the purposes of the process standards was to provide
ways for teachers to think about how to engage students as they learned
mathematics. The process standards were adopted by many states, including
Virginia, in designing their mathematics standards. In an effort to identify the kind
of mathematical understanding we want students to possess, Adding It Up (2001),
a publication by the National Research Council, describes five strands of
mathematically proficiency: (1) conceptual understanding, (2) procedural fluency,
(3) strategic competence, (4) adaptive reasoning, and (5) productive disposition.
(Handout 8) The strands of mathematical proficiency are interwoven and
interdependent so the development of one aids the development of others (Figure
1). Below are brief descriptions of each of these stands:
Figure 1
Conceptual understanding. Conceptual understanding is knowledge about the relationships or foundational ideas of a topic.
Procedural fluency. Procedural fluency is knowledge and use of rules and procedures used in carrying out mathematical processes
and also the symbolism used to represent mathematics. A student may choose to use the traditional algorithm or an invented approach
whichever is most efficient given the numbers in the situation.
The ineffective practice of teaching procedures in the absence of conceptual understanding results in a lack of retention and
increased errors. Think about the following problem: 40,005 – 39,996 = ___. A student with weak procedural skills may launch into
the standard algorithm, regrouping across zeros (this usually doesn’t go well), rather than notice that the number 39,996 is just 4 away
from 40,000, and 5 more mean the different is 9.
Strategic competence. In solving problems, a student who demonstrates strategic competence can use a variety of representations
and strategies to explore a task and is able to determine if an approach is not productive he can shift to try something else.
Adaptive reasoning. A student who demonstrates adapative reasoning is able to reflect on his work, evaluate it, and then adapt, as
needed.
Productive disposition. Someone who has a productive disposition is committed to making sense of and solving tasks, knowing
that if he keeps at it, he will be able to solve the problem at hand.
Understanding requires the development of all the strands as each strand includes different ideas and skills that together help students
think about mathematics. Understanding in the form of mathematical proficiency is our primary goal in the Number and Computation
courses.
Part 6: Constructivism
Ask facilitators what they know about constructivism and engage in a discussion with the group. What follows is a brief description of
constructivism that may be helpful in facilitating your discussion.
Constructivism is a theory of learning that embraces the notion that learners are not blank slates but rather creators (constructors) of
their own learning. All people, all of the time, construct or give meaning to things they perceive or think about. To get a notion of
what it means to construct an idea, consider Figure 2, a diagram that models a small section of our cognitive makeup. The red and blue
dots represent ideas. The lines joining the ideas represent our logical connections or relationships that have developed between and
among ideas. The red dot is an emerging idea, one that is being constructed. Whatever existing ideas (blue dots) are used in the
construction will be connected to the new idea (red dot) because those were the ideas that gave meaning to it. The more existing ideas
that are used to give meaning to the new one, the more connections will be made, creating integrated networks of ideas.
Figure 2
One way that we can think about understanding is that it exists along a continuum (Figure 2) from a relational understanding—
knowing what to do and why—to an instrumental understanding—knowing something rotely or without meaning (Skemp, 1978). You
can imagine, in terms of constructivism, that with relational understanding each new idea (red dot) is connected to many existing ideas
(blue dots) so there is a rich set of connections. The understood idea is associated with many other existing ideas in a meaningful
network of concepts and procedures. With instrumental understanding, at the other end of the continuum, ideas are completely isolated
or nearly so. Here we find ideas that have been rotely learned. Due to their isolation, poorly understood ideas are easily forgotten and
are unlikely to be useful for constructing new ideas.
A mathematical example of instrumental understanding of the idea 7 x 8 = ?.
A student knows the numbers 5,6 and 7,8 go in that order. So, he remembers that those numbers “go together” so he can
remember that 7 x 8 = 56.
A mathematical example of relational understanding of the idea 7 x 8 = ?.
A student knows several different ways to determine that 7 x 8 = 56:
I know five 8’s is 40 and two 8’s is 16.
3 times 8 is 24, double that to get 48. I need one more 8 to get 56.
40 and 16 are 56.
10 x 8 is 80. Take away 3 8’s or 24 is…60, then 56.
7 x 8 = 8 x 7 and is the same as finding 8 7s.
Seven 7’s are 49, so all I need is one more 7.
7 x 8 = 8 x 7 and is the same as finding 8 7s.
4 times 7 is 28. Double that would be 56.
7 x 8 = 8 x 7 and is the same as finding 8 7s.
7 + 7 = 14 (2 7s)
2 x 14 = 28 (4 7s)
2 x 28 = 56 (8 7s)
Our goal is for children to develop relational understanding of the ideas they are learning so that each new mathematical idea is
embedded in as rich a web of related mathematical ideas as is possible. Our example above clearly illustrates such a rich web of
connected ideas.
Part 7: Dealing with Resistant Teachers
A common concern for new facilitators of professional development is dealing with resistant teachers. One way to deal with this issue
is to consider what might be the origin of the resistance. Sometimes teachers are fearful of change because it is uncomfortable not
knowing. Sometimes teachers find making any changes overwhelming. Be accepting of whatever causes teachers to be resistant and
encourage them to talk about it. This provides you with more information to better understand the causes of the resistance and so can
allow you to identify possible solutions. Possible approaches to help teachers move beyond the resistance are:
1. Make it less overwhelming.
• Ask teachers to choose a part of their practice to focus on one semester.
• Describe change as a process, not an event.
• Describe the process of professional development and change as a marathon not a sprint.
2. Share what we know from research and international studies about focusing on teaching and learning mathematics for
understanding.
3. Keep the focus on the students and what is best for them in terms of learning for understanding.
Part 8: Professional Standards for Teaching Mathematics
Professional Standards for Teaching Mathematics (NCTM, 1991) (Handout 9)
Worthwhile Mathematical Tasks – This is not just finding a good problem to solve. Teachers must base their decisions on 3 areas
of concern – the math content, the students, and the ways in which students learn mathematics.
Teachers complete a task and share various ways they solved it. Then share student work and examine and analyze student responses.
What do the students’ responses reveal about students’ mathematical understandings and misconceptions?
Give the Hiking Club Problem (Handout 10a) and ask teachers to solve it anyway they would like. If manipulatives are not available,
encourage teachers to draw a picture. Ask participants to share their solutions. Next go back and consider the following questions:
What is the purpose of the problem?
Highlight specific processes – Problem Solving – a problem to be solved – Communication – must communicate mathematical
ideas with other folks in our group – Reasoning and Proof – must justify our responses to our group members, the members of
the class, and the teacher – Represent – the problem using concrete materials.
What prior knowledge and experiences can students draw on to solve the problem?
Are the students members of any clubs? Are they in the band? Could you change the problem to make it fit something more
interesting to them? When I taught at the College of Charleston, climbing a mountain would not have been of interest to them,
but a surf club might.
What mathematics do the students need to know to solve the problem?
Fractions represent part of a whole. If students were asked to only teach this problem procedurally, and they had only been
taught a specific procedure, they would have to have the math knowledge of adding and subtracting fractions with unlike
denominators, finding common denominators, least common multiples, determining equivalent fractions, subtracting a fraction
from a whole, solving for a variable, and multiplying a fraction by a whole number.
How will I present this problem?
Don’t just give the problem and expect a miracle to happen. ‘Use any of the materials provided to demonstrate how you could
find the answer to this problem.’ In particular I would like students to solve the problem without using conventional
procedures. How might we use the tools available to solve the problem?
What questions will I ask struggling students?
Think ahead of time about what difficulties the task might present for students and how they could help common
misunderstandings. Difficult to figure out what students might do without considering multiple representations. Clarifying
questions for fraction bar problem:
1. What does your 1 whole represent?
2. Do the subsequent fractions (⅔, ½, and ¼) represent parts of the same whole?
Fraction Circles:
3. What happens if we add ⅔ and ½? Can we have more than 1 whole?
4. Is the ¼ part of the ½ or ⅔?
How might students solve the problem?
Share 3 different methods for how students might solve the problem (Handouts 10a and 10b). Included with these documents is
the work of 2 different students who were ‘stuck’. They should be used to provide an opportunity for participants to address
the questions they could ask to help get the student ‘unstuck’ without simply telling the student what to do.
Teacher’s and Student’s role in Discourse
The discourse of a classroom - the ways of representing, thinking, talking, agreeing and disagreeing - is central to what students learn
about mathematics as a domain of human inquiry with characteristic ways of knowing. Discourse is both the way ideas are exchanged
and what the ideas entail: Who talks? About what? In what ways? What do people write, what do they record and why? What
questions are important? How do ideas change? Whose ideas and ways of thinking are valued? Who determines when to end a
discussion? The discourse is shaped by the tasks in which students engage and the nature of the learning environment; it also
influences them.
The teacher's role is to initiate and orchestrate this kind of discourse and to use it skillfully to foster student learning. In order to
facilitate learning by all students, teachers must also be perceptive and skillful in analyzing the culture of the classroom, looking out
for patterns of inequality, dominance, and low expectations that are primary causes of nonparticipation by many students. Engaging
every student in the discourse of the class requires considerable skill as well as an appreciation of, and respect for, students' diversity.
(NCTM 1991 p. 35)
Learning Environment – Create a classroom where proof was important but justification and validation came from the students,
not the teacher. Pose a problem that provides an opportunity for students to think. Teachers begin to wonder how students would solve
problems for which they did not have rules or procedures available. Make sure not to interpret what students said in light of their own
understanding of a problem. If we want students to learn to make conjectures, experiment with alternative approaches to solving
problems, and construct and respond to others' mathematical arguments, then creating an environment that fosters these kinds of
activities is essential. What do they do (or could they do) to create this type of learning environment?
Analysis of Teaching and Learning – Determining what students know and can do mathematically is a critical aspect of
teaching, since it allows teachers to make instructional decisions before, during, and after teaching. Examining students’ work affords
the opportunity for teachers to move away from their own understanding of a task to a broader consideration of what students’
responses might reveal about their thinking, what difficulties such tasks might present, and how teachers might help students address
(or avoid) common misunderstandings. Student work can help teachers realize that children’s ways of interpreting, representing, and
solving problems are different from the teacher’s, but their methods may be equally valid. They facilitate teachers’ learning about
strategies that were unfamiliar to them. Share student work (Handout 11) and ask participants what they can determine about the
students’ understanding from their solutions to the problem.
Part 9: Practice-based Instruction
Smith (2001) suggests the following cycle when considering practice-based instruction.
Planning Reflec.ng Teaching Planning – The teacher decides what mathematical knowledge and processes she wants students to learn; determines the relevant
prior knowledge and experiences on which students can draw to construct new knowledge; and creates, finds, or adapts tasks or
activities that build on prior knowledge and experiences and have the potential to foster intended learning. Thoughtful planning
requires that teachers understand what mathematics children need to know in order to solve the task, recognize the mathematics
embedded in a task, and match their goals for the students’ learning with tasks that have the potential for achieving the goals. The task
provides and interesting context for discussion. Tasks can be selected to specifically highlight process standards. We want to move
away from adult ways of doing math to think about how children construct mathematical meanings and inform instruction.
Teaching – Teachers must understand the math concepts that they teach. During the act of teaching the teacher must engage students
in the task or activity, make mid-activity alterations as needed to fit the needs of the students, provide ‘scaffolding’ for students’
learning to sustain their engagement in the math activity and formally and informally assess what students are learning. Must analyze
and critique episodes of our own teaching.
Reflecting – The teacher must consider the level and kind of thinking in which the majority of students engaged during a lesson and
what students did and said that suggested understanding of important mathematical ideas. They also must consider the ways in which
the teaching may have supported or inhibited students’ engagement with the tasks as intended. Did the teacher provide sufficient
opportunity for developing thinking, reasoning, and problem solving skills as well as other, more basic skills.
Planning – The cycle begins again with planning the next lesson.
Questions to consider when planning:
1. What opportunities to learn mathematics are afforded by the task?
2. What prior knowledge and experience would students need in order to engage in the task successfully?
3. How would you expect students to go about solving the task?
Questions to consider when thinking about teaching (as if viewing a videotape of teaching):
1. What decisions did the teacher make during the lesson?
2. What decisions were made by the students?
3. Who validated answers?
4. Who asked the questions?
5. What was the nature of the questions asked by the students? The teacher?
Questions to consider when analyzing what students seemed to be learning and how they learned it:
1. What were the mathematical ideas with which students appear to grapple?
2. What do students’ solutions tell about what they know and understand?
3. What factors appeared to support students’ engagement in the mathematical activity?
4. What factors seem to hinder such engagement?
Questions to consider when planning the subsequent lesson:
1. What should be the mathematical target of instruction of the next lesson?
2. What knowledge have students demonstrated that will serve as a foundation for constructing new knowledge?
3. What task would accomplish the learning goal?
(Smith, M.S., 2001, pgs. 8-10)
Part 10: Revisit Joys/Concerns/Questions
Take some time at the end to give your teacher-facilitators an opportunity to request more support for their questions/concerns/joys
they shared on index cards during the initial session or to identify additional ones that have emerged in subsequent sessions. Your
teacher-facilitators need to feel prepared and supported so they can in turn provide support for their teacher-learners. Allowing them
opportunities to share concerns as well as joys can help develop a cooperative community within your facilitators who will look to
each other for ideas and support.
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