Saturday, September 27, 2014
1. Asking questions
2. Developing and using models
3. Planning and carrying out investigations
4. Analyzing and interpreting data
5. Using mathematics, information and computer technology, and computational thinking
6. Constructing explanations and designing solutions
7. Engaging in argument from evidence
8. Obtaining, evaluating, and communicating information
Tuesday, September 16, 2014
Lesh & Clark (2000) - Formulating operational definitions of desired outcomes of instruction in mathematics and science education
p127: Cognitive objectives function similarly to the ways cyclotrons, cloud chambers, and vats of heavy water are used in physics. That is, they are defined operationally by specifying: (a) situations that optimize the chances that the targeted construct will occur in an observable form; (b) observation tools that enable observers to sort out signal from noise in the results that occur; and (c) quality assessment criteria that allow meaningful comparisons to be made among alternative possibilities.
Wednesday, September 3, 2014
Instructional Activities Using Routines as Tools for Teacher Education
• Choral counting: The teacher leads the class in a count, teaching different concepts and skills by deciding what number to start with, what to count by (e.g., by 10s, by 19s, by 3/4s), whether to count forward or backward, and when to stop. The teacher publicly records the count on the board, stopping to elicit children’s ideas for figuring out the next number, and to co-construct an explanation of the mathematics that arises in patterns.
• Strategy sharing: The teacher poses a computational problem and elicits multiple ways of solving the problem. Careful use of representations and targeted questioning of students are designed to help the class learn the general logic underlying the strategies, identify mathematical connections, and evaluate strategies in terms of efficiency and generalizability.
• Strings: The teacher poses several related computational problems, one at a time, in order to scaffold students’ ability to make connections across problems and use what they know to solve a more difficult computational problem. This activity is used to target a particular strategy (as compared to eliciting a range of strategies). For example, posing 4 × 4, then 4 × 40, and then 4 × 39 is designed to help students consider how to use 4 × 40 to solve 4 × 39, developing their knowledge of compensating strategies in multiplication (Fosnot & Dolk, 2001).
• Solving word problems: The teacher first launches a word problem to support students in making sense of the problem situation, then monitors while students are working to determine how students are solving the problem, gauges which student strategies are best suited for meeting the instructional goal of an upcoming mathematical discussion, and makes judgments about how to orchestrate the discussion to meet those goals.
[a fifth IA in recent articles is Quick Images: The goal of this activity is to build students' ability to visualize a quantity.]
A Focus on Instructional Dialogue
A relatively recent focus of Leinhardt’s work on teaching routines has been how they are used in “instructional dialogue” (Leinhardt & Steele, 2005), a practice we would consider to be the centerpiece of ambitious mathematics teaching. In this kind of teaching, an explanation is co-constructed by the teacher and students in the class during an instructional conversation. Maintaining a coherent mathematical learning agenda while encouraging student talk about mathematics is perhaps the most challenging aspect of ambitious teaching. In their study of teaching through instructional dialogues, Leinhardt and Steele (2005) demonstrated the use of eight kinds of “exchange” routines in this kind of teaching to accomplish explanatory work, including maintaining mathematical clarity in the face of student inarticulateness, fixing the agenda of the class on a single student’s idea, making it safe for students to revise incorrect contributions, and honing students’ contributions toward mathematical accuracy and precision. The exchange routines that Leinhardt and Steele (pp. 143–144) identified include the following:
• The call-on routine, which is initiated by a rather open invitation to discussion and has two separate components: the initial identification of a problem and the speaker who responds, followed by a second part in which the class is prompted to analyze, justify, or critique the statement given by the first speaker or another speaker in the discussion.
• The related revise routine in which students were asked to rethink their assertions and publicly explain a new way of thinking about their solutions.
• The clarification routine “which was invoked when a confusion arose regarding an idea or conjecture volunteered into the public space, which in turn involved understanding the source of confusion.”
Cite This Article as: Teachers College Record Volume 113 Number 7, 2011, p. 1361-1400
http://www.tcrecord.org/library ID Number: 16072
Organizational Resources in the Service of School-Wide Ambitious Teaching Practice
by Magdalene Lampert, Timothy A. Boerst & Filippo Graziani
Challenges of Ambitious Teaching (pp 1364-7)
1. Teaching students to perform on authentic tasks needs to happen at the same time as teaching the basics (Kilpatrick, Swafford, Findell, National Research Council, 2001; Snow et al., 2005). Consequently, one challenge of ambitious teaching that occurs across subject matters is keeping different kinds of content on the table at the same time.
2. Assessment is a second challenge. Teachers with ambitious learning goals must do more than act reflexively on judgments of separate elements of students’ work as right or wrong according to an answer key. ... In mathematics, teachers need to know a broad spectrum of methods that students might invent to solve problems and what mathematical understanding is embedded in their inventions in order to assess competence and promote sense-making (Franke, Kazemi, & Battey, 2007).
3. A third challenge of making academically demanding work available to diverse students is adjusting teaching to what particular students are currently able to do (or not). Teachers teach a variety of different individuals in a common social setting, and in order to succeed with diverse learners, they need to find ways to “microadapt” based on continually assessing and learning about students as they teach (Corno, 2008).
4. In addition to these intellectual challenges, this kind of teaching also requires teachers across subject matter domains to manage more complex and risky forms of social organization (Cohen, 1998; Kennedy, 2007). ... adapting teaching to learning requires working in the relational space where students’ anxieties and fears can intrude on the learning process (Corno, 2008). Ambitious teachers need to lead discussions in which students learn from talking about ideas and enable students to engage productively in collaborative investigation with partners they might not chose as friends (Chapin, O’Connor, & Anderson, 2003; Kazemi, 1998; Rex, Murnen, Hobbs, & McEachen, 2002).
Social practice theory - pp. 1367-9
To understand how the challenges of ambitious teaching could be managed regularly at the level of the school, we needed to investigate not only what individual teachers could do to address them, but also what they can do routinely with their colleagues as part of a structured social system working on a joint enterprise using a common set of resources to meet common objectives. We employed the concept of a “social practice” to name this kind of system and to analyze the link between the existence of organizational resources in a school and the effects of the common work that occurs across individual teachers as they use them. A social practice takes shape as people interact with one another using the tools of their trade, developing a shared repertoire they can call upon to get their work done (Engeström & Middleton, 1998).
In clarifying how “social practice theory” is different from other kinds of cultural theories, Reckwitz (2002) emphasizes its focus on how the coordination of individual action with commonly available resources enables the coherent use of those resources.
In social practices, Reckwitz observes, these different kinds of resources interact to form a ‘block’ that cannot be reduced to a set of single elements. Social practice is more than just “talk”. It is built in the multidimensional terrain where practitioners interact in particular places in particular ways using particular objects.
Teachers who work on problems of practice together interpret what they see students doing through their common values, norms, rules, beliefs, and assumptions, and given that shared interpretation, individuals decide what to do (Little, 1982; Weick & McDaniel, 1989)
Three inter-related dynamics:
Teachers’ common commitments to ambitious goals
Teachers’ individual and collective use of resources to scaffold the practice of ambitious teaching
Teachers' social use of resources in planning and evaluations of lessons and students
SOCIAL, MATERIAL, AND INTELLECTUAL RESOURCES IN SUPPORT OF AMBITIOUS PRACTICE
In analyses of the lessons we observed, we examined in detail how individual teachers used the school’s resources to mediate the challenges of ambitious teaching in ways that were similar to what Lampert observed as a language-learning student. We noted the prevailing use of resources across routine phases of the work of teaching. Our analysis took into account three phases:
• planning: i.e., how teachers prepare for ambitious practice;
• instruction: i.e., what teachers do in interaction with subject matter and diverse students across time; and
• reflection: i.e., how teachers think about, talk about, learn from, evaluate and capture their insights about students and content from enacted practice.
During instruction, teachers use resources in social interactions with students to maintain and accomplish ambitious goals (Cohen, Raudenbusch, & Ball, 2003).
Fig 2 & 3
Listening | | |
Speaking | | |
Reading | | |
Writing | | |
In this article, we use the term “teaching” to refer to what teachers do in relationships with students and subject matter in environments. It is the teacher’s contribution to a phenomenon we will call “instruction” as defined by David Cohen, Steve Raudenbusch, and Deborah Ball: “Instruction consists of interactions among teachers and students around content in environments. . .‘Interaction’ refers to no particular form of discourse but to teachers’ and students’ connected work, extending through, days, weeks, and months. Instruction evolves as tasks develop and lead to others, as students’ engagement and understanding waxes and wanes, and organization changes (Lampert, 2001). Instruction is a stream, not an event, and it flows in and draws on environments — including other teachers and students, school leaders, parents, professions, local districts, state agencies, and test and text publishers.”(Educational Evaluation and Policy Analysis, Summer, 2003, Vol 25, no.2, p.122)