Van Enger's advice: Show the students (or let them perform) the actions implied by operations on objects. As you talk, do not expect the students to learn without observing the actions on the objects. Then help students symbolize the actions, and later generalize to larger numbers for which the actions on objects become awkward. At this stage the structure of mathematics, or the generalization of the operation, should allow the student to work symbolically (xi).
What is UNDERSTANDING?
- Knowing a subject means getting inside it and seeing how things work, how things are related to each other, and why they work like they do (2)
- Understanding is one of the most intellectually satisfying experiences (not understanding is one of the most frustrating)
- Creates internal rewards and engagements (not understanding creates little sense of satisfaction and likely withdrawal from learning)
- Understanding breeds confidence and engagement (not understanding leads to disillusionment and disengagement)
- Complex - it is not something that you have or do not have. It's a continuum.
- DEFINITION: we understand something if we see how it is related or connected to other things we know. The more relationships we can establish, the better we understand.
- Two processes that plan an important role in making of connections: reflection (thinking) and communication (talk, listen, write, demonstrate, watch, etc)
- By communicating we can think together about ideas and problems...so we often can accomplish more than if we worked alone.
- Communication works together with reflection to produce new relationships and connections.
- Evidence Required: reasoning clearly, communicating effectively, drawing connections without mathematics and between mathematics and other fields, and solving real problems. (Often in the form of explanations of why things are like they are; explanations are filled with useful connections)
Two types of understanding:
- Relational Understanding: knowing what do to and why (goal)
- Students often rebel when we try to show them why something works or attempt to develop relational understanding. We must show the rationale. Top reasons to shoot for understanding:
- gives assurance of retention
- equips with a means to rehabilitate skills quickly that are temporarily weak
- increases the likelihood that arithmetical ideas and skills will be used
- contributes to ease of learning by providing a sound foundation and transferable understandings
- reduces the amount of repetitive practice necessary to complete learning
- safeguards from answers that are mathematically absurd
- encourages learning by problem solving in place of unintelligent memorization and practice
- provides a versatility of attack which enables to substitute equally effective procedures for procedures normally used by not available at that time
- creates relative independence to face new quantitative situations with confidence, allows for transfer to new tasks
- presents the subject in a way which makes it worthy of respect (x)
- MORE: things learned with understanding...
- can be used flexibly
- can be adapted to new situations
- can be used to learn new things
- are most useful in a changing and unpredictable world (1)
- Instructional Understanding: knowing what to do or the possession of a rule and ability to use it.
- Quick Answers
- Remember steps of a algorithm
Questions to Ask:
- What do I mean by understanding?
- How do I know my students understand?
- What connections am I looking for?
- What communication to I expect?
- Does this task encourage reflection and communication?
- How does the social culture of my school affect the culture in my classroom?
- Are there other mathematical tools needed in my classroom?
- How do I use mistakes as sites to encourage learning by everyone?
- How do I encourage students to share essential information?
- How do students determine correctness of their mathematics?
- How do I involve each and every student in the sharing of their development of mathematical knowledge?
Dimensions and Core Features of classrooms that create understanding:
- The nature of Classroom Tasks define for students the nature of the subject and contribute significantly to the nature of classroom life (7). Appropriate tasks have at least 3 features: 1) problematic for students = meaning students see the task as an interesting problem; 2) tasks must connect to where students are; 3) tasks must engage students in thinking about (reflecting and communicating) important mathematics. = Tasks leave behind something of mathematical value.
- The Role of the Teacher is shaped by the goal of facilitating conceptual understanding (8).
- sets the task with the goal in mind
- selects and poses appropriate sequence of problems as opportunities for learning
- shared information when it is essential for tackling problems
- facilitates the establishment of classroom culture in which students work on novel problems individually and interactively, discuss and reflect on their answers and methods
- knows that understanding is constructed by students through solving problems
- knows that intervening too much and too deeply can easily cut off students' initiative and creativity and can remove the problematic nature of the material.
- sets the classroom culture
- The Social Culture of the Classroom is one of a community of learners (9).
- Norms and expectations are clear
- Interacting is not optional - it is essential (communication is necessary for building understanding).
- Four features (9):
- Ideas are the currency of the classroom. Ideas have the potential to contribute to everyone's learning and consequently warrant respect and response.
- Autonomy of students with respect to the methods used to solve problems.
- Appreciation for mistakes as learning sites...places that afford opportunities to examine errors in reasoning, and thereby raise everyone's level of analysis. Mistakes are used constructively.
- Correctness resides int he mathematical argument, rather than in the social status of the participants. Persuasiveness of an explanation or the correctness of a solution depends on the mathematical sense it makes...
- Mathematical tools (oral language, written notation, and any other tool with which students can think about mathematics) are seen as learning supports. Students must construct meaning with them. Tools must be used to accomplish something...to solve problems. Ex: provide a convenient record, can be used to communicate more clearly, and used as an aid for thinking... Many tools should be available to support all levels of understanding (10).
- Equity and Accessibility:
- Tasks are accessible to all students
- All students are heard
- Ever student contributes
- Listening, really listening, is one of the best ways of encouraging this (11).
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