Technology Reading – #3 (continued…)

Source:
 
National Research Council of the National Academies.  (Ed.). (2005). How Students Learn. Washington D.C.: The National Academies Press.  (Chapters Introduction, 5-8).

Chapter 5 – Mathematical Understanding: An Introduction (Karen C. Fuson, Mindy Kalchman, John D. Bransford

• RQs
  1. How do the principles of How People Learn apply to mathematics?
• Notes

1.      How Students Learn Mathematics

1.1.   Rarely taught with the 3 principles in mind.

1.2.   Currently students are learning mathematics as mastering procedures with no meaning behind it.

1.2.1.      Instead of connecting, and building upon, and refining mathematical understandings (principle 1), instruction overrides reasoning processes, replacing them with rules and procedures that disconnects problem solving from meaning making.

1.2.2.      Instead of organizing skills and competencies around core concepts (principle 2), the skills and competencies are the center of and/or whole of instruction.

1.2.3.      Procedural knowledge is divorced from decision making, students do not use metacognitive strategies (principle 3) when solving problems.  (p.217)

1.3.   “Mathematical Proficiency” instead of mastery of procedure, constituted by the following strands.  (AddingIt Up Report from the National Research Council)  These strands map to the principles.

1.3.1.      Conceptual Understanding – comprehension of mathematical concepts, operations, and relations.

1.3.2.      Procedural Fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

1.3.3.      Strategic Competence ­– ability to formulate, represent, and solve mathematical problems.

1.3.4.      Adaptive Reasoning – capacity for logical thought, reflection, explanation, and justification.

1.3.5.      Productive Disposition ­– habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one’s own efficacy.  (p.218)

2.      Principle #1: Teachers must engage students preconceptions

2.1.   At a very early age, children begin to develop the awareness of number and it is universal though the rate differs depending on environmental influences.

2.2.   People possess resources in the form of informal strategy development and mathematical reasoning that can serve as a foundation for learning more abstract mathematics.  But the connections are not automatic, meaning there is no bridge between informal and formal mathematics.

2.2.1.      Brazillian street children can perform mathematics when making sales in the street, but were unable to answer similar problems when presented in a classroom context.

2.2.2.      Californian housewives are able to price compare and solve mathematical problems when shopping, but unable to solve similar problems presented abstractly in the classroom. 

2.2.3.      Men who handicapped horses could not apply the same skill to the stock market.

2.3.   Need to build on existing knowledge and engage students’ preconceptions, especially when they interfere with learning.

2.4.   Some preconceptions are fostered in early childhood and are very counterproductive and may lead to students believing mathematics is “not for them”. (p. 219)

2.4.1.      Preconception #1: Mathematics is about learning to compute.

2.4.1.1.Example: What, approximately, is 8/9 plus 12/13?

2.4.1.2.Most people want to find the common denominator, which makes the problem difficult, instead of just realizing that both numbers are close to one and thus the approximate answer is 2.

2.4.1.3.Mathematics is about problem solving and computation is a tool for use to that end when it is helpful.  This always “sense making” to take place.

2.4.2.      Preconception #2: Mathematics is about “following rules” to guarantee correct answers

2.4.2.1.  Mathematics is viewed as a cut-and-dried discipline that specifies rules for finding the right answers. (p. 220)

2.4.2.2.  In reality, mathematics is a constantly evolving field involving systematic pattern finding and continuing invention.

2.4.2.3.History of mathematics illustrates that what is taught in one point in time as a set of procedures really was a set of clever inventions designed to solve pervasive problems of everyday life.  (Examples: abacus and calculators)

2.4.2.4.If mathematics procedures are understood as inventions designed to make common problems more easily solvable, and to facilitate communications involving quantity, those procedures take on new meaning. 

2.4.3.      Preconception #3: Some people have to ability to “do math” and some don’t.

2.4.3.1.Very widespread in the United States, but not necessarily other countries.

2.4.3.2.Can easily become self-fulfilling prophecy. (p. 221)

2.4.3.3.The United States is more likely to assume that ability, not effort, is more important to learning mathematics, and that is socially acceptable not to put forth effort. 

2.4.3.4.Teachers in the United States are less likely than their counterparts to let students struggle for a while with problems, and simplify them so that the students don’t struggle at all.

2.4.3.5.Without conceptual understanding of the nature of problems and strategies for solving them, failure to retrieve learned procedures (rules forgotten after summer vacation) can leave a student completely at a loss.

2.4.3.6.Students also feel lost when they don’t “get it” from the beginning.

2.4.3.6.1.      If students learn to memorize procedures, they can be baffled by the unexplained. 

2.4.3.6.2.      Providing a secure conceptual understanding of mathematics that is linked to students’ sense-making capabilities is critical so that students can puzzle productively over new material, identify the source of their confusion, and ask questions when they do not understand.  (p. 222)

2.5.   Engaging students’ preconceptions and building on existing knowledge

2.5.1.      Perspective includes an understanding that the rules for computation and solution are a set of clever human inventions that in many cases allow us to solve complex problems more easily, and to communicate about those problems with each other more effectively and efficiently.

2.5.2.      How can we link formal mathematics training with students’ informal knowledge and problem-solving capacities?

2.5.3.      Features include:

2.5.3.1.Allowing students to use their own informal problem-solving strategies, at least initially, and then guiding their mathematical thinking to more effective strategies and advanced understandings.

2.5.3.2.Encouraging math talk so that students can clarify their strategies to themselves and others, and compare the benefits and limitations of alternate approaches.

2.5.3.3.Designing instructional activities that can effectively bridge commonly held misconceptions and targeted mathematical understandings.

2.5.4.      Allowing multiple strategies (p. 223)

2.5.4.1.If students believe there are multiple ways to solve a problem their engagement strategy is kept alive.

2.5.4.2.It does not mean that all strategies are equally good, but students can learn to evaluate different strategies for their advantages and disadvantages.

2.5.4.3.A wrong answer is usually partially correct and reflects some understanding, so finding the part that is wrong and understanding why it is wrong can be a powerful tool to understanding and promotes metacognitive competencies.

2.5.4.4.Understanding new methods can be worthwhile mathematical project for a class, and others can be involved in trying to figure out why the method works.

2.5.4.4.1.      Illustrates how a classroom community can function.

2.5.4.4.2.      Demonstrates that not all mathematical issues are solved or understood immediately; sometimes sustained work is necessary. (p. 224)

2.5.5.      Encouraging math talk

2.5.5.1.Students and teachers actively discuss how they approached various problems and why.

2.5.5.1.1.      Can help everyone in the classroom understand a given concept or method because it highlights contrasting approaches.

2.5.5.1.2.      Also facilitates other sorts of learning.

2.5.5.1.3.      Helps teachers become more learning focused and make stronger connections with each student.

2.5.5.1.4.      Can provide springboard for further instruction, enabling them to extend thinking more deeply of understand and correct errors.

2.5.5.2.Students need to feel comfortable expressing their ideas and revising their thinking when feedback suggests the need to do so.

2.5.5.3.Allows teachers to draw out and work with preconceptions students bring with them to the classroom and helps students learn how to do this sort of work for themselves and others.

2.5.5.4.Making drawings (and explanation of drawing) also helps and is a form of communication. 

2.5.5.4.1.      Can help students reapply concepts and methods.

2.5.5.4.2.      Can help students figure out problems when errors creep back into their methods.

2.5.5.5.Teachers need to use carefully designed visual, linguistic, and situational conceptual supports to help students connect their experiences to formal mathematical words, notations, and methods.  (p. 228)

2.5.5.5.1.      Modeling language and help students use it in their discussions.

2.5.5.5.2.      More advanced students help less advanced students to learn by modeling, asking questions, and helping others.  (p. 229)

2.5.6.      Designing bridging instructional activities

2.5.6.1.Research has uncovered common student preconceptions and points of difficulty with learning new mathematical concepts that can be addressed preemptively with carefully designed instructional activities.

2.5.6.2.A teacher or curriculum designer can make a framework for a math domain by selecting conceptual supports that will help students make links among math words, written notations, and quantities in that domain. 

2.5.6.3.Identifying real-world contexts whose features help direct students’ attention and thinking in mathematically productive ways is particularly helpful in building conceptual bridges between students informal experiences and new formal mathematics.

3.      Principle #2: Understanding requires factual knowledge and conceptual frameworks

3.1.   Conceptual understanding and procedural fluency, as well as effective organization of knowledge that facilitates strategy development and adaptive reasoning.  (p. 231)

3.2.   Recognition in the weakness of conceptual understanding of students in the U.S. has resulted in increased attention to the problems involved in teaching mathematics as a set of procedural competencies.

3.3.   At the same time, students with too little procedural knowledge do not become competent and efficient problem solvers.

3.4.   Both factual and procedural knowledge are critical and must be closely linked, especially because mathematics will become more complex as a student progresses through the school years and students will need to build on those previous skills.

3.5.   Teacher’s challenge is to help students build and consolidate prerequisite competencies in a network of knowledge.  They also must provide sustained and spaced opportunities to consolidate new understandings and procedures.

3.6.   Networks of knowledge are organized as learning paths from informal concrete methods to abbreviated, more general, and more abstract methods. 

3.6.1.      Discussing multiple methods in class can help provide students a conceptual ladder that helps them move in a connected way from where they are to a more efficient, abstract approach.

3.6.2.      Students can adopt an intermediate procedure that they can explain so they are comfortable until they are able to move to the next step.

3.7.   Developing mathematical proficiency

3.7.1.      Requires students to master both concepts and procedural skills needed to reason and solve problems effectively in a particular domain.

3.7.2.      Deciding which advanced methods students should learn to attain proficiency is a policy matter involving judgment about how to use scarce instructional time.  (p. 232)

3.7.3.      Deciding methods must also take into account which methods are clearer conceptually and procedurally.

3.7.4.      Teachers do not always need to teach multiple methods, but they will arise frequently in the classroom, especially because students bring them from home or think differently about mathematical problems.  Discussing the different methods can help produce deeper understanding and flexibility.

3.7.5.      For less-advanced students, it might helpful to select an accessible method that can be understood and is efficient enough for the future, and for those students to concentrate on learning that method and be able to explain it.  It is possible to do this while facilitating problem-solving with alternate methods.

3.7.6.      Instruction to support mathematical proficiency

3.7.6.1.To support conceptual understanding and procedural fluency, primary concepts underlying an area of mathematics must be clear to the teacher.

3.7.6.2.Because math has been taught with emphasis on procedure, adults taught this way might have difficulty identifying or using core conceptual understandings in a math domain.  (p. 233)

3.7.6.3.Major shifts occur in whole numbers (quantity), rational numbers (proportion and relative number), and functions (dependence in quantitative relationships).  That is the basis of this book for mathematics.  (p. 234)

3.7.6.4.Balance must be maintained between learner-centered and knowledge-centered needs and must continue to relate to individual learner knowledge.  (p. 235)

4.      Principle #3: A metacognitive approach enables student self-monitoring.

4.1.   Learning about oneself as a learner, thinker, and problem solver is an important concept of metacognition.  (p. 236)

4.2.   Sometimes wisdom can not be simply parted from teacher to student.

4.3.   Students’ experiences have strong effects on their beliefs about themselves and their abilities to remember info and use it to solve problems.

4.4.   If a student is primarily frustrated, simply telling him he can do it will not have a great effect. 

4.5.   Instead, help student experience their own abilities and find patterns in problems, invent solutions, and contribute to and learn from discussions with others.

4.6.   For optimal learning, students need to reflect on their experiences and begin to see their ideas as instances of larger categories of ideas.

4.7.   “Number Sense”  (p. 238)

4.8.   Instruction that supports metacognition

4.8.1.      Making students’ thinking visible can be thought of as ongoing assessment.

4.8.2.      Assessment can include students so they become involved in thinking about their own mathematical progress.  This can be internalized as metacognitive self-monitoring. 

4.8.3.      Classroom communication about students’ mathematical thinking greatly facilitates both teacher and student assessment of learning.

4.8.3.1.Teachers can discern primitive solution methods.

4.8.3.2.Teachers can see particular difficulties.

4.8.3.3.Teachers can see how students are advancing in their helping and explaining abilities.

4.8.3.4.Students can learn some general problem solving strategies that apply to many different problems, such as “make a drawing” and “ask yourself questions” which are a means of self-monitoring.

4.8.3.5.Students can offer teachers a view into their thinking and provide info about how to better help along a learning path to efficient problem solving methods.

4.8.4.      Emphasis on debugging

4.8.4.1.Metacognitive functioning is also facilitated by shifting from a focus on answers that are right or wrong to a focus on “debugging” a wrong answer.

4.8.4.1.1.      Useful in complex problem solving (such as computer programming).

4.8.4.1.2.      Technological advances mean that more adults will need to do more complex problem solving and error identification.

4.8.4.1.3.      Good skill to have that can be learned in a mathematics classroom.

4.8.4.1.4.      Focus on understanding can help students debug their own errors.  (p. 238)

4.8.5.      Internal and external dialog as support for metacognition

4.8.5.1.Very important for students to communicate about mathematics and for teachers to help them learn to do so.

4.8.5.1.1.      Students can learn to reflect on and describe their mathematical thinking.

4.8.5.1.2.      They can learn to compare advantages and disadvantages of different methods of solving a problem.

4.8.5.1.3.      They can learn to ask thoughtful questions of other student thinking.

4.8.5.1.4.      They can learn to help each other in informal and formal ways.

4.8.5.2.Teachers must help students learn to interact properly and in a manner that is useful to the class.

4.8.5.3.The goal in all of this discussion is to advance everyone’s thinking and monitoring of their own understanding and that of other students.

4.8.6.      Seeking and giving help

4.8.6.1.Students must have enough confidence to not only to engage with problems and try to solve them, but also seek help when they are stuck.  (p. 241)

4.8.6.2.Students helping each other is helpful in developing metacognitive awareness of all parties.

4.9.   The Framework of How PeopleLearn: Seeking a balanced classroom environment

4.9.1.      Effective teaching and learning depend on balance between learner-centered, knowledge-centered, assessment-centered, and community-centered classroom environments.

4.9.2.      Overemphasis on learning-centered teaching results in insufficient attention to connection with valued knowledge networks, crucially important guiding roles of teachers and of learning accessible student methods, and the need to consolidate knowledge.

4.9.2.1.Some suggest that students invent all mathematical ideas and that we should wait until they do so rather than teach ideas. 

4.9.2.1.1.      Ignores the fact that all inventions are made within a supportive environment and that providing appropriate supports (scaffolding) can speed such inventions.

4.9.2.1.2.      Can hold students back and they need to be helped to more generalized “good enough” methods.

4.9.2.2.Balance means focus should be on sense making and understanding of the methods, not students inventing.  (p.242)

4.9.3.      Classroom discussions may not be sufficiently guided by the teacher through the learning path and go on aimlessly, and thus balance and direction is needed.

4.9.4.      Use of real-world situations and conceptual supports may not be connected enough to standard math notation.  Need a careful approach to the choices of core representations or bridging contexts that might guide students through a coherent learning path.

4.9.5.      Learning may not be consolidated enough because of excessive focus on initial learning activities.  Must make time for consolidation of learning with feedback loops (for errors) in order to attain mathematical fluency. 

5.      Teaching Communities

5.1.   The U.S. teaching style is still overwhelmingly traditional.  Making a move toward the principles in How People Learn can be aided by evaluating ones own teaching practices and participating in forms of teaching communities of like-minded colleagues.

5.1.1.      Can help in the creation of learning paths from traditional teacher to being more focused on conceptual understanding.

5.1.2.      Can organize discussions around issues that arise from teaching a curriculum that supports the conceptual manner.

5.1.3.      Other examples: video clubs, lesson studies.  (p.243)