Finals

• Journal writing


  A systematic way of documenting learning and collecting information for self-analysis and reflection
• Journal writing in mathematics

  A platform where students write a mathematical-related situation in a way that reveals learning of mathematical concepts, how it was taught, the difficulties encountered, and attitude towards mathematics
• Ernig (1977) emphasized that writing processes information at the motor level, sensory level, and cognitive level
• Studies have documented positive effects of journal writing in mathematics classroom in terms of achievements, mathematical reasoning, and problem solving
• Journal writing is an effective way to increase teachers' understanding of student's learning, attitude, and disposition towards mathematics

  
• Two benefits of journal writing according to Mok (2010)


  • Mathematics journal is a cognitive training
  • Mathematics journal can help early reflection of one's mistakes
• Two types of journal writing

  • Free writing - students are free to write any of their learnings in mathematics and teachers may not provide any instruction or guide
  • Writing from a prompt - students write their learnings in mathematics through carefully chosen prompts
• Three categories of prompted writing

  • Affect/Attitudinal - invokes how students feel
  • Mathematical Content - engages students to talk about their learning in mathematics concepts
  • Process and Application - talks about the skills and the applicability of what they had learned
• Two types of grading journals

  • Analytic Rubric - allows separate evaluation of areas such as mathematical concepts, organization of ideas, and expression
  • Holistic Rubric - occurs when there is an overlap of criteria set for evaluation of different ideas
• Pitfalls of journal writing in the classroom

  • The potential for teachers to hurt students' feelings
  • Possible loss of instructional time to teach syllabuses
  • Tremendous increase in the marking load of the teacher
  • What to grade? Language or mathematics concept
• Open-ended math problems


  Math problems that have more than one possible answers
• Features of open-ended math problems

  • There is no fixed answer
  • Solved in different ways and on different levels
  • Empower students to make their own mathematical decisions and make room for their mathematical thinking
  • Develop reasoning and communication skills
• Creating open-ended tasks


  1. Identify a mathematical topic or concept

  2. Think of a closed question and write down the answer

  3. Make up a new question that includes or addresses the answer
• Closed problems


  • Routine close problems - usually multi-step challenging problems that require the use of a specific procedure to arrive at the correct answer
  • Non-routine close problems - imply the use of heuristic strategies to determine the correct answer
• Open-ended problems


  Also named ill-structured problems because they require a higher degree of ambiguity and may allow several answers
• Types of open-ended problems

  • Converted closed problems
  • Applied problems
  • Mathematical investigations and projects
• Strategies to convert closed problems/questions

  • Turning around a question
  • Asking for similarities and questions involves choosing two concepts and how are they alike and different
  • Asking for explanations
  • Creating a sentence involves asking students to create a mathematical sentence that includes numbers or words
  • Using "soft" words requires the use of the words "close" or any equivalent to have richer discussion
• Considerations in creating an open-ended problems or tasks

  • Know your mathematical focus
  • Develop a right degree of ambiguity
  • Plan for two types of prompts: Enabling Prompt - a prompt fit for those students who struggled to start working, Extension Prompt - a prompt fit for those students who finishes quickly
• Features of high quality responses

  • Systematic
  • All solutions can be found (if the solutions are finite)
  • Evident in response (if the patterns are found)
  • Shows complex examples
  • Connections to other content areas
• Benefits in using open-ended tasks in classrooms

  • Open-ended problems encourage higher-order thinking skills
  • Open-ended problems build confidence in your students
  • Open-ended problems are engaging
  • Open-ended problems encourage creativity
  • Open-ended problems make it easy for teachers to look what levels students working at
• Pros of using open-ended tasks in classrooms

  • Provides valuable and specific information to the teacher about student understanding and application of learning
  • Allows the teachers to assess accuracy in computation and abilities
  • Permits the teacher to see students' flexibility in learning
  • Gives student the opportunity to practice and fine tune their problem solving, reasoning, critical thinking, and communication skills
  • Creates opportunites for real-world application
  • Empowers students to extend their learning and reflect on their thinking
  • Fosters creativity, collaboration, and engagement in students
  • Facilitates a differentiated learning where all students can access the tasks
• Cons of using open-ended tasks in classrooms

  • Increases time in collecting data
  • Provides higher complexity of the data
  • Requires the implementation and practice of routines
• Lester (1983) refers mathematics problems as tasks, where a good tasks does not separate mathematical thinking from mathematical concepts, captures students' curiosity, and invites them to pursue their hunches
• A problem is considered a task if it elicits activity on the part of the students and through which they learn by doing the activity
• Perceptions of mathematical tasks according to Mason and Johnson-Wilder (2006)


  • Imagined by the task author
  • Intended by the teacher
  • Specified by the teacher-author instructions
  • Constructed by the learners
  • Carried out by the learners
• Components of a word problem (Verschaffel et. al., 2000)

  • Mathematical Structure - involves the nature of the given, unknown quantities, and operations
  • Semantic Structure - describes the interpretation of text points in relation to mathematics concepts
  • Context - talks what the problem is all about
  • Format - involves how the problem is presented and formulated
• Kulm's categories of task variables (1979)

  • Describe the problem syntax (similar to semantic structure)
  • Characterize the problem's mathematical and non-mathematical context (similar to context)
  • Describe the structure of the problem (similar to format)
  • Characterize the heuristic process evoked by the problem (similar to mathematical structure)
• Errors in mathematical assessment items


  • Language Related Errors - Unclear Instructions, Missing Keywords or Phrases, Using the Incorrect Direction Verb, Incorrect Description of the Context
  • Content Related Errors - Over-defined Conditions, Mathematical Concepts, Errors Related to Diagrams as Support, Context-Related Errors
• Causes of errors in mathematics assessment items according to Carter (1984)


  • Insecurity of the pre-service teachers in writing good assessment items
  • The pre-service teachers copy or paraphrase similar terms from a textbook
  • The pre-service teachers have little time in revising or edit their previous items
• Five cornerstones in reaching curricular goal of developing mathematical skills of the students (Singapore Ministry of Education)

  • Concepts
  • Skills
  • Attitudes
  • Process
  • Metacognition
• Attitude


  The affective aspects of mathematics learning, such as beliefs in mathematics and its usefulness, interest and enjoyment in learning mathematics, appreciation in the power and beauty of mathematics, confidence in using mathematics, perseverance in solving a problem
• Popham (2006) argues that the students' affective status clearly predicts students' future behavior and their future poses concern for teachers
• Beliefs, attitudes, values, dispositions, and interests

  • Attitudes is always directed at an object or idea
  • Beliefs focuses on how the object or an idea is perceived
  • Values deal with the worth of an object or idea
  • Interests focus on the inclination towards an object or idea
• Affective assessments


  May involve students' preferences or opinions in connection in mathematics, a topic in mathematics, the mathematics teacher, or the learner itself
• Cognitive assessments


  Focus mainly on the optimal performance in a task
• Popham (2006) emphasizes that affective assessments should be done as a class rather than as individuals
• Techniques in assessing affect

  1. Crafting your own summative scale - Gather or write the number of affective statements we have in mind, Present the statements along with the agreement-disagreement scale to the students, Obtain the total score by computing the students' score to each statements, The total score indicates the student's affective status, Decide to assess other aspects of affect and so put the statements in self-report forms

  2. Generating your own interest inventory

  3. Crafting your own semantic differential
• Four types of variants in the agree-disagree category

  • Frequency - asks students to report how often they do a certain task
  • Potency - asks students to report how fast they do a certain task
  • Recency - asks students to report the latest time they performed a certain task
  • Utility - asks students how to utilize resources in spare time
• Self-assessment


  One assessing himself or herself
• Three methods in self-assessment

  • Structured Self-Assessment - self-assessment by the teachers using pre-designed self-assessment survey forms
  • Integrated Self-Assessment - embedding self-assessment into other assessment methods
  • Instructional Self-Assessment - integrating self-assessment into classroom instruction