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| Introduction |
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| Can Thinking Skills Be Taught? |
Can thinking skills be taught? Perhaps, but there is no question that they can be learned. Thinking skills are one of the most important, yet inadequately implemented areas of the curriculum. They are attributes many of us want students to possess and many times they are found in our standards and benchmarks and in our "Student Learning Goals". "Critical thinking skills figure prominently among the goals for education, whether one asks developers of curricula, educational researchers, parents, or employers." (Beyer, 1985).
Certainly a part of helping students is to help them develop and improve their thinking skills. Thinking skills and reasoning processes are critical elements to designing purposeful questions and tasks. Questions and tasks designed around thinking skills invite students to think mathematically, formulate and communicate new ideas, justify procedures, and defend the reasonableness of their answers. Students' responses to those questions and tasks can give us great insight into their thinking and mathematical knowledge. So, we need to give students opportunities to practice thinking skill thinking. In order to do this we need to write good questions that reflect the type of thinking we value.
Good questions, tasks, and problems require students to make decisions or judgments based on facts, information, logic and/or rationalization. Students should be required to justify all decisions and reasoning based on the principles being learned. Questions, tasks, and problems should require students to define what assumptions are needed (and why), what information is relevant, and/or what steps or procedures are required in order to solve the problem. Good questions, tasks, problems …
- Challenge students to develop higher levels of thinking, moving them beyond Bloom's (1956) lower cognitive levels (recall of a fact or reproduction of a skill) to the higher Bloom cognitive levels;
- Educate; that is, the pupil will learn from attempting it, and the teacher will learn about the pupil from the attempt.
- Allow for multiple entry levels where everyone can make a start and where students may approach the problem in multiple ways;
- Create the potential for further investigation to challenge and expand student understanding through related questions or extensions;
- Motivate students to probe for deeper understanding of the concepts;
Relate to important mathematical ideas and involve sound and significant mathematics;
- Access multiple solution methods (multiple representations) and may have several acceptable answers.
- Develop students’ mathematical understandings and skills so students develop a better concept of the content under study;
- Promote communication about mathematics through student engagement and discourse;
- Afford opportunities to interpret, justify, and conjecture
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| Thinking Skills in the Clasroom Overview |
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I. Create the Climate (For Students and Yourself)
Resources to consider
- Strategies For Advancing Children’s Mathematical Thinking (TCM, April 01)
- Using Questions to Help Children Build Mathematical Power (TCM, May ’98)
- Designing Questions to Encourage Children’s Mathematical Thinking (TCM, Feb. ’00)
- What Kind of Questions Best Support Math Learning (Math Solutions Newsletter)
- Mathematics Assessment Myths, Models, Good Questions, … - NCTM publication
II. Determine Your Content Focus
III. Coach/Teach
A. Try out the questions on students
B. Use their responses and ideas for suggestions of improvement and for other questions.
IV. Test – "Celebrations of Learning"
V. Feedback and Accountability
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| Writing Thinking Skills Questions |
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There are two ways you can approach the writing of thinking skills questions.
Approach 1: Start With The Content
- Determine your Content Focus - Content Specification
- Determine the Specification sub-skills: Propostion(s)
- Plan Your Question
- What Thinking Skill is your Focus
- Write Your Question
Approach 2: Start With The Thinking Skill
- What Thinking Skill is your Focus
- Determine your Content Focus - Content Specifications
- Determine the Specification sub-skills: Propostion(s)
- Plan Your Question
- Write Your Question
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II. Starting With The Content
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A. Write/Develop Table of Specifications
1) Look for natural subdivision in the material you intend to teach;
2) Be sure the categories have clearly marked limits and are large enough to contain a number of important elements of knowledge within each.;
3) Use subdivisions of content that are likely to make sense to students as a result of their studies.
4) Examples - NCTM
- work flexibly with fractions, decimals, and percents to solve problems;
- compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line;
- understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
- develop and analyze algorithms for computing with fractions, decimals, and integers and
- develop fluency in their use;
- develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results;
B. Write a proposition-general statement (refer to content specifications, content focus: students will, students should)
1) Choose concepts or central ideas from a unit you are teaching or plan to teach.
2) Choose concepts that you consider important rather than trivial.
3) Align the concepts with your Table of Specifications
4) District Document Examples
- Understands the order relationships of rational numbers in the same form and in a variety of forms (e.g. 1/2, 0.34, 7, 42%)
- Applies knowledge of addition, subtraction, multiplication, and division of whole numbers
- Applies knowledge of addition, subtraction, multiplication, and division of fractions, mixed numbers, decimals, and integers.
C. Plan Your Question: Writing Different Questions Around A Proposition
1) Getting Questions
a) Use Existing Questions
- Internal: Take existing question from the end of a chapter, homework assignment, quiz, or test.
- External: Web Sites, Mathematics Journals, Professional Development Workshops, …
b) Write Your Own Questions (It’s not hard!)
2) Planning Tips
a) When planning your questions try to anticipate possible student responses.
b) Think about how students might solve the problem.
- What are some typical misconceptions that might lead students to incorrect answers?
- What are some typical mistakes that students make
What do research and experience show you or tell you
- The data showed considerable confusion in students’ algorithms for working with fractions. For example, an important source of error in adding fractions was simply adding numerators and denominators (Lankford, 1981)
- Students have trouble with, need work on, comparing the size of fractions.
- Understanding of the relationship of the size of the product compared to the factors when multiplying fractions.
- Understanding of the relationship of the size of the quotient when compared to the divisor and dividend when multiplying fractions.
- How have students in the past solved this problem?
c) What type of response do I expect from students, a definition? Example? Solution?
d) Anticipating student responses should help in your planning by forcing you to consider whether phrasing is accurate, whether questions focus on the goal you have in mind, and whether you have enough flexibility to allow students to express ideas in their own words.
D. Determine What Thinking Skill(s) You Want To Address
1) What is your goal or purpose with the concepts? Your goal might help you determine what thinking skill questions you will ask.
2) Read through the descriptions of the thinking skills format
For more thinking skills visit the web site http://edservices.aea7.k12.ia.us/framework/thinking/index.html.
3) Decide which thinking skill process you want your students to demonstrate.
E. Write Your Question
1) Write a question(s) to reflect the thinking skill you chose.
2) Use your statements about how students might solve the problem to help you phrase the question and to think about different ways this concept can be represented in that thinking skill format. For example, from the Curriculum and Evaluation Standards, operations sense should be expanded with such examples as, "Is 2/3 x 5/4 more or less than 2/3? More or less than 5/4?"
- Determine which thinking skill(s) best suits this question or choose which thinking skill you wish to focus on.
- Modify the question to reflect the thinking skill. Use your statements to help you phrase the question.
3) Where can you get questions?
a) External: Getting your problems elsewhere
- Get a problem idea: Web Sites, Mathematics Journals, Professional Development Workshops, Textbook and Ancillary textbook resources, …
- Align it with your Table of Specifications
Think about how students solve the problem, the misconceptions they have, the mistakes they make, etc and write these ideas as a statement.
- Determine which thinking skill best suits this question or choose which thinking skill you wish to focus on.
- Modify the question to reflect the thinking skill.
- Use your statements to help you phrase the question.
b) Internal: Writing your own questions
- Select a math concept you wish to address
- Align it with your Table of Specifications
Think about how students solve the problem, the misconceptions they have, the mistakes they make, how have students in the past solved this problem, etc, and write these ideas as statements.
- Determine which thinking skill best suits this question or choose which thinking skill you wish to focus on.
- Modify the question to reflect the thinking skill.
- Use your statements to help you phrase the question.
G. Other Things to Consider
1) Will I accept the answer in the student's language or am I expecting the textbook's words or my own terms?
2) What will my strategy by for handling incorrect answers?
3) What will I do if students do not answer?
4) Having a prepared list of questions will help to assure that you ask questions appropriate for your goals and representative of the important material.
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