## Critical Thinking Questions In Mathematics

** Welcome to Education World's Work Sheet Library. In this section of our library, we present more than 100 ready-to-print student work sheets organized by grade level. Click on a grade level folder below to find a library of work sheets that you can use with your students to build a wide variety of critical thinking skills. All the work sheets in this library were provided to Education World by our partners at CriticalThinking.com. Click on the small banner advertisement above for a complete catalog of CriticalThinking.com teacher-ready activities.) **

Visit Education World's Work Sheet Library for a wide variety of free printables for use across the curriculum and across the grades.

**Geometry**

Students will love this analytical sheet. (Grafes 6-8)

**Mathmatical Reasoning**

Taky your expectations to the next level. (Grades 6-8)

**Balance Benders**

Given certain facts, which objects weight will even off the scales? (Grades 6-8)

**Balance Benders (#2)**

Given certain facts, which objects weight will even off the scales? (Grades 6-8)

**Matching Figures: Figural Similarities**

Among four similar figures, which two are exactly alike? (Grades 6-8)

**Verbal Similarities and Differences: Antonyms**

Which of the three words means the opposite of the first word in the line? (Grades 6-8)

**Oh, Fur Goodness Sake!**

Read the true story. Then make an inference based on the evidence in the story. (Grades 6-8)

**Obie One, Obie Two**

Read the true story. Then make an inference based on the evidence in the story. (Grades 6-8)

**Quit That Rockin**

Read the true story. Then make an inference based on the evidence in the story. (Grades 6-8)

**Do Size, Shape, or Weight Make a Difference?**

High density objects fall at about the same rate of speed regardless of size, shape, or weight. (Grades 6-8)

**Rhyme and Reason (#1)**

Can you figure out the subjects of these simple rhymes? (Grades 6-8)

**Rhyme and Reason (#2)**

Can you figure out the subjects of these simple rhymes? (Grades 6-8)

**Rhyme and Reason (#3)**

Can you figure out the subjects of these simple rhymes? (Grades 6-8)

**Uses of Peanut Oil**

Find the 10 errors in this brief article. (Grades 6-8)

**The Missing Cookie Caper**

Find the 10 errors in this brief article. (Grades 6-8)

**Temperature Tale of Two Cities**

Study the temperature graph. Use it to fill out the temperature charts. (Grades 6-8)

**The Roberts Family Reunion**

Be a math detective: use clues in the story to answer the questions. (Grades 6-8)

**The Amazing Mayans**

The story and diagrams help you learn about a Mayan number system. (Grades 6-8)

**Who Works Where?**

Use the clues to match each womans name with her kind of work. (Grades 6-8)

**Married People**

Use the clues to figure out what last name goes with each persons first name. (Grades 6-8)

**Ice Cream**

Use the clues to figure out each persons favorite flavor. (Grades 6-8)

**Picnic in the Park**

Use the clues to figure out the peoples full names and what they brought to eat. (Grades 6-8)

**Alls Fair in Science**

Use the information to figure out which science fair project each kid did. (Grades 6-8)

**Unusual Animals**

Use the clues to figure out what makes these animals unique. (Grades 6-8)

**Career Women**

Use the clues to figure out each womans job and her income. (Grades 6-8)

**Novel Thinking: Charlie and the Chocolate Factory**

Use the vocabulary words to complete the crossword puzzle. (Grades 6-8)

**Novel Thinking: Charlottes Web**

Draw a line from each important event to the detail that tells more about it. (Grades 6-8)

**Novel Thinking: Georges Marvelous Medicine**

Use the clue to help you unscramble each vocabulary word. (Grades 6-8)

**Novel Thinking: Shiloh**

Use the vocabulary words and definitions to help you fill in the puzzle. (Grades 6-8)

**Novel Thinking: In Their Own Words: Abraham Lincoln**

Write a vocabulary word and its part of speech next to each definition. (Grades 6-8)

**Defining Geometry**

Use evidence presented in the passage to answer each of the questions. (Grades 6-8)

**Galileos Vision**

Use evidence presented in the passage to answer each of the questions. (Grades 6-8)

**Foods as Medicine**

Use evidence presented in the passage to answer each of the questions. (Grades 6-8)

**Invertebrates**

Use information in the story to answer the questions and complete the diagrams. (Grades 6-8)

**Earth Materials and Their Uses**

Use information in the story to answer the questions and complete the flow chart. (Grades 6-8)

**Algebra, Exponents, and Using Formulas**

Figure out which number or letter should replace each of the question marks. (Grades 6-8)

**Think Quick!**

Three fun math challenges from Dr. Funster. (Grades 6-8)

**Matrix Fill-Up**

Fill in each box with a word that begins with the letter indicated and belongs under the heading. (Grades 6-8)

**Skating Party**

Use the clues to enter the correct digits in the puzzle. (Grades 6-8)

**Table Logic**

Use the clues to write the name of each person at the place he or she is seated. (Grades 6-8)

**Root Words: geo and More**

Use the word root chart to help you match each words to its meaning. (Grades 6-8)

**Word Meaning Worksheet**

Use what you know about word roots -- or your dictionary -- to complete this matching activity. (Grades 6-8)

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## Using Questioning to Stimulate Mathematical Thinking

##### Stage: 1, 2 and 3

Article by Jenni WayPublished October 2001,February 2011.

Good questioning techniques have long being regarded as a fundamental tool of effective teachers. Unfortunately, research shows that 93% of teacher questions are "lower order" knowledge based questions focusing on recall of facts (Daines, 1986). Clearly this is not the right type of questioning to stimulate the mathematical thinking that can arise from engagement in open problems and investigations. Many Primary teachers have already developed considerable skill in good questioning in curriculum areas such as Literacy and History and social studies, but do not transfer these skills to Mathematics. Teachers' instincts often tell them that they should use investigational mathematics more often in their teaching, but are sometimes disappointed with the outcomes when they try it. There are two common reasons for this. One is that the children are inexperienced in this approach and find it difficult to accept responsibility for the decision making required and need a lot of practise to develop organised or systematic approaches. The other reason is that the teachers have yet to develop a questioning style that guides, supports and stimulates the children without removing the responsibility for problem-solving process from the children.

### Types of Questions

Within the context of open-ended mathematical tasks, it is useful to group questions into four main categories (Badham, 1994). These questions can be used be the teacher to guide the children through investigations while stimulating their mathematical thinking and gathering information about their knowledge and strategies.1. Starter questions

These take the form of open-ended questions which focus the children's thinking in a general direction and give them a starting point. Examples:

How could you sort these.......?

How many ways can you find to ....... ?

What happens when we ......... ?

What can be made from....?

How many different ....... can be found?

2. Questions to stimulate mathematical thinking

These questions assist children to focus on particular strategies and help them to see patterns and relationships. This aids the formation of a strong conceptual network. The questions can serve as a prompt when children become 'stuck'. (Teachers are often tempted to turn these questions into instructions, which is far less likely to stimulate thinking and removes responsibility for the investigation from the child).

Examples:

What is the same?

What is different?

Can you group these ....... in some way?

Can you see a pattern?

How can this pattern help you find an answer?

What do think comes next? Why?

Is there a way to record what you've found that might help us see more patterns?

What would happen if....?

3. Assessment questions

Questions such as these ask children to explain what they are doing or how they arrived at a solution. They allow the teacher to see how the children are thinking, what they understand and what level they are operating at. Obviously they are best asked after the children have had time to make progress with the problem, to record some findings and perhaps achieved at least one solution.

Examples:

What have you discovered?

How did you find that out?

Why do you think that?

What made you decide to do it that way?

4. Final discussion questions

These questions draw together the efforts of the class and prompt sharing and comparison of strategies and solutions. This is a vital phase in the mathematical thinking processes. It provides further opportunity for reflection and realisation of mathematical ideas and relationships. It encourages children to evaluate their work.

Examples:

Who has the same answer/ pattern/ grouping as this?

Who has a different solution?

Are everybody's results the same?

Why/why not?

Have we found all the possibilities?

How do we know?

Have you thought of another way this could be done?

Do you think we have found the best solution?

### Levels of Mathematical Thinking

Another way to categorise questions is according to the level of thinking they are likely to stimulate, using a hierarchy such as Bloom's taxonomy (Bloom, 1956). Bloom classified thinking into six levels: Memory (the least rigorous), Comprehension, Application, Analysis, Synthesis and Evaluation (requiring the highest level of thinking). Sanders (1966) separated the Comprehension level into two categories, Translation and Interpretation, to create a seven level taxonomy which is quite useful in mathematics. As you will see as you read through the summary below, this hierarchy is compatible with the four categories of questions already discussed.1. Memory: The student recalls or memorises information

2. Translation: The student changes information into a different symbolic form or language

3. Interpretation: The student discovers relationships among facts, generalisations, definitions, values and skills

4. Application: The student solves a life-like problem that requires identification of the issue and selection and use of appropriate generalisations and skills

5. Analysis: The student solves a problem in the light of conscious knowledge of the parts of the form of thinking.

6. Synthesis: The student solves a problem that requires original, creative thinking

7. Evaluation: The student makes a judgement of good or bad, right or wrong, according to the standards he values.

### Combining the Categories

The two ways of categorising types of questions overlap and support each other.For example, the questions:

Can you see a pattern?

How can this pattern help you find an answer? relate to Interpretation, and;

the questions:

What have you discovered?

How did you find that out?

Why do you think that? require Analysis, and;

the questions:

Have we found all the possibilities?

How do we know?

Have you thought of another way this could be done?

Do you think we have found the best solution? encourage Evaluation.

In the process of working with teachers on this topic, a table was developed which provides examples of generic questions that can be used to guide children through a mathematical investigation, and at the same time prompt higher levels of thinking.

*You may also find Jennie Pennant's article Developing a Classroom Culture That Supports a Problem-solving Approach to Mathematics useful, which includes a section on questioning.*

### References

Badham, V. (1994) What's the Question?. Pamphlet 23. Primary Association for Mathematics (Australia)Badham, V. (1996). Developing Mathematical Thinking Through Investigations. PAMphlet 31. Primary Association for Mathematics (Australia)

Bloom, B. (1956). Taxonomy of Educational Objectives Handbook 1: Cognitive Domain. New York: David Mackay

Dains, D. (1986). Are Teachers Asking the Right Questions? Education 1, 4 p. 368-374.

Sanders, N. (1966). Classroom Questions: What Kind? New York: Harper and Row.

Team-building. Shape, space & measures - generally. Problem solving. Pedagogy. Mathematical Thinking. Averages. Rich Tasks. Questioning. Learning mathematics. Investigations.

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