Return to Math Homepage

 

Higher Order Thinking Skills

 

HOTS   HOTS Strategies

Questioning for HOTS

 

..........

Suggestions for HOTS

 

These question stems from NCTM Professional Standards for Teaching Mathematics help promote students' higher order thinking while requiring students to engage in the standards for mathematical practice and probing for understanding.

 

Questions that help students work together to make sense of mathematics:

Cooperate

 

 

 

 

  • What do other think about what _____ said?
  • Do you agree? Disagree?
  • Does anyone have the same answer but a different way to explain it?
  • Would you ask the rest of the class that question?
  • Do you understand what they are saying?
  • Can you convince the rest of the class that that makes sense?

 

Questions that help students rely more on themselves to determine whether something is mathematically correct:

Confidence1

  • Why do you think that?
  • Why is that true?
  • How did you reach that conclusion?
  • Does that make sense?
  • Can you make a model to show that?

 

Questions that help students learn to reason mathematically:

Reason

 

 

  • Does that always work?
  • Is that true in all cases?
  • Can you think of a counterexample?
  • How could you prove that?
  • What assumption are you making?

 

Questions that help students learn to conjecture, invent, and solve problems:

Conjectures

 

 

 
  • What would happen if …? What if not?
  • Do you see a pattern?
  • What are some possibilities here?
  • Can you predict the next one? How about the last one?
  • How did you think about the problem?
  • What decision do you think she should make?
  • What is alike and what is different about your method of solution and his/hers?

 

Questions that help students to connect mathematics, its ideas, and its applications:

Connecting

 

 

  • How does this relate to…..?
  • What ideas that we have learned before were useful in solving this problem?
  • Have we ever solved a problem like this before?
  • What uses of mathematics did you find in the newspaper tonight?
  • Can you give me an example of…?

 

 Additional HOTS Questioning Resources

 

HOTS Verbs

Questioning Clues and Verbs for the Development of

Higher Order Thinking Activities

 

 

HOTS Key Words

Questions and Key Words for Critical Thinking

 

HOTS Questioning

Improving Learning Through Questioning

 

HOTS Def Strategies Assessment

Higher Order Thinking Skills

 

 

 

 Related Articles

 

Math Questions Worth Asking

A discussion of the characteristics of questions that call on higher order thinking skills and how to infuse math class with open questions and activities targeting higher order thinking skills.

 

Teaching Problem Solving Skills in Math by Using HOTS

Students at all levels of mathematics should be expected to think about deep questions about the content, and they should be tasked to engage in positive collaborative problem solving activities regularly.  This paper explores strategies for developing higher order thinking skills using open-ended tasks, questioning at the right level, "mathematizing" real world situations, requiring students to prove their answers, and engagaing students in mathematics discovery and discussions.

 

Open-Ended Questions and the Process Standards 

This article from the Mathematics Teacher publication of NCTM emphasizes that educationing students - for life, not for tests - requires the use of open-ended questions to develop higher-order thinking.

 

HOTS Promoting

Promoting Higher Order Thinking

 

 

 

 

 

 

 

 

Use already-created or develop "Three-Act" tasks.

(Dan Meyer is credited with this idea.)

 

Act1

 


 

Act One is the video or image of a situation that generates questions that can be answered mathematically. In Act One the student asks a question and then determines what information is needed to answer the question.

 

Another idea is to take a problem and strip it down to a minimum of information.  Have students provide the question.  Some of the questions generated may need further information. Have students identify the additional information that will be needed. This act engages all students - everyone has an entry point. No one is wrong.

  


 

 Act2

Act Two gives further information that may help answer the question.

 

Give students a little more information. (In a true Three-Act, the teacher gives students information only as they request it. This gets them thinking about the problem and the math necessary to solve it.) Again, have students provide the question.

 


 

 Act3

Act Three shows a video or image that answers the question.

 

Act Three provides the "reveal."  If you started with a "stripped down" problem, this is when you give students the question that was part of the original item. At this point, students have had lots of exploration with the problem and they will be interested in the intended question for the item. 

 

This is an opportunity, too, for students to develop a "sequel" to the problem. What other questions can be asked and answered?

 


 

 

Incorporate PBL (problem-based learning).

 

PBL

 

Students meet an actual or simulated situation (based upon a real-world model) at the opening of a unit. The situation is the envelope containing a problem to be solved.


The problem to be solved is ill-structured. It must be analyzed through inquiry and investigation before it can be resolved. Ill-structured problems provide an effective learning environment because they:

  • lack important information when first encounted requiring the learner to hypothesize, question, collect data, and think,
  • only reveal their complexity through investigation and are liable to change as inquiry progresses,
  • defy solution by simple formula requiring the application of reason, and
  • require action (solution) even when the problem solver is 100% sure of the "right" answer because data might be missing, in conflict or able to be interpreted from different perspectives.

Students must solve real problems, not just learn procedures: teachers coach for growth in metacognition and critical thinking.

 

Resources for PBL

 

 

Use open-ended problems.

 

Question-Mark-Box 

 

Resources for open-ended problems

 

Fermi questions emphasize estimation, numerical reasoning, communicating in mathematics, and questioning skills. Students often believe that word problems have one exact answer and that the answer is derived in a unique manner. Fermi questions encourage multiple approaches, emphasize process rather than the answer, and promote nontraditional problem-solving strategies. The Questions Library features classic Fermi questions with annotated solutions, a list of questions for use with students, questions with a Louisiana twist, and activities for the K-12 classroom.

 

Created for students in grades 6 to 8, the site offers math challenges that focus on everyday life, such as how fast your heart beats, what shape container holds the most popcorn, and how much of you shows in a small wall mirror.

 

Over 2,000 questions are archived. Online tools allow you to search the collection by content area, grade level, and difficulty. The site also shows what students at each achievement level are likely to know and how NAEP questions are scored.

 

Prepared by the Ohio Resource Center, this collection of problems includes not only test items on proportion but also access to performance data by subgroups of students, a scoring key, and discussion of the tested content.

 

The Ohio Resource Center collection of intriguing, inquiry-based problems for grades 3-12 can be browsed by topic and grade level.

 

This resource, available online for a small fee, provides more than 450 open-ended questions. All involve significant mathematics, are solvable in a variety of ways, elicit a range of responses, and enable students to reveal their reasoning processes. The site also offers samples of student answers, a scoring rubric, and additional narrative material that addresses the nature, construction, and reasons for using open-ended items.

 

The purpose of this site is to prepare middle school students for open-ended problem solving on the Philadelphia standardized tests. Many sample items are given.

 

Classroom practice questions for grades 4, 5, and 8 as well as algebra, geometry, probability and statistics. In addition to addressing content standards, the questions also address process standards and require students to explain or justify their answers and/or strategies.