Math Questions Worth Asking
By Wendy Petti
We all know the difference between closed and open questions: Closed questions typically involve recalling a fact or performing a skill; open questions stimulate deeper thought. In a typical math classroom, the majority of questions are closed. Let's look at the qualities of questions that call on higher order thinking skills and consider how we can infuse our math classes with open questions and activities targeting higher order thinking skills.
QUALITIES OF GOOD QUESTIONS
Good Questions Extend Learning
Math questions worth asking are likely to have one or more of these qualities or intents:
- There might be more than one acceptable answer.
- Students are asked to apply what they know to a new and challenging situation.
- Students might be asked to bring together and synthesize a range of math concepts and skills in solving one problem.
- The question might trigger a student-driven investigation or exploration.
- Students might be able to learn something new about math, mathematical connections, or their own approach to learning as they tackle the problem.
- The teacher learns about the students' conceptual understanding and problem-solving skills.
Good Questions Are Self-Differentiating
Many questions worth asking can be answered on various levels according to a student's conceptual understanding. Thus, they are self-differentiating. Students with stronger skills or a deeper understanding can probe the question in depth or uncover all the possible answers, while other students might respond on a more basic level. The teacher also can prepare one or more follow-up questions to extend the challenge.
Let's contrast a closed question and an open question on the same topic: finding factors.
Closed question: "What are the factors of 36?"
Open question or task with self-differentiating potential: "Find a number under 100 that has lots of factors."
A student approaching that task at a basic level might be content to find that 12 has six factors -- 1, 2, 3, 4, 6, and 12 -- or that 36 has nine factors -- 1, 2, 3, 4, 6, 9, 12, 18, 36. Or that student might find most -- but not necessarily all -- of the factors of a number; enough to verify that it has "lots of factors."
A student responding on a more advanced level might feel challenged to find the number under 100 with the greatest number of factors, or he or she might choose to start listing the factors of all the numbers under 100. Such a student might notice relationships between some of the numbers while undertaking that investigation -- noticing the similarity in factors between 12, 24, and 36, for example -- or stumble onto a new question worth exploring: "Which numbers have an odd number of factors, and why?"
If you "call time" after three minutes, both students will have accomplished the task successfully with something to share, and both will have been challenged at an appropriate level.
Benjamin S. Bloom's ground-breaking 1956 Taxonomy of Educational Objectives outlines six categories along a continuum from more concrete to more abstract, from lower order thinking skills (knowledge and comprehension) to higher order thinking skills (application, analysis, synthesis, and evaluation). A 2001 collaborative revision of Bloom's Taxonomy -- A Taxonomy for Learning, Teaching, and Assessing -- updates this taxonomy and extends its applications. The six categories are expressed not as nouns but as verbs:
- Remembering: Retrieving, recognizing, and recalling relevant knowledge from long-term memory.
- Understanding: Constructing meaning from oral, written, and graphic messages through interpreting, exemplifying, classifying, summarizing, inferring, comparing, and explaining.
- Applying: Carrying out or using a procedure through executing or implementing.
- Analyzing: Breaking material into parts, determining how the parts relate to one another and to an overall structure or purpose through differentiating, organizing, and attributing.
- Evaluating: Making judgments based on criteria and standards through checking and critiquing.
- Creating: Putting elements together to form a coherent or functional whole; reorganizing elements into a new pattern or structure through generating, planning, or producing.
(Anderson & Krathwohl, 2001, pp. 67-68)
The hierarchy of the categories has been altered as well, as shown in this inverted pyramid:
The Revised Bloom's Taxonomy extends into the knowledge dimension factual knowledge, conceptual knowledge, procedural knowledge, and meta-cognitive knowledge - as well as the six processes of the cognitive dimension. A http://oregonstate.edu/instruct/coursedev/models/id/taxonomy/#tabletaxonomy table indicating actions related to the intersection of these two domains can be useful as we seek to tap the full scope of knowledge and cognitive processes in our students.
Let's use the topic of fractions to illustrate how the taxonomy can help us devise a range of instructional tasks tapping those dimensions. Here they are written as tasks (and quite a few of the tasks are closed, not open) in order to highlight verbs associated with various dimensions. Can you think of open questions to target the same cognitive skills?
Factual: Identify which part of this fraction is the denominator.
Conceptual: Describe an improper fraction.
Meta-Cognitive: Use fraction circles to show 2/3.
Factual: Summarize the steps for simplifying a fraction.
Conceptual: (Interpret) -- Determine which fraction is in lowest terms.
Meta-Cognitive: (Execute) -- Simplify 12/16.
Factual: Classify the following numbers as fractions, improper fractions, or mixed numbers.
Conceptual: (Experiment) -- Find some fractions between 2/3 and 3/4.
Procedural: (Calculate) -- If four friends want to share 6 candy bars, how much will each friend get?
Meta-Cognitive: (Construct) -- This recipe will serve 12 people. Come up with a way to adjust the list of ingredients for 8 servings.
Factual: (Order) Write five fractions with different denominators, then order them from greatest to least.
Conceptual: Explain why we might want to simplify a fraction.
Procedural: (Differentiate) Which of these drawings illustrate a fraction as a part of a set and which illustrate a fraction as a part of a whole?
Factual: Rank these fraction word problems by level of difficulty.
Conceptual: Assess which method of comparing equivalent fractions is easiest to use.
Meta-Cognitive: (Take Action) -- Show a classmate how to use your preferred method for comparing equivalent fractions.
Factual: Combine what you know about equivalent fractions, adding fractions, improper fractions, and simplifying fractions to add and simplify two mixed numbers.
Conceptual: Plan a learning activity to help your classmates practice identifying equivalent fractions.
Procedural: (Compose) -- Make a poster showing how to subtract fractions with unlike denominators.
Open questions by their very nature tend to stimulate higher order thinking skills... but which ones? As you develop open questions, consider which knowledge and cognitive processes will be engaged.
SETTING THE STAGE FOR QUESTIONING
Prepare questions before class.
Our questions and open-ended tasks are more likely to stimulate higher order thinking skills if we prepare them ahead of time. With a little thought, we can turn closed questions into open questions and ensure that we are tapping a range of higher order thinking skills.
Create open questions.
Two easy-to-implement strategies for creating open questions are described in Good Questions for Math Teaching:
- "Work Backward:
- Identify a topic.
- Think of a closed question and write down the answer.
- Make up a question that includes (or addresses) the answer."
- Adapt a standard closed question from a text or other resource to transform it into an open question. You might think in terms of Jeopardy: "Here's the answer; what could my situation be?"
Good questions promote dialogue.
Math questions worth asking lend themselves well to exploration in student pairs, or in small groups arranged according to ability, or in mixed-ability groupings. After each group has had a chance to agree on one or more good answers, they can share their reasoning with the whole class.
Good questions lead to more good questions!
As students learn to explore open questions in math class on a daily basis and learn to reflect on their own learning processes, they will grow as inquirers, posing their own questions worth asking and exploring. In asking students good questions, we help them become better mathematical thinkers and engaged, lifelong learners.
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