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NYC Helpline: How To: Teach Math

Creating Open Questions- One Way to Differentiate Instruction In Mathematics
by Luzviminda “Luchie” B. Canlas

“Children already come to us differentiated. It just makes sense that we would differentiate our instruction in response to them”—Carol Ann Tomlinson

“Excellence in mathematics education requires equity –high expectations and strong support for all students.” —NCTM (2000 p.12)

Having high expectations means believing that all our students will attain success academically. However, it is not enough to have high expectations. We should have a deep understanding of the subject we teach. It is also our responsibility to find and apply effective ways to teach students well as they learn mathematics. Additionally, we must continually refine our instructional strategies and learn to provide appropriate accommodations so that everyone will have equal access to understanding whatever we want to teach them.

One way to achieve this is to differentiate our instruction. Carol Ann Tomlinson states, “To differentiate instruction is to recognize students’ varying background knowledge, readiness, language, preferences in learning, interests, and to react responsively. Differentiated instruction is a process to approach teaching and learning for students of differing abilities in the same class. The intent of differentiating instruction is to maximize each student’s growth and individual success by meeting each student where he or she is, and assisting in the learning process.*

Lillian Katz says, “When a teacher tries to teach something to the entire class at the same time, chances are, one-third of the kids already know it; one-third will get it, and the remaining third won’t. So two-thirds of the children are wasting their time.”**

We should avoid this. As educators we need to ask ourselves, “What do I need to do so that I do not waste anyone’s time and so that each of my students gets the support that he or she needs and deserves?” We must realize that some of our students may need extra support in terms of understanding the English language so that they can actively participate in mathematical discourse. We should know that students with disabilities need certain accommodations such as time and additional resources to engage them fully. Likewise, students who are gifted or are above benchmark may need more challenging resources to push them forward. In any case, we are responsible for exercising the equity principle in mathematics.

Differentiation in instruction can come in many forms. There are several ways to do this. Is it possible to differentiate instruction for the whole class? The answer is yes. One strategy is to create open questions or tasks that teach the same big ideas or math concepts to our students who have diverse developmental levels and needs. How is this done?

Examine these two questions that are presented to the whole class:
Question A: What is 2/3 of 9?
Question B: Describe what you see in mathematical terms:

  

Which is an open question? Which one do you think will all students be more likely to attempt to solve?

By definition, a question is open if there are many ways that children can try to answer or approach it. If a student doesn’t understand fractions, then question A may be difficult to solve for him or her. However, in question B, even if the child does not know fractions, she or he can participate and tell that there are 6 yellow counters in a set of 9 counters. Some may state, “There are three more yellow counters than red counters.” Others can see that 1/3 of the set of counters is red and 2/3 of the set of counters is yellow. Others can see this as a multiplication sentence: 3x3=9, or a division sentence: 9÷3=3. Some might say “There’s twice the number of yellow counters than the red ones.” In other words, all of them have access to the question because they are allowed to provide their own mathematical interpretation. As you can see, every one of these responses are acceptable. The children will have self-assurance and will believe that they are valued contributors in the classroom discussion. This strategy will definitely have a positive impact and influence on students’ beliefs about how they learn and what learning is. The responses reveal a lot about the developmental level of the students so teachers can readily use these as a form of informal assessment. Moreover, open questions are more like “invitations to communicate” rather than “problems to solve,” thereby making mathematics less intimidating and inhibiting. Additionally, students will learn from each other from the responses they share. Hopefully, they will discover that a problem can be solved from multiple perspectives.

Marian Small, in her book “Good questions: great ways to differentiate mathematics instruction (2009) suggests five ways to create open questions. These strategies are :

    • Question turn around
    • Finding similarities and differences
    • Replacing a number with a blank
    • Asking for a number sentence
    • Altering a question

How to turn a question around: You need to give the answer. The student will provide the question. Here is an example. The original question is: I bought two books that cost $11.75 and $4.25 respectively. If I paid the salesclerk with a $20 bill, how much change will I get? To turn this question around, I’ll ask : $5.00 is the change I received after buying two items from a store. What are my two items, how much does each cost and how much money do I have in the beginning?

Finding similarities and differences: You will choose two math concepts (polygons, polyhedrons, shapes, numbers, operations, measurements, functions, graphs, etc) and ask students to compare and contrast the two. They need to explain how the two are alike and how they are different. For example, you can ask students how a pyramid is like a prism and how a prism is different from a pyramid.

Replacing a number with a blank: Giving your students the chance to choose the numbers to use to create the problems can be empowering for them. For example, tell them to pick two two-digit numbers less than 20 to multiply and then develop their own multiplication situational problem. You are giving them freedom to choose the numbers. This is different from asking them to find out how many eggs are in 13 crates if each crate has a dozen eggs.

Asking for a number sentence: You will give your students numbers and words to include in their sentence. For example, ask them to create a sentence with ½ and ¼ and the words “add” and “unit”. They may come up with the following sentence:

  • ½ is a unit fraction that is more than ¼. If I add ¼ to ¼, I get 2/4 or ½.

Changing the question: You can modify the questions that are found in textbooks. For example: your textbook may include the problem: A rectangle has a length of 6 inches and a width of 4 inches. What is the perimeter of the rectangle? You can change this by asking: I have a perimeter of 24 inches. What are some possible lengths and widths of rectangles?

Creating open questions is one effective strategy to meet the needs of all our learners. This strategy will help stimulate thinking and develop reasoning abilities while building ownership and self-esteem. By doing this, the students generate “products” that are truly reflective of their abilities. As a result, assessment results are more reliable and valid.

Creating open questions will also help promote communication and foster a culture of inquiry and wonderment as we teach math concepts. We need to make sure that the classroom community is inclusive and every answer or response is valued and appreciated. We need to know our students well. Our screening, diagnostic, and formative assessments will reveal that our children have varied stages of math content knowledge and abilities. Students have different backgrounds, interests, performance readiness levels, and learning styles. Some may be visual, auditory, linguistic, kinesthetic, or tactile learners. No matter where they are and how they learn, our primary goal is to make math accessible to all of them by making them feel like valuable consumers and contributors to learning. We need to make them feel that what they are learning or about to learn is significant and what they are about to say, do, make, or contribute is important. This may seem a monumental task but it can be done.

Building community is a beginning. We must promote collaboration and team building in the classroom. We should create a risk-free environment where mistakes are considered opportunities to learn. Then, we need to understand and accept the fact that children have differences and similarities in learning. Next, we need to use this knowledge to use effective strategies to support all of them. These strategies must be manageable and “handy” so that we can easily apply its use. Differentiating instruction doesn’t mean creating seven different lessons for seven different groups of students. We can plan to differentiate instruction for the whole class by designing one open task or one open question that allows everyone to respond confidently. One open question that will invite all of them to actively participate in the discussion can go a long, long way.


* Tomlinson, Carol Ann. (2001). How to differentiate instruction on mixed ability classrooms (2nd edition). Alexandria, VA. ASCD.

** Katz, L.G. and Chard, S.C. (2000). Engaging children's minds: The project approach (2nd edition). Stamford, CT. JAI Press.

 

If you have a question or comment about this article e-mail Luchie.

 

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