One way to differentiate in math class is creating **open-ended tasks and questions** (I talked about several *differentiation* strategies I use here – Mathematically Speaking).

I think it is useful to clarify the scheme of mathematical problems – below I used Foong Pui’s research paper:

“Problems in this classification scheme have their different roles in mathematics instruction as in teaching ** for **problem solving, teaching

**problem solving, or teaching**

*about***problem solving.”**

*via***1. ****CLOSED**** problems** are well-structured problems in terms of clearly formulated tasks where the one correct answer can always be determined in some fixed ways from the necessary data given in the problem situation.

** A.****Routine** closed problems – are usually multi-step challenging problems that require the use of a specific procedure to arrive to the correct, unique, answer.

**B. Non-routine** closed problems – imply the use of heuristics strategies * in order to determine, again, a single correct answer.

*Problem-solving heuristics: work systematically, tabulate the data, try simpler examples, look for a pattern, generalize a rule etc.

*EXAMPLES*

** Routine problem**: Minah had a bag of rice. Her family ate an equal amount of rice each day. After 3 days, she had 1/3 of the rice left. After another 7 days, she had 24 kg of rice left. How much rice was in the bag at first?

*Non-routine problem***: **How many squares are there in a chess board?

**2. OPEN –ENDED**** problems** – are often named “ill-structured” problems as they involve a higher degree of ambiguity and may allow for several correct solutions. Real-life mathematical problems or mathematical investigations are of this type – e.g. “How much water can our school save on a period of four months?” or “Design a better gym room considering the amount of money we can spend.”

**FEATURES of open-ended problems**:

- There is no fixed
*answer*(many possible answers) - Solved in different
*ways*and on different*levels*(accessible to mixed abilities) - Empower students to make their own mathematical
*decisions*and make room for own mathematical*thinking* - Develop
*reasoning and communication*skills

**HOW do you create open-ended tasks?**

Usually, in order to create open-ended questions or problems, the teacher has to **work backwards**:

- Indentify a mathematical topic or concept.
- Think of a closed question and write down the answer.
- Make up a new question that includes (or addresses) the answer.

**STRATEGIES to convert closed problems/questions**

**Turning around a question**

CLOSED: What is half of 20?

OPEN: 10 is the fraction of a number. What could the fraction and the number be? Explain.

CLOSED: Find the difference between 23 and 7.

OPEN: The difference between two numbers is 16. What might the numbers be? Explain your thinking.

CLOSED: Round this decimal to the decimal place 5.7347

OPEN: A number has been rounded to 5.8. What might the number be?

CLOSED: There are 12 apples on the table and some in a basket. In all there are 50 apples. How many apples are in the basket?

OPEN: There are some apples on the table and some in a basket. In all there are 50 apples. How many apples might be on the table? Explain your thinking.

**Asking for similarities and differences.**

Choose two numbers, shapes, graphs, probabilities, measurements etc. and ask students how they are alike and how they are different.

Example: How are 95 and 100 alike? How are they different?

Possible answers:

They are alike because you can skip count by 5s, both are less than 200, both are greater than 90 etc.

They are different because one is a three-digit number, only one ends in 5, only one is greater than 99 etc.

Example: How are the numbers 6.001 and 1.006 alike? How are they different?

**Asking for explanations.**

Example: Compare two fractions with different denominators. Tell how you compare them.

Example: 4 is a factor for two different numbers. What else might be true about both numbers?

**Creating a sentence**

Students are asked to create a mathematical sentence that includes certain numbers and words.

Example: Create a sentence that includes numbers 3 and 4 along with the words “more” and “and”.

Possible answers:

- 3 and 4 are more than 2
- 3 and 4 together are more than 6
- 34 and 26 are more than 34 and 20 etc.

Example: Create a question involving multiplication or division of decimals where the digits 4, 9, and 2 appear somewhere.

Example: Create a sentence involving ½ and 64 and the words “less” and “twice as much”.

**Using “soft” words.**

Using the word “close” (or other equivalents) allows for a richer, more interesting mathematical discussion.

Example: You multiply two numbers and the product is *almost* 600. What could the numbers have been? Explain.

Example: Add two numbers whose sum is close to 750. What can the numbers be? Explain.

Example: Create two triangles with different but close areas. (*instead of, “Create a triangle with an area of 20 square inches.”)

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A few important **considerations **are to be made when creating open-ended problems or questions.

- Know your
**mathematical focus**. - Develop questions with the
**right degree of ambiguity**(vague enough to be interesting and to allow for different responses, but not too vague so as students get frustrated). - Plan for two
**types of prompts**:

*enabling*prompts (for students who seem unable to start working)*extension*prompts (for students who finish quickly)

**High quality responses** from students have the following features:

- Are
*systematic*(e.g. may record responses in a table or pattern). - If the solutions are finite,
*all*solutions are found. - If patterns can be found, then they are
*evident*in the response. - Where a student has challenged themselves and shown
*complex*examples which satisfy the constraints. - Make
*connections*to other content areas.

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References:

*Designing Quality Open-Ended Tasks in Mathematics*, Louise Hodgson, 2012

*Using Short Open-ended Mathematics Questions to Promote Thinking and Understanding*, Foong Pui Yee, National Institute of Education, Singapore

*Good Questions – Great Ways to Differentiate Mathematics Instruction*, Marian Small, 2012

Thank you for posting this. I appreciate how there is a comparison between the two (closed and open ended) types of questions and the considerations that go along with each. Thanks!

You are welcome!

Wow, well-written, thank you. I’m excited that my teaching is getting great, clear, and organized at the level that I’m at. But this article reminds me there are many higher levels I can get to, including this area of more open-endedness. Thank you!

I am happy to have helped even in a small way!

Thank you so much