# How to Write Higher-Order Math Questions

- 1 November, 2012 //
- How To //
- Tags : math, test designer
- 5 Comments

As educators, we expect our students to learn and apply math concepts using higher-order thinking skills that go beyond rote learning. With the adoption of the Common Core Math Standards, many of us must do just that, by focusing more in-depth on fewer math concepts. However, writing math problems that require higher-order strategies can be almost as difficult as solving them.

To get started, try writing a lower-level math problem then apply one or more of the following techniques:

**Tips for Writing Higher-Order Math Problems**

- Have students determine and extrapolate a mathematical process or pattern and apply it to an unfamiliar problem or scenario
- Ask students to identify and evaluate missing or incorrect information
- Challenge students to solve one problem using multiple methods
- Consider asking, “Given ____, what would happen if ____ changed?” questions
- Give an answer and a mathematical concept, have students write their own questions or equations that produce the given answer
- Ask students to justify their solutions or identify and justify the “best” or “most correct” solution from a selection of plausible choices
- Write problems that ask for connections between more than one set of information, this could include charts, tables, equations, graphics, and data sets
- Watch that answers for multiple-choice questions are logical and that the correct choice is not structurally different from the incorrect answers
- If you are having difficulty writing a problem, start by constructing questions that incorporate these higher-order key words and concepts:
*analyze, justify, explain, apply, interpret, compare, estimate, predict, prove, formulate, modify*

Consider the Common Core Math Standard 7.G.4

*Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.*

A lower-order question for this standard could be:

**What is the area of a circle with a radius of 5?**

**a) 5π**

**b) 10π**

**c) 25π**

**d) 100π**

As presented, this question strictly tests student knowledge and application of the required formula. Instead, consider structuring the question using a “real-life” scenario that requires multiple steps to solve:

**Official tournament play of Ringer marbles requires a circular game ring with a diameter of 10 feet. Alexis needs to construct multiple rings for a tournament using rope to mark the circumference of each ring. If she has 100 feet of rope, what is the maximum number of rings Alexis can make?**

**a) 2**

**b) 3**

**c) 4**

**d) 5**

Finally, try constructing an open-response question that requires students analyze and evaluate the information in a non-routine manner:

**Jackson explains to his classmate that doubling the circumference of a circle results in the doubling of the circle’s area. Is Jackson correct? Use the formulas for area and circumference to justify your answer. **

This problem still satisfies skills posed in standard 7.G.4, however it now requires students not only demonstrate knowledge and application of the formulas, but also analyze the relationship between them.

Writing higher-order problems takes time, but ideally, the additional time will help students further develop the critical-thinking skills we strive to nurture as educators.

For more examples of higher-order math questions, view these questions created by TestDesigner.com members:

*Example 1: * Problem Solving Strategies

*Example 2:* Fractions

*Example 3:* Logical Thinking

*Example 4:* Algebraic Expressions

*Example 5:* Exponents

*Example 6:* Factors

Read “How to Design a Good Test” and “Five Essential Thinking Skills to Teach in September” for more tips on creating meaningful questions and assessments.

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