Using a Journal Article as a Professional Development Experience

**Number and Operations**

**Title:** Mental Mathematics beyond Middle School: Why? What? How?

**Author:** Rheta N. Rubenstein

**Journal:** *Mathematics Teacher*

**Issue:** September 2001, Volume 94, Issue 6, pp. 442 – 446

**Rationale for Use**

This article emphasizes the importance of mental mathematics in the high school curriculum. In high school, it is important for students to be able to estimate solutions as well as judge the reasonableness of the solutions they compute. Mental mathematics allows students to share problem solving strategies, investigate the depth of number and operation, and use the properties of number systems flexibly. Participants link mental mathematics strategies with critical thinking strategies.

This article offers teachers the opportunity to:

- Consider the appropriate mental mathematics curriculum for their students
- Integrate mental mathematics strategies into their current lesson
- Add to their tool kit of instructional strategies

This reflection guide will take at least two sessions to complete.

**Procedures/Discussion Questions**

*Session One*

*Goal:* Participants will explore the role of mental mathematics in high school mathematics. This discussion should center on mental mathematics and not just the appropriate use of calculators (this can be a topic for another session).

- Have the participants read the article [
*Note*: this could occur prior to the session].
- Have a short whole group discussion about the strategies the participants used to solve the original six problems in the article.
- Why is it important for people to be able to solve problems mentally?
- How often do you give assignments in your classes that should be done without a calculator?
- How do you promote mental mathematics in your classroom?
- Why is mental mathematics important and useful for high school students?

- Break the participants into pairs or small groups by course-alike content to address the following questions specific to the course they are discussing: (Have them record the main ideas from their discussion on chart paper).
- In your course, how do you expect students to note the reasonableness of their answers? How do you promote this in your lessons?
- What strategies, either from the reading or from personal experience, help students develop mental mathematics?
- How would you use and/or create developmental practice sets as described in the article?
- In what ways do you promote students sharing mental mathematics strategies in the classroom? How do you share/model your mental mathematics strategies?

- Bring the participants together and share the work from their small groups by posting charts and doing a gallery walk. Were there any common generalizations about mental mathematics?
- Look at the list of objectives and sample items from one of the three included courses: number sense (general mathematics), beginning algebra, and precalculus. Why is it important to develop a clear set of objectives for what content students should be able to do mentally? Does everyone agree on the selections? What might you change or add?

*Homework:* Select one of the curricula you teach and develop clear and specific mental mathematics objectives for the course. These objectives could be in list form or a set of problems you believe represent a comprehensive understanding of the mental mathematics for the course (similar to the ones in the article). In other words, what do you expect your students should be able to do in their head? Please develop these lists individually – the sharing of them will occur in the next session.

*Session Two*

*Goal:* Participants start to develop a scope and sequence of mental mathematics in their high school.

- Have the participants re-gather in their course-alike teams and share their lists of mental mathematics objectives. What were there common objectives? What objectives did others include but you missed? How much common ground was there?
- Have the small groups make two lists on chart paper. The first chart should list the objectives everyone in the group agrees are important for the designated course. The second chart should list the objectives that the group does not concur about at this point. Give each group a unique color (algebra writes in blue, geometry writes in red, etc.) to record their thoughts.
- Post both pieces of chart paper on the wall for each course-alike group and have the group conduct a “gallery walk,” with everyone looking at the lists of “shared” and “pending” objectives. Allow other groups to make notes on the chart paper using their course-specific color. These notes should be connections to other course content. This will aid in the process of connecting mental math objectives across multiple mathematics courses.
- Have a group discussion noting any generalizations or patterns. Discuss with the group what benefits there are in this type of activity. Gather the lists and publish for future reference and discussion.

**Next Steps/Extensions**

Develop a framework of mental mathematics skills for the curricula in your district.

Use “A Dialogue between Calculator and Algebra” (*Mathematics Teacher*, February 2006, vol. 99, no. 6, pp. 391 – 393) to spur a discussion of the role of the calculator in the learning of mathematics. Also reference the NCTM position paper on “The Role of Technology in the Teaching and Learning of Mathematics.”

**Connections to Other NCTM Publications**

- Bennett, C. D., DeYoung, M. J., Rutledge, J. J., & Young, E. (2006, September). Con-fusing pairs: An intriguing investigation of LCMs.
*Mathematics Teacher, 100*, 140-144.
- Edwards, M. T. (2002, November). Symbolic manipulation in a technological age.
*Mathematics Teacher, 95*, 614-620.
- Gould, L. S. (2005, November). The tellem weavers meet the graphing calculators.
*Mathematics Teacher, 99*, 230-236.
- Herman, M., Milou, E., & Schiffman, J. (2004, November). Activities for students: Unit fractions and their ‘basimal’ representation: Exploring patterns.
*Mathematics Teacher, 4*, 274-284.
- Milou, E., & Schiffman, J. L. (2007, December). The spirit of discovery: The digital roots of integers.
*Mathematics Teacher, 5*, 379-383.
- Quintanilla, J. (2002, October). Ascending and descending fractions.
*Mathematics Teacher, 95*, 539-542.
- Reinstein, D. (2005, April). Delving deeper: Multiplying everything using the differences of two squares.
*Mathematics Teacher, 98*, 550-556.
- Schultz, H. S., & Schiflett, R. C. (2005, March). Delving deeper: Reducing the sum of two fractions.
*Mathematics Teacher, 98*, 486- 490.
- Wanko, J. J. (2009, March). The infinite hotel.
*Mathematics Teacher, 102*, 498-503.