Using a Journal Article as a Professional Development Experience

**Number and Operations**

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**Title:** 10 Practical Tips for Making Fractions Come Alive and Make Sense

**Author:** Doug M. Clarke, Anne Roche, and Anne Mitchell

**Journal:** *Mathematics Teaching in the Middle School*

**Issue:** March 2008, Volume 13, Issue 7, pp. 373 – 380

**Rationale for Use**

The opening sentence in this article says, “Fractions are difficult to teach and learn, but they should not be viewed as a lost cause”. The authors then proceed to make ten suggestions, based on research and classroom practice that make fractions make sense for middle school students.

*Principles and Standards for School Mathematics* (PSSM) NCTM, 2000, suggests that students in the middle grades should deepen their understanding of fractions, decimals, percents, and integers. This article gives practical suggestions how this might happen.

**Procedures**

- Working in groups, present the following task: Represent ¾ (words, drawings) in as many ways as you can. (Note that answers could be written on acetate, scanned to PowerPoint, or drawn on chart paper to facilitate sharing with the large group.)
- Participants share with the group their different representations. Discussion could center on different interpretations of fractions implied by each.
- Read the section in the article, p. 373-374, “What Makes Fractions so Difficult to Teach and Learn”. Discuss whether all five interpretations of fractions identified in this section were included in the representations presented. Were there some interpretations included but not listed in the article? Discuss.
- Take each representation made by Darcy in Fig 1, p. 374, and classify according to fraction interpretation.
- Suggest participants give the Darcy assignment to their own students and bring student work to the next session for discussion. Discuss what was learned about students’ understanding of fractions.

- Assign one each of Tips 1, 2, 3, 7 from the article to groups (arranged according to number of participants in the group). Each group should read the tip assigned and report back to the large group, addressing the following questions:
- What was the main suggestion made by the authors to make teaching/learning fractions easier?
- What else would you like to know about this tip?

- Related to Tip 4 - Prepare the Color in Fractions worksheet (p. 380) for each participant. Have participants play the game with a partner. When finished discuss the following in the large group:
- Fractional concepts and skills encountered while playing the game;
- Strategies used to be a winner; and
- Would you use this with your class? Why or why not?

- Participants read Tip 5, and do the model tasks in Figs 3 and 4. Participants should share experiences they have had with the model tasks involving partitioning with unequal parts, and perceptual distracters, as described in Tip 5. Suggest participants design similar tasks and assign to their students. Bring student work to the follow-up session and share experiences of what was learned about students’ understanding of fractions.
- Participants do the task in Figure 5. Answers are found mentally and a pencil is only used to record answers. Share answers and strategies with the whole group.
- As in #4, suggest using task in Figure 5 with students. When students have discussed strategies for finding solutions assign a similar sheet of fraction comparisons. Bring student work and record of discussion to the next session.
- The authors, p. 376, Tip 6, say that when students share their strategies for comparing fractions during class, other student s may be convinced to use the two strategies of “benchmarking’ and “residual thinking” to tackle relative size and ordering of fraction problems. Discuss if this was so from experiences of doing the activity described above with students.

- Prepare the set of fraction cards as suggested in Tip 8. Participants do the activity as suggested.
- Discuss strategies used to order themselves.
- Ask participants about adaptations that could be made to the activity to use it with their students.

- Prepare the cards and board for the Construct a Sum activity in Figure 7.
- Suggest participants use the activity in a one-to-one interview with a select number of students. Based on each student’s thinking, actions, and dialogue while doing the task, write an evaluation of what was learned about each student’s attempt to make sense of fractions and each student’s understanding of improper fractions.

- When all or most of the activities above are completed, ask teachers to discuss if the article has, as suggested by the authors in the Conclusion, p. 378, given ideas and suggestions for making fractions come alive and make sense in the middle years. What leaves you wondering? What else would you like to know more about in regard to the teaching and learning of fractions?

**Next Steps/ Extensions**

- On page 374 of the article cited above, the authors say, “Curriculum documents sometimes give the sense that the ultimate goal of teaching fractions is that the students will be able to carry out the four operations with them. We believe that students need to be given time to understand what fractions are about (rather than moving too quickly to computation)…”
- In Canada students first learn operations with fractions in Grades 7 and 8. The earlier grades (grades 2 to 6), are used developing the meaning of fractional numbers and developing fractional sense.

Discuss the advantages and disadvantages of delaying learning operations with fractions until grades 7 and 8. Do you think it would be easier for your students to learn fractional concepts and operations?
- The Number and Operation Standard for Grades 6-8 in
*Principles and Standards for School Mathematics* (NCTM, 2000), p. 215 states: “In the middle grades, students should become facile in working with fractions, decimals and percents. Teachers can help students deepen their understanding of rational numbers … that call for flexible thinking”. The problem below is suggested as an example of such a problem.

a. Work each problem above. Discuss strategies used to find solutions.

b. Would you use these problems with your students? Why, or why not?

c. As noted in the excerpt above, taken from PSSM, problems that require flexible thinking deepen students’ understanding of rational numbers. Discuss why the three problems above are examples of using flexible thinking.

d. Make up another problem, involving rational numbers that may require students to use flexible thinking.
- The problem below is taken from Addenda Series, Grades 5-8, Developing Number Sense (NCTM). The problem is intended to help teachers gain a sense of students’ understanding of the effects of certain operations on fractions.

e. Give each participant a copy of the number line, as drawn in the problem. Present the questions to the group, discuss answers explaining and justifying the reasoning used.

f. Encourage participants to use the activity with their students. Share the experiences with students at a follow-up session. Discussion at this session can center around what was learned about students’ fractional operation sense.

**Connections to Other NCTM Publications**

- Adams, T. L., & Murphy, K. (2005, February). A look at some numbers of old: Perfect, deficient, and abundant.
*Mathematics Teaching in the Middle School, 10*, 309-313.
- Azim, D. (2002, April). Understanding multiplication as one operation.
*Mathematics Teaching in the Middle School, 7*, 466-471.
- Barlow, A. T. & Drake, J. M. (2008, February). Division by a fraction: Assessing understanding through problem writing.
*Mathematics Teaching in the Middle School, 13*, 326-332.
- Bay, J. M. (2001, April). Developing number sense on the number line.
*Mathematics Teaching in the Middle School, 6*, 452-455.
- Bay-Williams, J. M., & Martinie, S. L. (2003, February). Thinking rationally about number and operations in the middle school.
*Mathematics Teaching in the Middle School, 8*, 282-287.
- Brahier, D. J. (2001, September). Understanding mathematics and basic skills.
*Mathematics Teaching in the Middle School, 7*, 8-9.
- Cheng-Yao, L. (2007, December). Teaching multiplication algorithms from other cultures.
*Mathematics Teaching in the Middle School, 13*, 298-304.
- Clarke, D. M., Roche, A., & Mitchell, A. (2008, March). Ten practical tips for making fractions come alive and make sense.
*Mathematics Teaching in the Middle School, 13*, 372-380.
- Davis, B. (2008, September). Is 1 a prime number? Developing teacher knowledge through concept study.
*Mathematics Teaching in the Middle School, 14*, 86-91.
- Earnest, D. & Balti, A. (2008, May). Instructional strategies for teaching algebra in elementary School: Findings from a research-practice collaboration.
*Teaching Children Mathematics, 14*, 518-522.
- Friedmann, P. (2008, November). Quick reads: Another good idea: The zero box.
*Mathematics Teaching in the Middle School, 14*, 222-223 .
- Golembo, V. (2000, May). Writing a PEDMAS story.
*Mathematics Teaching in the Middle School, 5*, 574-579.
- Hodges, T. E., Cady, J., & Collins, R. L. (2008, September). Fraction representation: the not-so-common denominator among textbooks.
*Mathematics Teaching in the Middle School, 14*, 78-84.
- Jarvis, D. (2007, April). Math roots: Mathematics and visual arts: Exploring the golden ratio.
*Mathematics Teaching in the Middle School, 12*, 442-473.
- Kribs-Zaleta, C. M. (2008, April). Oranges, posters, ribbons, and lemonade: Concrete computational strategies for dividing fractions.
*Mathematics Teaching in the Middle School, 13*, 453-457.
- Leonard, J., & Campbell, L. L. (2004, February). Using the stock market for relevance in teaching number sense.
*Mathematics Teaching in the Middle School, 9*, 294-299.
- Lewis, L. D. (2007, April). Irrational numbers can “in-spiral” you.
*Mathematics Teaching in the Middle School, 12*, 442-446.
- Parker, M. (2004, February). Reasoning and working proportionally with percent.
*Mathematics Teaching in the Middle School, 9*, 326-330.
- Peck, S., & Wood, J. (2008, November). Elastic, cottage cheese and gasoline: Visualizing division of fractions.
*Mathematics Teaching in the Middle School, 14*, 208-212.
- Ponce, G. A. (2008, November). Using, seeing, feeling, and doing absolute value for deeper understanding.
*Mathematics Teaching in the Middle School, 14*, 234-240.
- Suh, J., Johnston, C., Jamieson, S., & Mills, M. (2008, August). Promoting decimal number sense and representation fluency.
*Mathematics Teaching in the Middle School, 14*, 44-50.
- Taber, S. B. (2007, January). Using Alice in wonderland to teach multiplication of fractions.
*Mathematics Teaching in the Middle School, 12*, 244-250.
- Weidemann, W., Mikovch, A. K., & Hunt, J. B. (2001, December). Using a lifeline to give rational numbers a personal touch.
*Mathematics Teaching in the Middle School, 7*, 210-215.
- Yalles, A. (2001, October). Making connections with prime numbers.
*Mathematics Teaching in the Middle School, 7*, 84-86.
- Yang, D. -C, & Reys, R. E. (2001, November). One fraction problem, many solution paths.
*Mathematics Teaching in the Middle School, 7*, 164-166.
- Zambo, R. (2008, March). Percents can make sense.
*Mathematics Teaching in the Middle School, 13*, 418-422.