Skip to main content
SpringerLink
Log in

RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND ALGEBRAIC REASONING: THE CASE OF WILLA

  • Published:
International Journal of Science and Mathematics Education Aims and scope Submit manuscript

Abstract

To investigate relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, the first author conducted a case study of a seventh grade student, Willa. Willa’s fractional knowledge—specifically her reversible iterative fraction scheme and use of fractions as multipliers—influenced how she wrote equations to represent multiplicative relationships between two unknown quantities. The finding indicates that implicit use of powerful fractional knowledge can lead to more explicit use of structures and relationships in algebraic situations. Curricular and instructional implications are explored.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Behr, M. J., Harel, G., Post, T. R. & Lesh, R. (1993). Rational numbers: Toward a semantic analysis—emphasis on the operator construct. In T. P. Carpenter, E. Fennema & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 13–47). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D. & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 235–272). New York: Lawrence Erlbaum.

    Google Scholar 

  • Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 547–589). Hillsdale: Erlbaum.

    Google Scholar 

  • Corbin, J. & Strauss, A. (2008). Basics of qualitative research (3rd ed.). Thousand Oaks: Sage Publications.

    Google Scholar 

  • Driscoll, M. J. (1999). Fostering algebraic thinking: A guide for teachers, grades 6–10. Portsmouth: Heinemann.

    Google Scholar 

  • Ellis, A. B. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.

    Google Scholar 

  • Empson, S. B., Levi, L. & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai & E. J. Knuth (Eds.), Early algebraization (pp. 409–428). Berlin: Springer.

    Chapter  Google Scholar 

  • Hackenberg, A. J. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 26, 27–47.

    Article  Google Scholar 

  • Hackenberg, A. J. (2009). Relationships between students’ fraction knowledge and equation solving. Paper presentation at the Research Pre-session of the annual conference of the National Council of Teachers of Mathematics, Washington, D.C.

  • Hackenberg, A. J. (2010). Students’ reasoning with reversible multiplicative relationships. Cognition and Instruction, 28(4), 383–432.

    Article  Google Scholar 

  • Hackenberg, A. J. & Tillema, E. S. (2009). Students’ whole number multiplicative concepts: A critical constructive resource for fraction composition schemes. Journal of Mathematical Behavior, 28, 1–18.

    Article  Google Scholar 

  • Kaput, J. (2008). What is Algebra? What is Algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5–17). New York: Lawrence Erlbaum Associates.

    Google Scholar 

  • Kaput, J. J., Carraher, D. W. & Blanton, M. L. (Eds.). (2008). Algebra in the early grades. New York: Lawrence Erlbaum.

    Google Scholar 

  • Kieren, T. E. (1995). Creating spaces for learning fractions. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 31–65). Albany: State University of New York Press.

    Google Scholar 

  • Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170–193.

    Article  Google Scholar 

  • Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. J. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–667). Charlotte: Information Age.

    Google Scholar 

  • National Mathematics Advisory Panel (2008). Report of the task group on conceptual skills and knowledge. Washington, DC: U.S. Department of Education.

    Google Scholar 

  • Norton, A. (2008). Josh’s operational conjectures: Abductions of a splitting operation and the construction of new fractional schemes. Journal for Research in Mathematics Education, 39(4), 401–430.

    Google Scholar 

  • Norton, A. & Wilkins, J. L. M. (2009). A quantitative analysis of children’s splitting operations and fraction schemes. Journal of Mathematical Behavior, 28, 150–161.

    Article  Google Scholar 

  • Russell, S. J., Schifter, D. & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai & E. J. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 43–69). Berlin: Springer.

    Chapter  Google Scholar 

  • Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In M. J. Behr & J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp. 41–52). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • Smith, J. & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–132). New York: Erlbaum.

    Google Scholar 

  • Steffe, L. P. (1991). Operations that generate quantity. Learning and Individual Differences, 3(1), 61–82.

    Article  Google Scholar 

  • Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 43, 259–309.

    Article  Google Scholar 

  • Steffe, L. P. (2001, December 9–14). What is algebraic about children’s numerical operating? Paper presented at the Conference on the Future of the Teaching and Learning of Algebra, University of Melbourne, Australia.

  • Steffe, L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior, 20, 267–307.

    Article  Google Scholar 

  • Steffe, L. P. & Olive, J. (2010). Children’s fractional knowledge. New York: Springer.

    Book  Google Scholar 

  • Steffe, L. P. & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 267–306). Hillsdale: Erlbaum.

    Google Scholar 

  • Thomas, G. (2011). How to do your case study: A guide for students and researchers. Thousand Oaks: Sage.

    Google Scholar 

  • Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany: SUNY Press.

    Google Scholar 

  • Thompson, P. W. (1995). Notation, convention, and quantity in elementary mathematics. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 199–219). Albany: State University of New York Press.

    Google Scholar 

  • Thompson, P. W. & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), Research companion to the principles and standards for school mathematics (pp. 95–113). Reston: National Council of Teachers of Mathematics.

    Google Scholar 

  • von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning (vol. 6). New York: Routledge Falmer.

    Book  Google Scholar 

  • Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Thousand Oaks: Sage.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Education, Arizona State University, Farmer Education Building, 1050 S Forest Mall, Tempe, AZ, 85282, USA

    Mi Yeon Lee

  2. Indiana University, Bloomington, IN, USA

    Amy J. Hackenberg

Authors
  1. Mi Yeon Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amy J. Hackenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mi Yeon Lee.

Appendices

Appendix 1: Fraction Interview Protocol

  1. F1.

    The drawing (a line segment) below shows my piece of string. Think of your piece of string so that mine is five times longer than yours. Can you draw what you’re thinking of? Can you show for sure that mine is 5 times longer than yours?

  2. F2.

    Sara has a stack of CDs that is 65 cm tall. That’s 5 times the height of Roberto’s stack. Draw a picture of this situation. How tall is Roberto’s stack of CDs? How did you solve this problem?

  3. F3.

    The drawing (a rectangle) is a picture of a candy bar. Draw a separate candy bar that is 9/7 of that bar. If the student completes this problem, ask her or him to shade one piece and tell how much it is of the original bar. (After the interviewer shading one piece from the student’s picture) “How much is that piece? How much of the top bar is it? How much of the bottom bar is it? Is it possible for the same piece to have different names?

  4. F4.

    This candy bar (a rectangle) is 3/5 the size of another candy bar. Make a separate drawing of the other candy bar. How did you make your drawing? What is the fraction name of another bar? Which bar represents the unit?

    1. a.

      If difficult, try: This candy bar (a rectangle) is 1/5 of another candy bar. Make a separate drawing of the other candy bar.

    2. b.

      If easy, try: This candy bar (a rectangle) is 4/3 of another candy bar. Make a separate drawing of the other candy bar.

  5. F5.

    Tanya has $84, which is 4/7 the size of David’s amount of money. Draw a picture of this situation. How much does David have?

    1. a.

      If difficult, try: Tanya has $15, which is 1/5 the size of David’s amount of money. Draw a picture of this situation. How much does David have?

    2. b.

      If very easy, try: Cassie earned $48 babysitting. That’s 4/3 the amount of money Serena earned. Draw a picture of this situation. How much money did Serena earn?

Appendix 2: Algebra Interview Protocol

  1. A1.

    Do you have a cord with earplugs for listening to music? How long do you think it is? It’s not a value that we know exactly, right? But we could measure it to find the exact value. Okay, so Stephen has a cord for his iPod that is some number of feet long. His cord is five times the length of Rebecca’s cord.

    1. a.

      Could you draw a picture of this situation? Describe to me what your picture represents.

    2. b.

      Can you write an equation for this situation? Elicit what the letters represent to them. Can you tell me in words what your equation means?

    3. c.

      As necessary. Can you check your equation with your picture?

    4. d.

      As necessary. Check your equation using this question: Who has a longer cord, Stephen or Rebecca?

    5. e.

      Can you write more than one equation? As necessary (if they have only written something like t   =   5*q, where t represents Stephen’s cord length and q represents Rebecca’s cord length): Can you write an equation to express Rebecca’s cord length in terms of Stephen’s?

    6. f.

      As necessary (if they have written something like t   =   q ÷ 5): Can you write this equation using multiplication?

    7. g.

      Let’s say Stephen’s cord is 15 ft long. Explain how to find the length of Rebecca’s cord.

  2. A2.

    There are 5 identical candy bars (show picture) and each candy bar weighs some number of ounces. Let’s say that h   =   the weight of one bar. How much does 1/7 of all the candy weigh?

    1. a.

      If this question is hard, start with 2 or 3 bars and ask about 1/3 or 1/5.

    2. b.

      If still hard, use sharing language to find out about whether the student can make fair shares.

    3. c.

      Could you draw a picture of 1/7 of all the candy?

    4. d.

      Can you write down an expression for the weight of 1/7 of all the candy?

    5. e.

      If part (d) is hard, then drop back to notating quantitative situation of sharing 5 bars fairly among 7 people.

    6. f.

      If okay, move onto how much 3/7 of all the candy weighs. What would a picture of that look like? Can you write down an expression for the weight?

    7. g.

      To test out improper fractions, could try asking for 9/7 of all the candy.

  3. A3.

    Theo has a stack of CDs some number of cm tall. Sam’s stack is two-fifths of that height.

    1. a.

      Draw a picture to represent this situation.

    2. b.

      Can you write an expression for how tall the height of Sam’s stack is? Elicit what the letter(s) represent(s) to them. Can you tell me in words what your expression(s) mean(s)?

    3. c.

      Can you write an equation based on your expression in (b)? Can you tell me in words what your equation means?

    4. d.

      Can you write another equation for the situation? Can you tell me in words what your equation means?

    5. e.

      (If students write the second equation using inverse of 2/5), Can we switch the numbers like that? What relationship do you see in these two equations?

    6. f.

      Ask them to test their equations with particular numbers, such as 10 cm as the height of Theo’s stack.

    7. g.

      If this is hard, start with: Sam’s stack is 1/5 of that height.

    8. h.

      If this is easy, start with: Michael’s stack of CDs is 7/5 of the height of Theo’s.

  4. A4.

    Christina earned some money babysitting. That’s 4/3 of what Serena earned.

    1. a.

      Draw a picture to represent this situation.

    2. b.

      Can you write an equation that relates the amount of money Christina earned to the amount of money Serena earned? Can you tell me in words what your equation means?

    3. c.

      Can you write another equation? Can you tell me in words what your equation mean(s)?

    4. d.

      Ask them to test their equations with particular numbers, such as $36 for Christina’s money.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lee, M.Y., Hackenberg, A.J. RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND ALGEBRAIC REASONING: THE CASE OF WILLA. Int J of Sci and Math Educ 12, 975–1000 (2014). https://doi.org/10.1007/s10763-013-9442-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10763-013-9442-8

Key words

Search

Navigation