Hendrik.VanSteenbrugge@UGent.be PME 34 – 18-23/7/2010 Belo Horizonte

# Hendrik.VanSteenbrugge@UGent.be PME 34 – 18-23/7/2010 Belo Horizonte

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## Hendrik.VanSteenbrugge@UGent.be PME 34 – 18-23/7/2010 Belo Horizonte

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1. A wizard at mathematics as teacher? A study into the knowledge of fractions of preservice primary school teachers Support committee: Prof. dr. A. Desoete (co-promotor, UGent) Prof. dr. K.P.E. Gravemeijer (ESOE) Prof. dr. J. Grégoire (UCL) Prof. dr. M. Valcke (promotor, UGent) Prof. dr. L. Verschaffel (KULeuven) H. Van Steenbrugge, M. Valcke, A. Desoete Hendrik.VanSteenbrugge@UGent.be PME 34 – 18-23/7/2010 Belo Horizonte

2. Overview • Introduction • Current study • Methodology • Results • Conclusion

3. Introduction: fractions • Fractions: important though very difficult topic • students’ performance results (NCES, 2000) • Teachers (Van Steenbrugge, Valcke, & Desoete, 2010) • Important: percentages, decimals, and algebra (Lamon, 1999)

4. Introduction: elementary school students • Gap procedural – conceptual knowledge of fractions (Aksu, 1997; Post, Cramer, Behr, Lesh & Harel, 1993) • Conceptual K: multiplicity of meanings (Kilpatrick, Swafford, & Findell, 2001) • 5 subconstructs

5. Introduction: elementary school students • Five subconstructs • Part-whole: • Ratio: “John and Mary are making lemonade. Whose lemonade is going to be sweatier, if the kids use the following recipes? John: 2 spoons of sugar for every 5 glasses of lemonade; Mary: 4 spoons of sugar for every 8 glasses of lemonade” • Operator: “By how many times should we increase 9 to get 15?” • Quotient: “Five cakes are equally divided among four friends. How much does anyone get?”

6. Introduction: elementary school students • Measure: number and interval • Number: “Write for every number in the left column, the corresponding fraction in the right column.” • Interval: “Locate 9/3 and 11/6 on the following number line”

7. Introduction: elementary school students • Students most successful on tasks about the part-whole sub construct & In general they have too little knowledge of the other sub constructs • Especially students’ knowledge on the sub construct measure seems to be problematic (Charalambous & Pitta-Pantazi, 2007; Clarke, et al., 2007; Hannula, 2003).

8. The current study • Teachers’ knowledge of fractions?? • Scarce and limited (Newton, 2008) • Deep understanding of school mathematics by elementary school teachers  educational practices & students’ learning (Borko, et al., 1992; Hill, Rowan, & Ball, 2005; Ma, 1999) • How some procedures work & WHY these procedures work (Newton, 2008)

9. The current study • Deep knowledge: preservice teachers and inservice teachers ?? (Tirosh, 2000; Zhou, Peverly, & Xin, 2006) • Serious concerns regarding the readiness of some student teachers to teach mathematics to elementary school children (Conference board of the mathematical sciences, 2001; Verschaffel, Janssens, & Janssen, 2005)

10. The current study: research questions • To which extend do preservice teachers master the procedural and conceptual knowledge of fractions • To which extent do preservice teachers master a deeper knowledge concerning fractions?

11. Methodology • 290 preservice teachers • First year trainees: 184; Third year trainees: 106 • Male: 43; Female: 247 • General oriented secondary education: 197 • Practical oriented secondary education: 93 • Instrument: • Conceptual K: existing instruments (elementary school students) • Procedural K: textbooks

12. Results: procedural – conceptual K • Grand mean: .80 (.11); PK: .86 (.15); CK: .79 (.12) • 2*2*2*2 mixed ANOVA design: • Gender: M > F • Sec: GSE > PSE • Type: PK > CK • Gender*Type • Gender*Type*Sec

13. Results: procedural – conceptual K • Gender*Type • CK: M > F; PK: M = F • F: P > C; M: P = C • Gender*Type*Sec • GSE & PSE: CK: M > F; PK: M = F • GSE & PSE: F: P > C; M: P = C • CK: F(GSE) > F(PSE) ; M(GSE) = M(PSE) • PK: F(GSE) = F(PSE) ; M(GSE) = M(PSE) • => M>F on CK; PK>CK for F; GSE>PSE for F&CK

14. Results: conceptual K • 2*2*2*2 mixed ANOVA design: • Subconstruct • RATIO > rest • P-W > rest minus ratio • OP > mi; = q; < ratio; p-w; mg • Q > mi; = op; < ratio; p-w; mg • MG > op; q; mi; < pw, ratio • MI < rest • Gender: M>F • Sec: GSE>PSE • Gender*Subconstruct: M>F except for Q • Year: not significant

15. Results: Deep knowledge • 2*2*2* ANOVA design: • 5/6 – 1/4 = …; 2/6 + 1/3 = …; 5 : 1/2 = …; 2/5 x 3/5 = …; 3/4 : 5/8 = … • Mean: .42 (.20)  /2 ! • Sec: GSE >PSE; p = .045 • Gender*Year; p = .047 • Year: …

16. Conclusion • Partial overlap results preservice teachers & elementary school students • Chicken – egg? • .80/1: ok? • Deep knowledge ... • Teacher education?

17. Wrong WrongWrong Some remarkable results ... 70% 63.10% • Locate number 1: • “Is there a fraction located between 1/8 and 1/9? If yes, give an example.” • 1.33 = ... • “By how many times should we increase 9 to get 15?” • “Which of the following are numbers? Put a circle around them: A 4 * 1.7 16 0.006 2/5 47.5 1/2 \$ 1 4/5” • “Peter prepares 14 cakes. He divides these cakes equally between his 6 friends. How much cake does each of them get?” 52.07% 43.45% 35.86% 35.52%