Secondary school students are known to face a range of difficulties in learning about proof and proving in mathematics. This paper reports on a study designed to address the issue of students’ cognitive needs for conviction and verification in algebraic statements. Through an analysis of data from 418 students (206 from Grade 8, and 212 from Grade 9), we report on how students might be able to ‘construct’ a formal proof, yet they may not fully appreciate the significance of such formal proof. The students may believe that formal proof is a valid argument, while, at the same time, they also resort to experimental verification as an acceptable way of ‘ensuring’ universality and generality of algebraic statement

Fujita, Taro

Jones, Keith

Kumakura, Hiroyuki

Kunimune, Susumu

English

Southampton (e-Prints Soton)

Cite as: Kunimune, S., Kumakura, H., Jones, K. and Fujita, T. (2009) Lower secondary school students’ understanding of algebraic proof. In, Tzekaki, M., Kaldrimidou, M. and Sakonidis, H. (eds.) Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (PME33), Volume 3, pp441-448.  3- 441     LOWER SECONDARY SCHOOL STUDENTS’ UNDERSTANDING OF ALGEBRAIC PROOF Susumu Kunimune Shizuoka University, Japan Hiroyuki Kumakura Shizuoka University, Japan Keith Jones University of Southampton, UK Taro Fujita, University of Plymouth, UK  Secondary school students are known to face a range of difficulties in learning about proof and proving in mathematics. This paper reports on a study designed to address the issue of students’ cognitive needs for conviction and verification in algebraic statements. Through an analysis of data from 418 students (206 from Grade 8, and 212 from Grade 9), we report on how students might be able to ‘construct’ a formal proof, yet they may not fully appreciate the significance of such formal proof. The students may believe that formal proof is a valid argument, while, at the same time, they  also  resort  to  experimental  verification  as  an  acceptable  way  of  ‘ensuring’ universality and generality of algebraic statements. INTRODUCTION Evidence from a range of research studies indicates that across the world, secondary school students have difficulties in following and constructing formally presented deductive proofs, in understanding how such proofs differ from empirical evidence, and in using deductive proofs to derive further results (for recent reviews, see, for example, Mariotti, 2007). As part of a wider research initiative, we have researched such issues in the case of geometrical proofs (see, for example, Kunimune, 1987; Kunimune, Fujita and Jones, 2009). In this paper we address the issue of students’ natural cognitive needs for conviction and verification in algebraic statements.  In what follows, we first provide some background from related existing research. We then outline our theoretical framework which seeks to capture secondary school students’  understanding  of  algebraic  proof.  This  leads  to  the  presentation  of  our results  in  terms  of  how  students  in  lower  secondary  schools  perceive  ‘proof’  in algebra through an analysis of data from 418 students (206 from Grade 8, and 212 from Grade 9) collected in Japan in 2005. STUDENTS’ UNDERSTANDING OF ALGEBRAIC PROOF Of the range of research studies on students and algebraic proof, we highlight two studies  that  are  particularly  pertinent.  Healy  and  Hoyles  (2000)  surveyed  high-attaining 14- and 15-year-old students about proof in algebra and found that students simultaneously held two different conceptions of proof. On the one side, the students Kunimune, Kumakura, Jones & Fujita   3- 442   PME 33 - 2009 viewed algebraic arguments as those they considered would receive the best mark from their teacher. On the other side, empirical argument predominated in students’ own proof constructions, although most students were aware of the limitations of such arguments.  Similarly, Groves and Doig (2008) report that while over 35% of Year 8 students (aged  13)  recognised  the  need  for  a  logical  mathematical  explanation  to  prove Goldbach’s conjecture, over 60% “believed that it was enough to show it true for at least 1000 randomly chosen numbers or as many as possible, or to find one number for which it was not true” (p 345).  Such studies illustrate the need to continue studying students’ cognitive needs for conviction and verification in algebraic proofs. THEORETICAL FRAMEWORK Extensive research in algebra education (for recent reviews, see, for example, Kieran, 2006)  suggests  the  following  issues  are  relevant  to  students’  understanding  of algebraic proof: •  Cognitive gap between arithmetic and algebra  •  Understanding of algebraic symbols For example, consider the statement: ‘Sums of three consecutive whole numbers such as 2, 3 & 4 or 7, 8&9 are always multiples of 3’. To ‘prove’ this statement (that is, to verifying its generality) a secondary school student might do one of the following: •  Use arithmetic examples, sometimes with large numbers, and check results. For  example,  the  student  might  say  ‘I  tried  4+5+6=15=3x5, 12+13+14=39=3x13,  23+24+25=72=3x24,  and  so  one,  and  I  found  the answers are always multiples of 3. •  Use algebraic symbolisation to provide an argument which might say three consecutive  numbers  can be expressed  as  ‘x’,  ‘x+1’  and  ‘x+2’;  the  sum  is ‘3x+3’. Now ‘3x+3 = 3(x+1)’. This shows that the sum is always a multiple of three. Given our focus on students’ understanding of proof, and the knowledge that the transition  from  experimental/empirical  verification  to  formal  proof  is  not straightforward, in our research we capture students’ understanding of proof in terms of the following two components: ‘Generality of proof’ and ‘Construction of proof’ (see, Kunimune, 1987; Kunimune, Fujita and Jones, 2009). In our work on students’ understanding of algebraic proof, we refer to these two aspects of proof and proving as: ‘Construction of algebraic proof’ and ‘Generality of algebraic proof’. The first one of these, ‘Generality of proof’ recognizes that, on the one hand, students have  to  understand  the  generality  of  proof  (including  the  generality  of  algebraic symbols,  with,  for  example,  ‘x’  as  generalised  number),  the  universality  and Kunimune, Kumakura, Jones & Fujita  3- 443  PME 33 - 2009 generality of proved algebraic statements, the difference between formal proof and verification  by  examples,  and  so  on.  The  second  of  these  two  components, ‘Construction of proof’, recognises that, on the other hand, students also have to learn how  to  ‘construct’  deductive  arguments  in  algebra  by  knowing  sufficient  about definitions, assumptions, proofs, theorems, logical circularity, and so on. In Table 1 we characterise the nature of the two aspects of student proof and proving in  algebra:  ‘Construction  of  algebraic  proof’  and  ‘Generality  of  algebraic  proof’ using ideas related to the cognitive gap between arithmetic and algebra, and student understanding of algebraic symbols.  Construction of algebraic proof  Generality of algebraic proof To follow, or construct, algebraic proof, students might have to: •  Understand  what  is  required  to show/explain in given problems •  Understand  assumptions  and conclusions in statements •  Represent  given  word  problems by  using  algebraic  symbols, interpret algebraic results etc. •  Undertake  fundamental  algebraic manipulations; for example: 3x+3 =  3(x+1),  2x+y+3x-6y  =  5x-5y and so on To appreciate or understand why formal proof is necessary, students might have to: •  Understand the universality and generality of statements which are represented by algebraic symbols •  Understand the universality and generality of algebraic symbols •  Understand the universality and generality of proof •  Understand difference between formal proof and experimental verification (inductive approach) Table 1: The two aspects of students’ understanding of algebraic proof Given these two aspects of student proof, our theoretical approach, informed by our work on proof in geometry (see, for example, Kunimune, 1987; Kunimune, Fujita and Jones, 2009) is to characterise four levels of students’ understanding of algebraic proof. This characterisation is presented in Table 2. We argue that this framework captures the increasing complexity in students’ attempts at construction of algebraic proof and generality of algebraic proof. METHODOLOGY The research design involved a survey of secondary school students’ understanding of algebraic proof. A sample of relevant questions, and the corresponding marking scheme, is provided in Appendix A.   Kunimune, Kumakura, Jones & Fujita   3- 444   PME 33 - 2009   Construction of algebraic proof  Generality of algebraic proof Level 0  At  this  level,  students  do  not understand  what  they  have  to explain.  At  this  level,  students  do  not understand  what  they  have  to explain. Level I  At this level, students explain their argument  without  using  any algebraic symbols At  this  level,  students  do  not understand  neither  why  algebraic proof  is  necessary  nor  empirical verification is not enough to verify the universality and generality of algebraic statements Level II  At this level, students start using algebraic  symbols  in  their argument, but their use is incorrect At this level, two things occur: a)  students  start  recognising  that empirical  verification  is  not enough,  but  do  not  understand why  they  have  to  use  algebraic symbols b)  students  start  understanding why algebraic proof is necessary, but do not recognise that empirical verification is not enough Level III  At  this  level,  students  use algebraic  symbols  properly  to prove statements At  this  level,  students  can understand why algebraic proof is necessary Table 3: levels of students’ understanding of algebraic proof FINDINGS AND DISCUSSION  Our data is from 418 students (206 from Grade 8, and 212 from Grade 9) surveyed in Japan in 2005. The results for ‘Construction of proof’ are given in Table 4.   Level 0  Level I  Level II  Level III  N G8 G9 64% 29% 6% 4% 10% 14% 20% 53% G8=206 G9 = 212 Table 4: results for ‘Construction of proof’ As can be seen from Table 4, some 70% of Grade 8 students are at the Level I or below; that is, these students use empirical examples to verify the statements (Level I) or do not know what to do (Level 0). The results of Grade 9 students are superior. This is likely to be because students study more algebraic manipulation and proof in Kunimune, Kumakura, Jones & Fujita  3- 445  PME 33 - 2009 Grade 9. Nevertheless, 33% of students remain at either Level 0 or I, which implies that the teaching of algebraic proof could be improved in Grades 8 and 9.  The results for ‘Generality of proof’ are given in Table 5.   Level 0  Level I  Level II(a) and (b) Level III  N G8 G9 15% 11% 36% 23% 4%&26% 3%&24% 19% 39% G8=206 G9 = 212 Table 5: results for ‘Generality of proof These results suggest that at Grades 8 and 9, students begin pondering the difference between  empirical  verifications  and  proof.  This  is,  as  indicated  above,  because students study more algebraic manipulations and proof in Grades 8 and 9. In general, more students are at Level II-b) than Level II-a). This implies that the students start understanding why algebraic proof is necessary in Grade 8, yet, at the same time, they  do  not  recognise  that  empirical  verification  is  insufficient  for  mathematical proof.  Furthermore,  58%  (11+23+24)  of  Grade  9  students  remain  at  Level  II-b), Level I or Level 0. These findings are very similar to the findings for a parallel study on geometrical proof (Kunimune, 1987, 2000; Kunimune, Fujita and Jones, 2009).  Thus, Grade 8 and 9 students are achieving in terms of ‘Construction of proof’, but not necessarily in terms of ‘Generality of proof’. There is a gap between the two aspects. This means that students might be able to ‘construct’ a formal proof, yet they may not appreciate the significance of such a formal proof. They may believe that formal  proof  is  a  valid  argument,  while,  at  the  same  time,  they  also  believe experimental verification is equally acceptable to ‘ensure’ universality and generality of algebraic statements.  We now compare students’ Construction of proof (CoP) and their Generality of proof (GoP) at Grade 8 and Grade 9, see Table 6. Grade 8 totals  15%  36%  30%  19%  100%  N=206 CoP III  0%  1%  6%  12%  19%   CoP II  0%  3%  4%  3%  10%   CoP I  0%  4%  2%  1%  7%   CoP 0  15%  28%  18%  3%  64%   Levels  GoP 0  GoP I  GoP II  GoP III  Total      Kunimune, Kumakura, Jones & Fujita   3- 446   PME 33 - 2009 Grade 9 totals  11%  23%  27%  39%  100%  N=212 CoP III  0%  4%  14%  35%  53%   CoP II  1%  4%  6%  3%  14%   CoP I  1%  2%  1%  0%  4%   CoP 0  9%  13%  5%  1%  29%   Levels  GoP 0  GoP I  GoP II  GoP III  Total   Table 6: compare students’ Construction of proof (CoP) and their Generality of proof The results in Table 6 show that, on the one hand, progressions from CoP I and CoG I to CoPII and CoG II are observed in Grade 9, when students study more algebra than in Grade 8. In addition, students are introduced to ideas of ‘proof’ in geometry, and this is likely to contribute to students’ awareness of formal proof. On the other hand, in Grade 9 some 18% (=14+4) of students at CoP Level III are, at the same time, at GoP Level II or I (indicated in gray). This suggests that the teaching of algebra that the students might have experienced might have particularly emphasised the ‘Construction of proof’ aspects of algebra.  In general, more students are at Level II-b) than Level II-a). This implies that students start understanding why algebraic proof is necessary in Grade 8, but do not recognise that empirical verification is not enough. Furthermore, half of the Grade 9 students remain at Level II-b) or below. These findings are very similar to those that we have found with geometrical proof (Kunimune, 1987, 2000; Kunimune, Fujita and Jones, 2009).  CONCLUDING COMMENT The Grade 8 and 9 students that we studied are achieving in terms of ‘Construction of proof’, but not necessarily so in terms of ‘Generality of proof’. There is a gap between the two aspects. This means that students might be able to ‘construct’ a formal proof, yet they may not appreciate the significance of such formal proof. They may believe that formal proof is a valid argument, while, at the same time, they also resort to experimental verification as an acceptable way of ‘ensuring’ universality and generality of algebraic statements. References Groves,  S.  and  Doig,  B.  (2008),  Students’  understanding  of  what  constitutes  a  proof. Proceedings of PME32; Morelia, Mexico. Healy,  L.,  &  Hoyles,  C.  (2000).  A  study  of  proof  conceptions  in  algebra,  Journal  for Research in Mathematics Education, 31(4), 396-428. Kieran,  C.  (2006).  Research  on  the  learning  and  teaching  of  algebra:  A  broadening  of sources of meaning. In A. Gutiérrez & P. Boero (Eds.), Handbook of Research on the Kunimune, Kumakura, Jones & Fujita  3- 447  PME 33 - 2009 Psychology of Mathematics Education: Past, present and future (pp. 11-49). Rotterdam: Sense Publishers. Kunimune, S. (1987). The study of understanding about ‘The significant of demonstration in learning geometrical figure’, Journal of Japan Society of Mathematical Education, 47&48, 3-23. [in Japanese]. Kunimune, S. (2000). A change in understanding with demonstration in geometry, Journal of Japan Society of Mathematics Education, 82(3), 66-76 [in Japanese]. Kunimune, S., Fujita, T. & Jones, K. (2009). “Why do we have to prove this?” Fostering students’  understanding  of  ‘proof’  in  geometry  in  lower  secondary  school.  Paper accepted for ICMI study 19 on “Proof and proving in mathematics education”. Mariotti, M. A. (2007). Proof and proving in mathematics education. In A. Guitierrez & P. Boero  (Eds.)  Handbook  of  Research  on  the  Psychology  of  Mathematics  Education. Rotterdam: Sense Publishers.  Appendix A Survey questions Q8 Consider three consecutive whole numbers and their sums, e.g. 2+3+4=9=3x3, 7+8+9=24=3x8, and they are always the multiples of 3. In fact, if you consider any three consecutive numbers, then their sums are always the multiples of 3. Explain this. Q9. See the calendar below carefully. You might notice that the sums of the three numbers in the boxes are as three times as the middle numbers (e.g. 2+9+16=27 = 3x9). Explain this is always true.  Q10. Read carefully the following three statements which explain a statement ‘A sum of two odd numbers is an even number’. Student A: 1+1=2, 3+3=6, 1+3=4, 3+7=10. So, I think a sum of two odd numbers is an even number. Accept/Not accept Student B: Let one odd number be ‘m’, and the other ‘n’. The sum is ‘m+n’, and ‘m+n’ is an even number. I think a sum of two odd numbers is an even number. Accept/Not accept Kunimune, Kumakura, Jones & Fujita   3- 448   PME 33 - 2009 Student  C:  Let  ‘m’  and  ‘n’  be  whole  numbers.  Two  odd numbers are ‘2m+1’ and ‘2n+1’ The sum is (2m+1)+(2n+1) =  2m+2n+2=2(m+n+1).  As  ‘m+n+1  is  a  whole  number, therefore 2(m+n+1) is an even number. Accept/Not accept  Table 4: survey questions of students’ understanding of algebraic proof   Construction of proof  Generality of proof Level 0  Q8 & 9 •  No answer •  Does not make sense •  Copy questions •  Wrong explanation Q10 •  No answer Level I  Q8 & 9 •  Explanations  with  concrete examples, figures or words •  Incomplete  explanations with words •  Explanations  with  concrete examples  and  arithmetic calculations Q10 Answers such as: A:  Accept;  B:  Accept;  C:  Not accept or A:  Accept;  B:  Accept;  C: Accept Level II  Q8 & 9 •  Incomplete  or  incorrect explanations with algebraic symbols Q10 Level II(a) A:  Not  accept;  B:  Accept;  C: Accept Level II(b) A:  Accept;  B:  Not  accept;  C: Accept Level III  Q8 & 9 •  Explanations with algebraic symbols •  Explanations with algebraic symbols with examples  Q10. A:  Not  accept;  B:  Not  accept; C: Accept  

A change in understanding with demonstration in geometry,

A study of proof conceptions in algebra,

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Lower secondary school students’ understanding of algebraic proof

Lower secondary school students’ understanding of algebraic proof

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