For Groups Q and R:

10. Sarah argues that if 3k 2+2k-1=0 and 3+k-2k 2=0 then it is true that

3k 2+k-1=3+k-2k 2 . Is she right? Explain your answer.

For Groups Q and R:

10. Sarah argues that if 3k 2+2k-1=0 and 3+k-2k 2=0 then it is true that

3k 2+k-1=3+k-2k 2 . Is she right? Explain your answer.

Common Responses Group P: For lower secondary school (age 14-15 years)

There were 29 students in Group P and students answers to questions 1a-1e are: (1a) 10 students out of 27 (34.4 %) said Yes k=5 is an equation; there were 2 no responses; (1b) 5 out of 27 (17.2%) said YES, 2 no response; (1c) 21 out of 27 (72.4%) said YES, 2 no response (1d) 23 out of 27 (79.3%) said YES, 2 no response (1e) 24 out of 27( 82.8%) said YES, 2 no response. Students reasons for answering in such ways were categorized in 3 groups (20 students out of 29)*:

(Category 1) Equation needs and “=” sign (1a, 1c, 1d, 1e were put into this group by 8 students because of this reason)

(Category 2) Equation needs an operation to carry out (1b, 1c, 1d, 1e were put into this group by 3 students)

(Category 3) Equation needs an “=” sign and an operation to carry out (1c, 1d, 1e were put into this group by 9 students)

*Note that 6 students’ answers did not fit into any category of these three categories and the rest of the students (3 of them) did not give a reason or an answer.

These 20 students’ answers to question 2 (transitivity question) were investigated to see how the answers fit into the three categories. 7 out of 8 students in category 1 answered question 2 correctly; 1 out of 3 student in category 2 answered question 2 correctly; and 3 out of 9 students in category 3 answered the question correctly.

Group Q: For upper secondary school (age 16-18 years)

For this group the equation in 5d replaced the equation in 1d, because 5d equation was more appropriate for this level of students. Also instead of transitivity used in question 2, for this group reflexive property of equivalence were used in 5f.

There were 76 students in this group and the answers for 5 were:

Question 5 Yes(Equation) No(not equation) No Response

a) a = 5 31 (40.8) 43(56.6%) 2 (2.6%)

b) 7w- w 21 (27.6%) 53(69.7%) 2 (2.6%)

c) 5t-t = 4t 62(81.6%) 12(15.8%) 2 (2.6%)

d) 0 = x2+2x-5 72 (91.1%) 2 (2.6%) 2 (2.6%)

e) 3w = 7w - 4w 61 (80.3%) 13(17.1%) 2 (2.6%)

f) a = a 27 (35.5%) 47 (61.8%) 2 (2.6%)

The reasons that students gave were also investigated further, and categorized. The above three categories were also seen but there were 2 more categories.

67 students out of 76 were put into the below categories:

(Category 1) Equation needs and “=” sign (5a, 5c, 5d, 5e were put into this group by 24 students because of this reason)

(Category 2) Equation needs an operation to carry out (5b, 5c, 5d, 5e were put into this group by 10 students)

(Category 3) Equation needs an “=” sign and an operation to carry out (5c, 5d, 5e were put into this group by 20 students)

(Category 4) Equation needs an operation but is not an identity or an assignment (5b, 5d were put into this group by 7 students)

(Category 5) Equation needs an operation and an “=” sign but is not an identity or an assignment (5d was put into this group by 4 students)

The last two categories were more apparent in Group Q students’ answers and it could be because as students get older they seemed to emphasise the solving aspect of equations since they are exposed to such process more and systematically.

Category 1 students’ (24 of them in Group Q) answers to question 3 seem to indicate that these students use ‘surface structure view’ of equation (Laborde, 2003).

Category 2 students’ (10 students) answers to question 3 clearly show that these students prioritize “solving”, “finding” or “calculating” an answer.

Group R:For first year university students

For this group as well as reflexive property(9e), symmetric property(9f) was used.

Question 9 Yes No No response

a) a = 5 21 8 1

b) 7w -w 4 26 0

c) x(x + 2) =x2 + 2x 27 2 1

d) 0 = x2+ 2x-5 29 1 0

e) a = a 18 12 0

f) a + b = b + a 22 8 0

In the above table we see that first year university students started to develop the concept of equivalence relation by making progress in understanding reflexive property (9e- 60% versus 35.5% in Group Q). However 26.7% of the students seemed to think in symbolic world terms rather than formal when we investigate their responses to 9f.

Question 10 for Groups Q and R:

Yes No No response

Group Q (76 students) 33 (43.4%) 23 (30.3%) 20 (26.3%)

Group R (30 students) 13 (43.3%) 8 (26.7%) 5 (16.7%)

The level of reasons given by Group R students for and against using transitivity changes, for example students, who said yes, stated that two things equal to the same thing must be equal to each other. One student state that these equations share a common solution rather than being equal to each other.

Some students, who said no, stated that k would be different in each equation.

It seems that organizing the properties of the concept of equation into a coherent schema is hard for students.