Students’ mathematical misconceptions Sally Jordan 28 th April 2007

1. Students’ mathematical misconceptionsSally Jordan 28th April 2007

2. My aims in this workshop • To tell you about my work in analysing student responses to S151’s online assessment questions; • To discuss some of the issues raised.

3. This work happened for two reasons: • Longstanding interest into why science students make the mathematical errors that they do. • The development of S151 Maths for Science and its assessment system. • S151 was developed to build the mathematical confidence of Science Faculty students between S103 and Level 2 courses.

4. Content of Maths for Science • Starting points (includes arithmetic, negative numbers, fractions, powers) • Measurement in science (includes SI units, scientific notation, significant figures) • Calculating in science • Algebra • Using graphs

5. Content of Maths for Science cont. • Angles and trigonometry • Logarithms • Probability and descriptive statistics • Statistical hypothesis testing • Differentiation • Appendix: Resolving vectors

6. S151 : Maths for Science • Has been studied by around 7000 students since it was first presented in September 2002; • Has been very well received by students; • Has increased retention rates on level 2 Science Faculty courses.

7. S151’s Assessment System We wanted to provide students with feedback on their work that was • Rapid • Targeted • Detailed So we developed an interactive online system for both formative and summative assessment. The system, now called OpenMark, is now used by several OU courses, and is being integrated into our VLE. The S151 Practice Assessment is at https://students.open.ac.uk/openmark/s151.pa/

8. The Maths for Science Assessment System Has succeeded in its aims; As a side-benefit, an analysis of the data-files is providing insight into students’ mathematical misconceptions. • Most of the analysis has been done on student responses to the summative End of Course Assessment; • Most of the questions are not multiple choice; • Results are similar whatever the variables in the question.

9. What mistakes do students make? • What do you think?

10. What mistakes do students make? • Students are surprisingly good at rearranging algebraic equations… • But they are surprisingly bad at substituting values for physical quantities into the equations.

11. Problems with substituting values into equations are caused by • Problems with units; • Problems with precision; • Problems with precedence, so students find

12. So why is the answer always 243?

13. So what are the problems with units? • Students give units of s1, s2 or s2 when finding the time period of a simple pendulum using the equation:

14. Students are not good at simplifying algebraic expressions Sometimes they oversimplify:

15. Students are not good at simplifying algebraic expressions Sometimes they fail to simplify

16. Students are not good at simplifying algebraic expressions Negative powers and dividing by fractions cause particular problems…

17. Students are not good at simplifying algebraic expressions Negative powers and dividing by fractions cause particular problems…

18. Students have particular difficulty in linking a straight line graph to its equation • Is this linked to problems with calculus?

19. When asked to find the gradient of a straight-line graph: • Students ignore the fact that not all straight-line graphs go through the origin; • Students find the reciprocal of the correct value – and have particular problems with negative gradient; • Students fail to read the labels on the axes; • Students ‘count squares’.

20. Taking this work forward • I have around 30 questions still to analyse • I would like to investigate some of the findings further, using observational and interview techniques.

21. A very different example • A question requiring a free text response: • ‘What does an object’s velocity tell you that its speed does not?’ • The correct answer is along the lines of ‘it tells you the direction in which the object is moving’; • But a number of students answered ‘It tells you about the object’s change in direction’. • Why?

22. Some text from the course…

23. Implications of the work • Changes are being made to the assessment questions; • Changes are being made to the course material; • New Open University courses are being written from a position of greater knowledge of students’ mathematical misconceptions.

25. How do these findings relate to your students mathematical misconceptions? • These findings relate to students who typically came to the course very lacking in confidence; • How do they relate to your experience? • Some of the findings have been very much what I expected (from many years as an OU tutor), but there have been some surprises; • But the findings are indisputable!

26. The misunderstanding is frequently at a simpler level than we imagine… • Problems with rounding, fractions, finding the gradient of a graph; • Sawyer (1964) ‘If we imagine that a pupil understandings something, when in fact he does not, then we are like a man trying to build on a foundation of air?’.

27. Sally JordanThe Open University in the East of England s.e.jordan@open.ac.uk