**JOURNAL OF RESEARCH IN NATIONAL DEVELOPMENT VOLUME 8 NO 1, JUNE, 2010**

**ANALYSIS OF STUDENTS’ PROBLEM SOLVING SKILLS IN MATHEMATICS PROBABILITY BY SECONDARY SCHOOL STUDENTS IN NIGER STATE**

**A.B.C. Orji and Sumbabi T. Umoru**

**Department of Science and Environment Education, University of Abuja**

**E-mail:sutumoru@yahoo.com**

**Abstract **

*This study assessed problem solving skills in mathematics probability by secondary school students with a view to identifying faulty steps that lead to poor performance. A total of three hundred (300) male and female students were randomly selected from ten (10) schools in the three (3) education zones of Niger state. The study is a survey type of research. The instrument used for data collection was an essay test item on probability (ETIP). Three mathematics educators validated the instrument (face and content). The reliability coefficient of the instrument was obtained as 0.87. One null hypothesis was stated at 0.05 level of significance. Data collected were analyzed using means, standard deviation and anova. The study revealed that: operational steps 4 and 5 posed more problems to the students than operational step 1 and 2 and 3; and students demonstrated most accurately the skills in operational steps 1 followed by step 3 and demonstrated lack of comprehension in steps 4 and 5. Based on these findings recommendations were given accordingly.*** **

**Keyword***:* Analysis, operational steps, probability, problem solving

**Introduction **

The development of any nation depends on advancement in science and technology (Ball, 1990). Shawn and Roger (2001:2) asserted that the people of the world are living in a changing world where science and technology have been part of the world’s tradition and any country that fails to recognize this fact does so at the risk of remaining backward technologically. It is for this reason that the National Policy on Education emphasized that, students should be well trained to be able to meet demands of modern age of science and technology.

To achieve this objective, students should have a good background in science subjects especially mathematics. It is the basic tool for science and technology and a pre-requisite for students’ admission into tertiary institutions to pursue any science or technological related courses. Yet students’ performance in mathematics is not encouraging (WAEC, 2003).

Probability which is the focus of this study is a branch pf mathematics that deals with calculating the likelihood of a given event occurrences, which has a number between 0 and 1 (Orage, 2001). This area of mathematics is of great importance to every day life and to the technological development of our nation. As increased use of technology and empirical science spread throughout the society, the use of data to communicate information and in decision making are influenced by projected outcomes based on estimated probabilities. In spite of the importance of probability to man and national development students continue to see the concept as abstract and difficult to understand (Gould, 1999). It is not easy to majority of our secondary school students to solve probability related problems in mathematics (Wilensky, 1994). The commonest area of difficulties of probability calculation among secondary school students lies in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events, and inability to draw meaningful conclusion from the probabilities calculated (Edwin, 1996).

This study therefore, analyzed problem solving skills in mathematics probability by secondary school students with a view to identifying faulty steps that led to poor performance.

**Purpose of the study **

- To ascertain if operational steps used in mathematics probability calculation affects students’ performance.
- To identify faulty steps that lead to poor performance and the mathematical operations that are involve in those steps.

**Research question**

What is the extend of difference in the mean performance of secondary school students regarding operational steps then adopt in solving mathematics probability problems?

**Hypothesis **

Ho: There is no significant difference in mean performance of secondary school students regarding operational steps they adopt in solving mathematics probability problems.

**Method **

This study used the survey research design. Three hundred (male 192 and female 108) students randomly selected from ten co-educational government owned secondary schools in Niger state, were used for the study. The target population for the study was the year III senior secondary (SS) III mathematics (25, 228) students in Niger state.

The data for the study was collected directly from the students in the selected schools in Niger state. The primary data collected was to analyze the operational steps used in solving mathematics probability problems with a view to identifying faulty steps that lead to poor performance.

Essay Test Items on probability (ETIP) drawn from past questions of senior school certificate Examination (SSCE) conducted by West African Examination Council and National Examination Council (NECO) was the main research instrument for this study. The instrument was subjected to face and content validity by three experts with a post graduate degree in mathematics and at least ten years of professional experience. The reliability coefficient of the test instrument was obtained as 0.87, which shows a test of high reliability.

The researcher personally administered the test instrument on the three hundred male and female students at the selected schools. The response rate achieved was 100%, while mean, standard deviation and anova were employed for data analysis.

**Data analysis **

Research question:

What is the extend of difference in the mean performance of secondary school students as regards operational steps which they adopt in solving probability related problems? To answer the question mean and standard derivation were used to analyze data as shown in table 1.

Table 1: Analysis of mean and standard deviation scores of five operational steps adopted by secondary school students in solving probability problems.

Description of operational steps |
Means score |
Standard deviation |

The ability to understand the probabilities of occurrence of events |
11.10 |
5.12 |

Interpretation of “OR” as sum of probability |
9.38 |
4.93 |

Interpretation of “AND” as product of probability |
5.55 |
3.90 |

To understand that pr (E) |
3.62 |
2.26 |

Evaluation of fractions |
3.08 |
1.71 |

The table above shows the mean scores and the standard deviation on the operational steps adopted by secondary school students in solving probability problems. The mean scores of operational steps 1, 2, 3, 4 and 5 are 11.10, 9. 38, 5.55, 3.62 and 3.08 respectively, while the standard deviations from the mean of each operational steps are 5.12, 4.93, 3.90, 2.26 and 1.71 respectively.

From the mean scores, it is observed that operation steps 1 and 2 (i.e. the ability to understand the probabilities of occurrence of events and interpretation of “OR” as sum of probability) are well understood by the secondary school students in solving probability problems (table 1). On the other hand, operational steps 3 4 and 5 (i.e interpretation of “AND” as product of probability, to understand that

pr (E) = No of expected outcome/ No of possible outcome and evaluation of fractions)

were not quite comprehensible to the students. The faults identified were associated with steps 3, 4 and 5 which affect their problem solving skills in probability calculations.

**Testing of hypothesis **

Ho1: There is no significant difference in the mean performance of secondary school students as regards operational steps which they adopt in solving probability problems.

To test the hypothesis, analysis of variance (ANOVA) was used and results are as shown in table 2a

Table 2a: ANOVA Results for Mean Performance of Students in Probability Problems Based on operational Steps

Source of Variation |
sum of square |
Df |
mean square |
F-value |
Std. Error |
Sig. (Two-tailed) |
Decision |

SSw |
33880 |
30 |
99.41 |
6.00 |
0.3456 |
0.0000 |
Rejected |

SSb |
2386 |
4 |
59.46 |

The ANOVA value for all items was obtained as shown in table 2a. Result displayed on the table showed that there was significant difference students’ mean performance in probability problems based on the operational steps. The null hypothesis was therefore rejected based on these results. It then implies that the students’ performance in probability problems using operational steps different significantly. In order to determine the source of the significant difference, a Multiple Classification Analysis was used as shown in table 2b.

Table 2b: Multiple Classification Analysis of Mean Performance Based

On Operational Steps Grand Mean = 38.30

variable/Category |
Number (N) |
Unadjusted Deviation |
ETA |
Adjusted for Independence Covariates Deviation |
BETA |

Step 1 |
30 |
12.0 |
0.21 |
- 0.13 |
1.51 |

Step 2 |
30 |
9.80 |
0.47 |
||

Step 3 |
30 |
11.30 |
0.14 |
||

Step 4 |
30 |
4.01 |
-0.29 |
||

Step 5 |
30 |
7.11 |
0.304 |

Table 2b shows that in the groups the adjusted mean response of each group expressed as a deviation of the grand mean (38.30) indicated the highest mean score of 50.3 for step 1 followed by Step 3 with 49.6 and then Step 2 with mean score of 48.1 as well as Step 4 and 5 in that order. It then means that students demonstrated most accurately the skills in Step 1 followed by Step 3. On the contrary, they demonstrated lack of comprehension and application of the skills involved in Steps 4 and 5

**Discussion of findings**

This study analyzed student problem solving skills in mathematics probability by secondary students. The findings of this study indicate that:

- Operational steps 4 and 5 posed more problems to the students than operational step one (1) and two (2) x3 in essence students do not appear to have full grasp of skills in evaluating fraction when solving probability problems
- Students demonstrate most accurately the skills in operational step or (1) follow by steps 3 and demonstrate lack of comprehension steps 4 and 6 (table2b)

The results of the study reported are similar to the earlier findings in the related studies (Gould, 1991 and Edwin, 1996). From these findings, it is clear that students could not do well in probability calculation due to their inability to understand operational steps 4 and 5 (i.e to understand that

p (B) = No of expected outcome / No of possible outcome and evaluates fraction)

student could have due to their inability to understand the questions and interpret some of the concepts that are needed to be applied in the various steps that could lead to the correct solution. Likewise Wilensky (1994) asserted that it is not easy to majority of our secondary school students to solve probability related problems in mathematics. From these findings, it implies that lack of exposure to mathematical problem solving skills through previously acquired knowledge could have contributed to their inability to understand the skills required in the operational steps.

**Conclusion **

It has been established through this study that operational steps used in solving mathematics probability problems affects students’ performance in mathematics (Table 1) and that the mathematical operation that are difficult to the student in the area of mathematics probability calculations are addition and multiplication.

Some of the implications of this study are: students performance in mathematics is determined by the operational steps used in solving the problem. If students do not understand the operational steps that could lead to the correct solution of the problem, it will always lead to poor performance in mathematics. If students poor performance in mathematics calculation persist as a result of inability to apply the correct operation steps that leads to the solving of any mathematical problems, it could also affect their performances in other science related courses.

As a result, advancement in science and technology can be hindered. In a situation, where students are performing poorly in mathematics, as a result of their inability to apply the operational steps correctly, it brings a challenge to mathematics teachers to revisit their methodology of teaching, paying attention to the operation steps used in solving the problems.

**Recommendations**

In view of the findings of this study, the following recommendations are deemed pertinent:

First and foremost, the enormous nature of the task performed by a mathematics teacher in secondary schools is quite commendable. There is still need for the improvement of mathematics teaching in our secondary schools.

Teachers should give more assignments to students in probability problems for better understanding of the concept. The teacher should endavour to check the assignments whether they are properly done.

During classroom instruction, teachers should check the operational steps used by the students in solving mathematics problems with a view to correcting their mistakes to enhance better performance in mathematics. Other topics on mathematics can be researched into with a view to considering the operational steps used so as to help the students overcome the difficulties they encounter in those topics.

**References**

Ball, D. (1990). With an eye on the mathematical horizon: Dilemmas

of teaching. Paper presented at the annual meeting of the American educational research association. Boston M.A

Edwin, T.J. (1996). *Probability theory:* The logic of science. Preprint:

Washington University (HTMC) press.

Gould, B. (1991). On proof and progress in mathematics. *Bulletin* of

the *American Mathematical society* 30 (2). 10-15

Orage, C.O. (2001). *Further mathematics*. Second edition Kaduna Nigeria: Clemol publishers.

Shawn, S. and Roger, S. (2001). Spatial visualization: Fundamentals

and trends in engineering graphics. *Journal of industrial technology* 18 (1), 2-4.

WAEC (2003). West – African Examination Council, chief examiners

report for May/June mathematics in Nigeria.

Wilensky, U. (1994). Learning probability through building

computational models. Proceedings of the nineteenth international conference on the psychology of mathematics education. Reafe, Bratil, July, 1994.