Comparing Grades 10 – 12 Mathematics Learners ’ Non-routine Problem Solving

The study, which is derived from a larger study, compares grades 10 – 12 mathematics learners’ non-routine problem solving. An exploratory study was conducted on a convenience sample drawn from three high performing high schools located in Tshwane North District, Gauteng province of South Africa. Learners wrote a non-routine problem solving test. Findings revealed that the 11 grade learners obtained the highest mean score while that of the 10 grade learners was the lowest. High school learners’ level of strategy use on solving non-routine problems improved significantly as they progress from grade 10 to higher grades. No significant difference was discovered as learners progress from grade 11 to 12.


INTRODUCTION
Mathematics is hierarchal and gets increasingly complex when progressing through the grades.Conversely, learners need requisite competencies to learn mathematics successfully in each grade.In this regard, the South African (SA) school curriculum sets out to expose learners to mathematical experiences that can develop their essential mathematical skills so that they can identify, investigate and solve problems creatively and critically, and develop mathematical reasoning and creative skills in preparation for more abstract and complex mathematical content (Department of Basic Education [DoBE], 2011, p9 &10).Also, in SA, mathematics is compulsory till grade 9 and generally schools only allow above-average learners to continue with it into senior grades.A learner who achieves poorly in mathematics in either 10 th or 11 th grade he/she will be made to switch to the less cognitively demanding mathematics literacy.In the senior grades however teaching is still generally examination driven where learners are subjected to drill and rote learning with emphasis on mastery of algorithms and procedures rather than developing conceptual reasoning strategies and highorder thinking skills (cf.CDE, 2013).Suffice it to state that SA learners'problem solving ability is compromised due to such teaching such that, for example, grade 12 learners could not successfully solve problems that involved high-order thinking and reasoning (AMESA, 2013) and TIMSS 2011 showed that SA learners lagged behind their counterparts in other participating countries with their mathematics competency (HSRC, 2012).Furthermore, Bansilal, Brijlall and Mkhwanazi (2014) measured mathematics teacher's content knowledge and found that on average teachers scored 29% on problem solving questions.Mathematics problem solving is essential in developing thinking and reasoning skills, promotes high-order thinking and facilitates conceptual understanding and meaningful learning (see Mogari & Lupahla, 2013).Webb (2010) notes that engaging learners more on cognitively demanding mathematical tasks requiring critical thinking and reasoning tend to enhance deeper knowledge and understanding which are requisite for developing problem solving competencies.Such competencies, according to OECD (2010), are learners' capacities to engage in cognitive processes in order to understand and resolve tasks where a method to discover the solution to a problem is not obvious.This study is derived from a larger study that sought to establish a link, if any, between learners' non-routine problem solving and their belief systems, and seeks to compare non-routine problem solving strategies of grade 10 -12 mathematics learners by addressing the questions: 1. Which non-routine problem solving strategies do grades 10 -12 learners use in mathematics?
2. Is there any significant statistical difference among the scores obtained by grades 10 -12 learners in non-routine problem solving?
Despite, efforts to promote problem solving learning in schools evidence shows that learners are still not competent enough to solve cognitively demanding problems such as non-

State of the literature
 Varying results produced in various countries on the level of development of non-routine problem solving ability and amount of (textbook) mathematics learnt. Non-routine problem solving continues to pose difficulties to learners across many countries  Generally, the quality of mathematics teaching remains a challenge in South Africa where emphasis is still on drill and rote learning.

Contribution of this paper to the literature
 Successful routine problem solving does not guarantee any success on solving non-routine problems  Successful non-routine problem solving seems not to be a function of amount of mathematics content acquired  Affirmation that conceptual learning under girds successful non-routine problem solving.
routine problems (e.g.Elia, Van den Heuvel, & Kolovou, 2009;AMESA, 2013;DoBE, 2014).CDE (2013) attributes the deficit to poor teaching in lower grades that has left learners with limited understanding of the basics and insubstantial foundational competencies.However, there are schools in SA, such as those in this study, that have displayed a consistent above average performance in grade 12 mathematics.This study therefore seeks to determine whether there is any improvement in the non-routine problem solving capacity of learners of high performing schools as they progress through grades.Van den Heuvel-Panhuizen, Kolovou and Robitzsch (2013) compared the grades 4 -6 learners' non-routine algebraic problem solving and found an improvement as learners' moved up the grades.Arslan and Altun (2007) investigated the success of grades 7 and 8 learners in using non-routine problem solving strategies and found better success in grade 8; learners displayed pre-knowledge of some non-problem solving strategies which they might have learnt from routine problem solving; and that grade 7 learners learnt some strategies more rapidly than those in grade 8.In TIMSS 2011, SA participated with grade 9 learners but was still outperformed by countries that participated with grade 8 learners (HSRC, 2012).By implication, it does not necessarily mean that learners in senior grades are always more mathematically capacitated than those in lower grades.Moreover, that generally there is emphasis on rigid, recurring and repetitive use of known procedures to solve routine problems in SA mathematics lessons (cf. CDE, 2013).
The South African 2015 and 2016 national senior certificate examinations diagnostic reports (see DoBE, 2015DoBE, , 2016) ) highlight a concern on learner poor performance on NRMPs.The reports indicate that learners copied with lower order questions that required application of routine procedures taught in the classrooms.They failed to cope with non-routine problems that required independent or creative thinking.The assumption was that learners were not adequately exposed to the NRMPs.They were not provided with adequate and appropriate classroom exercises that require application of basic knowledge to unfamiliar problem situations.Because of the focus on quantity and quality of mathematics examination results by most teachers as they teach mathematics, and the possible challenge of teaching and learning non-routine problems to both teachers and learners, this study examines high school learners' knowledge of problem solving (PS) strategies and their ability to effectively apply the strategies to PS.Despite effort put on teaching PS, a study by Aslan and Altun (2007) discovered that learners faced difficulties on mastering the non-routine PS skills.In addition, they lacked confidence on selection and application of the strategies to PS.Some challenges on teaching learners PS possibly result from learners failing to effectively learn and apply some heuristics (e.g., 'make a diagram', 'use a simple case') (Schoenfeld, 1992).Instead of teaching general heuristics to PS, Schoenfeld (1992) recommends teachers to teach learners specific strategies that relate or link directly to specific problem categories.
For different reasons, teachers tend to expose non-routine problems only or more often to the most able learners in mathematics; those who finish classwork earlier than others, or simply neglect to teach and adequately cover the problem solving content areas.This practice is against the expectation of CAPS that recommends all learners to be exposed to non-routine problems in order to develop higher thinking and reasoning skills.Exposing learners to nonroutine mathematical problems (NRMPs) can improve their ability to search for and apply strategies that are effective in solving problems and appreciate the beauty of mathematics (e.g., there is more than one way to solve a NRMP; mathematics is a product of human creation).Rather than being worried of some possible negative effects (e.g., demotivation), all learners should be encouraged to solve NRMPs to reap the possible benefits attached to it.The need for employees with high proficiency levels in non-routine problem solving by the current work industry (Gilfeather & Regass, 1999) could not be over emphasised.
This study was conducted assuming that all learners are exposed to non-routine problem solving as stipulated in CAPS document (see DoBE, 2011).Therefore, this study was pursued against this backdrop.The study seeks answers to questions: What are the nonroutine problem solving strategies the 10 th , 11 th and 12 th graders know?What is the learners' level of using strategies to problem solving?PREVIOUS RELATED STUDIES Arslan and Altun (2007) studied the level at which the 7 th and 8 th grade learners learn and use strategies to solve NRMPs.They discovered that the level of learning and effective application of a strategy to PS was dependent on the age of a learner.For example, as compared to the 8 th graders, the 7 th graders showed a greater improvement to applying the 'make a list' strategy to PS.On the other hand, as compared to the 7 th graders, the 8 th graders showed a rapid improvement to learning and application of 'look for a pattern' strategy to PS.A study in Turkey by Yazgan (2015) on the role of each PS strategy on explaining learners' success in solving NRMPs and in discriminating between successful and non-successful 6 th graders revealed that success in solving NRMPs depends on knowledge of PS strategies.Specifically, Yazgan (2015) discovered that strategies explained 65% of the learners' success in PS.On this regard, Gilfeather and Regass (1999) contend that learners can be able to discover solutions to NRMPs by being able to use procedures and strategies effectively.The strategies, as ordered in terms of their significant contribution to success in solving NRMPs were 'make a drawing', 'look for a pattern', 'guess and check', 'make a systematic list', 'simplify the problem' and 'work backward'.Yazgan (2015) also discovered that the level of strategy use possibly discriminated between high achievers and low achievers in NRMP solving.For instance, high achievers could use 'look for a pattern' and 'make a drawing' more successful in PS than the low achievers.
This study attempts to fill the following gaps in the literature: Most studies reviewed fail to describe or analyse in full the high school learners' non-routine problem solving abilities.In addition, there is a limited number of studies that analyse, describe and compare high school learners' levels of NRMP solving abilities.

THEORETICAL BACKGROUND
It is argued that in order to explore the learners' non-routine mathematical problem (NRMP) solving capacity it is crucial to get insight into 'What are non-routine problems?';'What do NRMP entail?' and 'What does solving NRMP require and involve?'

Non-routine problems
These can be described as mathematical problems that are more complicated and difficult, do not have straightforward solutions, require productive thinking, are approached in more or less sophisticated ways, are nonstandard, involve unexpected and unfamiliar solutions, require an insightful approach and strategic thinking and involve the use of various mathematical concepts (see Mogari & Lupahla, 2013, p95) Brunning, Schraw and Ronning (1999) view NRMPs as ill-defined problems with several acceptable solutions that can be obtained using several unique strategies.Elia, van den Heuvel-Panhuizen and Kolovou (2013, p607) consider NRMPs 'more complicated and difficult than routine problems' and similarly, according to Mogari and Lupahla (2013, p95), they are 'more cognitively challenging and demanding than routine problems'.NRMPs are tasks based on real life circumstances or models thereof that cannot be solved using familiar methods by replicating known procedures (see Muis, 2004;Arslan & Altun, 2007;Mogari & Lupahla, 2013).To this end, in the view of Arslan and Altun, NRMP reflects the relationship between mathematics and reality.This study hopes to establish whether a link exists between the capacity to solve NRMPs and the amount of mathematics learned when moving up the grades.

Non-routine problem solving
Non-routine problem solving depends considerably on higher-order thinking and reasoning and, according to Carson (2007) and Lester (2013), it requires learners to have capacity to synthesise and coordinate their knowledge and skills and apply them to novel problem situations.Arslan and Altun (2007) consider skills and knowledge consisting, among others, of concepts, formulae and algorithms essential in NRMP solving.Elia et al (2009) report that non-routine problem solving is a non-straightforward process that requires creative thinking and use of heuristic strategies to understand the problem situation as NRMPs are complicated and difficult.Arguably, NRMPs have no definite or rehearsed procedure or pathway to follow when solving them.Seemingly solving NRMPs require thinking, flexibility, creativity, inventiveness and resourcefulness and these are some of the attributes aspired by the SA school curriculum.This study intends to determine whether these attributes develop as learners go up the grades.That is, is there any link between the amount of mathematics learned and the capacity to solve non-routine problems?Lester (2013) contends that for learners to succeed in problem solving, they need adequate and relevant previous experiences in learning how to solve problems, a strong mathematical knowledge base, knowledge of various mathematical models or representations, and have the ability to model or represent mathematical situations and construct or draw patterns of inferences.Kolovou (2011) indicates that the solution process of NRMP requires productive and strategic thinking, unravelling complexity of the problem situation, and insights into how quantities in a problem situation relate to one another.Given that prior instruction plays insignificant role in developing NRMP solving ability (Mabilangan, Limjap & Belecina, 2011), it thus posited that solving many of such problems is essential in this regard and, according to Schoenfeld (1992), success in NRMP solving hinges on cognitive resources, strategy use, beliefs and control process.But, successful NRMP solving depends on proper use of problem solving strategies (Elia et al, 2013).The study therefore seeks to determine whether the degree of use of strategies improves with progression through the grades.

Non-routine mathematical problem solving strategies
Unlike routine problems that require algorithms, solving NRMPs involve the use of strategies (see Polya, 1985;Schoenfeld, 1992;Elia et al, 2009) even though strategies don't guarantee solution to problems.Instead, strategies help to establish procedures or pathways to solutions.We argue that arguably the process is based on learners' thinking and understanding of NRMP.Strategies are defined in Yazgan (2013, p572) as procedures used to explore, analyse and probe aspects of non-routine problems with a view to devising a pathway to the solution.There are two types of strategies, namely, cognitive and meta-cognitive strategies.Cognitive strategies include general strategies, such as trial-and-error; working backwards; finding a pattern; using analogies; considering extreme cases; visual representation; intelligent guessing and testing; systematically accounting for all possibilities; and deductive reasoning (see Muis, 2004;Russell, 2007;Depaepe, 2009;Logsdon, 2007;Malouff, 2011).Learners don't normally use strategies spontaneously when solving NRMPs (Schoenfeld, 1992).They instead glance at NRMP in trying to decide what reckoning to perform and this is a shortcoming on their part (Arslan & Altun, 2007).Also, the cognitive demands of respective strategies vary and this affects the rate of use of each strategy, for example, trial-and-error is set to have minimum cognitive demands and it is commonly use (Elia et al;2009).Some problems are solved with more than one strategy.
In terms of picking on strategies to use in NRMP solving, learners tend to prefer some strategies to the other albeit with varying outcomes.For example, Mogari and Lupahla (2013) found that Namibian learners most preferred the algebraic strategy to other strategies, even though they successfully solved problems that were illustrated with diagrams.The Filipino learners in Mabilangan, Limjap and Belecina (2011) used the 'making a model or diagram' strategy the most.Learners frequently used trial-and-error strategy with high rate of success in Elia et al. (2009).Hegarty and Kozhevnikov (1999) discovered that Ireland learners performed poorly in problems accompanied with diagrams or pictorial representations, probably, because the diagrams diverted the learners' attention from the main relationships in the problem statements to irrelevant visual details in the diagram or picture.Meta-cognitive strategies include self-regulatory actions such as decomposing the problem, monitoring the solution process, evaluating, and verifying results (Schoenfeld, 1992).This study will explore the cognitive strategies learners use and also determine whether their ability to use the strategies improves as they go up the grades.
In sum, to succeed with NRMP solving, it is argued that it is important for learners to know the features of NRMP so that they can distinguish it from routine problems.As this enables learners realise that NRMP requires different procedures to solve.NRMP solving involves strategy use instead of algorithms, and learners have to understand the potential and efficacy of each strategy.Learners should also know that solution pathways of some NRMP may entail more than one strategy and the solution pathway of each NRMP may not necessarily work for the other NRMP.Also, appropriate strategy use is function of creative thinking.In turn, success in NRMP solving depends on strategy use while ability to think creatively in NRMP solving is informed by one's mathematical prowess.

METHOD
An exploratory study was conducted to determine 10 th to 12 th grades learners' nonroutine problem solving strategies.The first author administered a NRMP solving test to a convenience sample of 395 learners (grades 10; 11 & 12 = 173; 180 & 42 1 , respectively) drawn from three high performing high schools located in Tshwane North District, Gauteng province of South Africa.The schools were 9, 7 and 4 kilometres apart, respectively.To ensure that learners in these schools wrote the test in the same way and did not exchange any information, arrangements were made to have the test written on the same day albeit at different times since the first author had to travel between the schools.Learners were made aware of the instructions of the test (see Appendix A).The idea of providing rough work and alternative solutions was informed by the notion that NRMP solving is seemingly a 'haphazard' process and hinges on high-order thinking and reasoning.It was therefore thought it will be worthwhile to note the learners' reasoning and thinking processes invoked in solution pathways hence learners were encouraged to also provide rough work and alternative solutions.Mathematically correct workings were duly credited.
The test had six non-routine problems that did not require aspects of the mathematics school curriculum to solve (see Table 1).Problem 1 (P1) was an arithmetic sequence word problem that required derivation of general rule from the pattern and use the rule in the solution process.Problem 2 (P2) was on simple proportion and required logical reasoning.Problem 3 (P3) was an inequality problem that required manipulation of variables.Problem 4 (P4) could be solved by either logic (reasoning) or systematic trial and error.Problem 5 (P5) could be solved by an algebraic method (forming an equation) or by systematic trial and error.Problem 6 (P6) had no clear mathematics referents (i.e. had no numbers in its formulation) but required reasoning and algebraic method to solve.The test was content validated by two heads of mathematics and three experienced mathematics teachers.Cronbach alpha yielded a value of 0.79.
To avoid giving specific grade(s) (especially higher grades) advantage over other grades, the test items were not composed of specific mathematics content taught in the different grade levels but were real life problems that were within reach of learners of grades 10, 11 and 12. Learners could apply the mathematics learnt in school to resolve the problems.The expectation was that learners had not previously encountered and solved similar problems set.The problems could be resolved in many different approaches.As such, learners were to search for and apply strategies they think could possibly solve the problems.The problems enabled the researchers to examine the different strategies the learners used to find a solution.

ANALYSIS
To analyse the learners' solutions, a coding scheme (see Table 1) derived from Elia et al. (2009) and Mabilangan, Limjap and Belecina (2011) was used.The strategies learners used were identified by matching each learner's solutions against specifications in Table 1.
The way a strategy is used in the solution process was categorised into Thorough/insightful use of strategy; Partial use of strategy; and Limited use of strategy as was done in Mabilangan et al. (2011).Details of each category are listed in Table 2.The solutions were scored by awarding 5 points for thorough/insightful use, 3 for partial use and 1 point for limited use of strategy.
Where the assessment of a solution fell between limited and partial use of strategy 2 points were awarded.Similarly, 4 points were awarded for between partial and insightful use of strategy (Mabilangan et al., 2011).For blank or wrong answer zero was awarded.The first author scored the learners' solutions and the second author verified the scoring by checking through 50% of randomly selected scored learners' answer sheets.The general practice in SA is to select 20% of the answer sheets for scoring verification.
The respective mean scores on the 6-items problem solving test of the three grades were compared using one-way analysis of variance determined using SPSS.The grades' levels of strategy use in solving NRMPs were compared to determine if the learners' insight in resolving non-routine problems improves as they progress through the grades.

Learners' strategies
The following are strategies learners used: Systematic Listing (SL); Modelling (MD); Trial-and-error (TE); Use a Formula (F); Systematic Guess, Check and Revise (GCR [sys]); Unsystematic Guess, Check and Revise (GCR [unsys]); Consider a simple case (SC); Logical reasoning (LG); No logical reasoning (NLG); and Look for patterns (LP).However, how strategies were used and their frequency of use differed across the grades and learners.
10 th Grade: Table 3 provides percentage numbers of 10 th grade learners per strategy used per problem.
In total, 9 strategies (i.e.GCR (unsys); GCR (sys); SL; MD; SC; LG; NLG; F & TE) were used in varying degrees.GCR (unsys); NLG; SC & MD were used in all the problems.Of these strategies, on average, GCR (unsys) appears to be the highly used strategy, followed by NLG, then SC and lastly MD.On the other end of the spectrum, LP was not used in any of the problems, TE; GCR (sys) & SC were used only in one problem.TE & GCR (sys) were used in problem 4 by 1% & 2% of learners, respectively.SC was used in problem 1 by 35% of learners.

Partial use of strategy Limited use of strategy
There is some evidence of insightful thinking in problem exploration.
The learner's work is clear and focused.
The strategies are appropriate and demonstrate some insightful thinking.
The learner gives possible extensions or generalizations to the solution or the problem.
There is some focus but with limited clarity.
The learner applies a strategy which is only partially useful.
The learner starts the problem appropriately, but changes to an incorrect focus.
The learner recognizes the pattern or relationship, but expands it incorrectly.
There is no central focus and the details are sketchy or not present.
The procedures are not recorded (i.e., only the solution is present).
Strategies are random.The learner does not fully explore the problem and look for concepts, patterns or relationships.
The learner fails to see alternative solutions that the problem requires.
LG was used in problems 2 & 6 by 17 % and 8% of learners, respectively.Strategy F was not used only in problems 5 & 6.All learners attempted problem 1 and the highest number of learners, i.e. 20%, did not attempt problem 3.
Figures 1, 2 and 3 exemplify how a typical learner used strategies to solve 3 different randomly selected problems.
11 th grade: Table 4 shows % numbers of 11 th grade learners per strategy used per problem.
The group used all the strategies in varying degrees.The most commonly used strategy was GCR (unsys) and was used the highest of all the strategies, i.e. 67%; 59% & 58% of learners, in problems 5; 4 & 6, respectively.In terms of frequency of use in the problems, it was followed, on average, by MD and then SC.NLG was not used only in problem 6 even though it was used the highest, i.e. 46%, in problem 2.
LG was not used in problems 3 & 5, but was the second highest, i.e. 39%, used strategy in problem 2. In addition to not being used in problems 5 & 6, the use of F in other problems did not exceed 1%.GCR (sys) and TE were used only in two   The learner continued guessing until a solution that satisfied the algebraic statements was obtained 12 th grade: Table 5 shows % numbers of 10 th grade learners per strategy used per problem.  1, the learner first rewrote the information given in the problem and then determined the number of gallons of fuel the car uses per year.Unlike figure 1 where the number of gallons as the cost of using the car per year was considered, the learner successfully converted the number of gallons to the cost of using the car per year.The learner then considered the whole picture of the problem and used all the necessary information in the problem.However, the statements presented were not mathematically correct due to use of a 'piece-wise approach' to problem solving.The learner equated quantities of different values.3, the learner used GCR (unsys) where she made a single guess of the number of hutches and rabbits and this did not make sense.Rightfully, the number of rabbits should be a multiple of 9 because the rabbits were to be grouped in nines The learners used 8 strategies where GCR (unsys), MD, SC and NLG were used in all the problems with GCR (unsys) being the highly used and SC the least used among the four strategies.GCR (unsys) was used by 48% of learners in problem 1, by 52% in problem 3 and by 60% in problem 5.The next highest used strategy, on average, was MD, which was used by 48% of learners in problem 6, by 29% in problem 1 and by 17% in problems 4 & 5. Strategy F was not used in problems 5 & 6, LG was only used problems 2 & 6 and LP was only used in problems 1 & 5. SL was only used in problem 1.There were blanks in all the problems with the highest percentage (i.e.43%) recorded in problem 5.  Similar to 10th and 11th grades learners, a 12thgrade learner solved the problem by first rewriting the given information and then determined the cost of using the car per year.He solved the problem the same way as an 11thgrade learner by using logical reasoning and piece-wise approach.Even though the solution was correct, the steps were not mathematically correct because the learner equated quantities of different values Figure 10.Similar to the 10th grade learners (see Figure 2), the 12th grade learner, initially, guessed that each person had one cup of tea and one piece of cake and, thereafter, presented a fractional number of people who took the tea.The learner could not develop a mathematically acceptable solution Figure 11.Unlike the 10 th and 11 th grades, the learner solved the problem using MD and GCR (unsys) strategies.The learner considered all the conditions in the problem and successfully identified the correct number of hutches and rabbits that were there.The learner then verified the correctness of the answers by modelling the situation using diagrams.However, the learner presented on the answer space only one set of guessed values that satisfy the given conditions and no other values that might have been tried in the process of solving the problem were presented 12 th grades.LP was not used by the 10 th grades while it was the least used by the 11 th grades and the second least strategy by the 12 th grades.The 11 th grades used all strategies with varying degrees to solve all the problems.
All the grades used SC and LG strategies to solve P2.Learners' presentations of incomplete solutions and non-sensible mathematical statements might be due to failure to monitor the solution process.When solving P4, the 10 th and 12 th grades resorted to guessing and could not present mathematically sensible solutions.An 11 th grade learner modelled the problem situation using algebraic expressions.But, the learner did not consider the whole picture of the problem and this led to incorrect models.When solving P5, the 10 th and 11 th grades failed to consider the constraints described in the problem situation that limited the possible number of hutches and rabbits.They could not look back to check if their answers made sense or satisfy the conditions stated.Though the 12 th grade could solve P5, all the mental processes undergone on solving the problem were not put down on the paper.When solving P6, the problem that involved no numbers, the 10 th grade did not only model the situation and present a conclusion as was done by 11 th and 12 th grades, but the learner justified the conclusions by calculations.On the overall, learners tended to use learned and practiced procedures they normally use in solving routine problems when they solved non-routine problems.

How strategies were used?
Analysis of the level of strategy use per question across the grades (see Table 6) reveals that 44,5% of the 10 th graders showed 'limited use of strategies' on P1.The percentage rate of use of limited strategies on P1 by the 11 th and 12 th graders were 27,8% and 33,3%, respectively.The 10 th graders (20,8%) and the 12 th graders (16,7%) had the highest percentage use of 'partial strategies' on P1, while the 11 th graders (13,9%) had the lowest percentage use of 'partial strategies' on P1.The 11 th graders (58,3%) and the 12 th graders (50%) had a highest percentage rate of 'thorough use of strategies' on P1, while the 10 th graders (34,7%) had the lowest percentage rate of 'thorough use of strategies' on P1.
The 10 th and 11 th graders' percentage rate of 'limited use of strategies' on P3 were 99,4% and 97,8%, respectively.The 12 th graders (97,6%) had the lowest percentage rate of 'limited use of strategies' on P3.There was no 12 th graders who showed 'partial use of strategies' on P3.The 11 th graders (1,11%) had a highest percentage rate of 'partial use of strategies' on P3.There was no 10 th graders who showed a 'thorough use of strategies' on P3.The percentage rate of 'thorough use of strategies' on P3 for the 11 th and 12 th graders was 1,11% and 2,38%, respectively.The 10 th graders (92,5%) and the 11 th graders (75%) showed the highest percentage rate of 'limited use of strategies' on P6.The 12 th graders (40,5%) had the highest percentage rate of 'partial use of strategies' on P6.The 11 th graders (3,33%) and the 12 th graders (2,38%) had the highest percentage rate of 'thorough use of strategies' on P6.
There is evidence of limited use of strategy in learners' solutions (see, for example, Figures 2, 3, 4, 6, 7 and 10).The percentage rate of this category for the 10 th , 11 th and 12 th grades was 84.6%, 74.1% and 73.4%, respectively.In terms of partial use of strategies, as in Figures 1, 8 and 12, the percentage use by 10 th , 11 th and 12 th grades learners was 8.9%, 11.1% and 14.7%, respectively.For thorough/insightful strategy use, as in Figures 4, 5, 9 and 11, the percentage use for the 10 th , 11 th and 12 th grades was 6.6%, 14.8% and 11.9%, respectively.Table 7 provides breakdown of how strategies were used.
Overall, there was considerable 'limited use of strategy' in all grades but % frequency decreased when moving up the grades.There was, however, an increase with 'partial strategy use' when moving up the grades.For 'thorough/insightful strategy use', lowest % frequency occurs in the 10 th grade while the 11 th grade produced the highest % frequency.The mean scores obtained are all out of 30.One-way analysis of variance (ANOVA) was used to determine whether there was a significant difference between the mean scores at 100 LS = Limited use of strategies (scored 0, 1 or 2 points per question) PS = Partial use of strategies (scored 3 or 4 points per question) TS = Thorough/insightful use of strategies (scored 5 points per question) p < 0.05.ANOVA F-statistic was highly significant [F(2,392) = 17.856; p < 0.05] (see Table 9).This means that there exists at least one statistically significant difference among the problem solving mean scores.

Comparing mean scores across the grades
The results also showed that Levene's test was significant [F(2,392) = 5.137; p = 0.006 < 0.05].This confirmed that the variances were significantly different.However, due to the violation of the assumption of homogeneity of variance, the equality of means was tested using more robust procedures, e.g., the Welch and Brown Forsythe F tests.Both the Welch and Brown Forsythe's F statistics were highly significant [F(2,111) = 18.686; p < 0.05 and F(2,142) = 16.412;p < 0.05, respectively] (see Table 10).This ascertained the existence of at least one significant difference among the means.Games Howell's post hoc test was conducted to identify mean pair(s) which are most significantly different (see Table 11).
Table 11 shows that the mean difference of the 11 th and 10 th grades was highly statistically significant ( ≈ 0.000 < 0.05).Similarly, the difference between the problem solving mean scores of 10 th and 12 th grades was statistically significant ( = 0.057 ≈ 0.05).The mean difference of the 11 th and 12 th grades was not statistically significant (p = 0.552 > 0.05).A statistically significant mean difference between the 10 th and 11 th grades and 10 th and 12 th grades implies that the 11 th and 12 th grades used more insightful strategies to solve problems than the 10 th grades.A statistically insignificant mean difference between the 11 th and 12 th grades may mean that these classes have comparable insights in the use of problem solving strategies.However, on average, all grades obtained low scores meaning that the learners had difficulties solving the non-routine mathematical problems.

DISCUSSIONS
Data show that learner performance varied across the grades, with 10 th grade learners obtaining the least mean score and the mean score of 11 th grade learners being the highest.An increase in learner scores from 10 th to 11 th grade is consistent with the findings by Van den Heuvel-Panhuizen et al (2013) and Arslan and Altun (2007).The decrease of scores from 11 th to 12 th grade is somehow compatible with the results of TIMSS 2011 (see HRSC, 2012).According to the results, the learners' non-routine problem solving skills, unsurprisingly improved between the 10 th and 11 th grades and astonishingly deteriorated between the 11 th and 12 th grades.It may very well be that, in 10 th and 11 th grades, ostensibly the above-average learners pursuing mathematics will do everything to ensure that they continue to perform well lest they will be made to switch to the career-limiting mathematics literacy if they perform poorly.Learners therefore solve as many problems as possible with the hope of becoming more mathematically competent and thus succeeding.This view is conforming to theory presented by Webb (2010) that one's problem solving capacity improves by engaging in many mathematical tasks as possible.And, according to Arslan and Altun (2007) more routine problem solving tends to have positive effect on NRMP solving.In 12 th grade however much time is dedicated to revision and rehearsing by working through previous years' examination question papers where anything that is not examinable is ignored and as implied in CDE (2013) such practice is not beneficial.
All grades used a number of problem solving strategies that include Systematic Listing (SL); Modelling (MD); Trial-and-error (TE); Use a Formula (F); Systematic Guess, Check and Revise (GCR(sys)); Unsystematic Guess, Check and Revise (GCR(unsys)); Consider a simple case (SC); Logical reasoning (LG); No logical reasoning (NLG) and Look for patterns (LP).This is anomalous given that non-routine problem solving tends to receive minimal attention in most SA schools (see AMESA, 2013).The current study fell short to verify AMESAs observation in the 3 participating schools.Nevertheless, the finding accords with that by Mabilangan et al (2011) who discovered that Filipino secondary school learners could apply strategies to solve non-routine problems without any relevant teaching.Albeit learners could apply the strategies to non-routine problem solving, as evident in their solutions, their understanding and effective use of the strategies was problematic.In particular, learners could not plan and effectively execute the strategies.The results also show that GCR (unsys), NLG and SC were frequently and widely used in 10 th grade with GCR (unsys) being mostly used.
In 11 th grade, it was GCR (unsys), MD and SC that were widely used with GCR (unsys) being the most used strategy.For 12 th grades, it was GCR (unsys), MD, NLG and SC that were widely used and GCR (unsys) was still the most used strategy.Elia et al. (2009) found TE to be the mostly used strategy and attributed this to the rudimentary state of the 4 th grade learners' NRMP solving.In Yazgan (2013, p575), most 12 th grade learners benefited from 'use of equation' (or formula as referred to in this study).
The low mean scores indicate that learners battled with NRMPs.Unlike routine problems that require procedures and algorithms to solve, NRMP solving hinges on reasoning and thinking (Kolovou, 2011) coupled with proper strategy use (Elia et al. 2013).Furthermore, success in NRMP solving is a function of mathematical capacity.The low scores thus depict the low level of learners' mathematical capacity and this is consonant with the report by AMESA (see AMESA, 2013).

CONCLUSION
The study shows that it is not a forgone conclusion that learners' NRMP improves progressively as more mathematics is learned or as more routine problems are solved.It seems NRMP depends on the type of teaching learners are exposed to.The teaching that focuses largely on drill and rote learning tends to compromise learners' NRMP solving capacity.It thus concluded that NRMP solving is independent of one's ability to solve routine problems.In terms of strategies learners used, it is evident that learners mostly used strategies that may be considered less sophisticated and are 'haphazard' in nature or follow a 'hit-or-miss' approach, i.e., GCR (unsys), NLD and SC even though MD is the second popular strategy used in grades 11 & 12.The choice of these strategies may be attributed to the learners' underdeveloped essential mathematical competencies that are crucial in solving NRMP.Another possible thinking is that learners seem to think that what works for routine problems (i.e.algorithms can be replicated on other problems) also works for NRMP.The choice of MD by grades 11 & 12 seems to be influenced by significant amount of word problem solving learners are exposed to.However, pervasive instances of use of ineffective problem solving strategies were noted whereby learners created models without use of all conditions or restrictions in the problem situation.Some created models and, thereafter, abandoned them in favour of strategies set to be 'haphazard ' or 'hit-or-miss' oriented, e.g., GCR (unsys) and NLD.This could be, probably, due to learners' low level complexity of their mathematical knowledge that is characterised by lack of creativity, reasoning and high-order thinking.It is thus concluded that prowess use of problem solving strategies tends to be associated with the level of complexity and sophistication of mathematical knowledge and skills.

NOTES
1 Generally schools are reluctant to avail their grade 12 learners for research studies as attention is devoted to preparing for end-of-examinations.At the time of the study, only 42 grade 12 learners were availed for the study and these were considered the above average cohort.

Figure 1 .
Figure 1.The figure shows that a learner first rewrote the given information and then used logical reasoning to determine the cost of driving the car for a year.Notably, though, the problem was partially solved because the learner did not use the R2 cost of travelling either 16km or 26km by a new and an old car, respectively

Figure 2 .Figure 3 .Figures 4 , 5
Figure 2.A learner guessed that the bill per person is the sum of the cost cup of tea and cost piece of cake, and then worked out the number of people who had tea by dividing total bill by bill person.The quotient is a fraction and this does not make sense

Figure 4 .
Figure 4.She used SC, MD and LG where she attempted first to express the problem algebraically, then guessed the solution and verified its correctness by substituting numbers in the algebraic statements.The learner continued guessing until a solution that satisfied the algebraic statements was obtained

Figure 5 .
Figure 5.As in Figure1, the learner first rewrote the information given in the problem and then determined the number of gallons of fuel the car uses per year.Unlike figure1where the number of gallons as the cost of using the car per year was considered, the learner successfully converted the number of gallons to the cost of using the car per year.The learner then considered the whole picture of the problem and used all the necessary information in the problem.However, the statements presented were not mathematically correct due to use of a 'piece-wise approach' to problem solving.The learner equated quantities of different values.

Figure 6 .
Figure 6.A learner usedMD and GCR (unsys)  strategies to solve the problem.The learner introduced x and y variables to represent the number of tea cups and number of pieces of cakes per person, respectively.However, the algebraic expressions and guesses made were incorrect.The learner did not consider the third variable (i.e. the total number of people who took the tea) in the algebraic expressions.The learner overlooked ensuring that the values of x and y are cardinals

Figure 8 .
Figure 8. Similar to Figure 4, the learner provided algebraic representation of the problem.Unlike in Figure 3, the learner in Figure 6 used names as variables and then provided answers without solution pathways

Figures 7 ,
Figures 7, 8 and 9 are examples of solutions of a typical learner.

Figure 9 .
Figure9.Similar to 10th and 11th grades learners, a 12thgrade learner solved the problem by first rewriting the given information and then determined the cost of using the car per year.He solved the problem the same way as an 11thgrade learner by using logical reasoning and piece-wise approach.Even though the solution was correct, the steps were not mathematically correct because the learner equated quantities of different values

Table 1 .
Coding scheme for problem solving strategies two trials of which the last value given is the answer.The steps are not of the same size, and the 'movement' of the trials does not necessarily go in one direction.Making a guess and lack checking or revision to improve the guess.Making one trial only.Giving the answer only.
No logical reasoning NLG Statements lack logic or does not make sense.Unreasonable (absurd).Naïve, impulsive or unthinking.Not answering the question asked.Unable to detect method used.

Table 2 .
Rubric to classify strategy use

Table 3 .
Frequency of strategy use per problem

Table 4 .
Frequency of strategy use per problem by grade 11 learners

Table 5 .
Frequency of strategy use per problem by grade 12 learners Figure 7. Similar to Figure

Table 8
presents the descriptive statistics of the scores obtained by the grades.

Table 6 .
Frequency of level of strategy use per question

Table 7 .
Breakdown of strategy use

Table 10 .
Mean equality test a Asymptotically F distributed.