CHAPTER TWO LITERATURE REVIEW
2.3 THE TEACHER AND PROBLEM SOLVING
Teaching mathematics in an interesting and challenging way has always brought anxiety to mathematics teachers. More of a concern to teachers was the changes to the mathematics curriculum content for the various grades and also how to teach it. The changes made to the
38
mathematics curriculum content between the various grades have result resulted in many learners being „held back‟ due to them performing poorly as they were not able to adjust to master and understand the content taught at that grade level. To ease the anxiety of these changes the focus was placed on mathematics problem solving.
One of our goals as teachers is to assist learners turn into better problem solvers, and to reflect on problem solving as a common, even exciting and engaging process. It was envisaged that using problem solving as a foundation in mathematics will provide learners with a deeper understanding of mathematical concepts and how solutions can be reached. Another associated objective is to recognise and discuss with learners the problem solving strategies that they will utilize in their problem solving process. By applying problem solving techniques allows the learners to build their understanding of the mathematical concepts while increasing their level of their confidence. The teachers are expected to be the agents of change in this mathematics curriculum transformation.
Duru et al (2011:3464) stated that the teacher plays an important role in how learners solve problems. Improving learning is dependent on the teaching abilities of the teacher. Since one of the most important objectives in teaching mathematics is to expand learner‟s mathematical problem solving skills, mathematics teachers must make sure that they are provided with opportunities to struggle with mathematics. Understandably not all teachers know how to teach in an efficient and effective manner. To change this scenario in schools, the pre-service teachers, as future teachers, should be introduced to a diverse assortment of strategies with a focus on the development of creativity and collaborative problem solving skills. By being creative will allow the teacher to teach the content material practically thus allowing the learners the freedom of discovery. Whereas previously the focus was on memorizing formulas and methods, an understanding of the problem solving strategies will make it possible for the teachers to assist make the problem clearer, simpler and more manageable. The pre-service teachers should learn more about problem solving strategies and be able to expose their learners to these mathematical skills.
Effective teachers are those who stimulate classroom relationships that permit learners to deliberate for themselves, to ask questions and to make rational risks when solving problems (Anthony and Walshaw, 2009). Problem solving skills do not develop within a few weeks of schooling but it is a slow and progressive way to becoming a skilled problem solver as the learner progresses through the grades. The teacher as a facilitator in the realm of problem solving must assist the learners by providing them with challenging age appropriate problems; encouraging and accepting learners own strategies; supporting and extending learners learning
39
abilities; using questioning techniques and prompting learners in the correct direction; observing and assessing learners during the problem solving process; identifying learners who encounter conceptual blocks and assisting them to recognize and rectify these misconceptions.(Ministry of Education, 2007:26).
In order to assist the learners the pre-service teachers themselves need to be coached to implement the mathematical strategies and techniques in the classroom. They need not be experts but rather make an attempt to make it part of their practice by integrating the relevant problem solving strategies to maximise teaching and learning in the classroom. This must be part of their teacher training modules at their tertiary institutions. In their theory learning they need to have access to the various teaching and learning tools which will enable them to train learners to improve their attitude towards problem solving. In practice they need to engage the learners with the mathematical problems to develop their critical thinking.
Besides learning how to use problem strategies to promote effective learning the pre-service teachers need to learn how to implement the teaching techniques in the classrooms. According to the Ministry of Education (2007:32) questions and prompts are critical when providing guidance to learners. Draper (2002:527) stated that teachers can provide learners with metacognitive prompts while they read and learn mathematics. The idea will be for teachers to provide most of the prompts first and then wean learners from relying on them. The type of questions and prompts that are used can assist the teacher to scaffold support for the learners. According to a teacher, from the Lorantffy Zsuzsanna Reformed Church School in Oradea, who uses the Varga method to teach mathematics, “if it is needed we help them with prompts/questions that may lead them to the solution” (Debrenti, 2013:90). Providing too much of information during the prompting phase or asking lead questions can lead directly to the solution to the problem and defeat the purpose of the task at hand. The teachers must also know when to prompt or ask questions as to not derail the learners thought processes. When asking a question or providing a prompt, a teacher needs to give the learners a reasonable period of time to allow them to comprehend and reason further.
A teacher together with good questioning techniques and prompts can also use probing. According to Killen (2013:153-154) “probing is the process of seeking clarification or more information when a learner attempts to answer a question”. Probing can be successfully used during the discussion phase of the lesson especially when solving a problem. This is when learners discuss and justify their strategies on how they arrived at their solutions. The learners are asked to elaborate on the methods they used and demonstrate their solutions in order to provide a better understanding to other learners in the classroom. Thus probing can be used as a
40
teaching and learning tool by the teacher in the classroom in order to get learners attempt to clarify an answer. This is to compel learners to raise their thinking levels and to also gain a clearer picture of learners understanding (Killen, 2013). Probing through redirection in the classroom can lead to an intense discussion when a teacher seeks further information from the learners. I have used probing in my mathematics lessons and often ask questions that lead learners to „doubt‟ their solutions. This forces them to revisit the problem and check the validity of their solutions. When the learners are confident they will justify their answers or they will back to verify their answers or collaborate with others in the classroom. Whilst probing has its benefits, Killen (2013) cautions its use in the classroom.
Teachers may use probing and prompting in order to push learners towards a solution but Killen (2013:154) stated that they must be wary as not to embarrass the learners because they cannot read and comprehend the initial problem. In such circumstances, especially in the lower grades, teachers are known to read the problems to the learners. The teachers read in a manner such that they draw the learners‟ attention to search for key words (clue words) in the problem so that it keeps their concentration on the said problem and exclude unwarranted information that distracts them. Whilst many will agree that giving learner‟s guidance in finding and using the magic words (clue words) in a problem is beneficial, I out of experience, oppose such a strategy. It is a known fact that mathematics as a language is difficult to understand and many a word, due to having dual meaning, can leave learners stranded especially when deciding what operation to use to find the solution to the problem. van de Walle, Karp and Bay-Williams (2014) cautioned against using key words or clue words as they can be misleading. Many problems may have no clue words but the learners will look for words as any easy way of solving the problem.
I identify a few distractors such as „altogether‟, „in all‟, „difference‟, „give/gave‟, „larger than and greater than‟, „less and more than‟, „share and divide‟ that can be problematic.
Let me put forth the following examples:
Tony had 250 marbles. He gave Anthony 120. How many did he have left altogether?
Whilst as a learner I remember the teacher stressing that „when you see the word altogether you must add‟ and „when you see the word difference, less or give you must subtract‟. If this style of teaching continues in the modern day classrooms then learners will be found wanting in their choosing of operations in given problems.
41
In the above example „greater than‟ is used in the problem. This problem created two misconceptions. Firstly, I found that my learners had used the symbols
‹
and›
in their answer due to them learning in the previous grades that greater than and less than can be represented by a symbol. Secondly, the learners wrote out their solution as 247 – 550. This problem actually required learners to add both the number to find the solution, namely, 797.I have witnessed situations where the teacher gave the learners the solution out of sheer frustration or compassion when they discovered that the learners were not making progress towards a solution. According to National Research Council (2001:335) the learners may start pressing the teacher to reduce the challenge by specifying the procedures for them to perform. In certain circumstances the learners often ask the teacher on how to solve the problem when they themselves cannot solve the problem. The teacher leads the learners to an answer by “telling them what to do” (National Research Council, 2001:335). . Bauersfeld (1988) referred to this as funnelling. Funnelling is when the teacher assists the learners by asking simple questions thus pushing them towards the answer and when the expected answers are not exactly to the teacher‟s answer he provides the answer (Bauersfeld, 1988). In this situation the learners copy the teacher‟s solution without giving the learners an opportunity to use their own solution strategies. This type of „learning‟ (giving and asking for the answers) continues in a vicious cycle in „teaching‟ grade after grade. This prevents the learners from developing their skills and strategies to becoming independent problem solvers. When learners realise that the teacher will provide them with the answer they will have no motivation to work through the problem on their own accord. The teachers should therefore resist the impulse to give the learners the answers. Therefore the learners ought to be given a chance to decide for themselves what the problem is about and how to solve them.
Mathematics teaching and learning problem solving is entwined. Therefore it is important for pre-service teachers to become acquainted with the problem solving approach and it may be necessary to provide them with knowledge of the problem solving approach that will support their teaching and learning of mathematics. The problem solving approach should allow pre- service teachers to make the connection between the strategies and problem solving. Bahtiyar and Can (2016:2109) conferred that “through this way, pre-service teachers can be trained for considering the teaching of problem solving skills as one of the most effective ways”.
42
Figure 5 Pre-service teachers explaining their solution/problem solving strategy
To expose the current cohort of pre-service teachers to the problem solving approach, I introduced them to problem solving during their lectures. They participated in a gallery walk as a novel way of learning problem solving strategies. Initially they worked in groups of five to solve the given problems. I encouraged them to show all the steps involved in finding the solutions to the problems. When they completed the task I displayed the charts in the lecture room. They all walked around the room checking the solutions and making notes of the different strategies used by their peers. In this manner they learn by seeing. Once the walk was completed they moved into a discussion session with their colleagues. The pre-service teachers were given an opportunity to use their chart and the white board to display their effort (Figures 5 and 6). They made constructive input, in some cases, offering alternate solutions or indicated how solutions could be improved on what they saw.
43
A crucial area of concern of the mathematics lesson is the conclusion which in my opinion is sadly and badly neglected. Most often at the end of the lessons the teachers put up the answers on the board and request the learners to do their corrective work. No discussion occurs. In getting the pre-service teachers to discuss their solutions and strategies, I needed to make a statement. I used the pre-service teacher‟s discussion to demonstrate to them how important the conclusion aspect is in a mathematics lesson. I stressed to them that it is imperative that the teachers engage their learners to show how they found their solutions. It is only in this manner that the teacher can identify misconceptions and errors made by the learners. He must be able to focus, reinforce and summarize the explanation of concepts that were misunderstood. This to me is sound educational practice as it prevents the „carrying on‟ of misconceptions in the learner‟s scholastic career.