Aims and Objectives
Based on Meyer, Turner and Schweinle (2006) students classroom’s experience both motivationally and emotionally play a very crucial role in shaping their attitudes, actions as well as achievements. This action research is therefore designed to explain how engagement in my junior classroom affected the students’ experiences. A research by Moch in 200l made use of active engagement with the elementary students. The result was that the use of active engagement gave the students a chance to feel and touch mathematics. This takes them out of the normal seeing and hearing that often surrounds mathematics’ learning. The experience gained through touching and feeling mathematics shifted the traditional mathematics environment. On this study the primary focus will be allowing students to engage in mathematics’ concepts actively. The task will be concentrated around the multi-digit addition and subtraction topic.
Typically, mathematics class within the United Kingdom have been described as one which entails practice and drill which gives little emphasis to the use of hand and active participation (Kutz,1991). In the past studies it has been proven that majority of the teachers consume a lot of time teaching students’ definitions and procedures at the expense of emphasising on technicality and applications surrounding the procedures (Stigler and Hiebert,1999). In these cases, students are merely taught to solve problems whose contents’ meaning they have no idea about. Vinson (2001) indicated that students in the UK only possess a moderate of procedural knowledge accompanied by a very low level of mathematics concept knowledge. Stigler and Hiebert attribute this to the culture of teaching mathematics in Britain.
The two have proven that students use most of their time to acquire isolated skills through repeated practice. This way there is a very minimal time to master problem-solving procedures. The students are only taught to learn mathematics using worksheets and rote memorisation.
Even though teachers have undergone training and professional improvement they still prefer to teach their students using the techniques that were used to teach them in the past years (Quinn, 1998). On top of this some elementary teachers who happen to possess little confidence in their mathematics skills tend to rely on experiences gained throughout their education system to teach mathematics.
The teachers in this category prefer to solely rely on the traditional methods of teaching during their classroom sessions (Gresham, 2007). With this the procedure is simple teaching lectures followed by memorisation then workbooks and finally worksheets.
Due to the limited mathematics ability of some teachers or insecurity on their conceptual knowledge they tend to stick to the traditional teaching techniques during their mathematics lessons (Gresham, 2007). There are several modern techniques that can be applied in teaching mathematics some examples include: use of games, individual and small group instructions, cooperatively working in groups, application of manipulatives, class discussions among students’, justification and presentation of their work. The testing criteria that the state mandate often plays decisive role instructions applied in the classrooms.
The pressure of testing does trickle down the hierarchy line to fall on the teachers and students at the end. Being that the teachers often carry the burden of the students’ scores McReynolds (2006) suggested that schools have been made to restrict their learning areas to those that are most likely to be tested. With this several teachers are comfortably training students how to pass the tests other than applying the use of non-traditional methods of instruction. During my graduate studies I managed to learn a lot regarding the importance of allowing students a room to acquire a deeper understanding of mathematics procedures as well as concepts. This has allowed me to embrace the idea of constructivist learning approach and so I support the notion that learning should be student-centered.
During my time as a student I did endure hardships in classrooms that applies traditional teaching styles. In this case the criteria was a sit and listen approach where I had to solely listen to teachers’ lectures and learn how to solve a problem.
My aim is to give my students an opportunity to access quality education. In the NCTM (2000) it is claimed that reducing mathematics to a memorisation and a copying process will kill students interest in the discipline. In addition, by making learning understandable and attractive the students focus will be retained. A combination of such research and my personal ambitions of becoming a better teacher is what made me undertake the investigation of the practice of applying manipulative and active engagement in my year 11 classroom. My objective was to investigate the nature of the impact of the use of manipulative on the students’ engagement, class participation and academic excellence.
The following will entail some of the terminologies which will be used in the study:
Academic Performance: this indicates whether learning is taking place. Some of its indicators involve: fluctuations in the students’ pre-test and post-test results as well as records of work applied in problem-solving.
Conceptual Understanding: is what the students perceive as the meaning of procedures and concepts in mathematics.
Constructivism: this is anchored on the notion that learners build knowledge from personal experience and beliefs. The learning experiences under constructivism entail explorations, interactive group work, student centred activities as well as taking part in group discussions (Snider and Roehl, 2007).
Engagement: this entails active, flexible, goal-directed, persistent, constructive and focused interaction with social and physical surroundings (Furrer and Skinner, 2003).
Direct Instruction; is a teaching model which is teacher-centered and often involves the use of lectures as the mode of teaching. Its common is the traditional teaching methods where the teachers tell the students ways and where to apply the taught concepts. The students therefore have to memorise the teachers ‘lectures (Kroesbergem, Van Luit and Maas, 2004)
Guided Instruction: A student centred teaching model that make use of explorations, taking part in discussions and group work as the main modes of teaching. The lesson as well as class discussions are designed around student’s contributions and strategies (Kroesbergem, Van Luit, & Maas, 2004). The method is common in constructivism.
The manipulative and active engagement materials in mathematics: this are materials with visual and tactile appeal and so represent mathematical ideas in an explicit and concrete manner. They are manipulatable by the students through experiences (Moyer, 2001). They include counters, pattern blocks dice, colour tiles Cuisenaire rods, hundreds board as well as cm cubes. Some basic household items like coins, toothpicks, beans and checkers can also be used.
Tools in mathematics: in this case will refer to mathematical tools that are utilised that don’t fall under the category of manipulative and active engagement tools. Some of the items under this category are measuring tapes, rulers, protractors as well as calculators.
Participation: is defined as active involvement in the activities taking place in the classroom. Asking questions, giving examples and class presentations are part of it.
Procedural understanding: is an understanding that can be traced to the steps that were used to solve the math problem.
Standard Algorithm: the commonly applied and memorised technique for solving problems.
Traditional teaching; entail teacher-centered teaching techniques which make use of the independent practice, directed guided practice as well a continues assessments. The teacher in this case is the message conveyer and the students learn mainly by listening to his/ her lectures.
In my role as a classroom teacher I aspire to improve my students’ mathematics academic excellence together with their involvement in mathematics participation. I undertook this study to investigate the impact of applying manipulative and active student engagement materials on the participation, student engagement as well as their academic performance.
The study aims at reflecting my teaching practices using manipulative and active engagement to determine how it has influenced the students’ classroom participation and performance. The primary question that this research is designed to answer is what impact do teaching mathematics using manipulative and active engagement have on 11 students class participation and engagement. In addition, the study will seek the solution of: the impact of manipulative and active engagement on the third-grade academic excellence in multi-digit addition and subtraction. In chapter 1 of this document I will describe the research settings and methods applied in acquiring the data used to answer the research questions.
Based on the definition of Johnson (2008) action research is stated as planned, methodical observation that has an interrelationship with teaching. The study made use of qualitative data collection techniques. The data collected was on students’ engagement, their participation and academic excellence. Students work samples, field notes prepared by teachers as well as pre-test and post-test results were some areas from which data was gathered.
The study was conducted among the year 11 classroom. A total of 21 students within the age range 8 to 10 years were considered in the study courtesy of being members of the sample classroom. The school head designed the 11-year classes based on diverse gender range, race, reading and mathematics ability of the students. In each classroom the students were taught reading, science, social studies and mathematics daily. The lessons were designed so that mathematics was taught at 9:45 am and the class lasted for an hour.
From the 22 students in the classroom, one was autistic and was mainstreamed in the class just to undertake mathematics instruction. Other students were under gifted services while four students were grouped as ESOL making them receive special accommodations. Prior to participation in the study all the students took home parental consent forms which were signed by their parents and returned to the school. In this list two students were omitted from participating as their parents failed to consent. In this case data on those students was only limited to work samples, pre and post-tests as well as teacher observation field notes. The set of students who fully participated in the entire study were 21; 11 males and 10 females. The research group was composed of 6 whites, 2 Asians, 1 Black, 11 Hispanic as well as 1 mixed. The students were each allocated a number which they used in the classroom within the year. This was a normal classroom procedure in the school.
The use of traditional teaching method is primarily teacher-centred. The teacher is tasked with the role of providing directed guided practice, continuous assessments and applications of the learned skills to the students. The rote memorisation techniques are the most applied method. After the students are taught the procedures of performing a task they are expected to memorise them and use them to solve problems. The teaching method relies on the use of standard algorithms to make students memorise the taught concepts after which they will later be tested through completion of worksheets and time drilled. The teaching is all about memorising steps needed to solve a problem.
The NCTM (National Council of Teachers of Mathematics) strongly advocates for the teaching of mathematical reasoning and understanding. This unfortunately has not been accepted by the majority of mathematics teachers, instead they prefer to spend time learning, teaching and practising computation procedures. Most of the teaching time is spent in the use of lecture, rote memorisation, worksheets and standard algorithms (Gresham,2007).
Teachers stress basic skills at the expense concepts and devote more time to seatwork class instruction. The teaching of the classroom problems one by one while insisting on only one way of solving the problems deny the students room to develop mathematically (Broody, 2006). Quality experience in mathematics should entail more than this.
In Alsup and Sprigler (2003) the impacts of the traditional instruction on the achievements of 335 eighth graders were investigated. The researchers used the comparison of direct instruction curriculum with a more reformed curriculum that made use of hands and laboratory procedures instead of direct instructions. From here they recorded the scores of all the students who attended the researchers class during the year. In the first year the direct instruction method was used for teaching. In the second the teaching was shifted to the use of direct curriculum, finally, in the third year the two methods were used in mixed proportions. In the results of the study the students who were previously taught using the reformed instruction recorded a significant improvement. This though was not the case with the ones who were previously taught using the traditional teaching techniques. They demonstrated a recognisable improvement on the procedural tasks.
In Baroody (2006) the children master and learning of mathematics was studied. Based on the findings of Baroody conventional teaching styles makes it hard to learn basic mathematics. By focusing the entire task to memorisation of individual combinations, the children ‘s mathematical proficiency is just restricted.
This teaching technique is simply procedural. Students are presented with quick facts and tasked with their memorisation. Given that there is no meaning behind the taught concepts there is a likelihood that the students will misapply the information when the teaching is done. Being that the students fail to truly master the meaning behind the memorised facts Sousa (1995) claim that lecture results in a lower degree of retention is just proven by Baroody.
In cases where the students can understand the concepts nevertheless their chances of forgetting remain high. Isaacs and Carroll (1999) in their work with students in the elementary schools to teach basic number facts concluded that use of traditional rote approach of learning basic mathematics by using drills and timed tests could lead to the creation of serious weaknesses in the students understanding. The students are forcefully fed facts and are demanded to regulate the facts very fast. In 1996 a study was designed by Cain-Caston to compare the achievements of grade 3 students taught using manipulative techniques with those who were taught using worksheets. The achievements of 70 third grade students were assessed using the California Achievements Test Form E. from the findings of the study Cain Caston concluded that the use of worksheets discourage the students thinking and use of applications to solve a problem instead they just had to remember the previously taught algorithm.
The students do not bother to gauge the meaning behind the concepts. With this they end up facing problems when tasted with more advanced problems. The advantages of using the traditional techniques to handle students with learning disabilities was questioned by Kroesbergem, Van luit and Maas in 2004.
In their study they made a comparison of three sets of conditions: traditional explicit instruction, control group based on the curriculum and constructive instruction. The comparisons were focused on identifying the advantages of the students’ fact automaticity and problem tackling. The participants involve those from elementary schools undertaking general education as well as elementary schools for special education. The selection of participating students was based on low performance in mathematics.
In the conclusion it was stated that the use of either explicit and constructivist instruction were superior to that of regular curriculum automaticity. The students who were given explicit instruction had little different from those who received constructivist instruction in terms of automaticity.
The students who had low learning ability demonstrated an improvement in problem-solving compared to their counterparts. From these results we can argue that academically weak students can benefit greatly when taught using explicit teaching strategies (Kroesbergem, Van Luit, & Maas, 2004).
Teaching students to memorise the algorithms is not enough for them to master the concepts and apply them appropriately in solving problems. Constructivism sets in to offer a remedy to this weakness, students learn better from individual experiences and beliefs. This is what drives the concept of constructivism. The students need a reason behind the algorithms. Excellence performance in mathematics entails more than acquiring procedural and computational knowledge (Dec Corte, 1995). As research on students learning intensifies modern ideas of teaching are coming up which need to be implemented.
The best approach is to begin by considering the students thinking (Isaacs and Carrol, 1999). To encourage independent thinking among the students a problem-based approach should be utilised. Through solving a problem by sharing ideas among themselves the students are taught several ways to solve a single problem. The presentations by their mates will enable them to evaluate the meaning of various concepts. This way students can select the strategy that they are more comfortable with while solving the problem.
This situation makes students think conceptually regarding the mathematics problem and can understand why it can be tackled in several ways. In 2006 Baroody studied instructional techniques that determine how elementary students learn number computations. In addition, he also studied the number-sense view which emphasizes the need for conceptual understanding.
The learning and practice of number combinations should be done in a purposeful way (Baroody, 2006). This way the students are empowered to discover their own patterns and strategies hence they gain a greater conceptual understanding. Giving students a room to think independently other than over-rely on textbooks and teachers allows them to find their own ways of tackling problems.
This method was discussed by Carpenter, Jacobs, Fennema, Franke and Empson in 1997 as they studied invention and students understanding regarding multi-digit addition and subtraction (Carpenter, et al., 1997). The progression of the mathematics concepts was studied among 82 students as they progress in their years from grade 1 all the way to 3. The objective was to highlight if an understanding difference exists between students who use invented strategies and those who solve problems by applying the learned maths algorithms.
The teachers involved in the students received no guidelines on the instructions to apply. Most of the teachers allowed the students to apply a diversity of strategies to solve the problems. Afterward, the class discussed the strategies used which entail sharing of ideas in terms of strategies applicable in solving the task. They found that after the three years most students preferred the use of the standard algorithms other than the invented strategies. The largest jump who proffered the standard algorithms were the grade 2.
Even though most jumped to the use of algorithms there was a discovery regarding those who applied invented strategies before they learned the algorithms. They showed mastery of baseten concepts before the ones who heavily relied on the algorithms. In addition, invented strategies showed a hallmark of characteristics in understanding, the children were able to fluctuate the use of the invented strategies to meet the demands of new situations. This way they were well set to solve more advanced problems compared to their counterparts who only used the standard algorithms.
Also, the students who used invented strategy had fewer systematic errors (p. 160. From this several advantages arise when students learn to use invented strategies prior to being taught the standard algorithms. Giving the students a chance to develop their meaning and ways of solving the problem makes the students to master the concepts better than when they are directly told how to solve a problem. Hiebert and Wearne investigated how instruction influences understanding of children when learning multi-digit numbers as well as skills of computation.
In their work the researchers worked with 70 students over the course of their three years in school. the students were either given the textbooks instruction or an alternate instruction by the use of active engagement. From their results the two concluded that the instructions availed to the students should have the aim of supporting the understanding of the students instead of perfecting procedural proficiency. They also studied alternate and conventional instruction and their impact on the conceptual understanding. The use of alternate instruction gave the students a room to construct relations of several issues as well as generate their own ways of understanding and solving the problem. On the other hand, conventional approach concentrated on textbook-driven instruction. The lessons were simply taught by demonstrating techniques of solving problems and giving practice questions on the same topics
The student's work was enduement and were encouraged to apply the standard algorithms taught. Th students who were taught using the conventional techniques managed to perform highly though they could not demonstrate a higher level of understanding. From this Hiebert and Wearne concluded that the use of alternative instruction empowers the students in terms of the level of understanding and skill development.
Teachers should possess a firm understanding of the math’s concepts. In the study by Kamii, Kari and Rummelsburg which was designed to investigate the practice of applying traditional as well as constructivist techniques to teach maths to low performing and low ESE students in first grade. Throughout the entire year in school one set of students were taught using an instructor who made use of constructivist methods stressing on physical activities of gathering knowledge. This showed that students taught using these techniques tend to perform better than the other group taught using the traditional teaching techniques.
The Constructivist had a more stable mathematical foundation and could remember numerical facts hence they performed at a higher level (Kamii, Rummelsburd and Kari, 2005). The findings in (Mccaffrey et al., 2001) supports the findings by the three researchers (Kamii and group). In the research by McCaffrey they explored instructional practices in a high school classroom. They discovered that students taught using the student-centred approaches were able to perform better than their counterparts taught using the conventional teacher-centred approaches (McCaffrey, et. al., 2001).
From previous studies learning have been shown to be an active process and students are able to do better when they are actively involved (Petress, 2006). He has listed student’s involvement in the class discussions and activities to be an integral part of learning. In his findings he realised students’ capacity to generalise what they have learned via class activities surpass that of when they learn through listening and reading. The students also tend to retain more information whenever active learning is applied in introducing new concepts. For the students to optimise the learning outcome, there is need to involve active and true participation of the students.
In 2004 Turner and Patrick investigated the interactions of students and teachers in a classroom to gauge the impact of participation in the students understanding. They focused on two students during their learning of mathematics in 6th and 7th grade.
The students had to fill several maths-oriented surveys. To enable them to gauge individual student’s participation the two researchers noted all the occurrences: students talk and behaviours in class. Also, the teacher's talk and behaviours that might have impacted the students’ outcome were recorded.
They realised that through classroom participation the students were able to learn and practice new ideologies more effectively. In addition, the students were also able to explain there reasoning making it easy for the teacher to examine their thinking. They further showed that intense participation allowed the students to think, examine, understand, practice and solve problems in their own. Skinner and Belmont (1993) undertook a study which observed the behaviour of the teacher in addition to the engagement of the students. They found that the attitude of students involved in active learning was very positive and they were more engaged especially when the learning method was sustained for a long time. On the other hand, Skinner, Connell and Wellborn (1999) also confirmed that whenever students are engaged intensively in a total of 16 academics they earn higher grades and even scored higher in the state standardised tests.
The conclusion of Tuirner and Patrick (2004) was that classroom environment should be conducive to allow students participation. Whenever the students participate in classroom learning they take the risk of being rejected whenever they share their ideas with their classmates and the teachers. Also, they often fear getting the wrong answer and undergoing public humiliation. Through the teacher's discourse and pre-set classroom norms the students’ participation can either be exhibited or motivated.
The two indicated that students are more willing to participate in classroom learning activities whenever the teachers showed enthusiasm for learning and let them believe that they can all learn. Also, the teacher needs to offer academic and emotional support so that the students can improve their understanding. The teachers possess the ability to create an environment that can either encourage or discourage students participation.
The nature of instructions given to the students in addition to the enthusiasm and support of the teachers can create an environment that encourages the students to get involved in the classroom participation. The use of manipulative materials as part of instructions given in classroom increase the participation environment. The materials often act as motivational means. Marzola (1987) researched on the applications of active engagement in mathematics’ instruction. From the research outcome it was concluded that the appropriate application of engagement materials can lead to an increase in students awareness level.
Evaluation of personal practice in subject-specific
When active engagements are applied appropriately with the help of concrete instructional materials in the classroom, the students will tend to understand the maths concepts presented better (Vinson, 2001). Even though kinaesthetic experience enhances perception, conceptual understanding and thinking, Ball (19920 indicated that understanding cannot be transferred from the fingertips up the arm to the brain. He raised concern that some teachers are viewing active participation as a magical way of learning meant to cure all the problems affecting the students when it comes to acquiring concepts. This is not the case. Active engagement only works best when introduced appropriately and supplemented by other learning practices.
From Sanders (1993), active engagement selected should be in line with the objectives of learning. For instance, the use of fraction circles when teaching multiplication of whole numbers will only confuse the students. Also, just availing the material and leaving the students to play with them will not result in positive learning of the intended concepts, the teachers need to oversee lessons through utilisation of active engagement. The students should be allowed room to discuss and share techniques and strategies that are in accordance with the use of manipulative methods.
Development from personal findings
The research by Furrer and Skinner (2003) on participation and engagement showed that active involvement signs involve asking questions, giving examples and making contributions in class. The students were offered participation surveys which they filled based on their understanding of the meaning of class participation. The surveys include class participation and what it means to follow the below instruction:
Should no discourse exist between the teachers and the students, then the children will simply follow rote procedures for the use of materials. The use of engagement materials while at the same time applying traditional teaching methods is entirely possible. The teachers need to front appropriate discourse that gives an emphasis on the conceptual understanding as shown by the use of manipulative materials.
Through true participation students will be able to derive maximum benefits from the active involvements (Petress, 2006). The most common response to the two questions (what participation in a maths lesson mean and what participation in maths mean) can be given by the phrase: answering questions, listening to teachers as well as presenting a sample of the responses.
To gauge what it looks like to participate in maths lessons, the students were required to describe the nature of activities that take place whenever participation in a maths lesson is taking place. The goal was to have a glimpse of what the students perceive to be active participation in learning. The students below represent some of their responses.
- Listening attentively to the teacher and paying attention
- Sitting quietly, doing maths while listening to the teacher.
- Listening and doing what the teacher says.
- Answering several questions
To gauge the effects on the students’ academic performance a pre-test was availed at the beginning of the research. From the test it was shown that students lacked understanding and proficiency in multi-digit addition and subtraction. Majority of the students used little or no written techniques to solve the problems. No single student applied the use of manipulative materials.
Analysing of the students work indicated a similar trend. First, the students failed to apply any written technique to add and subtract especially in techniques that required regrouping.
With time their work samples began to show written record of what they were thinking and the process they applied in obtaining the solutions. A few of them even drew the blocks to assist them solve the problem.
Results of the Post-test
The results obtained in this section were significantly different from those of the pre-test. Some students made use of the engagement materials. Of those students, 50 % recorded an increase in the scores of the test. The other half had their scores decrease. The other students made use of written techniques and recorded the processes they followed to solve the problems. There was a total of 23 occurrences in which the students’ pre-test showed no work but showed the work on the post-test.
Out of the 23 cases 22 questions were wrongly answered on the pre-test but were correctly answered in the post-test. Of all the students who took part in the pre-test and post-test 70% recorded an increase in the test scores. Petres (2006) declared that students learn best when they are actively involved in the learning process.
The students who seemed to be actively involved through exhibiting features of participation and engagement as shown in a video recording all managed to improve their scores. Only student #7 was exempted as the score remained same.
The use of engagement material is not the sole factor which contributed to the improvements. As stared by Isaacs and Carrol in 1999, students learning should first accommodate the thinking of the students and let them compose their own strategies for solving unfamiliar problems. This was one feature of the instructions applied in this study. Thus, it made it possible for the students to share diverse strategies during the class discussions which were not limited to active engagement. This exposure to diverse ways of solving the multi-digit addition and subtraction propelled the improvement in the students’ test scores. From the data collected and analysed, the application of active engagement in learning have the potentiality to improve the student's scores but is not the only factor at play in this scenario.
The findings if this study is not generalisable to all other classroom populations. The study has several limitations which need to be noted. First the study population was too small to warrant making conclusive assumptions.
The third graders who participated in the study were assigned to the researcher by the school administration and reflect the make-up of the school population. Students are individuals who tend to possess wider varieties of learning styles as well as preferences. Moreover, the parental support offered to the students may differ, the actions of the students’ parents at home may greatly impact on their classroom performance. In addition, the teachers are also individuals they tend to have varied teaching styles and preferences.
The results if this study indicated potentiality. The use of active engagements materials made the students more willing to take part in class activities. After undertaking this study, I have realised a loop in the academic performance, participation and engagement that need to be studied in relation to the use of mathematical active engagement.
From the result of the study all make two recommendations. First due to the short length of the unit, guided by the state guidelines set forth by the district, it will have been better if the research topic was studied over a lengthy period. the topic in this case was a bit narrow. Researching the question with no limit to the multi-digit additional subtraction would have given a wider insight into ways through which active engagement truly impact students’ academic participation, engagement and performance.
The next recommendation is that if student interviews were conducted then more insight would have been gained as to why students preferred or disliked the use of active engagement and ways in which they do so. Doing this would give an in-depth information of the students thinking when using the engagement materials.