Strengths and Weaknesses Identification
Teaching younger students mathematics is an incredibly important and complicated task that demands much responsibility and analysis. Since children shape their understanding of mathematics and their analytical skills at their early stages of development, it is crucial that such an important issue as money value should be taught with due care. In the given case study, the effects of the provided system of students’ performance evaluation are being considered and further avenues for its improvement are offered.
The provided system has a number of strengths, the key one being the ability to check a very single step of the students’ progress. Instead of having a big final quiz in the end of the session, the students are provided with several mini-quizzes, which are bound to help spot the emerging issues at the earliest stages and allow for students’ developing the right idea about money value.
One of the basic weaknesses, however, stems from the test system that has been used as the basis for the students’ progress evaluation. It seems that some students might have trouble understanding the principles of addition and developing their number sense. However, for some reason, these students hesitate to ask questions and, therefore, are likely to become completely confused about the topic.
Therefore, it can be assumed that at certain points, the assessment plan could use some improvements. In order to improve the current assessment plan and to introduce a supplementary one, it will be required to take a closer look at the current version of the assessment strategy.
Assessment plan and is advantages: a multi-level evaluation strategy
When it comes to speaking of the doubtless advantages of the provided testing system, one must give credit to the latter for allowing checking the students’ progress gradually. As it has been mentioned above, the earlier a teacher figures out that the students or a particular student have issues with a specific concept, the more efficiently the problem is going to be addressed. Therefore, the fact that tests are run on a regular basis after a new topic is introduced to the students speaks in favor of the adopted assessment system.
In addition, it is important that the provided evaluation system integrated different types of tests, including not only the written ones but also the ones that allow for a practical operation with money, which is especially important for the students. Since the application of theoretical concepts to a practical task is always hard, the given type of tests serves as a perfect mediator between the classroom theory and the real-life tasks.
Assessment plan and its weaknesses: lack of teacher-student communication
Despite the fact that the given assessment plan has a number of advantages and seems to be working rather well with the students, it is still obvious that it could be improved considerably. At present, it seems that the provided system of evaluation merely tests the students’ ability to answer the questions on the topic. While the given feature is important for a test, a good evaluation system should also help define the reasons for the students who have done poorly to fail their test – and the given assessment does not.
Moreover, misunderstandings and the lack of understanding is typical for any students, which means that in the course of the lesson, the students should be asking the questions that help them learn the issue better. The application of the testing system, however, does not seem to be helping the students realize the necessity to ask questions – instead, they figure out the ways to answer the test questions correctly. Therefore, it seems that the key problem with the given system of assessment is that it has replaced completely the process of communication between a student and a teacher in the course of an exam.
Indeed, while an oral examination not only provides the student with an opportunity to increase his/her score, but also to interpret the acquired information in his/her own unique vision, the teacher is able to see the possible gaps in the student’s knowledge and, thus, identify the ways to address the emerging problem. A series of quizzes, on the contrary, deprives a teacher of the given opportunity, which the results of the case study in question have proven.
Additional Assessment as the Means to Improve the Teacher’s Plan
As it has been stressed above, the current system of the students’ knowledge evaluation leaves much to be desired in terms of defining the obstacles in the students’ way of understanding the topic. While the students are provided with an ample opportunity to practice their newly acquired skills and apply their theoretical knowledge to practical assignments, they still clearly lack the teacher’s guidance, which the test results, sadly enough, show quite explicitly.
Therefore, the additional assessment, the plan to which is provided below, must be focused on the communication between a student and a teacher and, therefore, the exploration of each student’s individual progress, as well as the definition of the individual obstacles on the way of each student to understanding the course material and to develop the ways in which these gaps in students’ knowledge can be filled. To achieve the given goal, it is required that the assessment should be tailored to suit the students’ individual needs and at the same time to provide the tutor with feedback, which will allow the tutor to locate the source of the problems.
Given the fact that the topic of the lesson presupposes that the students should be able to switch from theory to the operation with the material element, i.e., the money, therefore, learning its value, it is crucial that the link between the abstract mathematical principles and their practical application should be established firmly. It is clear now that the previously used system of evaluation did not allow checking whether the students have passed the stage of concrete thinking (i.e., the identification of the money as an object) to abstract thinking (i.e., the ability to link the money value to the bills and carry out calculations). Therefore, the new assessment system should help handle the given oversight. In order to define the issues that prevent the students from understanding the new material, the following task was designed:
Each student is given a box of plastic coins and bills. The box contains examples of each type of coin and bill. Students will use various coins and bills to demonstrate the sum of two given amounts. In addition, students will use coins and bills to show two different ways to make a given value.
The given assignment allows the students to both apply their knowledge to practice and to communicate with the teacher, who, in his/her turn will be able to observe the progress of each student separately.
The proper time to use the additional assessment
Taking a closer look at the provided system of students’ performance evaluation, one might notice that the schedule is very accurate and that a new evaluation might not factor into the course of a lesson well enough. The process of testing being rather time-consuming, it can be assumed that the lesson plans provided in the assessment should be shaped in order to fit in the new activity.
To figure out how the new assignment can possibly factor in the lesson plan, it is important to split the lesson into several stages. For instance, the first section, i.e., “Number Sense,” shows that the students “mode and solve problems by representing, adding, and subtracting amounts of money” in the course of the lesson. Therefore, it is logical that the students’ skills should be evaluated in the process of them dealing with the tasks. To be more exact, the practical task described as a supplementary assessment should precede the theoretical solution. For instance, when asked how much $4 plus $2 is, a student is supposed to take the money or the models of the money and place them together, and then do the sum, i.e., $2 +$4 = $6. Thus, the students will be able to recognize the value of the bills and develop their mathematical reasoning at the same time.
The key goals: defining the possible misconceptions
Judging by the results that the previous assessment returned, some of the students – or, to be more exact, most of the students – have problems with identifying the value of the money, especially when it comes to coins and bills. The given problem triggers the students’ inability to provide mathematical reasoning for their actions when they attempt at conducting operations with numbers and relating these numbers to bills.
With that being said, the key goal of the new assessment strategy is to figure out at which point the students stopped understanding the subject matter of the lesson and what issues they need to revise in order to be able to adopt an analytical approach towards the suggested problems, as well as to teach the students to apply their mathematical thinking skills to problems solution. Therefore, the subsequent goal is to help the students apply the theoretical concepts, which they have already learned, into practice. Once the students understand the essence of mathematical thinking and are able to use the latter in order to solve mathematical problems, they will be able to embrace the principles according to which the calculations are carried out.
Key goals:
- Define the reasons for students’ lack of understanding of the subject matter;
- Develop the means to help the students overcome the obstacles and acquire the necessary skills;
- Implement the new program so that it could integrate successfully into the students’ learning program.
Picking the type of assessment: oral activity
As has been mentioned above, the students clearly have issues with their understanding of the principles of mathematical thinking. Even though the exact points at which the students started having troubles with understanding the material are yet to be defined, it is crucial that the chosen type of assignment should address the given problem. At this point, the issue of communication between a student and a teacher should be brought up once again. Ideally, as soon as a student feels that (s)he fails to understand a certain issue, (s)he raises his/her hand and asks a question concerning the problem. Since, during the class sessions, none of the students asked the teacher about the details that they failed to grasp, it can be deduced that the students’ hesitation to ask questions was predisposed by a series of external and internal factors. The internal being the students’ shyness and fear of being laughed at, which should be handled as an issue related to the organizational behavior, the external factors concern the way in which the classes and evaluations are organized, which can and must be shaped by the teacher.
Reconsidering the way in which the class activities are organized and conducted, one must notice that there is a considerable lack of a) interaction between the students and a teacher; b) time for students to reiterate the newly acquired knowledge. Therefore, it is desirable that the new type of activity should be predominantly oral, i.e., a detailed explanation of the students’ train of conclusions when solving a problem, etc.
Concerning the purpose: checking students’ understanding
One of the key specifics of the given assessment, which makes the latter especially complicated, is that it has not one, but two purposes. The first purpose concerns defining the difficulties that each of the students has with the topics of the lesson. Since in the given situation, the teacher will have to deal not with one or two students who have misunderstood the theoretical issues, but with an entire group of students, each having an individual problem with the basic mathematical operations and mathematical logics, the key challenge is in the quantity or the issues to deal with. With that being said, the first purpose of the assessment is to figure out which student at which point has started having problems, and with which issues in particular.
Another obvious purpose of the given assessment is to help the students realize that they have to make an effort in order to understand the principles of mathematical operations. Often students believe that learning something by heart will suffice for solving the basic problems and maintaining a passing grade; in mathematics, however, it is crucial that students should understand the logic behind the operations. With that being said, it is crucial that the students should recognize the problem and make an effort by trying to understand the topic better.
Finally, the third goal of the given evaluation is for the teacher to obtain informing instructions. At present, the key issue is that the teacher has no idea what prevents the students from using the pattern provided for solving problems. Therefore, it is crucial that the teacher should have some information on the problems faced by the students and, therefore, be able to come up with adequate means to address these problems.
The purpose of the given assessment is that a) the teacher should identify the misunderstandings emerging in the process of the lesson, and b) the students should be able to understand the logic behind the actions taken in order to solve a problem instead of repeating the pattern blindly. Therefore, the key purpose of the given assessment is to work on the improvement of the learning process so that the students could acquire the necessary knowledge and train their skills efficiently.
It could be argued, however, that in a broader sense, the purpose of the given evaluation is to figure out what hinders the students from applying the acquired skills efficiently in order to solve the suggested problems. Taking the given idea to an even wider context, one can assume that the purpose of the assessment is to make certain that the students have passed the stages from concrete thinking to abstract thinking to problem-solving.
Assessment implementation: practice and theory
Since the given assessment has been designed in order to detect the gaps in the students’ knowledge and the mistakes that have been made in designing the learning process, it is important that the assessment in question should be implemented with both the practical and the theoretical implications in mind. Since the given type of activity is going to be mostly oral, it will be relatively hard to come up with a detailed grading system. Therefore, the six-grade A-B-C-D-E-F system is acceptable in the given case. The grade is going to be given based on five major parameters, which are:
- Participation;
- Ability to interpret the theoretical material;
- Ability to use theoretical material to solve a practical task;
- Ability to relate the bills to their numerical value;
- Ability to create a train of logical conclusions, e.g., “If two minus one is one, then two dollars minus one dollar equals one dollar.”
Defining feedback strategies: what students have to say
Needless to say, the previous experience, i.e., the case study in question, shows that collecting and analyzing feedback from the students is essential for the students’ performance, as well as for drawing the plans for further lessons. With that being said, it is important that the new feedback strategy should provide more fruitful results than the one used previously.
As has been stated already, it is crucial that the students should feel free to ask questions during the lesson. However, the given approach will not suffice for changing the course of an entire group of students. In addition, the teacher should help the students voice their questions with the help of encouragement. It should also be kept in mind that there are active and passive learners in the class, and that, while the former is going to be proactive, the latter would prefer to keep silent. Thus, to encourage the latter to participate, the teacher could come up with activities like “who asks the smartest question,” etc. to see where the students’ gaps in knowledge are located.
As has been mentioned above, in choosing between the oral and written feedback, the teacher must give preference to the oral one. There are many reasons for the given choice, the key one being that in written feedback, it will be quite complicated to track down every student’s train of thoughts and specify the problems that each of the students has with the topic in question and its key concepts. Given the specifics of the subject, it will be relatively easy for the student simply to copy the answers from a book or from other students without the teacher’s noticing it. Even if a student does make a mistake, it will be very hard to figure out why the student came up with this very solution; in other words, the sources of particular problems will not be located.
Oral feedback, on the contrary, allows for tracing every single mistake or wrong logical operation that a student makes in the course of solving the problem. Since the feedback is going to be mostly oral and will demand that the teacher should gather it immediately after the students’ responses have been voiced, special techniques will be required.
To start with, it is crucial that the teacher should adopt such a technique of gathering feedback as discussions. As has been stated previously, getting feedback from students immediately is crucial, which is why online discussion boards cannot be used. Instead, it will be reasonable to ask the students specific questions and trick them into discussing the problems of addition, subtraction, and other logical operations with the teacher or with each other. The teacher will put the students’ answers (perhaps, shortening them) on the blackboard. Thus, the teacher will be able to track the students’ course of thoughts and spot the point at which they start getting confused about the topic.
Of course, it will be required that the teacher should be able to write down his/her notes on the students so that (s)he could be able to address the specific problems with specific students individually in an efficient manner. However, it is still crucial that the students should talk about the mathematical concepts that they have been operating in the course of the lesson.
The choice of the teacher’s feedback strategies, however, is nonetheless important. It is clear that the teacher should also participate in the discussion of the issues that students might find obscure. Therefore, it is essential that the teacher could not only spot the problem but also solve it as soon as possible. The key strategy in the given case will be providing the students with the opportunity to deal with a set of tasks concerning their individual problems. For example, if John Smith suddenly reveals the fact that he confuses numbers six (6) and nine (9), the teacher should immediately come up with a small exercise that will help John memorize these numbers and their value properly, e.g.:
- Activity A. The teacher starts drawing numbers 6 and 9 on the blackboard in random succession, while John immediately names them; or
- Activity B. John has to name all the objects in the classroom that are somehow related to 1) the number six (6); 2) the number nine (9).
While coming up with such assignments will definitely require considerable quick-wittedness, it will doubtlessly lead to not only spotting the problem easier but also addressing it properly. Therefore, it is crucial that the teacher should not only train the students’ skills in interpreting the material learned over the course of the lesson, but also to notice the gaps in the students’ knowledge or the problems that the students have with specific issues, and address these problems immediately.
Another type of feedback that the teacher could use to help the students deal with the problems that they are currently facing with the new topic is highlighting their progress. At the given point, a teacher might face a dilemma. On the one hand, there might be specific issues that a specific student might still behave with the topic. However, the teacher must provide not only corrections and comments on the student’s deficiencies but also on the progress made by the student to encourage the latter to study the subject further. Therefore, it is essential that the students should receive appraisal as often as they get criticized for their performance.
When speaking of the proper moment for the teacher to comment on the student’s progress, one can pick the several minutes that the teacher has at the end of the class, or the very moment at which a student is doing especially well. The latter seems the most appropriate. In addition, the praise should sound sincere; otherwise, the student will not accept it as a motivation. For example, even if a student still fails at adding two numbers to get a two-digit figure yet manages to get a one-digit sum, the teacher should address the latter fact as an improvement.
Informing instruction: how the learning process can be modified
As it has been stressed above, in the course of the assessment, the students’ weaknesses and gaps in the knowledge of the recently learned topic are going to be detected. By analyzing the students’ activity and level of participation, as well as their initiative in asking questions, one will be able to come up with ideas on how to make the learning process more productive. The informing instructions are going to be split into two major categories, i.e., the issues that have been covered yet have slipped the students’ understanding, and the manner in which the lessons should be generally conducted for the students of the given group. It is crucial to take the learning specifics of the children, both the proactive and the less active ones.
In addition, the new methods of evaluation of students’ performance offered above presuppose that the informing instructions will be obtained with the help of the analysis of the students’ oral performance, which makes the process of distilling the actual informing instructions a tad more complicated. It should be kept in mind that after the given strategy is applied, the informing instructions will be coming uninterruptedly as long as the students speak; therefore, the teacher’s task in terms of receiving the informing instructions will become considerably more difficult. In the environment of the lesson conducted by the newly provided strategy, the teacher will have to get the informing instructions as they come, analyze them immediately, spot the problem and, to top it all, provide an adequate response that will address the problem specified, as shown in the example above.
Indeed, in the given environment, the task of analyzing the informing instructions will turn into a very tense session. Therefore, several strategies that will allow minimizing the tension have to be provided. To start with, it will be necessary to consider the possible type of tasks that the students might need in the course of the future lesson; for example, seeing how the students are most likely to fail to identify a figure and the corresponding number that it denotes, it will be reasonable to use visual aids that will help the students understand the topic of the lesson better.
It should also be kept in mind that the informing instructions received in the process of the given assessment set the course for further lessons. As some of the problems that students might have with the topic can possibly take considerable time to deal with, it will also be necessary to analyze the informing instructions not only from the perspective of the areas that need to be covered, but also to figure out the time needed for improving the students’ performance.
With that being said, it will be required that before starting the assessment, the teacher should split the supposed informing instructions into two major types, i.e., the ones that will point at the issues to be fixed immediately, and the ones that will show what areas of mathematical reasoning will have to be explored during next several lessons.
Needless to say, obtaining the informing instructions will be rather hard. Unlike the written tests, in which the mistakes are obvious for the teacher to spot, during an oral assessment of the students’ skills, some mistakes made by students may slip the teacher’s radar once in a while. Therefore, the teacher might possibly involve the students into the process of specifying each other’s mistakes. For instance, by stating at the very start of the lesson that today, the students are going to play a game and spot all the mistakes that they and their peers do, getting points for each mistake named correctly, a teacher can possibly make the process considerably easier. The given activity will also help turn the inactive students more proactive and engaged into the learning process. Eventually, the given game can possibly help shed some light on the misconceptions that students have about certain mathematical concepts. The introduction of given activity, however, does not mean that the teacher should be any less attentive to the students’ responses and the slips that they make in the course of the lesson.
Additional Assessment as the Tool for Improving the Teacher’s Assessment Plan
Not only the students but also the teacher is going to benefit from the introduction of the new system of assessment. There is no need to stress the significance of the new assessment as the source of the information on the individual specifics of each student; by observing the students handling the newly assigned tasks, the teacher will be able to detect the points at which the students start hesitating or making mistakes, thus, realizing what issues must be revised. Thus, the assessment plan is going to be modified based on the individual learning patterns of the students, which is bound to increase the efficacy of the assessment and provide the teacher with ideas on further lesson structuring.
Specific Information Provided for the Students with the Help of the General and Additional Assessments
With that being said, it must be admitted that the key goal of the supplementary assessment in question will be to help the students feel confident about themselves and to stop fearing asking questions, as well as to help the teacher realize at what point the students became confused about the topic in question. Presumably, the pivoting point will happen to be the transfer from the concrete real-life objects to the abstract mathematical concepts and ideas; however, the true reasons for the students’ slackened progress are yet to be found. In his/her turn, the teacher of the given group of students will also be able to detect the problem and work on its elimination with the help of the given supplementary assessment. As soon as there are no misunderstandings between the teacher and the students, the learning process is bound to become productive.