During their stint in third grade, the students in the New Haven Public schools are expected to have proficiency in many areas within the realm of mathematics. They must first be proficient in their basic addition and subtraction facts and then upon that foundation much can be built. Multiplication and division are introduced during the course of this year as well as basic geometry. While the students venture through all of those operations, there is always one constant that reappears throughout it all. That constant is of course, the word problem.
When they enter the third grade, students are expected to have certain mathematical skills in place. They are expected to have mastered one-digit addition and subtraction without regrouping (facts to 18), as well as decoding and solving word problems. Ideally students would have enough knowledge and experience with word problems to understand the steps in which to solve them. These statements of course are the ideal, but to an educator, it becomes evident immediately within a classroom, that the ideal is not always reality. The reality is that students will enter a grade and be at varied levels, thus making a heterogeneously grouped classroom. Since every student is an individual and has individual needs, the teacher’s job is to first assess the student’s abilities and then differentiate the instruction according to his/her needs.
Within the domain of the word problem, the students have high expectations placed upon them. They must first decipher the vocabulary of the word problem while actively thinking about what the words involved mean. This can be done by reading the problem through carefully first without having an assumption of which function to use. An immediate assumption of which function to use can lead to an inadequate reading of the problem. Having sifted through the wording, the student must then identify what the end result of the problem is and what the question is asking. In many problems, the answer cannot be found unless a series of steps are completed and many students find themselves challenged by that process. It is the teacher’s duty to teach students how to tackle word problems through modeling, analytical group discussions, and much guided practice. It is after that in which students can be challenged to use their own knowledge of the methods and strategies to independently solve word problems. During teacher instruction, classroom discussions should be encouraged. Discussions on word problems can aid in a student’s understanding of the process of solving and deciphering problems, and can also be insightful for a teacher. When a student verbally expresses the processes used, he/she must be confident in the methods he/she utilized and must really think about how he/she arrived at the solution in order to articulate it. Teachers and other students can benefit by listening to an oral explanation. Teachers can understand the thought process of the student and pinpoint the area in which students are experiencing trouble. Other students can hear the methods their peers used to attack the problem and use those strategies to aid them if they encounter problems. In some cases, an explanation in “student lingo” can help a student more than an explanation given by a teacher. Discussions also allow the students to experience metacognition, or thinking about their own thinking. By doing this they can examine their processes to identify which are essential to solving the problems at hand.
In some instances the final step in a teacher’s assessment of a student’s proficiency of solving word problems is found by giving the student a problem and then examining the processes used and then the answer to the problem itself. For the sake of this unit on the other hand, this is not the last the student will see of the word problem. The goal of this unit is for students to use their accumulated knowledge of word problems and the processes that are needed to solve them, to synthesize their own complex, multi-step word problems.
The series of lessons within the unit are meant to teach students about word problems and to also immerse them in exploration and discussion. The reasoning behind this is to allow the students to become more familiar with the process of solving the problems and explaining how they arrive at their conclusions. Since scaffolding is the process of building upon information the students have previously learned, the lessons are designed to begin with simpler problems that then progressively and increase in difficulty.
It is also essential to peer into the psychological and developmental aspects of the students when it comes to mathematics. Piaget had strong views on mathematical word problems. There are great intellectual expectations that are placed upon students but are not facilitated within the classroom, thus leading to failure in mathematics. It is through the exploration of his writings that the realization of how reasoning in problem solving is often overlooked or misconstrued due to the inexperience of students. When the complexity of the problem increases some students will abandon logical thinking and practical methods to decipher the problem out of frustration. That then results in the students using incorrect operations to get a numeric answer. Piaget suggests working backwards when approaching problems by excluding the numerical values to have a narrow focus on the words themselves. After the numeric distractions have vanished, students can deductively reason what operation should be utilized and then the numbers can reappear in the problem.1
The rationale for creating this unit is to allow students to create and be actively involved in their own learning. The world problem is intimidating for some students and with that frame of mind they are hindered in their abilities. The goal is to take that scare-factor out of the word problem and have students feel confident to master solving them and then become fluent enough to create them. The process of learning must also be a process of doing. A student cannot fully comprehend unless he or she is immersed in doing. The formation of problems connects to this in such a way that students must be able to do and create a work in which others must also be able to understand and explore. It is then, of course, the teacher’s duty to provide those opportunities that will involve the students in actively creating and synthesizing.2
My desire to enable student synthesis of quality word problems was inspired by Benjamin Bloom and the educational taxonomy that he created. The following are the stages within Bloom’s taxonomy and a brief explanation of what they each encompass taken from “The Learning Skills Program” website that adapted the taxonomy from, Bloom, B.S. (Ed.) (1956) Taxonomy of educational objectives: The classification of educational goals: Handbook I, cognitive domain. New York ; Toronto: Longmans, Green:
- Knowledge: observation and recall of information.
- Comprehension: understanding of information and grasping the meaning.
- Application: utilizing information as well as methods, concepts and theories.
- Analysis: seeing patterns, recognition of hidden meanings and identification of components.
- Synthesis: using old ideas to create new ones, generalizing from given fact, and relating knowledge from several areas.
- Evaluation: comparing and discriminating between ideas, assessing value of theories, and making choices based on a reasoned argument.3
The beginning stages of his system focus on memorization and the application of knowledge. The latter stages delve into the more complex thought processes in which students must fully understand the topic and then create a work that applies that prior knowledge, and proves the proficiency and knowledge of the topic. Synthesis is the second to last stage (before evaluation) that requires students to demonstrate activities that exhibit their knowledge and skills in certain areas. Unless students can show their skills through synthesis, they cannot be deemed fully proficient in the skill they are practicing.4 In the context of this unit, the knowledge portion would be the setting up of mathematical equations and deciphering the solutions to those, and synthesis would be their own creation of the problem.
Finally, the process of students generating their own word problems can also be used as an assessment by the teacher. The teacher can judge by student work if the student is fluent and proficient in the objectives of the lessons. There are two types of assessment that can be done as a result of the lessons contained in this unit. The first type is a performance-based assessment where the student generates a product and is then evaluated on it. The other assessment is process-focused which will prove to be an integral part to this unit. Process-focused assessment concentrates on not just what the answer is, but how the student arrives at that conclusion.5 Finding out how a student arrived at an answer is just as important (on some occasions more important) than the answer is. By assessing the process in which the student goes through, it can be determined if errors are just computational or if they are errors in reasoning and abstract operations.