“Mathematics is not about numbers, equations, computations or algorithms: it is about understanding.”1 Aligned with this mathematical concept are the Common Core State Standards which state that in each of the different math levels children should have an established and implemented view of the ‘processes and proficiencies’ with regard to mathematical education. In short, according to the standards, children should have a balanced combination of procedure and understanding. This element of ‘understanding’ allows children to practically apply academic knowledge that they have obtained to practical applications throughout their daily schedules. There are three different articles; “Relational Understanding and Instrumental Understanding” (Skemp), “Making Sense: Teaching and Learning Mathematics with Understanding” (Heibert), and “Good Questions” that delve into the concept of what it means to understand within the context of a mathematics classroom.
The article written by Skemp, defines the term understanding in two different ways. One defined element of understanding is relational understanding – that is to say, children know (understand) what to do and why. An additional way Skemp defines understanding is instrumental understanding, meaning, children are taught and understand rules without reasons. The results of these vastly different definitions of understanding vary greatly. A child who is taught according to instrumental understanding will learn and memorize rules, but he will not have much experience in practical application. On the other hand, a child taught based on rational understanding will be able to associate mathematical rules with practical problems. He will have much less work and rules to memorize for he will understand both how and why he should go about solving each problem.
Although, at first glance, rational understanding seems to be the better instructional method of the two, many educational instructors implement the concept of instrumental understanding during their mathematics lessons. These reasons may vary and may include the following: Instrumental understanding is easier to obtain within its own subject context. When a teacher is giving a math lesson and the goal is to produce a page full of correct answers then the proper method to use throughout the lesson is instrumental math (instrumental understanding). Another advantage of instrumental math is that the rewards are immediate and clear. Although you may receive faster and more apparent results through instrumental understanding; relational understanding (mathematics) is the method that will have long lasting results. There will be more to learn and understand, but once the knowledge is obtained it is long-lasting!
The concept of relational understanding supports several elements that are mentioned within the Common Core Standards for Mathematical Practice. Within the standards it states that students successful in mathematics will be able to explain, analyze and understand different mathematical questions. A child taught through rational instruction will have the tools and skills in place to not only successfully solve math problems, but to additionally explain and analyze why and how he should solve the problems.
I believe that there is much to be gained from both elements of instruction and neither one should be disregarded. Children in first years of schooling often do not have the skills in place to analyze and critique mathematical questions and as a result the instruction should be a combination of rational and instrumental understanding. Basic mathematical skills should be taught through rational understanding as those are skills that children will need for life and consequently they should rationally understand what and why they are learning. However, I believe that if some mathematical applications are too complicated for children to process t hen they should be taught on an instrumental level and then when the correct skills of understanding are in places these concepts should be taught on a rational level. In other words, some lessons should be taught at first instrumentally and when the students are on the appropriate cognitive level the same lesson can be taught rationally and the children will obtain skills to apply math practically.
Parallel to Skemp’s article on understanding is Heibert’s article “Making Sense: Teaching and Learning Math with Understanding”. Heibert states in his article that when students learn they should learn with understanding! While Skemp introduce instrumental instruction as a method of understanding, Heibert believes that if a student memorize the math rules than perhaps he is learning something, but he is certainly not learning math. Heibert’s beliefs are aligned with the Common Core Standards; in order to really know a subject, students must analyze how the information works, how different elements within the subject are related and why these elements work as they do.
Teachers desire for children to understand math. They believe that understanding is a good thing and consequently they teach to ensure understanding. In a proper classroom that facilitates understanding teachers will maintain that there should be two elements in place: reflection and communication. Students should be able to reflect on what they learn repetitively from numerous perspectives. The students will result in establishing new relationships and they will analyze their old mathematical relationships. Subsequently, their understanding will increase rapidly. Students should also be encouraged to communicate and share their thoughts on different mathematical problems that they are analyzing and attempting to solve. Each student will have the opportunity to clarify and explain what they understand, and each will learn how to justify his or her answers and methods. Reflection and communication will work hand in hand and as a result will facilitate understanding in mathematics.
In Heibert’s article, he states that understanding and skill are meant to develop and work together resulting in a successful mathematics classroom. Neither element should be sacrificed for the other – they are both necessary to ensure success. Unfortunately, many mathematic classes focus solely on skill and not on understanding. Many teachers have one goal in mind – they must complete the selected amount of pages in the textbook each day. The teachers are so intent on fulfilling this goal that the students are being neglected and they are not understanding what they are learning. The skills that they are being taught do not have the opportunity to develop into understanding.
A method that will allow skills to develop into understanding is to ask good questions. The article “Good Questions” defines good questions as open-ended questions. As opposed to close-ended questions which require students just to give an answer based on memory and skills, an open-ended question asks for the student to think deeply and critically before coming to an answer. A teacher who uses open-ended questions throughout the lesson allows the student to use their potential for thinking and reasoning.
Open-ended questions deepen and further understanding and additionally they allow children to become aware of what they do and do not know. Furthermore, the teacher will be able to see whether the students fully understood the concepts that were taught in class. The teacher will observe who is successful in answering the open-ended questions and from that she will be able to see which students need further help in the math relating to the questions. I was able to put this concept into practice. I work in a fourth-grade classroom and part of my job is going over their math homework. Although a girl appeared to fully understand long division when it was taught in class, she had immense trouble solving problems that were out of context and were practical to daily life. Her homework that was full of open-ended questions was a clear assessment of what the girl was successful in and what she did not understand. It is important to instill good question within every math lesson for the questions ensure higher level thinking and they allow students to develop problem expertise and important math skills that will be needed long after they have left the math classroom.
The three math articles expounded upon above have instilled within me a deeper understanding of how to teach math. I realize that the goal of math lesson should not just be arriving at the correct answer; the students should be able to understand and explain how they went about getting to their answers. While these skills and elements of understanding are taught and introduced in a mathematics lesson, they are skills that these children will be able to use permanently in their lives!
1 William Paul Thurston