Memo #1 Understanding

Part I-

Understanding is key when teaching mathematics, and too often we have turned our classrooms over entirely to procedural fluency. While solely relying upon procedural fluency is detrimental to a child’s learning, procedural fluency does have its place in the classroom. Hiebert outlines the fact that each of us have a limited amount of mental effort that can be expended at any given time. This means that “the more efficiently a procedure is executed, the less mental effort is required." With less of our mental capacity being tied up doing mundane tasks, the more we can use our brains to focus on meaningful mathematical knowledge. If students become bogged down by difficult computations or frustrated at the first step, they will never move on to grasp the larger scheme of things. In this way, procedural fluency is crucial in building conceptual understanding.

In alignment with the idea that the strands of learning should be taught en masse, procedural fluency and conceptual understanding are inseparable. For example, if two students want to add the numbers 1458 and 267, one may proceed with procedural fluency and begin adding the numbers like so:
            1 4¹9¹8             where the other may “borrow” 2 from                 1 5 0 0
            + 2 6 7             267 in order to round 1498 to 1500,                       + 2 6 5

             1 7 6 5             then add the remaining 265 like so:                      1 7 6 5

The second student may use his conceptual understanding of how to add, but he or she is still relying on the knowledge of how to compute 1500+265, a procedure simpler than the former, yet still a procedure.

How can we teach mathematics for understanding?  

Mathematics is difficult to teach as it is, but to teach for understanding is a whole new ballgame. In order to teach for understanding, we would need to shift our focus from procedures and practices to concepts and content. The most difficult part of this process is that the design of education places 25-40 students in every class, which makes encouraging critical thinking in each and every students mind a difficult task. Even more difficult is providing feedback to these students, whether in the form of informal or formal assessment. I personally think that another element of teaching for understanding involves genuine student inquiry, which is difficult to plan, especially when each student comes from a different level. Math is often taught in a superficial sense, emphasizing procedure over concepts and hurting our students in the long run. 

Part II:

Questions:

1.  What is an external versus internal representation?

External and internal representations are just like they sound. An internal representation is something that one might think about, like the concept of gravity, namely that things fall and we know and have experienced this over our lifetime. An external representation is something more concrete that expresses out internal representation. In this case, it may be anything from a graphic organizer detailing the things that we frequently think about falling, like balls, or people sky diving from planes, maybe even apples from a tree to a verbal explanation about what happens when you drop . An external representation could also be a diagram, or the equation for an object accelerating towards the ground. Representations are different for every individual, depending on their experience with the concept called to mind. (The graphic organizer maybe for small children, and the velocity/acceleration equation for a skilled high school or college student).

2. What are the benefits or results of mathematics understanding?

Students benefit from mathematical understanding rather than procedural fluency because information that is learned with understanding, in the sense that it is tied to other concepts and beliefs and has a firm foundation in a students mind, is called to memory quickly and more readily accessible to the student. The result of this is that a student with understanding is more able to reason and problem solve on their own, specifically when coming up against problems they might have not seen before. Even problems that are more abstract can be better understood by those with a true understanding of the concepts, instead of knowledge of procedures which are of no help if a student does not know how to use them. Both understanding and procedures are important, but they must work together to reach the outcome. 

3. What is ‘conceptual knowledge’ and ‘procedural knowledge’ and how are they connected in terms of math understanding?

In mathematics education, we describe two different approaches to solving an equation as either demonstrating procedural fluency or conceptual understanding. Procedural fluency is the ability of a student to apply a procedure, that is to say, the student has mastered an algorithm for solving a given type of problem, and is now applying it to a similar problem. An example of procedural fluency is when a student uses long division on 152/80928 to change it into decimal form. The student will setup the problem and begin chugging away at the answer in the manner that he or she was recently taught. The student will most likely arrive at the correct answer, but they may take some time and if they make any mistakes along the way, they might not be able to catch them.

Conceptual understanding is the converse of procedural fluency. Conceptual understanding involves a student being comfortable with the problem at hand and their ability to think critically about its meaning, applications, and the best strategy to solve it. The same example of 152/80,928 being changed into decimal form can be tackled using conceptual understanding. A student using conceptual understanding may reduce the fraction first in order to find a simpler solution, they may also obtain an answer such as .937 but quickly realize that this cannot be so since (roughly rounding up) 200/81,000 is nowhere near .937. A student using conceptual understanding also uses procedures, yet is able to think critically about which ones to apply in order to best obtain a practical answer.

There are key differences between conceptual understand and procedural fluency. Both aim to prepare students for the tasks ahead of them, but they tackle the issue in fundamentally dissimilar ways. Conceptual understanding puts at the forefront the student’s ability to think creatively and use logic in order to comprehend, interpret, and solve a problem. Procedural fluency is centered on the student’s ability to produce the correct answer, regardless of how they arrived at their response or they understood its significance. By Kilpatrick and Swafford’s definition, a student using procedural fluency has “skill in carrying out [a] procedure flexibly, accurately, efficiently and appropriately,” and a student demonstrating conceptual understanding will show “comprehension of mathematical concepts, operations, and relations”

While students using conceptual understanding or solely procedural fluency might show no difference in their answers, they may vary in their ability to explain how they obtained the answer and why it is significant. If the answer was wrong, students using understanding are will likely diagnose what went wrong because they can logically justify their steps in obtaining the answer. Students who possess procedural fluency may get frustrated at incorrect answers because they are simply following a rule and cannot provide the reasons why their formula or algorithm worked or not. As educators we strive to teach the former, as understanding also leads to a better grasp of the concept and its significance. 

4. Why is it difficult to assess mathematics understanding?

Understanding, specifically mathematical understanding is difficult to assess, because "understanding usually cannot be inferred from a single response on a single task; any individual task can be performed correctly without understanding" (89). Besides this, in classrooms we see today, it is difficult to assess each student's understanding specifically because they may express it in a variety of ways. For example, if you see a student has solved a problem wrong with no work, you might ask them how they got that answer, and they might tell you they did everything right but at the last moment made a simple mistake with subtraction, giving them the wrong answer. There is no way to know by simply looking at an answer to tell what the student was thinking. If a student shows work and you can tell by mistakes their misconceptions that they had the correct understanding but made a mistake, it is still hard to capture this in a multiple choice test that teachers use so often. The last scenario is where a student has the right answer, but no work showing how they came to that conclusion. They could have used the process of elimination, they could have guessed, used the wrong procedure, or even worse, they might have copied another student. In this case, it is next to impossible to tell if the student understands the content unless they showed their work or verbally explained it to you.