This week’s articles discuss the differences between understanding math relationally and instrumentally, as well as the implications for teaching with these different methods. I had not heard about the difference between these understandings before, but I can see that there is a clear difference. Thinking back on my own math education, most of what I learned has been taught instrumentally. However, I think that it is crucial that students develop a relational understanding of math, and the Skemp article provides important benefits. He notes that it is adaptable to new tasks and easier to remember, which both tie in with our previous discussions about memorization in science classes. Just as in physics and chemistry, if students understand how the formulas are developed, they will be better able to apply them to new situations. This understanding also reduces the need for so much memorization, because students could come up with the formulas while taking the test. Aside from these benefits, relational understanding can be “intrinsically satisfying” and “self-rewarding,” as Skemp puts it. This viewpoint turns the focus from learning math for a specific application (the “backwash effect of examinations”) to learning it for its own sake.
The Pesek and Kirshner article takes a look the implications of teaching using relational or instrumental methods. The article notes that teachers are encouraged by professional organizations to teach for “meaningful learning,” with the relational method, but that teachers are often pressured into using instrumental methods by administrators and parents. Time limits and focus on exams contribute to the high use of instrumental methods, but teachers often try to fit in some relational teaching as well. The article looks at the use of only relational methods compared with instrumental methods followed by relational ones. The study did find higher results for the relational only group, but the data did not meet the 95% confidence threshold. Three types of interference from first experiencing instrumental methods are suggested: cognitive, attitudinal, and metacognitive. In all subjects, students’ previous class experiences certainly affect their learning in new classes. I particularly found interesting the idea of metacognitive interference, which states that students may block out new instruction because it interferes with maintaining previous material. I had not previously heard of this fact for students not learning, but it shows that we need to pay attention to students’ focus.
The study did not compare the methods used to either instrumental only or relational followed by instrumental. I’d like to know what the results would be if students were introduced to topics relationally, but then still learned the instrumental methods in the end. This could still avoid interference, but ensure that students still learn the rules and prepar for the exams. It appeared as though the students in the relational only group in the study were never taught the area and perimeter formulas. While it will benefit the students to have a relational understanding of area, there are many situations where it will be more practical to know formulas. Additionally, while areas and perimeters sound like a topic that works well to teach relationally, there are other concepts that would be difficult to learn relationally. The articles mentioned multiplying two negatives or dividing by fractions. Both of these are probably hard to understand relationally at the grade level when they are introduced.
These articles brought up two forms of mathematical understanding that I have definitely been exposed without realizing the difference upfront. The Skemp article provided a number of useful examples to explain the key distinctions. With Instrumental Understanding, the goal is to learn math equations and theorems as a means to an end answer; it usually involves memorization of different rules and situations. Relational Understanding involves conceptually understanding the math to know what method to use and why to use it. Therefore, Relational Understanding appears to be the more beneficial type of understanding since it involves less memorization (which might otherwise cause problems if the student can’t remember each rule or if he/she tries to apply the same rules to situations that don’t warrant them). However, at the same time Instrumental Understanding can make complicated problems easier to understand in a shorter amount of time.
Two examples from my own experiences with these methods of understanding: I remember being taught Instrumental Understanding of the rules for handling operations with negative numbers (multiply two negative numbers together to get a positive number) and Relational Understanding to learn the signs and cosines of special angles (by conceptually understanding how to figure out these values using the unit circle and special triangles, instead of spending ours memorizing the answers like some of my peers did).
The Pesek article looked to figure out which was more beneficial: instrumental instruction followed by a relational instruction, or strictly a relational instruction. The research showed how students taught Relational Understanding from the start performed better on a posttest and retention-test than the students taught an Instrumental Understanding before the Relational. A difference in the conceptual understanding was also seen after interviewing some of the students from each group, where the students who had been in the Instrumental Understanding class first made a number of errors in applying the math to other life situations.
Both of these papers brought to attention the difference between instrumental and relational instruction. The Skemp article gives a clear explanation of the two different methods and his viewpoint on them was reinforced by the study explained in the Pesek article. Skemp describes that in mathematics these two types of instruction are very different and lead to different outcomes for the pupils. He believes that relational is by far the more beneficial type of instruction. Instrumental mathematics leaves the student with fixed plans of problem solving that are ineffective with change, no overall awareness of the topic at hand, and great dependence on guidance for each new situation. There are benefits with this type of mathematics, such as it being easier to understand, producing immediate rewards, and being able to get to correct answers more quickly and reliably. However, he believes the disadvantages outweigh the advantages by a large amount. Relation mathematics is described as giving students a deep conceptual structure for the subject, the ability to produce unlimited plans for different problems, and more independence at solving new problems using their deep understanding. Before reading about the study of fifth-grade mathematic students, I sided with Skemp’s opinion that relational instruction is the method able to produce substantial and conceptual understanding. If students have only learned one way to specifically complete a problem and think nothing of what their truly doing, a problem arises when the teacher slightly changes the question. The student does not have a conceptual understanding of WHY they are doing the problem in that manner, other than rote memorization. However, students who have had relational instruction understand the meaning behind the formula, so they are better able to adapt because they know the reasoning behind it. I have also seen the difference between instrumental and relational instruction firsthand in chemistry courses. There have been professors that only give the formula, without a clear explanation as to why this was used and what exactly was happening. However, I have also had professors that cared about how the content was being perceived and paid attention to make sure the students understood the deeper and conceptual meaning behind what was being done in class. Therefore, again I side with Skemp because I believe that the deeper conceptual learning attained from relational instruction has been more helpful to me. In my opinion, while instrumental learning may help students memorize and do well on individual exams, the final in cumulative courses usually contains a lot of material, making it hard to memorize. In this case sometimes having a more conceptual understanding will take the student further. Relational learning also has a higher retention rate, as seen by the Pesek study, 16.49 to 14.31 out of 34. The higher retention rate of material also plays a role in lessening the need for cramming for a final exam. Lastly, I thought of a question as I was reading both articles. If in instrumental mathematics the student still believes that he or she knows and understands the material, how is one always able to differentiate his or her own reasoning as instrumental or relational? Some of the I-R interviewees were able to answer questions, incorrectly, but they thought there was sound reasoning. Therefore, this goes back to the discussion we had earlier in the semester that most misconceptions are produced without realizing that they are misconceptions.
I found these two articles to be very well argued and I was content with the fact that their conclusions agree with my mentality as somebody who teaches and tries to keep teaching in an effective manner. More specifically, the Skemp paper makes very valid points on the importance of relational understanding for a student. The points 6 and 7 that he makes in page 10 are the ones that give the essence behind this method. As a student and as a teacher, I find the ability of a student to want to learn for the sake of learning without necessarily thinking of the end goal (grades, success etc), to be one of the most effective tools for actual conceptual understanding and personal growth. In addition to that, the possibility of such method to motivate a student to think about new areas and new ideas, based on the ability to create analogies and develop them further, is probably a very productive tool for science in general. In this way, with this method of teaching, and by promoting a relational way of thinking and understanding, we are creating capacity for future scientists that may discover something and contribute with that knowledge to the science in general. Skemp uses also the example of exploring a city and creating a mental map as a useful tool for someone to be able to find his/her way without knowing the information directly. I believe it summarizes perfectly the idea behind relational understanding, since “a person with a mental map has something from which he can produce, when needed, an almost infinite number of plans”.
The Pesek paper adds more into this idea by comparing the effect of promoting both relational and instrumental understanding interchangeably. In this case, besides the expected confusion that a student might have due to a radical change in teaching, relational understanding seems to be the winner again. However, the determination of an individual teacher to reform and design effectively his/her teaching according to what will make students learn better, is the most important factor into motivating students to learn that way and continuing to build up their knowledge according to that.
Finally, one point that I can’t fully agree with, can be found at Skemp’s paper. In page 8, in point 2, where he says that “it is nice to get a page of right answers, and we must not underrate the importance of the feeling of success which pupils get from this”. Although the psychological boost and the feeling of succeeding is a very important factor for a student’s psychology, in this case the apparent success can be deceiving just because of the fact that it was achieved through instrumental learning, and any added success based on this understanding cannot be maintained in the long term. Therefore, it is the responsibility of a teacher to clarify to students the gains and the losses that they will get from each method.
The articles for this week discussed the differences between instrumental and relational understanding and teaching of math. The Pesek article seems to come off with saying instrumental education is something that should be avoided. It also says that it is an approach that is favored by parents and politicians, because it is the idea of forcing students to learn how to solve problem by experience, but does not enforce a deeper understanding. After reading this article, I must agree that instrument is an easier approach than relational, but it is not better. It does not help foster the deeper understanding of the topic. But because class time is very constrained, it can be a useful method for teaching. The article later says that a better approach is one where teachers mix the two approaches, relational and instrumental, together to help save time, but also to help develop that important level of understanding.
The second article, by Skemp, raises a very important and interesting point. Because of the differences between these two teaching styles, there can be “mis-matches” between student and teacher. First, the pupil may want to learn instrumentally but the teacher uses the relational problem. This is not all too bad. But, it means the student is missing out on potentially important/useful material. The second is the student wants to learn relationally, but is taught instrumentally. This is more dangerous as it means the student is missing material that he/she desperately want to learn. This can lead to discouragement, frustration, discourse, etc. My one question raised from this article, is what are some effective ways to combat the problems created by these two possible mis-matches?
These articles discuss the differences between relational and instrumental understanding. I think that all students have come across this in several fields, namely math. Like the section in Skemp’s article called “Devil’s Advocate”, I believe that each of these types of understandings has a role in students understanding and use. As an example, I like to think about calculus 1 courses. The teachers spend a fair deal of time talking about derivatives and how to calculate them, then the students are taught the power rule. I think this is an ideal example of how students should learn about how things work before learning the shortcuts.
I really like the idea of students learning to gain a relational understanding, but sometimes it is not always practical. In a classroom where that is a goal, I think it important for the instructor to make it clear that that is what they are working towards. I think a fair measure of this understanding would be to include questions on a test that are similar to the questions students have seen in class, but expanding slightly more or applying it to a different situation. In other cases, I think that relational understanding is impractical; as an example, it’s not necessary to understand the complete workings of a computer program in order to utilize it.
I actually have thought about the differences between these understandings before, although without the language. The situation that got be thinking about this was when I was observing and helping in a 7th grade math class. I believe they were learning geometry, and when asked to calculate an area, almost every student turned to their calculators. I understand using a device such as a calculator to assist in difficult problems, but I was struck when they felt the need to use the calculator for what I consider easy calculations, such as multiplying by 10. I wonder whether previous classes that taught to instrumental understanding would inhibit a student from achieving relational understanding in a successive course. I also wonder how students’ attitudes and goals affect their level of understanding. In the second article, this is described as attitudinal interference. I also question whether a student would have more or less attitudinal interference after taking courses aimed towards instrumental or relational understanding.
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