Recently, we, in MTH 329-01, began our unit on proportions and fractions, and a related article is the following one by Skemp.

https://drive.google.com/file/d/0B8frwh-y1pyQRUVieTZubmY4Rm8/view?pli=1

This is a phenomenal article in which the author makes a strong case as to why all instructors need to begin teaching mathematics, and all subjects, in a relational manner. His biggest supporting point is the fact that research in related educational subjects reveals that when students are taught relationally, their long term cognitive abilities in that subject are drastically superior to students who were taught the same subject in an instrumental fashion. Furthermore, Skemp gives four reasons as to why relational teaching (aside from the primary mentioned above) is drastically superior to instrumental:

1. The means become independent of particular ends to be reached thereby.

2. Building up a schema within a given area of knowledge becomes an intrinsically satisfying goal in itself.

3. The more complete a pupil’s schema, the greater his feeling of confidence in his own ability to find new ways of ‘getting there’ without outside help.

4. But a schema is never complete. As our schemas enlarge, so our awareness of possibilities is there-by enlarged. Thus the process often becomes self-continuing, and (by virtue of 3) self-rewarding. (Skemp 8).

Clearly these are all goals that we, as future math educators, want to encourage and see develop in all of our students; however, as Skemp points out with his last section titled “Theoretical Formulation”, these ideals only hold weight in an abstract or experimental situation/classroom, and NOT in actual USA classrooms. Obviously this is a disappointment, but it is also not the least bit surprising. This is because Skemp gave four points as to why instructors choose to teach instrumentally. They are:

1. That relational understanding would take too long to achieve, and to be able to use a particular technique is all that these pupils are likely to need.

2. That relational understanding of a particular topic is too difficult, but the pupils still need it for examination reasons.

3. That a skill is needed for use in another subject (e.g. science) before it can be understood relationally with the schemas presently available to the pupils.

4. That he is a junior teacher in a school where all the other mathematics teaching is instrumental. (Skemp 6).

Skemp also gives some reasons as to why a curriculum or outside forces also push instructors into an instrumental teaching method:

1. The backwash effect of examinations.

2. Over-burdened syllabi.

3. Difficulty of assessment of whether a person understands relationally or instrumentally.

4. The great psychological difficulty for teachers of accommodating (re-structuring) their existing and longstanding schemas, even for the minority who know they need to, want to do so, and have time for study. (Skemp 6).

I would just like to point out that the rise and dominance of ACT, SAT, GRE, Common Core, and the like all dramatically increase the impossibility that any teacher, but especially new teachers, have a chance to even attempt instructing relationally due to the high demands of all the standardized tests and regulated curriculum. That being said, Professor John Golden told us that the most important thing we can do as teachers is to make the best choices for our students within the realm of things that we, as teachers, can control. Example being that if our district demands the “butterfly method” be taught to 5th-6th graders, then we as teachers should discover relational links between butterfly method and other fractional topics, thus ensuring or at least increasing [hopefully!!] the understanding of the covered mathematical topics.

Furthermore, I believe that all of Skemp’s reasons as to why teachers forgo relational instruction can be summed up as follows: instrumental is significantly easier and more time efficient to teach to students than instrumental; hence, since USA school curriculums become increasing “big test” oriented, instrumental becomes more and more attractive to all teachers, NOT just new ones. Additionally, Skemp’s 2nd points in both outside factors and primary factors directly relate to each other, and combined, they create a vicious cycle that creates a never ending need for more and more stuff in a more condensed time frame for standardized tests; thus, leaving only one option available to teachers whom want to cover all relevant material and (due to common core and higher review standards) keep our jobs. Finally, consider all of Skemp’s arguments for relational. Upon simple examination, it is apparent that all four look/sound like fantastic goals for teachers to have, but with closer inspection, we notice that a significant amount of effort might/can/will be needed to develop this in every student {approximately 30 per 3-6 classes} equates to 120 or so students that all learn in varied ways. It follows that, in only an 8ish hour day, that a teacher gets maybe 1 hour per class to “teach”, but that leaves a very small amount of time [even if all 60 min/class is spent in group/one-one discussion] it seems nearly physically impossible to achieve such a lofty goal given current resources/demands on single teachers in USA classrooms.

Personally, I feel that we {as [future] instructors} should endeavor to teach relationally all mathematical content to every single student that enters our classroom for the simple reason that it is our job to not only learn mathematics, but to obtain skills vital to survival and prosperity in a democratic society. This is perfectly aligned with Skemp’s reasons to teach relationally because, in USA, we tend to value individualism/uniqueness, innovative, confidence, independence, and creativity. These traits are directly developed and supported from all four of Skemp’s justifications to teach relationally. This is because a student who internalizes their desire to learn mathematics (and other academic topics) shows a drive to better themselves, and when they discover several different techniques to solving similar problems, this will have built up their creativity and innovative skills. Also, their skills will grow and the student will no longer need to be scaffolded in mathematics; thus increasing their confidence in both mathematics, and in life in general because mathematics is touted as “hard”, so when the student gains a powerful grasp of mathematics, then they feel emboldened to try other challenging things that may or may not deal with mathematics.

Overall, I really enjoyed reading this article and grasping/processing what the article is saying and how it will impact my teaching methods/choices when I am in the classroom. That being said, I feel it will be best to try and emulate what Skemp wants, but, until something changes, do not sacrifice material that is on ACT, SAT, etc. in loo of improving relational knowledge on fewer topics i.e. balance between quality and quantity must be maintained, but quantity, in this situation, beats quality.

P.S. We also played a game called “bizz-buzz”, which is an, allegedly, drinking game, which has been adapted to counting fractions and recognizing equivalent fractions. I found the game fun. Not much else to say on this.

Thanks for taking a few of your precious minutes to read this. I hope that you found this useful/interesting.

Good break down of Skemp! For completeness, I’d love to see some of your personal reaction to the four reasons to teach for relational understanding and the four for why we don’t, since some of the reasons we don’t are things you have seemed to be thinking about already.

clear, coherent, content (nicely), consolidated +

PS> Glad you like Bizz Buzz!

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I enjoyed this break down of the dense article that we read for our classroom discussion. As we progress through out studies, we need to implement the relational understanding rather than simply instrumental understanding within our classrooms. Although we may be told that we don’t have the time, the time needs to be there. Our main goal should be to better shape our students understanding and knowledge so that they can go out into the world and better themselves as well as others. I think that I would have liked to see more of your interpretation of the article in a more specific sense so that I could relate my own opinions to yours. Overall, I thought this was a great way to introduce and discuss relational teaching and understanding.

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I agree with you that there is a lot required of teachers and that trying to cover all of this and make sure our students reach a deep understanding of the topics will be difficult. As teachers, we have to find a balance and pick our battles so to speak. What should we go into more depth on and what will then have to be taught in a more instrumental fashion because of lack of time.

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