Conceptual and Procedural Understanding: Do I Really Understand the Math Problem?
MACSER: It’s about the math. (2018). Retrieved from https://www.anl.gov/mcs/article/macser-its-about-the-math
Hey everyone and welcome back to another blog!
Today our
class focused on the key differences between conceptual and procedural
(relational and instrumental) understanding, using resources for teaching and
learning mathematics and using manipulatives for teaching/learning math. The
first activity we did in class was a reading comprehension activity. We were
given a short paragraph to read and we had to answer a couple questions. The
activity is shown below in the image. This activity really opened my eyes to
the reality of “you don’t need to understand the content to get a good mark.”
No one in our class understood the context of this paragraph, however, we all
got 4/4 for answering the questions correctly. The paragraph below is written
using grammatical rules that you understand, therefore, understanding the rules
but you don’t understanding who, what, where, why and how. As a future
educator, we need to avoid shallow learning and memorization. This is a big
problem in both mathematics and science, we strive to drill in rules and then
give a test on memorization but not to focus on the understanding behind it. Therefore,
in my future classroom I want to focus on promoting more hands on
Additionally, in class we focused on the difference of relational
and instrumental understanding. Skemp (1987) describes the difference between
instrumental and relational understanding, and how the word understanding is
used by different people to mean different types of understanding. Instrumental understanding means a child knows
a rule or procedure, and has the ability to use it, where relational
understanding means a child knows what to do and can explain why. Therefore,
the traxoline activity directly relates to instrumental understanding because
we know how to apply the rules but we do not necessarily understand the deeper
meaning behind what traxoline is. Skemp (1987) compares and contrasts how much
of an issue this is in our educational system today when he explains what
happens with a child when they learn instrumentally vs. relational. The chart
below examines the short and long term effects on a child when a teacher
teaches instrumentally vs. relational. A quote that stood out to me in the
article was “an argument against instrumental understanding is that it usually involved
a multiplicity of rules rather than fewer principles of more general
application.” (Skemp, 1987, p. 90). Teachers usually use instrumental
understanding for math because it is easier to allow the student to memorize
the rule, test them and move on. I believe I was affected by this way of
teaching growing up because I recall the focus of memorization for math and
biology but some topics I still do not understand today. Therefore, as a future
educator I do not want to make those same mistakes, I would like to develop
critical thinkers, problem solvers and application skills.
Focus on Relational Understanding. (2018). Retrieved from
https://buildingmathematicians.wordpress.com/2016/07/31/focus-on-relational-understanding/
The last
topic we focused on in class today was using manipulatives to learn and teach
math. We explored this topic using base ten blocks and algebra tiles. As a
future high school mathematics educator I believe algebra tiles are an
excellent manipulative to use for visual learners. I had some trouble in grade
9 myself subtracting polynomials as I would get the negative signs mixed up. I
also struggled a bit with factoring polynomial expressions. I believe if my
teacher used algebra tiles the entire class would have grasped the concept much
faster than teaching it just using one method. A useful resource we were given in class today
was http://www.edugains.ca/resources/LearningMaterials/ManipulativesSupport/TipSheets/Manipulatives_AlgebraTiles.pdf
, which I can use in my future classroom, it includes sample activities too. I
also found an interactive adding and subtracting polynomial game of battle
ship. This is also something I could use in my future classroom as it will get
the students engaged and practicing adding and subtracting polynomials in a fun
way (https://www.quia.com/ba/28820.html)
This game also has three levels of difficulties which allows the students to
practice at their own pace. I also
learned in class today the drastic difference of hands on learning for math. I
have never experienced hands on learning in my previous math classes at a high
school level and it really promoted a deeper understanding of basic concepts.
For example, we were adding and subtracting polynomials, multiplying
polynomials and factoring polynomials. With pen and paper it is harder to
visualize the process and how it works, however with algebra tiles I believe I have
a better grasp on how factoring works and can use this in my future classroom. Overall
today’s class stressed the importance of understanding the meaning behind basic
math concepts which will directly correlate to how you teach the subject in the
future. Therefore, us as teachers need to shift the way we teach math for all
students to understand and to use different methods for different learners.
Remember,
Aftermath. (2016). Retrived from https://ppld.org/aftermath
References
Skemp, R. (1987). The Psychology of Learning Mathematics.
Hillsdale, NJ: Lawrence Erlbaum Associates
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