This week’s class began by answering some misconceptions about last week’s skyscraper activity. This activity was being confused as having an open-ended question style instead of the exploratory style it was. In order to show the difference, the class was put in groups and each person was given a random word or number. The goal was to create a sentence describing our group by using each word or number at least once. This was a great example to help clarify what constitutes an open ended question. From this activity I learned that the opportunities were endless. No two groups came up with the same answer. By using open-ended questions, students are given completely free reign to create whatever they desire. This differs from the skyscraper activity last week. There was a certain way to complete the activity; however, we were not given those instructions. The goal of this activity was to instead foster an exploratory or inquiry-based learning environment. Although this was only a small part of our class, it did help clarify two very different yet equally important teaching styles that I hope to use in my future classrooms.
For
most of the remaining class time, we discussed the readings assigned:
Relational Understanding and Instrumental Understanding by Richard Skemp (2006)
and An Alternative Reconceptualization of Procedural and Conceptual Knowledge
by Arthur Baroody, et al. (2007). In
groups, we discussed what we knew, what we learned, misconceptions, and
remaining questions (I learned a lot from these articles, which I will address
later on in this post). Lastly, in the
few remaining minutes at the end of the class, we were shown a few examples of
different manipulatives that could be used in high school classrooms and were
given a chance to explore them.
What did you learn/notice this week and how will it be useful to you?
The most significant ideas that I can take away from this week are the ideas from Skemp (2006) and Baroody, et al.’s (2007) articles about procedural and conceptual knowledge and understanding. Baroody (2007) defines procedural understanding as “knowledge of the procedure” and conceptual understanding as “knowledge of concepts and principles.” To put it even simpler, procedural understanding is knowing how to properly do the steps, whereas conceptual understanding is understanding why we do these steps.
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Growing up, I
never really knew that there were two different types of understanding. To me, understanding meant that I knew the
steps and got the correct answer. So looking
back, procedural understanding was the most important. As I began university and began taking
education courses, I learned about the importance of conceptual
understanding. It started to become
drilled in my head that conceptual understanding is the most important thing
that students must take from their education.
However, this contradicted the way that I had always been taught. I had always felt uneasy, because I still saw
the importance of procedural knowledge and understanding, however, now I was
supposed to focus more on conceptual understanding? These readings were like the light at the end
of the tunnel for me. Skemp (2006) and
Baroody (2007) taught me that although these are different concepts, they need
to work together in order for a student to learn mathematics. One is not better
than the other, nor should one be preferred over the other. By working in tandem, students will gain the
most from their education. By teaching
both procedural and conceptual understanding, not only will my future students
be able to use the proper steps to solve a problem, but they will be able to
look at their solution and answer the question, does this make sense?
Are
there any questions that you still have?
Although we did discuss it this week, I
do wish that we spent more exploring and using manipulatives. Thinking back to my elementary and high
school experiences, I cannot recall ever being given the opportunity to use
manipulatives in my math classrooms.
Therefore, I have very little knowledge and experience on when to use
manipulatives or even how to use them properly.
This was evident when we were given time to explore some manipulatives –
I had no idea how to start some activities.
I knew how to answer the question with a pencil and paper, but I had no
idea how to even attempt to answer it using the manipulatives. This is unfortunate because I believe that
manipulatives can be such a great tool for students to not only solidify their
procedural understanding, but also their conceptual understanding. It almost scares me that although I badly
want to have these resources available for my students to use, I have almost no
idea how to use them. For this reason, I
wish I was provided with more examples of manipulatives and the proper way to
use them in a classroom.
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Hi there,
ReplyDeleteLike yourself I had little experience using manipulatives during high school. I wish I had more experience because, as a visual learner, I would have benefited from using manipulatives. With such little experience I too hope that throughout our course we are provided with more activities and examples of how to use manipulatives in the mathematics classroom.
Melissa Broderick
Hi Heather,
ReplyDeleteI have also had changing opinions on whether conceptual or procedural knowledge is more important while going through school. Growing up I never really questioned why we used certain formulas for certain problems or what the answers really even meant; I was just taught that way so that is what I did. Now after completing my undergrad in mathematics I see the value in the understanding behind the computations, and this knowledge is just as important, if not even more important, than the procedural knowledge. I think as teachers we should try our best to teach both the conceptual and procedural knowledge to our students to help them develop the best understand of math that they can.
Karlee
Your post got me thinking; should schools with low ses have less funding and therefore cant afford as many/as nice manipulatives as other schools? Or maybe schools in low ses get more funding because the learning gap is evident so they have more or better manipulatives. I also agree that I used very few manipulatives in my education and thus it is difficult to understand how and when to best use them. I found it easy to google or pinterest highschool math manipulatives to get ideas. for example at the following site: http://teachingcommons.cdl.edu/noyce/newsevents/documents/GREISY_noyce_presentation_jan_09.pdf. That being said we have to be critical of the resources found online to ensure they aremaking full use of the manipulatives and it is teaching the topic we are teaching. We do not simply want to put manipulatives in students hands to say we used manipulatives but rather the manipulatives must have a purpose.
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