Reflection 2: Constructivist Approach to Learning
Throughout my studies and time spent in the classroom, I have implemented lessons and learning experiences that have allowed my students to develop schemas on their own. By scaffolding learning, I structured lessons in such a way that encourages students to continually build upon what they have already learned.
Many teachers plan lessons assuming that all of their students have the prior knowledge necessary to be successful during a unit. We know that all of our students have varying abilities; however, we continually teach new concepts without determining if all students have the necessary skills or schema to do so. When utilizing the constructivist framework, teachers must determine the experiences and cognitive structures of their students to ensure they are willing and able to learn (Bruner, 1996). The students’ readiness is detrimental to their potential for constructing new meanings.
As I went into student teaching, I did not know the skill set or knowledge base that my students had. Therefore, I had them take a pretest to determine what skills they were familiar with, and what areas they struggled in. The results from this test allowed me to structure my lessons in a way that clarified topics that they seemed to struggle with, and capitalize on their strengths throughout my lessons. For example, when creating my lesson on rational expressions, I looked at the pretests to see how my students performed with fractions. Most of them were successful in factoring and reducing fractions, however they struggled with finding common denominators. Therefore, before teaching rational expressions, a topic that required students to manipulate fractions that had algebraic expressions in them, I reviewed the basic rules of adding and subtracting fractions. Doing this review using fundamental, basic examples, provided my students with the prior knowledge necessary for them to construct meaning from the more complex examples we were to complete in upcoming lessons.
When reflecting on my use of the pretest, I can appreciate the insight it gave me into the students’ schema and cognitive structures. It allowed me to determine their readiness for lessons that I later taught them. If I were to do it again, I would give them a pretest before each unit that tested skills that are more specific to that particular unit, rather than one pretest that gave me insight to general skills my students would be using throughout my entire student teaching placement.
In every formal lesson plan I’ve written, I include a prior knowledge section in which I include objectives that my students must have mastered if they are going to be successful during this new lesson. Because I have considered this prior to entering the classroom, I informally test students on these prior knowledge requirements during the introduction part of the lesson. For example, when working at Temple Hill with a student named Lilly, I had to teach her how to compare and contrast two written passages similar to those that would be on the Regents Exam. Before developing a lesson with her, I determined that being able to compare and contrast specific items would be a prerequisite. If she was going to be able to compare and contrast whole narratives, then she would need to be able to do this on a smaller scale. Therefore, after explaining to her what I would be teaching her to do, I first asked her to compare and contrast Marc and Bobby, two of her fellow classmates. I observed Lilly think intently about the characteristics of the two boys then express confusion about how to express her thoughts on paper. I then asked her if she could tell me what compare and contrast meant, to which she responded correctly. Therefore, I could see that she knew what she needed to do, just was unsure how to represent it on a piece of paper. So I drew a Venn diagram, labeled it appropriately, and explained to her that she would put their similarities in the middle, and their differences on the outer parts of the circle. Once she successfully compared and contrasted her two classmates, I was now confident that she had the prior knowledge necessary to compare and contrast two stories.
If I could go back to this lesson with Lilly, I would have scaffolded her learning a bit more. Comparing and contrasting her two classmates was very simplistic compared to completing two narratives. Before jumping straight to two stories, I might have asked her to complete another compare/contrast task with a moderate degree of difficulty, such as comparing and contrasting two concepts such as friends and family. This would have given me the opportunity to see how well she performed a slightly harder task, before jumping to a difficult one that requires her to remember characteristics and make connections between two different short stories.
When implementing the constructivist framework it is essential that teachers scaffold learning in such a way that students can easily grasp new ideas (Bruner, 1996). If students are going to construct new ideas and concepts, it needs to be organized so they can make connections to prior knowledge and build upon it. Not all students naturally make mental connections; therefore the teacher must adequately organize learning, allow for independent discovery, and provide assistance when necessary.
While at my special education student teaching placement, I worked with a self-contained 7th grade special math class. These students exhibited a need for concrete manipulatives and pictures to help them to construct meaning from more abstract concepts. Therefore, I used boxes and circles to help them visualize fractions. First we would represent all fractions as a box divided into parts and the necessary parts shaded in to represent each example. As students mastered this, I would begin to ask them to add another fraction to it. I encouraged them to work together, try different representations of the fractions (ex: ½ vs. 2/4). As I walked around the classroom, I could see some students struggling at times, while others picked it up rather easily, however those who struggled were motivated to find the correct answer. One student in particular, Matt, knew that 1/4 + 1/2 would be 3/4; however he struggled with the drawing and shading. I offered him subtle hints without giving away the answer, and finally the light bulb went off. He managed to construct meaning from breaking up both boxes into the equal amount of pieces before determining the answer. Once Matt and his classmates figured this out, they were better able to find common denominators when working with the numerical representations of the fractions.
When reflecting on this practice, I can see that using the pictures helped my students construct their own meanings and understanding to the concept of fractions. Through independent thinking and my strategic guidance, Matt and his classmates successfully transferred their knowledge from the concrete, visual representation, to the symbolic and abstract concept of fractions.
Completing problem solving activities is another way to get students to construct meaning on their own. A problem I created, but have not been able to use, is the Check Mate problem. While this appears to be a straight forward probability task, it requires a lot of critical thinking skills and consideration on the students’ part. When I have a class of my own, I plan on using this to help my students practice their problem solving skills and discussions. Although it is a challenging problem, tasks such as this can be highly motivating once they are completed. The nontraditional aspects of this activity will allow students to use prerequisite knowledge in statistics, as well as incorporating the rigor that the state curriculum is enforcing.
Another lesson I created that allowed students to construct their own understandings was a discover a relationship lesson plan that I created for a 7th grade class. Unfortunately I did not get the chance to implement this lesson in a classroom; however, I did develop an inquiry based instructional strategy. Through observation and discovery the students would determine that the smallest angle of a triangle would be opposite to the smallest side and the same goes for the largest angle and side of the triangle as well. If I were given the opportunity to use this in a classroom I would give my students the freedom to come up with the angle/side relationship on their own. I would remain available if they have any questions; however I would allow them to construct their own hypotheses and made decisions on their own.
While at Cornwall Middle School for my special education student teaching placement, I carried around a bag of beans to help students who struggled with multiples and factors. Some of my 6th-8th grade students had trouble doing mental multiplication so I would allow them to count out beans and group them appropriately in order to solve problems. In my 6th grade class for example, my three students and I worked from a small round table each class. I left the beans in the middle of the table and as students needed them, they would take a handful of beans, and independently make groups to solve multiplication or division problems. Because the students had the freedom to do this on their own, and did not receive any guidance from me, they were able to transform their understanding from a concrete to a more abstract format using the beans.
When reflecting on this technique I am happy that I chose to make these tools available to my students. I could see that my students had difficultly bridging the gap between concrete and abstract concepts. Therefore, rather than me telling them the answers every time, I provided them with tools necessary to construct meaning for themselves. Whenever multiplication was used, my students would think to themselves, “Can I do this on my own?” If not, they would reach for the beans and create the groups and count them out to solve the problem.
Another activity I did with this group of students was a lesson on finding areas of irregular polygons. Rather than breaking up the unfamiliar figures into noticeable ones for my students, I encouraged them to figure out a strategy for doing this on their own. I asked the three girls “is this a shape you have a name or a formula for?” the answer to both was no. Then I asked “does it kind of look like a shape you know? Or does a piece of it look familiar?” These two questions got the girls thinking about breaking up this irregular shape into pieces. They then worked together and discussed strategies for finding the area. One girl Stephanie said “this piece here looks like a triangle” so they all shaded the triangle piece to let it stand out in their mind. Then Sam suggested that another piece looked like a rectangle. Through their conversation, and manipulation of the shapes, the girls were able to break them up into shapes they had area formulas for. Finally I asked them “well if we have the areas of all these pieces, how can we find the area of the whole thing?” After some consideration, the girls came up with the idea to add the formulas together to get the area of the whole shape.
Thinking back on this lesson, I can see how my questions scaffolded the thinking of my students. I allowed them to come up with hypotheses and strategies on their own, but my strategic questions guided their thinking in the right direction without giving them the procedure myself. In addition, because they came up with this strategy on their own, they easily recalled the procedure throughout the remainder of the unit and test. Their learning was evident through their completion of the task sheet and performance on the quiz offered later in the unit.
Another activity that I created and completed with this class was A Room Full of Candy. When developing the plan for this lesson, my goal was to require students to work on their problem solving skills. Complete the task without hints from me. As they completed the task, the girls couldn’t figure out the connection between finding the volume of the candy and the area of the room. They shared strategies and ideas with one another, tried various plans, but couldn’t get the hint. I then had to make suggestions regarding finding the measurements of the room, and the relationship it would have to the volume of the candy. Once the students understood what they needed to do, they effectively took measurements, and calculated their numbers. They were highly motivated to solve this problem, and shared their results with peers and teachers. I recall hearing Stephanie telling a student in the hallway “Guess what! It would take over 300,000 pieces of starbursts to fill our math classroom!” This lesson proved to be exciting, productive, and memorable for the students.
I ended up modifying the worksheet for this project. I do not have the original copy that I used with these girls because I immediately changed it after the project to include more structure. The original task sheet only had the directions and basic expectations for the assignment. As you can see, I’ve included the table and procedure to provide the students with more guidance when doing this project. This would allow me to walk around the classroom to observe behavior and discussions rather than answering questions for each group.
The final aspect of the constructivist frame work allows for students to extrapolate the information they’ve acquired in order to go beyond the information given to them (Bruner, 1986, 1990, 1996). After students have built upon their prior knowledge to understand a new concept, they should be given the opportunity to take that knowledge further by extrapolating more information.
While I was the president of Mount Saint Mary’s Math Club and Kappa Mu Epsilon honor society I organized a math day event at the Newburgh High School north and south campuses. Most of the students stopped at my table to check out the activity because it was on Facebook, a website they are very familiar with. As the students worked on problems that were not obviously related to mathematics, I heard some of them saying things like “hey can’t we figure this out by using that cross multiplication thing we do in math class?” Without my help, students worked together to solve authentic problems using math skills they had already acquired. Without even them knowing, they were extrapolating information using prerequisite skills.
When reflecting on this activity, I wish I was able to do it in a classroom setting, rather than a science fair type of setting so I could have students complete various activities and share what they completed with their classmates. It was remarkable to see them using math skills that they previously gave little value to, and apply it to an area of interest to them. It was clear that they were completing problem solving skills and higher order thinking to enhance their math skills while having fun. The best part is they completed these tasks with little or no help from a teacher. They were able to propose hypotheses and develop strategies on their own in order to construct meaning from the activity.
Another activity I created where students had to extrapolate information was in a Twitter group project. I did not have the opportunity to use this in a classroom, but I did create the worksheet and lesson plan for it. If I had the opportunity to use this in a classroom I would circulate the classroom to ensure the students had a clear understanding of the prior knowledge necessary for this activity, clarify any issues they may have, and pose questions that would appropriately scaffold their learning. It is essential that students have a firm understanding of the concepts if they are going to use constructed knowledge to extrapolate additional information.
When reflecting on my teaching, I can see that I have utilized all aspects of the constructivist approach to learning. From determining the readiness of my students, to scaffolding learning, and allowing students to go beyond and extrapolate information, I have allowed my students to be active in their learning process and develop their own understanding of concepts based on prior knowledge (Bruner, 1996). In the future I plan to continue to use this framework when it is applicable to my students because there is great value in giving them the independence to transform information, create hypotheses, and make decisions that enhance their own learning.
Appendix
Standard 2- Knowledge of Mathematics |
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Aspect |
Explanation |
Ways I Met the Standard |
Contexts for Mathematics |
Accomplished math teachers understand the connections between concrete math techniques and abstract concepts. |
- I had my students take a pretest to determine what skills they were familiar with, and what areas they struggled in. The results from this test allowed me to structure my lessons in a way that clarified topics that they seemed to struggle with, and capitalize on their strengths throughout my lessons. - Some of my 6th-8th grade students had trouble doing mental multiplication so I would allow them to count out beans and group them appropriately in order to solve problems. |
Core Mathematical Knowledge |
Accomplished math teachers understand the various domains of mathematics and ways in which to get students to develop understanding in all of these areas. |
- I allowed my students to use boxes and circles to help them visualize fractions. First we would represent all fractions as a box divided into parts and the necessary parts shaded in to represent each example. -The Facebook Activity used at NFA Math Day allowed students to use their math skills in a nontraditional, yet authentic and stimulating scenario. Because Facebook is an interest of theirs, they completed math problems and challenged one another without the laborious struggle that many experience during math. |
Standard 4- Knowledge of the Practice of Teaching |
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Aspect |
Explanation |
Ways I met the Standard |
Knowledge of Pedagogy Influences Instructional Strategies |
Accomplished math teachers use their knowledge of pedagogy, math, and student development to make informed, appropriate instructional strategies. |
- Determining the prerequisite skills in the comparing and contrasting lesson plan required deep understanding of the content and thought process of the students. By ensuring the student had mastered prerequisite skills, I ensured effective learning throughout the Power Lesson. -By knowing what questions to ask and hints to provide, I was able to allow students to construct meaning on their own for the discovering a relationship lesson plan. -Because social media is of interest to this generation, the Twitter Activity was an appropriate lesson that motivated students to learn, as well as challenge themselves to examine trends and extrapolate information based on twitter activity.
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Standard 5- Learning Environment |
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Aspect |
Explanation |
Ways I Met the Standard |
Classrooms are a Save, Productive, and Motivational Environment |
Accomplished math teachers create classroom environments in which students are actively learning, inclined to take risks, and develop self-confidence. |
-The Check Mate activity will test the students’ patience and self-efficacy in math. By motivating the students to continue working, and reminding them of their substantial knowledge of probability, I can maintain a safe, nonthreatening environment in which my students will take risks. -While trying to determine the volume of irregular polygons, the students’ discussions clearly indicated their guess and check procedure and determination to solve the problem despite various challenges. -During the Room Full of Candy Activity the students struggled with what they needed to do. However, they felt safe enough to take risks and try methods without giving up. They shared strategies and ideas with one another and motivated each other after a couple failed attempts. |
Bruner, J. S. (1996). The culture of education. Cambridge, Mass.: Harvard University Press.