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9.1M Ability to support and reinforce the instruction of
students in math following written and oral lesson plans developed by
Strategy: Using a Constructivist Approach-Guided Discovery
Some math courses are taught from a constructivist approach using the guided discovery method of teaching. This is a rare form of instruction due to the curricular challenge of designing a course with minimal teacher intervention. The lessons are constructed such that the students do activities in which, on conclusion, they have synthesized the goal concept from their prior knowledge. This is consistent with many cognitive theories in which state that knowledge is constructed in the mind by relating new stimuli to existing knowledge.
On completion of this strategy, the paraprofessional will be able to assist an instructor in teaching mathematics from a constructivist perspective.
The fact that students need to abstract and synthesize information to succeed in this type of class eliminates younger students from participating. Also, the self-discipline required to achieve the goals set suggests that an average to below average student would have problems completing the task as well. Younger students can be trained to participate in laissez-faire pedagogies, often extremely young students will be taught with a discovery method that includes no guidance. The guided-discovery technique can be taught in an individual work setting, but is mostly used collaboratively. This is due to the advantages that group discourse can provide to the discovery process. It is also characteristic of this teaching method for the instructor to encounter frustration from the students concerning their quest for this goal. This frustration is caused by students trying to overcome the confusion of understanding a new cognitive process. Facing and overcoming this confusion is a necessary process for the student to accomplish learning in guided-discovery.
The guided-discovery approach to teaching mathematics is known for positive results in research. The notion that students take ownership of their learning increases motivation (Middleton & Spanias, 1999). Retention and understanding also are known to increase when this learning style is used in the math classroom. The procedure is student-centered, which means the teacher has the difficult task of keeping a low profile in the classroom. Also, when teaching in this style, one must build on the knowledge that the student already has and help direct the student to the desired outcome through improvisations on the lesson plan. Jones & Tanner (2002) state that “underpinning these factors appears to be the teacher’s ability to anticipate the possible responses and errors that might arise, and their confidence to ‘go with the pupils,’ whilst still steering the lesson to achieve its objectives (p. 273).” Humphreys & Hyland (2002) also note that “the teacher must creatively and imaginatively improvise in the face of unexpected events: late students, alienated students, failed experiments, awkward questions, strange answers, and different levels of understanding within one group” (p. 515).
The lesson is designed for the students to discover a specific concept with minimal teacher intervention. The students virtually go through the same cognitive process that the mathematician who was first to discover the concept went through. The job of the instructor and the paraprofessional is to make sure that the groups of students are going through the proper progressions to find this discovery. The curriculum is designed for the students to make these discoveries without assistance, but at times, the student may stray from a predetermined path. This is where the teacher and paraprofessional must intervene and guide the students through the proper progressions.
In an 8th-grade math course, students are studying the mathematical concept of summations and series. In this introductory lesson, the teacher, Mr. Jenkins, who is a strong advocate of constructivism, has devised a lesson in which the students will discover a formula for solving the sum of a finite arithmetic sequence. The lesson starts by presenting the series 1 + 2 + 3 + … + 98 + 99 + 100. The students are given time to play around with this summation in their groups. The instructor and the paraprofessional, Miss Bishop, give the groups some time to explore, then check the means each group is using to find a sum. Some go directly to solving this with their calculators, some do a long-hand calculation, and others add the smallest term to the largest term, the second smallest to the second largest, and so on, to make an equivalent product of 50 x 101. The groups present their solutions and their procedures used. This concludes the first phase of this instruction.
The second phase begins with a related problem. The problem asks the students to come up with a formula to solve a series that does not necessarily have an even amount of terms. Mr. Jenkins explains that there is a formula that can be done to solve these types of problems without calculators, cumbersome calculations, or restrictions to an even number of terms. These are all the faults of the first methods that the groups tried when solving the original problem.
He recommends that the students set a series equal to a variable and find a way to solve for that variable. The groups begin working and after a while Mr. Jenkins and Miss Bishop begin checking their progress.
One group is analyzing the triangle numbers and its position in Pascal’s Triangle. Miss Bishop recommends that they stick with algebraic manipulation. Another group has multiplied each side by the isolated variable creating a quadratic. Miss Bishop compliments the group on their ingenuity, but then lets them know that they are making the problem more complicated.
Finally, the third group Miss Bishop observes has taken the equation 1 + 2 + 3 + 4 + 5 = x and added the equivalent equation 5 + 4 + 3 + 2 + 1 = x to it. The result is 6 + 6 + 6 + 6 + 6 = 2x or 5 x 6 = 2x. Mr. Jenkins has the group demonstrate this process to the other groups.
They are then asked to generalize this procedure so it will produce a formula for the sum of the whole number up to n with the solution, x in this case, isolated in the formula. It does not take long before all the groups are producing the formula x = n(n + 1)/2.
Constructivism using guided discovery
Questions to consider when implementing this strategy:
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