Math+Lab+1

= Algebraic Concepts and Bruner's Stages of Representation =

//Bruner's Stages of Representation// //Enactive (concrete)//  ﻿involving action - touching, feeling and manipulation //Iconic (pictorial)// //depending on visuals, such as pictures to summarize// //Symbolic (abstract)// //using words or symbols to represent information//

[[image:cognitive_revolution.jpg width="180" height="212" align="left"]]HISTORY
Jerome Bruner received his Ph.D. in Psychology from Harvard in 1941 during a time when "the field of psychology...was roughly divided between the study of perception and the analysis of learning"( []). Bruner did not adhere to the current trends in psychology which leaned toward behaviorism and psychophysics; but rather took interest in the "New Look" theory on perception. The "New Look" theory stated that perception was not something that occurred instantaneously, but rather through a process of information acquisition and selection. Jerome Bruner, along with other psychologists such as George Miller, Donald Broadbent and Noam Chomsky were essential players in the "Cognitive Revolution" in the late 1950's to 1960's. In 1960 Bruner and Miller opened the Center for Cognitive Studies at Harvard. Bruner's interest in cognitive psychology led to his involvement in the cognitive development of children in particular and ways in which this information could be used to develop appropriate methods of education.

APPLICATIONS TO TEACHING
To align with Bruner's stages of representation, it is crucial for teachers to remember that learning does not occur instantaneously through an immediate transference of information. Rather, learning is a process where the learner must receive, sort and mentally categorize and store information that he or she receives. Here are some methods in which to incorporate Bruner's Stages of Representation into a lesson: media type="youtube" key="2msVlhBtppo" height="315" width="300"media type="youtube" key="mxBoni8N70Y" height="315" width="300"
 * __Enactive__: Have students interact with manipulatives such as algebra tiles, base ten blocks, take measurements with a measuring tape, or counters
 * __Iconic__: Drawing pictures to represent the physical items in the enactive stage, graphing or charting data or presenting items in other visual formats such as video
 * __Symbolic__: Writing out information into words or symbols, whether it be writing equations or a paragraph summarizing the findings.

LESSON PLAN
__Establish Goals:__
 * Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (Common Core Standard 8SP-1)
 * Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (Common Core Standard 8SP-2)
 * Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (Common Core Standard 8SP-3)

__Stage 1 - Desired Results__
 * Students will understand how to plot their data on a scatter plot using the correct intervals in the axis.
 * Students will understand how to draw a line of best fit on a scatter plot.
 * Students will know how to calculate slope and y-intercept in order to determine the equation of their line of best fit.
 * Students will know how to apply what they have to learned to model other data such as comparing height with shoe size, length of the forearm with diameter of the head, etc.

__Stage 2 - Assessment Evidence__
 * Given a data set, students will be able to accurately plot the data onto a scatter plot.
 * Given a scatter plot, students will be able to draw a line of best fit
 * Given two points, students will be able to determine the slop and y-intercept to determine the equation of the line.

__Stage 3 - Learning Plan: Algebra - Fist size and Height__


 * SET UP REQUIREMENTS**
 * 1) Tape measurements in cm for each student.
 * 2) Graphing papers, 1 per student.
 * 3) Students are divided by pair.


 * DATA COLLECTION**
 * 1) Each pair of students uses tape measurement to collect data from each other.
 * 2) Measure distance across the knuckles when making a fist.
 * 3) Ask students what is the best way to measure across the knuckles? (from center of the first knuckle to center of the last, outer-side of first knuckle to last one, or inner-side of first knuckle to last) --> many ways to measure across the knuckles.
 * 4) Ask students which way is best to measure distance across knuckles? Does it matter? --> consistency in methodology is very important in collecting data.
 * 5) Emphasis on data consistency is crucial.
 * 6) Ask students to vote for common methodology in measuring the distance across the knuckles.
 * 7) Measure the height
 * 8) Use the same items 3 - 6 but use them for measuring the height.
 * 9) Start collecting the data among the students.


 * PLOT SCATTER-PLOT**
 * 1) Teacher collects all the data from students and tabulate it with students.
 * 2) With data table, teacher asks students for assigning X and Y terms, X min/max, and Y min/max for scaling.
 * 3) students to plot the data.


 * DATA REPRESENTATION**
 * 1) From the scatter plot, ask students to choose 2 plotted points to use for finding a line best-fit.
 * 2) Ask students to try out different plotted points to come up different best-fit lines.
 * 3) Ask students the meaning of best-fit line by looking at the graph.
 * 4) Ask students to vote for the best-fit line plotted points.
 * 5) Draw the best-fit line --> linear line.
 * 6) Ask students to define the linear line on the graph from the graph.
 * 7) Ask students to counting off the **slope** between the 2 chosen plotted points.
 * 8) Teacher to introduce **slope (m) equation** --> m = (y2-y1)/(x2-x1) = Rise/Run
 * 9) Ask students to compare results from (7) and (8) --> Are they the same?
 * 10) Ask students to extend the best-fit line until it intersects the y-axis.
 * 11) The point of intersection on y-axis is called **y-intercept**.
 * 12) Ask students to notice y-intercept coordinate as (0,b)
 * 13) Ask students to extend the other end of best-fit line until it intersects the x-axis.
 * 14) It is called x-intercept.
 * 15) Ask students to note the x-intercept coordinate as (a, 0).


 * DETERMINE LINE EQUATION**
 * 1) Teacher to introduce the general form of linear equation as: **y = mx + b** with m=slope, b=y-intercept
 * 2) Ask students using m value and one of the plotted point, solve for b (y-intercept).
 * 3) Ask students if the new calculated b value is equal to y-intercept.
 * 4) Ask students to find the x-intercept.


 * CONCLUSION**
 * 1) Linear equation can be represented by: graph, table, and algebraic equation.
 * 2) Real life problem can be solved mathematically.

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Application in music: []

Bruner - Constructivism Theory in Education []

Constructivism: []