Carole Greenes, Associate Vice Provost for STEM Education, Professor of Mathematics Education, and Director of the PRIME Center, ASU (firstname.lastname@example.org)
Mary Cavanagh, Executive Director of the PRIME Center (email@example.com)
Carol Findell, Visiting Scholar at ASU and Professor of Mathematics Education, Boston University (firstname.lastname@example.org and email@example.com)
This algebra module has been designed to introduce Pre-kindergarten students to basic concepts of algebra and to enhance their problem solving skills. Each of the five sections of problems focuses on a key algebraic thinking strategy. Within each section, problems are sequenced by difficulty. For each problem, there is a set of questions to help students focus on important features of the problem in order to solve it.
This module may be used to complement the existing instructional program. It is particularly useful to reinforce concepts of number (counting, quantifying and comparing) and measurement (weight comparisons), as well as to develop problem solving skills.
The Key Algebraic Concepts
Equality/Inequality: In this module, students explore equal and unequal relationships by interpreting and reasoning about pictures of teeter totters with animals, one at each end of the teeter totter. Their job is to determine which animal is heavier, lighter, or if the two animals weigh the same.
Proportional Reasoning: A major reasoning method for solving algebraic problems is by reasoning proportionally. Proportional reasoning is sometimes called “multiplicative reasoning,” because it requires application of multiplication. In this module, students are introduced to proportional relationships when they “fill” bags with shapes to look like a filled bag, and then count to figure out the total number of each type of shape.
Interpret Representations: Mathematical relationships can be displayed in a variety of ways including with words, tables, graphs, diagrams and symbols. Having students interpret these types of displays and use the data in the displays to solve problems is critical to success with the study of algebra. In this module, students interpret pan balances, pictographs, drawings of bags of blocks, trains with cars of different colors, and pictures of pumpkins, people and other objects.
Reason Deductively: Students compare pictures to clues in order to figure out the one picture that fits all of the clues. They then describe the matching picture. For all problems, three pictures are presented.
Reason Inductively: Presented with trains of cars of different colors in repeating patterns, students first identify the colors of the cars and their sequence in the repetition. Second they identify colors of missing cars, either at the end of the train (“What comes next?”) or somewhere before the end of the train (“What color is missing?”). Lastly, students color the cars to complete the pattern. Note that for each pattern, three repeats are given to establish the pattern.
The Problem Solving Five-Step Model
The model that we recommend to help students in all grades move through the solution problems has five steps. At the Pre-Kindergarten level, the focus is on Describe, Solve and Check.
Describe focuses students’ attention on the information in the problem display. In some cases, the display is a diagram. Other times it is a pictograph or model. Having students tell what they see will help them interpret the problem and figure out what to do to solve the problem.
My Job helps students focus on the task by having them tell what they have to do, that is, rephrase the problem in their own words.
Plan requires identification of the steps to follow to solve the problem and helps students focus on the first step. Knowing where to start is often the most difficult part of the solution process.
Solve is putting the plan to work and showing the steps.
Check is used to verify the answer.
We recommend that you “model” these steps in your instruction and that you encourage your students to follow the steps when solving the problems in this module and when relating their solution processes to others.