Mathematical Expressions Have Structure

Studies done in the 1980s discovered that a significant number of students did not realize mathematical expressions have an intrinsic structure. One of these studies determined that an effective way to explicitly teach mathematical expression structure was with expression trees (Thompson 1987).

This application is designed to help people learn about the structure that is associated with mathematical expressions. It does this by using expression trees and "LookSee" diagrams to show this structure in a visual way.

Modes Box

  • Explore

    This mode is for exploring various parts of an expression and its tree. Left clicking on a symbol in the expression tree selects it and also selects its match in the linear expression. Left clicking on a symbol in the linear expression selects it and also selects its match in the expression tree.

    "Explore" Box

    • "LookSee" If selected, a LookSee diagram is shown beneath the linear form of the expression.

    • "Clear Select" Clears any nodes that may be selected.

  • Evaluate

    Enables an expression to be evaluated step-by-step. Each step (before it is taken) is underlined in the linear form of the expression, and highlighted in the expression's tree.

    "Evaluate" Box

    • "Mode" The PEMDAS evaluation mode evaluates the expression tree using PEMDAS. The "Post-Order" mode evaluates the expression tree using post-order traversal.

    • "Highlight" If the "Highlight" checkbox is selected, the part of the expression tree to be evaluated next is highlighted. A LookSee diagram shows the part of the linear expression that matches the selected part of the tree. A short delay is added to the evaluation time if "Highlight" is selected to make it easier to see the selected part of the tree. The "Evaluate" button performs the next evaluation.

    • "Layout" If the "Layout" checkbox is selected, the tree will be laid out after each evaluation.

  • Find In Tree

    A random symbol is highlighted in the linear expression, and the goal is to select the same symbol in the expression tree.

    "Symbol Matching" Box

    • "Check" Indicates if the correct matching symbol is selected.

  • Find In Expression

    A random symbol is highlighed in the expression tree, and the goal is to select the same symbol in the linear expression.

    "Symbol Matching" Box

    • "Check" Indicates if the correct matching symbol is selected

Random Expression Generator Box

  • Operator Probabilities

    The operators that can be randomly included in an expression are listed in this box. Each operator's operands are indicated by "_" characters. The pulldown menu that is to the right of each operator is used to select the probability that the operator will be selected each time an operator is randomly chosen. The sum of the probability values for all of the operators must equal 1.

  • "Clear" And "Default" Buttons

    The "Clear" button sets all of the operator probabilities to zero. This sometimes makes it easier to set probabilities to desired values. The "Default Probabilities" button resets all operator probabilities to the values that were displayed when the application was first launched.

  • "Generate Random Expression" Button

    Generates a random mathematical expression using the options that are seleted in the "Random Expression Generator" box. The generated expression is placed into the text field that is in the "Expression" box.

Abbreviations Box

  • Negative Numbers

    When selected, the "Negative Numbers" checkbox shows negative numbers in abbreviated form. This means that -3 is shown as a single symbol instead of as a negation operator with an operand of 3.

Expression Box

  • Text Field

    Mathematical expressions can be entered into this text field either by typing them in or by pressing the "Generate Random Expression" button.

  • "Display" Button

    Displays the mathematical expression that is in the "Expression" box's text field using the exercise that is selected in the "Exercise Options" box. Each time a new expression is entered into the text field, or an option has been changed in the "Exercises" box, the "Display" button needs to be pressed to display the result of these changes.

    Zoom Box

  • "-" and "+" Buttons

    Decreases and increases the size of the linear expression and the expression tree.

Linear Expression and Expression Tree

When an expression is displayed, it is shown both in linear form and graphically in tree form. The linear form is placed above the tree form.

LookSee Diagrams

All operators (such as + and *) have one or more subexpressions that they operate on. These subexpressions are called "operands". The lines that are drawn beneath a given operator in an expression tree show which operands belong to this operator. An operand can be a single symbol (for example: "3" or "a"), or it can be a subtree.

If one imagines an operator "looking" at one of its operands, it only "sees" the topmost symbol in the operand, not the operand's whole subexpression. A "LookSee" diagram is used with a mathematical expression in linear form to highlight an operator's operands. The whole operand is underlined, and the part of the operand the operator can "see" is pointed to by an arrow.

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