Using Logo to Teach a Unit of Geometry to Adults
Logo, an educational programming language developed in the 1960s, offers powerful opportunities for teaching geometric concepts to adult learners. This approach combines mathematical reasoning with programming skills, creating an engaging learning environment that helps adult students visualize abstract geometric concepts. The following unit demonstrates how Logo can be effectively used to teach fundamental and advanced geometric principles.
Why Use Logo for Teaching Geometry to Adults?
Adult learners often bring unique challenges and strengths to mathematics education. Many adults who return to learning geometry may have had negative experiences with mathematics in the past or may struggle with abstract concepts without concrete visualization. Logo addresses these concerns through several key advantages:
- Immediate visual feedback: As learners type commands, they see the turtle trace shapes on screen, connecting abstract commands to concrete geometric results.
- Active learning: Adults are actively engaged in creating geometric figures rather than passively receiving information.
- Logical thinking development: Programming in Logo develops logical reasoning skills essential for understanding geometric proofs and relationships.
- Error analysis: Mistakes become learning opportunities as students debug their geometric constructions.
- Self-paced progression: Logo activities can be adapted to different skill levels, allowing adults to progress at comfortable rates.
Getting Started with Logo
Before diving into geometric concepts, adult learners need a foundational understanding of Logo commands. Initial sessions should focus on mastering basic commands that control the turtle's movement:
BK 30 (Backward 30 units)
RT 90 (Right turn 90 degrees)
LT 45 (Left turn 45 degrees)
PU (Pen up)
PD (Pen down)
REPEAT 4 [FD 100 RT 90] (Repeat commands 4 times)
Introductory Activity: Turtle Control
Have students experiment with making the turtle draw simple shapes like squares, rectangles, and triangles. Ask them to predict what a set of commands will draw before executing them, then adjust commands to correct any errors. This builds foundational skills while establishing that precise language leads to precise geometric results.
Teaching Geometric Concepts with Logo
1. Angles and Turns
One of the strongest connections between Logo and geometry is in the realm of angles. Adult learners often struggle with angle concepts, particularly interior and exterior angles. Logo makes these distinctions concrete through turtle movements:
Activity: Polygon Angles
Ask students to create procedures for different regular polygons. For example:
REPEAT 4 [FD 100 RT 90]
END
Then explore how the turn angle relates to the polygon's interior angles. Challenge students to derive the formula: Turn angle = 360/n, where n is the number of sides.
2. Properties of Triangles
Triangles are fundamental to geometry, and Logo offers excellent opportunities to explore their properties:
Activity: Triangle Exploration
Have students create procedures for different types of triangles:
REPEAT 3 [FD 100 RT 120]
END
TO RIGHTTRI
RT 90 FD 100
RT 126.87 FD 80
RT 143.13 FD 60
END
Ask learners to calculate the angles they need, then verify through testing that their procedures produce the intended triangles.
3. Symmetry and Transformations
Logo excels at teaching symmetry concepts and transformations (translations, rotations, reflections):
Activity: Pattern Creation
Guide students in creating symmetrical patterns using Logo:
RT 45 FD 50
RT 90 FD 50
RT 90 FD 50
END
REPEAT 8 [DESIGN RT 45]
Discuss rotational symmetry and have students modify procedures to create patterns with different lines of symmetry.
4. Circles and Arcs
Since Logo turtles move in straight lines, circles and curves require approximation, leading to rich discussions about the nature of circles:
Activity: Circle Approximation
Have students create a procedure to draw a circle:
REPEAT 360 [FD 1 RT 1]
END
Discuss why this works and how changing the number of repetitions affects the smoothness of the circle. This introduces concepts of pi, radians, and limits in an accessible way.
5. Area and Perimeter
Logo helps visualize relationships between perimeters and areas:
Activity: Perimeter vs. Area
Have students draw rectangles with the same perimeter but different shapes, then calculate and compare their areas:
REPEAT 2 [FD :width RT 90 FD :length RT 90]
END
Challenge students to find which shape maximizes area for a given perimeter, leading to the discovery that squares optimize this relationship.
Sample Lesson Plan: Exploring Polygons
| Time | Activity | Purpose |
|---|---|---|
| 10 min | Review basic Logo commands | Reinforce foundational skills |
| 15 min | Create a procedure for a square | Practice defining procedures |
| 20 min | Derive relationship between sides and turn angles | Develop mathematical reasoning |
| 25 min | Create procedures for various regular polygons | Apply understanding to new cases |
| 15 min | Calculate interior vs. exterior angles | Connect Logo to geometric terminology |
| 15 min | Challenge: Create a procedure that takes number of sides as input | Apply mathematical formula (360/n) in programming context |
Assessment Strategies
Assessing adult learners' geometric understanding through Logo can be accomplished through several approaches:
- Procedure creation: Ask students to write Logo procedures that demonstrate specific geometric properties.
- Reverse engineering: Provide a Logo program and ask students to identify and explain the geometric concepts it demonstrates.
- Debugging: Present Logo code with geometric errors and ask students to identify and correct them.
- Mathematical justification: Require students to explain why a particular set of commands produces a specific geometric figure.
- Creative applications: Assess understanding through projects that apply geometric concepts in creative ways.
Meeting the Needs of Adult Learners
Teaching adults differs from teaching children in several important ways that should inform how Logo is used in the geometry curriculum:
- Connect to practical applications: Show how geometric concepts are used in professional fields such as architecture, engineering, design, and construction.
- Better incorporate prior knowledge: Build on adults' existing understanding while correcting misconceptions in a non-threatening manner.
- Provide choice and autonomy: Allow adult learners some choice in which geometric concepts to explore in greater depth.
- Make time efficiency clear: Explain how learning geometry through Logo builds transferable skills in logical thinking and problem-solving.
- Create collaborative opportunities: Many adults learn well through peer interaction and discussion around mathematical concepts.
Implementation Tips
To successfully implement Logo-based geometry instruction for adults, consider the following recommendations:
- Start with clear, achievable tasks that build confidence early in the learning process.
- Use physical turtle manipulatives for students who benefit from tactile learning before moving to screen-based versions.
- Connect Logo activities to students' personal or professional interests where possible.
- Encourage students to verbalize their thinking as they write Logo procedures.
- Prioritize conceptual understanding over memorization of commands by focusing on why specific approaches work.
- Scaffold complex tasks by breaking them into smaller, manageable steps.
Conclusion
Using Logo to teach geometry to adult learners provides a dynamic, interactive approach that makes abstract concepts tangible and engaging. Through immediate visual feedback, hands-on experimentation, and logical problem-solving, adults can develop both geometric understanding and computational thinking skills simultaneously. This method respects adult learners' desire for relevance, autonomy, and practical application while addressing the common challenges many face with mathematical abstraction.
By carefully structuring Logo-based geometry units that build foundational skills before exploring increasingly complex concepts, educators can create transformative learning experiences that help adults overcome past mathematics frustrations and develop genuine confidence and competence in geometry.
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