Is mathematics discovered or invented? Some people believe that mathematics is the language of the universe, and it’s just waiting to be uncovered. Others believe that mathematics is a human construct, created to help us understand the things we observe. Regardless of where you stand in this philosophical debate, one thing is certain: Mathematics is a beautiful field, rich with observable patterns, governed by logic, and blossoming with examples. With the right scaffolding, our students can discover mathematical truths on their own.
Discovering math has a few components:
- Students explore examples and make structured observations with help from their teachers.
- Students reflect on their observations to develop conjectures, or mathematical hypotheses, about a pattern or broader truth.
- Students either generate or are given additional examples to test their conjectures.
- The teacher provides feedback and gives opportunities for the students to consolidate their new ideas.
When students discover math for themselves, they tap into their natural intuitions. They exercise pattern recognition and problem-solving skills. Most important, they build self-efficacy and a feeling of “I did this myself.”
6 Steps to Creating a Guided-Discovery Lesson
1. Decide what students should discover: An appropriate learning target for a guided-discovery lesson is one that states a rule or formula but holds a deeper meaning that is lost when you simply state the rule. Since the students will be making the discovery on their own, it helps if there are many tangible examples associated with the concept.
2. Scaffold the exploration: Create a bank of examples that students can investigate. Students should already have the requisite skills to explore the examples. Make sure that there are enough examples so that the patterns become apparent, and try to integrate a wide variety of examples, not just the most basic ones. It’s important that students think critically about edge cases.
3. Create a graphic organizer: The graphic organizer should be designed so that it directs students to pay attention to and make observations about the key aspects of what you want them to discover.
4. Scaffold a conjecture: Ask students to generalize their observations into a conjecture about the phenomenon they’ve been exploring.
5. Test the conjecture: Give students more examples, or ask them to generate their own, in order to test their conjecture.
6. Consolidate: Throughout the lesson, provide feedback and ask probing questions. At the end, bring the class together and make sure to tell them the truth behind the phenomenon. You don’t want students cementing a false conjecture into their memories.
Teaching A Theorem Through Guided Discovery
This example aims to help middle school students discover the triangle inequality theorem, which states that given three side lengths a, b, and c, where c is the longest, they can form a triangle only if the sum of the two shorter sides (a + b) is greater than the length of the longest side (c).
To begin, distribute 10 groups of three pipe cleaners, each group having varying lengths. Instruct the students to record the lengths of each pipe cleaner in each triad, arranging them from smallest to largest in a chart. Next, ask them to try to create a triangle using the three pipe cleaners in each triad and record whether their attempts were successful or not.
While they work on this, circulate around the room to ensure that they make accurate measurements and recordings and determine whether a triangle is feasible. If any mistakes are found, provide additional support and guidance as needed.
Once they have experimented with all 10 triads, prompt the students to formulate a rule that relates the side lengths to determine if a triangle is possible. Encourage them to write down their conjecture based on their observations. After this, offer a few more examples, including some trickier ones, for them to test and refine their rules further.
Toward the end of the class, clarify the true triangle inequality theorem to the students, using visual aids like pictures and referring back to the examples they worked on. Make sure they understand the concept and its application. Encourage them to recognize and practice using the rule on their own examples to reinforce their understanding. By the end of the activity, students should have a clearer grasp of the theorem and its significance in geometry.
Check out my blog post for a high school math example.
Through scaffolded exploration, all students are capable of discovering mathematics for themselves. As they observe patterns, generate conjectures, and construct meaning, they create memorable experiences for themselves and are left with not only a deep understanding of mathematics, but also a lasting sense of self-efficacy.