When you use an inquiry-based approach in your classroom, students discover mathematical properties for themselves. Instead of presenting students with a theorem, formula, or rule, you guide them through an investigation. Inquiry learning is often done in a hands-on way.
In my inquiry lessons, I like to have students write patterns and observations in complete sentences, then develop a rule for the property they have observed. This helps them to internalize what is happening. Then, I lead them into writing a mathematical formula or rule to represent their words. The students learn to look for connections. They also develop valuable skills such as the ability to self-direct and the ability to clearly express their discoveries in both words and mathematical language. |

## 1. Inquiry enhances independent problem solving skills.

The critical thinking skills that are required to ask questions that will lead to discovery are

**learned and acquired.**They develop through practice, just like any other skill. Students must practice the inquiry process, just like we as teachers must practice sitting back and letting them struggle. It's not easy, but begins to feel more natural over time.

As students get more and more accustomed to the inquiry structure that you use, they will slowly strengthen their own skills and gain independence in problem solving.

Here's an Algebra example:

I always teach exponent rules with guided inquiry. We work through the patterns and write out all the variables and cancel or expand until students develop product rule, power rule, and quotient rule for themselves. Then, when we get to the negative exponents later in the unit, their skills with this process have grown to the point that they know how to come up with their own concrete examples. They know how to look for patterns. I have them develop their own sets of rules for negative exponents.

**persistence in problem solving.**

## 2. Discovery leads to deeper understanding of a specific concept.

Let your students build a concept, not just follow a given process.

It's very rare that I give a formula as a part of notes. I like to have students find a formula for themselves. One example is for surface area of a cylinder. I have a dissection lesson for classes to "dissect" a cylinder "specimen."

They have to discover for themselves that the length of the rectangle is equal to the circumference of the circle, and then go from there. Most middle school students across the world REALLY struggle with truly understanding this formula. They have a hard time visualizing the base and why circumference plays a part in this formula. However, after this hands-on lesson, the students really "get it" and never forget how the rectangular face wraps around. You can read more about this lesson in my Cylinder Dissection blog post.

I do a similar structure for discovering circle theorems in High School Geometry. Students draw their own chords and have to tell ME how arcs, tangents, central angles, and inscribed angles are related, instead of me telling THEM these theorems.

## 3. Students discover a new level of math confidence & self-motivation.

I have noticed students have a sudden willingness to try a new challenge or approach a different type of problem instead of giving up. Kids believe in their

**ability to apply knowledge from one situation to another**.

I feel like this has given my students huge advantages on standardized tests. After I started incorporating more and more inquiry learning, the kids got more and more comfortable with being exposed to an unfamiliar problem type.

## 4. Retention increases HUGELY when **concept mastery replaces memorization.**

The student can reproduce the formula at any time because he "discovered" it. This is especially true if it was done in a hands-on way.

During a test, or later in life, the development of the idea is what will help a student recall how a property works or how to re-create the formula.

## So, here's how to make it happen in your classroom:

- Sit back. Don't jump in to help. Let students struggle through the first few inquiry lessons. It's so hard to resist giving hints, but do your best to choose when to give assistance and who to offer it to. They will get used to this process and it will really be best for them in the long run. Help those who truly need it, but otherwise, let them develop their own skills. Stick with it. It will get easier!
- Never (or very rarely) GIVE your students a rule or formula. Get out of the habit of feeding information and properties to them. For example, if you currently teach rules of divisibility, twist the lesson to have the kids
**find**the rules and present them to you. - Require your students to write up their observations in complete sentences. Follow up all inquiry lessons with a full written explanation. Give your kids the benefits that come from working to explain the properties that were discovered in their own words. Then, also have them translate into mathematical language to develop the formula as well.
- Have students share both their questioning process and discoveries aloud. They can work in pairs or small teams or they can present findings to the class. Allow them to collaborate and share the strategies they used to figure out the property. Encourage metacognition!
- Plan carefully so that your students do have enough guidance to succeed. Try a consistent structure like my SPORT method for structuring inquiry so that they know what to expect and can thrive in inquiry learning.
- When possible, incorporate materials and manipulatives that make the lessons more hands-on. This is a little more difficult in Algebra than it is in Geometry. Check out some specific examples for making your Geometry lessons into hands-on discoveries here.