• A shaded quarter circle
• The fractional form: 1/4
• The decimal form: 0.25
• As currency: $0.25

The Conversion icon promotes fluency with the concept of transforming values into different but equivalent forms.

Consider introducing the Conversion icon with measurement. You might ask your students how many equivalent ways there are to express one foot:

• 12 inches.
• 1/3 of a yard.
• 1/5280 of a mile.
• 1/6 the height of an average male.

Using currency, one dollar can be converted into:

• 100 pennies
• Twenty nickels
• Ten dimes
• Four quarters

But a dollar can also be converted to (at the time of this writing):

• About 0.8 euros
• About thirteen pesos
• About ninety yen

In this way, Conversion lends itself well to the study of ratios and proportions. If one dollar is about thirteen pesos, how many dollars do you have given fifty pesos? The Conversion icon promotes not just the idea of equivalent values, but the process of conversion as well. When reading a map, students need to convert the distance represented on the map into the real world distance.

Finally, take Conversion a step further by combining with the complexity tool of Multiple Perspectives. Ask students to consider why we need to convert between values. Ask when each equivalent value becomes useful.

• A foot is fantastic for measuring the height of a person, but useless in measuring the length of (most) insect.
• Quarters are preferable when I’m at the laundromat, but I’d rather have a twenty dollar bill in most other scenarios.
• Yen may be a mere curiosity to American friends, but in Japan it is indispensable.

Creating young mathematicians who are fluent in Conversion is essential to developing future scientists, economists, and business leaders.

Inquiry

The Inquiry Icon

To teach my students about inquiry, I ask questions that students do not have enough information to solve. They are required to carefully look at their existing information and then determine the right questions to receive the information they need. I ask my students to record their inquiry questions using the icon on their paper.

The inquiry icon allows students to become comfortable with uncertainty in math. They learn that asking the right questions leads them to the correct answers. Being able to confront uncertainty with confidence will help prepare students for the next icon…

Application

The applications icon answers the inevitable “when will we use this in real life” question. I use the application icon to introduce word problems but also ask students to create applications for new math concepts we learn in class.

The Applications Icon

After solving a one step algebra problem (such as 3x = 12), I ask my class to create an application for this problem. Initially, this stumps them and I create a sample application (for example: “My 12 foot house is three times as tall as my baby brother. How tall is my brother?”).

Using the application icon develops the correlation between abstract computation and math reasoning.

Continuing…

These two icons build on the tools we introduced last time, but there are still five more math icons to help your students develop a deeper understanding of math. We’ll explore these in upcoming articles.

Strategies

The Strategies Icon

The strategies icon invokes an image of getting from Point A to Point B. When using this icon, students should be thinking in terms of how they solve a problem. This may include:

• Finding a common denominator when adding fractions
• Isolating a variable in algebra
• Counting decimal places when multiplying two decimal numbers

By placing the strategies icon, students are recording the strategies they use to go from Point A (the question) to Point B (the solution). When you require the strategies icon, you are requiring students to record this information.

When introducing this icon, consider discussing the importance of maps to everyday life. Sure, some of us can remember how to get from place to place, but a written record gives us a way of communicating our directions and backtracking in case we get lost.

Proofs

The Proofs Icon

The proofs icon challenges students to prove the solution they arrived at using their strategy. The information under a proof icon might include:

• The steps involved in plugging in a value for a variable and solving using algebra.
• The geometric theorems used to arrive at a solution.

While similar to the idea of “checking work,” the proofs icon encourages students to see proving math as another element of being a mathematician. “Checking work” is a chore, but displaying a proof is providing evidence of correctness.

Only The Beginning…

While these two icons provide a useful starting point, there are seven more math icons to help your students understand math at a deeper level. We’ll explore these in upcoming articles.

Encourage your students to take a deep and complex look at themselves, historical figures, or literary characters. Designed for heterogeneous classrooms, A Deep and Complex Look at Me promotes powerful thinking and deep learning through the icons of depth and complexity. Your students will demonstrate self-understanding (an important component of social-emotional development) in both written and visual ways as they fill out the depth and complexity icon templates. Templates can be used with individual students, the entire class, or as a cooperative team project. Asking your students to think in deep and complex ways will help them develop important learning skills that will become habits of mind, and establish a strong foundation as they become life-long learners. Sample templates are provided, including a full example through the eyes of the main character of a Newbery Medal book