# Crippling with Compassion?

Strange title? It comes from teacher Ellen Berg’s article in Teacher Magazine, Teaching Secrets: Don’t Cripple With Compassion. From her perspective, “One of the major issues with American teachers especially is our predilection to rescue kids instead of letting them struggle with the content a bit. In essence, we’re too compassionate.” It is second nature for us as teachers to help our students, but do we rush in on rescue missions too often and too soon?

Berg writes, “I get how difficult it is to step back and let them struggle, but I also know that it’s in the disequilibrium that kids have to make sense of things and that’s when the learning happens. If we do it for them, why would they be persistent with a problem or give it more than 30 seconds? And how can they become confident, self-directed learners if we don’t ever let them have that experience? Finally, why would they ever believe that they are able to figure it out if we show them by our actions that we don’t believe they can, either?”

Thinking of how we math teachers might challenge students to tough thinking, I looked around for problems that would work in middle school classrooms. Here are a few below, but please share any of your favorites from the classroom in the comments section.

Balanced Assessment

A set of more than 300 assessment tasks actually designed for off-the-wall thinking. Most tasks, indexed for grades K-12, incorporate a story problem and include hands-on activities. Some intriguing titles include Confetti Crush, Walkway, and Hockey Pucks. Rubrics for each task are provided.

Understanding Distance, Speed, and Time Relationships

In these two lessons, students use an online simulation of one or two runners along a track. Students control the speed and starting point of the runner, watch the race, and examine a graph showing time versus distance. Students can use the activity to come to conclusions on the distance, speed, and time relationship. They can also use it to consider the graphical representation and the concept of slope.

Measuring the Circumference of the Earth

Through this online project, students learn about Eratosthenes and actually do a similar measurement that yields a close estimate of the earth’s circumference. It’s a challenge! Even with access to only one computer, students can obtain data from other schools that lie approximately on their own longitude. Careful instructions guide the students in carrying out the experiment and analyzing the data collected. The project also provides activities, reference materials, online help, and a teacher area.

Down the Drain: How Much Water Do You Use?

Students first collect data from their household members and their classmates and then determine the average amount of water used by one person in a day. They compare their average to the average amount of water used per person per day in other parts of the world. Through the Internet, they can collect and share information with other students from around the country and the world. A teacher’s guide is included as well as guidelines on how students can publish reports, photos, or other work directly to the project web site.

Accessing and Investigating Population Data

In these activities, students use census data available on the web to examine questions about population. They also formulate their own questions. For example, in one section they analyze statistics from five states of their choice, develop specific research questions using the data, and create three graphs to compare and contrast the information.

The Handshake Problem

This two-lesson unit allows students to discover patterns in a fictional but real-world scenario: How many handshakes occur when the nine Supreme Court justices shake hands with each other? Students explore—through a table, a graph, and finally an algebraic formula—the number of handshakes in any size group. A second pattern is explored, that of triangular numbers; again, students generalize the pattern with variables. The lessons are well illustrated and include background information for the teacher.

These problems require patience and analytical thinking, even the easiest of them. I would not give such problems without having prepared my students with the needed tools to do them, if not before they start the work, then as they’re doing it. As Ellen Berg put it, “I’m not talking about failing to scaffold instruction or give kids input. Of course we want to do that. What I’m talking about is resisting the urge to fix things for them instead of asking more questions to get them thinking. I’m talking about sometimes just telling them, ‘I know you can do this,’ and walking away.”

Another teacher who feels that we need to help math students less is Dan Meyer, a high school math teacher. This 11-minute talk, Math Needs a Makeover, begins with: “I teach high school math. I sell a product to a market that doesn’t want it but is forced by law to buy it.” From there he moves to actual examples of textbook math versus ways to present real, hard thinking problems. Worth watching!

Citation: From Teacher Magazine [Teacher Update], Wednesday, May 26, 2010. See  http://www.edweek.org/tm/articles/2010/05/26/tln_berg_compassion.html?tkn=URPFzAhx52nB4%2FOp1kNYkfQZs6eV8MJI9rtk&cmp=clp-edweek

# History to Enrich Mathematics Learning!

Although the NCTM Standards do not have student expectations for learning mathematics history, exposure to this history can help students see real-world connections in mathematics.

Approximating Pi
Mathematics and science owe a great deal to Archimedes, including a way to approximate pi. Here is historical information along with an applet to approximate pi using the perimeter lengths of polygons inscribed within and circumscribed around a circle.

The Beginnings of Probability
Teachers can share some of this mathematics history as they work with students to compute probabilities for simple compound events, an NCTM expectation for students in grades 6-8.

The Golden Ratio
This rich site connects linear measurement, ratio and proportion, art, and mathematics history.

Measuring the Circumference of the Earth
This Internet project is hands-on, real-world, and historical. Students join with classes around the world to repeat the experiment of Eratosthenes — measuring the shadow of a meter stick and making calculations to approximate the circumference of Earth.

Pythagorean Puzzle
The Pythagorean theorem is at the intersection of algebra and geometry. At this site, learn about the life of Pythagoras and the development of the Pythagorean theorem. And use an applet to explore the meaning of the most famous equation in algebra.

Manipula Math with Java: Pythagorean Theorem
Here is another applet offering a more sophisticated approach to affirm in a visual way the validity of the Pythagorean theorem.

Tortoise and Hare Race
Uing an applet, students can vary parameters for the race. There is information about Zeno’s paradox along with exploration questions for students that can lead to a discussion about infinity.