by Cindy Newton

I don’t know about you, but after watching this video a few times, I could almost agree with Ma and Pa’s conclusions. Where the Kettles and so many others falter in math is in viewing math as a set of facts, instead of as a way to make meaning of the world around them. Yes, there are certain mathematical facts that must be memorized, but if the facts have not been built on a foundation of trial, inquiry, and discovery, then math becomes no more than a routine set of calculations, where errors may go unrecognized. Rote memory cannot be transferred as a way of thinking that leads to active, purposeful problem solving.

My Journey
I grew up in the 1960s and ’70s when education was abuzz with “the new math.” I don’t remember much about that math other than that it left me confused and certain that if I could just memorize the steps, I could conquer this thing called “math.” I did well until I began algebra and the more advanced math disciplines, yet somehow memorizing routines pulled me through those courses as well. However, I was never comfortable with math, never connected math to problem solving, and avoided math at all costs!

I went to college for a few years, got married, had two children, and then sixteen years later enrolled in college to become an elementary teacher. As I entered the university doors, my greatest fear was the math courses. I enrolled in a refresher algebra and trig course, hoping to find a new perspective. Much to my surprise, my professor soon asked me to become a math tutor. I protested, “Me, a math tutor? I am almost twenty years older than most of the students in the class. They have just graduated from high school. Shouldn’t they be tutoring me?” But, it turned out that these struggling college students had spent their elementary and high school years just as I had done some twenty years earlier — memorizing facts and routines! I, on the other hand, was no longer satisfied with producing the “correct answers.” I wanted to know how and why math worked the way it did. I was discovering mathematical concepts. Perhaps for the first time, I was making meaning.

Today’s Journey
Today’s No Child Left Behind requirements, the NCTM’s Principals and Standards of School Mathematics, and state academic standards are redirecting schools, educators, and students toward a journey of meaning-making. But determining how to begin this journey can be confusing. Arthur Hyde, coauthor of the popular Best Practice, has recently written Comprehending Math, which may hold the keys to not only beginning the journey but also discovering how to arrive at the destination of meaning.

In his book, Hyde espouses that students need to develop clear ways of thinking about and understanding what they learn, no matter the content area. Hyde delivers a practical way of “braiding” together reading comprehension, language, and mathematical problem solving into broader ways of thinking. He shows us how students can practice math-based variations of K-W-L, visualizing, asking questions, inferring, predicting, making connections, determining importance, and synthesizing. Based on current cognitive research, Hyde’s book informs educators about how we can help students “do” math and make meaning at the same time.

Look at an example that Ellin Oliver Keene, author of Mosaic of Thought, provides in the foreword to Comprehending Math. Keene tells of a first grader who asked, “Why do you call it predicting when we’re talking about reading, hypothesizing when we’re in science, and estimating when we’re in math? Aren’t they really all the same thing?” Reading this was an ah-ha moment for me. For all intents and purposes, in elementary school these are the same. Why are we compartmentalizing content areas and losing meaning?

Beginning the Journey with Technology as a Vehicle
Students need a foundation of certain basic facts and vocabulary. In Comprehending Math, Hyde mentions the need for visualization and for braiding language with math. Digital media and technology can be perfect vehicles for students to drive on the meaning-making journey, because they provide for visualization, differentiated instruction, multiple intelligences, and social interactions.

A good example is NOVA scienceNOW’s, musical video on the Twin Prime Conjecture. This video assists the viewer in “seeing” the twin primes by using a moving number line with numerals in different colors to highlight the prime numbers. To further reinforce the concept of pairs of numbers, pairs of mathematicians add comments throughout the video. This video is a great introduction to the prime numbers. Students can then make their own number lines and explore the conjecture. As they find the prime numbers, students color code the primes. Students are then looking to determine whether the conjecture is accurate or flawed. They are becoming mathematicians. The site also offers song lyrics, a complete transcript, and links for further exploration.

There are numerous other Web sites that help students who struggle with foundations, and then there are sites that are just plain fun to use. My students use talking flashcards and Cyberchase’s talking calculator. These resources allow for experimentation and practice with numbers and number patterns. We also use an interactive 100’s chart to explore patterns and multiples for skip counting concepts. A favorite of mine and my students is A Maths Dictionary, an interactive dictionary that reinforces vocabulary through experimentation and visualization. Here students can engage in fun math-related activities, such as feeding fish to a pelican by adjusting the bill to different acute angles.

Other foundations sites to check out include:

• The National Library of Virtual Manipulatives
• Illuminations, an extension of the NCTM
• ReadWriteThink
• BrainPop and BrainPop Jr. (these are subscription sites, but they offer trial subscriptions and some free activities)

Along the Way
When putting together opportunities for your students, keep Hyde’s Braid Model in mind. For intermediate students, an inquiry into probability might begin with a basic probability video from BrainPop or Cyberchase to set the stage. Students can access the online Maths Dictionary for help with specific vocabulary, such as the terms “probability,” “coin toss,” and “graph.” Then they can experiment on their own with spinners and dice or coin tosses using online interactives. From there, students might want to try a game of interactive Yahtzee where they can apply their newfound knowledge and understanding.

When my first graders begin inquiring into measurement, they watch a Curious George video that introduces measurement with non-standard units. A printable growth chart from the George Shrinks Web site is placed by the classroom door and allows students to investigate their own measurements throughout the year. Sand Babies, a PBS Mathline lesson, instructs students to measure pounds with non-standard units and to convert bags of sand into babies that calculate their own birth weights. Instead of sand, I have found that one pound bags of rice and a five pound bag of sugar work well. Before the lesson, I send home a form for parents to record the students’ birth weights. Then, I have students convert their weight into bags of rice and sugar. They place the equivalent weight in a baby blanket and we wrap it up like a baby. Students enjoy holding their weights and comparing their “babies” to other classmates’ weights.

As we continue our study of measurement, my students preview an inches and feet video from BrainPopJr. The online Maths Dictionary can assist with vocabulary such as “inch” and “centimeter.” The Web site, Rainforest Maths, connects our rainforest theme study with many areas of math. Students use online paperclips, blocks, and rulers to measure rainforest insects. They can also use a scale balance to weigh the animals. Once students use the online tools, their confidence in using the actual measurement tools is much greater.

Other measurement sites to check include:

• Cyberchase’s Game Central, which offers games such as Wacky Ruler, Poddle Weigh-In and Can You Fill It?
• The PBS Parents site’s Early Math page, which offers activities and online games for PreK through 2nd grade

Nearing the Destination
Once students have solidified foundations and made connections and visualizations, they need to delve deeper toward predicting, inferring, synthesizing and using logic. I’ve already mentioned some of the Cyberchase games and videos from PBS that do this well. Cyberchase also has an easy search function that displays a title, the topic, and the NCTM standards addressed. And be sure to check out My Cyberchase Summer for math journals and other activities to keep math learning going this summer.

By spring, my students have a good grasp of addition and subtraction facts. They’re ready to enjoy Math Car Racing, a strategy game where students are asked to choose the largest sums or differences to fuel the car around the track. Students can play against the computer or other students. Socially-oriented students especially like the “real person” option, while the intrapersonal students enjoy playing against the computer.

Other logic and reasoning sites to check out include:

• The Math Playground
• Math Games
• The National Library of Virtual Manipulatives’ Numbers & Operations page, which houses games like MasterMind that make use of inference and logic

Exploring the Arrival
The NCTM Communication standard states that students need to communicate their mathematical thinking coherently and clearly to peers, teachers, and others using the language of mathematics to express mathematical ideas precisely. Online math journals and writing prompts are plentiful. Blogs, such as this 7th Grade Math blog and MisterTeacher, are a newer technology that facilitate communication about math learning — student to student, student to teacher, and teacher to teacher.

Of course, this month PBS Teachers has provided this site as a forum for educators to express their beliefs and learning about assisting students on their own mathematical journeys toward meaning. Please join in the dialogue.