PBS Teachers™
This month's expert is Elizabeth Ross Hubbell
Read more about Elizabeth or all the authors
Each month our guest experts discuss and invite you to share your ideas about using multimedia resources to address common instructional challenges. These practitioners live and work in your standards-based, resource-challenged world. They share your commitment to creating rich, engaging learning experiences for students and are pioneering methods for infusing their instruction with media to improve learning across grade levels and curriculum topics. Pull up a screen and join us!
Science & Technology
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: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: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.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.
More like this: Math, Grades 3-5, Grades K-2
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Cindy,
Your blog is fantastic! I love the video intro, and was amazed by the number of interesting hotlinks you included. I was already familiar with many of them, but there are some that I hadn’t met before. I will certainly use many of your ideas with my math methods instruction in the fall!
Diana
Diana Lambdin
Associate Dean for Teacher Education
Indiana University - Bloomington
Posted by Diana Lambdin, 2:49PM 06/01/07
Cindy,
What a great blog! I, too, grew up in the 60’s and 70’s and never understood what was so “new” about the New Math, and still don’t to this day. I could totally identify with how you felt about math growing up. Now, I am a teacher and have been basically teaching elementary students the way I learned…cook book formula style. Some get it, some don’t…and most of us don’t really understand it in a meaningful way. This year our school adopts math curriculum…specifically, Inverstigations, by SF. I’m told this curriculum will put the meaning and understanding into math. I’ll be a student along with my students. Thank you for all of the new sites you have shared on your blog. I plan to use them for me 1st…then pass them on to my students next year. By the way, our school subscribes to BrainPop each year of which I am so grateful. The kids and I both love it! Thanks again!
Tamara
Posted by Tamara, 4:55PM 06/02/07
Cindy, your blog is wonderful. Thank you for sharing the wonderful websites you use with your students allowing them to use technology to assist them throughout their mathematical journey. I plan on using some of them next year with my students.
Sandra
Posted by Sandra, 5:53PM 06/07/07
Tamara, I would love to hear more about Investigations. Is this a hands-on program? The title, Investigations, sounds like the kind of program students need to be “doing.”
It was interesting to read that you classify yourself as a student also. “Cookbooks,”as you put it, are great for baking pies, but not as effective for creating students who truly understand math concepts.I find that I am still a “student”along the journey.
Posted by Cindy Newton, 7:54PM 06/07/07
Sandra, technology plays a big part in our students daily lives. Therefore, it just makes sense to tap into this arena to assist student meaning-making. Technology goes hand-in-hand with math and science.True integration occurs when the use of technology is so seamless, students use technology as a tool as easily as they would pick up a pencil. Which web sites are you considering?
Posted by Cindy Newton, 8:05PM 06/07/07
Cindy,
Thanks so much for all the wonderful weblinks. I’ve had a blast playing with them and am certain my students will love them just as much.
One thing that I’ve noticed in teaching math to 6th grade special education students is that they are so bogged down with trying to remember basic facts that they don’t have the energy to do any problem solving.
One of my many summertime goals is to provide more problem solving, experimentation opportunities for my math students. Thanks for sharing your findings.
Posted by Missy, 9:52PM 06/12/07
Missy,
I have just transferred to the Middle School as the Media Specialist. I would be interested in learning more about how the experimentation opportunities work out for you and your students. Then I can pass your ideas on to the special ed. teachers in my new building.
Yes, math facts can get in the way of problem solving. However, not all problem solving deals with math facts. When fact knowledge is an obstacle, I have found the book, “Differentiated Instruction: Different Strategies for Different Learners” by Char Forster, to be helpful with intermediate students. Char has some unique ideas for using addition and multiplication charts and two different colored strips from transparency materials (I cut strips from thin colored plastic folders,). The highlighting strips reinforce fact-family relationships. The charts can be copied from the book’s Appendix and details of how students may use these materials are given.
Hopefully, these accomodation ideas will make problem soling more “do-able” for your students.
Cindy
Posted by Cindy Newton, 11:15PM 06/14/07
Cindy,
Congratulations on your new position! I’m sure you’ll be a terrific middle school Media Specialist, given your wealth of knowledge and experience. You’ll undoubtedly be missed at the elementary school, though.
Thanks for your great ideas here!
Jenny
Posted by Jenny Bradbury, 8:24AM 06/15/07
Cindy and Tamara,
Tamara wrote that her school has recently adopted “Investigations” as their new math curriculum, and Cindy replied that she’d like to learn more about this. This curriculum, originally written with funding by the National Science Foundation and now in its second edition, is built on the premise that students learn best when they are actively involved in figuring things out for themselves. The full name of the curriculum (K-5) is “Investigations in Number, Data, and Space” and it is published by Scott Foresman. For more information, consult their website: http://investigations.terc.edu/
Diana
Posted by Diana Lambdin, 10:00PM 06/18/07
Diana,
Thanks for the info and link. I took a look at Investigations. It reminds me of the old programs “Math Their Way”and “Math, A Way of Thinking” which my students loved and from which they gained so much “meaning” and problem solving abilities. I may have missed it, but is there a technology component in Investigations?
Cindy
Posted by Cindy Newton, 9:20AM 06/21/07
Cindy,
Your personal journey through mathematics is a story well worth telling! This should be required reading for all university-level students who are majoring in education. Preschool through high school level students need to make connections and are constantly seeking ways to make meaning of their learning. By sharing your personal history with a difficult subject, you have offered teachers and students a new avenue to continue making sense of the world around us. Thank you!
Susan
Posted by Susan Strube, 11:21AM 06/22/07
I’ve been in love with numbers from an early age. This blog makes teaching math sound exciting. The objectives are laudable.
I went to grade school in the early 60’s and we drilled math facts endlessly. To the best of my recollection, we worked on specific tasks until we had command of them so that extensive re-teaching did not dominate the subsequent years’ work. We moved on. I have vivid memories of doing division with decimals and taking square roots at the blackboard in 5th and 6th grade.
My perspective is from the other end of the education spectrum — the results I see in my college classroom.
I teach economics to Junior and Senior business students. These students, on average, scored above the national average on the SAT. The course pre-requisites include Calculus I, Introduction to Statistics, and the introductory information systems.
My students have been taught to rely on their calculators. They cannot perform the simplest arithmetic calculations, including converting a decimal to a fraction, dividing by a fraction, and graphing functions by hand.
Pay attention to your daily purchases and you’ll see that the typical high school graduate can’t make change unless a machine tells them how much to return to the customer. If the display shows the wrong amount, the person is unlikely to identify the problem. If confronted with the information by an honest customer, the person will be likely to argue that the display is correct.
To me and to many of us, mathematics has a certain underlying beauty. The ancient Greeks understood that. When it comes to purely practical everyday life, the ancient Egyptians appreciated the value of computation as they needed to re-establish property boundaries after floods and they needed to properly engineer the pyramids and other architectural wonders. At a minimum, we should ensure that our students are computationally competent.
While it is easy to denigrate rote memorization, I firmly believe that establishing a vocabulary of facts and a facility with processes is the essential foundation for creating understanding of the underlying mathematics ideas.
Who among us would expect a student to write a story without their having already learned the language and the basic structures of sentences and paragraphs? Teaching math must spend more time on the basics because we do not acquire mathematical knowledge in the same way we acquire language.
Early(or any)reliance on calculators robs students of the ready knowledge of arithmetic and higher mathematical concepts that is essential to solving problems. Drilling facts is not exciting, but it is a necessary step to equipping our students for life in the real world. Working knowledge of the facts form a solid basis for discovery.
Posted by Caroline Swartz, 8:27AM 06/24/07
Caroline,
I absolutely agree that facts are a foundation that cannot be overlooked. Many problem solving outcomes are dependent on correct calculations. Hopefully, students will build foundational concepts, get facts to memory, and be able to problem solve. I think the key in math education is being sure to include all of these components without overemphasizing one while neglecting the others.
As a youngster, I marveled at my elders who could “cipher.” I often went to the grain elevator with an uncle who could correctly “cipher” totals in his head faster than the clerk could total the figures on an adding machine. If he needed paint for a barn, he could quickly “figure out” how many gallons of paint were needed for the square footage, without buying too much or finding out half way through the project that he had not purchased enough. My uncle had the right combination of conceptual foundations, fact memorization, and problem solving. His mathematical ability is what most educators would consider the ultimate goal - he understood how math worked and could apply it automatically and correctly in his everyday living.
Cindy
Posted by Cindy Newton, 9:21PM 06/24/07
I had my usual frustration with most of the first websites and “multimedia” examples. Just as in the Ma & Pa Kettle scene, they simply provided a view of the symbols. At least the chart has all the numbers so there’s a closer connection to the meaning of the number, but is 41 really more t han 39? Some may not think so, since it had to go back to the beginning of the line and start over.
I think there is a whole lot of room for development of multimedia materials to help connect meaning with math - but it shouldn’t be the same confusing symbol manipulation wiht bells and whistles.
Posted by SueJ, 9:16AM 06/25/07
Thank you to all who participated in our discussion through blogging or reading this month’s blog. While this blog was about making meaning in math, our discussion touched on the need for differentiation while on that journey. I encourage you to follow the discussion into July’s blog, One Size Fits Few: A Look At Individualizing Learning, with Wade Whitehead.It is my hope that June’s journey has led each reader toward making more meaning in their own journey as a mathematician and as a math educator.
Posted by Cindy Newton, 9:04AM 06/30/07
Thank you to everyone who contributed to June’s discussion! We hope you’ll join our July conversation about individualized learning.
Jenny Bradbury
PreK-12 Education
PBS
Posted by Jenny Bradbury, 10:25AM 07/02/07