Click on the question to leap to the answer, or just scroll down the page.
Are there computer games (no Internet required) that you recommend for math?
Should I discourage my child from using his fingers when adding and subtracting?
What are the greatest challenges you face in evolving a math curriculum for the Lower School?
What are you most excited about introducing into the LS Math program in the coming year?
What are the greatest changes to the LS Math Program over the past 1-2 years?
How do you make a child curious enough to want to go to your math page and work the challenges?
The United States lags behind several other developed and developing countries with regard to math. Typically in those countries, math drills are introduced first (learning the rules) and then once that’s mastered and children are further along the developmental curve, the concepts are introduced when it’s believed that children are more receptive to understanding the meaning behind the concepts. What studies are you referring to that indicate that doing the reverse (introducing concepts) is more advantageous?
Do the curriculum materials provide challenge/enrichment in class exercises (non-written) and written exercises for the kids who are working at a level beyond the basic lesson (those who finish quickly)? Or is it the teacher’s responsibility to come up with those materials?
Yes, there are some computer games we can recommend, which do not require Internet connections. We have these accessible to kids in the Lab before school and during computer recess times.
Here are two programs we recommend:
Logical Journey of the Zoombinis – Fun problem solving that is recommended as part of the TERC Investigations in Number, Data, and Space program that we are piloting this year.
There are three levels of the program. You can find them at: http://www.broderbund.com/ProductGroup.asp?CID=584
Ice Cream Truck Math – this program supports NCTM standards while providing a stimulating activity kids love- running an ice cream stand! This program stretches from 2nd through 6th grade.
http://www.sunburst.com/resources/productcomm/ICT/
Logo Programming Resources - Students at Lower School begin programming in LOGO in first grade. At school, they use a version designed by TERC that is part of the math curriculum. There are many programs designed to give students access to programming LOGO at different levels of challenge. Here are a few resources to explore for home use:
http://www.logo.com/cat/browse/logo.html - The Simple Logo, Imagine Logo and Super Logo programs offer challenges for a range of ages and abilities. Take a look at the examples on this website.
http://www.microworlds.com/solutions/mw.html - Microworlds is a logo-based program with a wide range of interesting tools and features.
http://mindstorms.lego.com/eng/default.asp - Lego Mindstorms is a robotics system that includes a building set, a programmable device to run a robot, motors and sensors. It is a wonderful, high-powered system for those who are excited about building with LEGO and programming, too.
Many teachers give supplemental computational homework regularly for this purpose. The Everyday Math Journals feature “Math Box” pages, which feature up to six different strands of mathematics, requiring the students to regularly practice many different skills. We also use games and number tiles activities to reinforce facts and to ensure more practice.
Children differ in their ability to acquire, retain and apply basic addition, subtraction, multiplication and division facts. Some are quick to memorize while others struggle throughout their years at Lower School. Developing computational fluency requires a multifaceted teaching approach including everything from songs and movement routines to flashcards and old-fashioned drill.
Our report forms list specific skills and competencies for each grade level in mathematics. Yes, we can share these with parents (in their blank state) at the beginning of the year. Thanks for the idea. If you are interested in seeing these, feel free to contact your children’s teachers or request them from the Math Lab.
The Lower and Middle Schools historically have taken different approaches; however, just this year, the 5th grade teachers adopted the Everyday Learning program, which has been the mainstay of the LS curriculum for the past 10 years.
This question would most likely be answered different ways by different teachers at the Lower School, and I hope they will feel free contribute their voices to this webpage, especially if they disagree with me. However, I do not believe in discouraging children from counting on their fingers. In my mind, our ten fingers and ten toes are the reason why our number system is base 10. It seems completely natural, acceptable and intelligent to use the manipulative materials we have literally ‘on hand’ at all times. In time, most kids develop other strategies and grow out of counting on their fingers, but the truth is there are plenty of adults who still occasionally rely on this strategy!
Choosing instructional materials and curricula with their inherent teaching approaches can be controversial. As the Math Coordinator one of my greatest challenges is finding the best materials and approaches for our particular community. This process needs to be collaborative and it involves getting the teachers together to talk and work through the problems they face. There are strengths and weakness in all curricular materials. Furthermore, different curricula encourage different approaches to teaching. Using new materials and new approaches demands that the teachers have sustained, ongoing professional development. Teachers need mathematical knowledge as well as pedagogical knowledge; they must continually adjust their practices and extend their knowledge. New curricula and new technologies are being developed constantly. To support our students we need to learn how to become even more alert to our students’ thinking processes and development of understanding, which entails reading everything from neurophysiologic research on the brain to the latest children’s books on math.
For a math curriculum to evolve successfully not only the teachers, but the entire community, including administrators and parents, must focus on mutual goals for mathematics education. It is our hope that the Math Nights will serve as important vehicles in this process. By focusing on the Standards, created by NCTM, we can create a vision for teaching and learning mathematics as a community. By constantly considering the NCTM Principles (equity, curriculum, teaching, learning, assessment, technology), we can build an exemplary program.
I get very excited about the new materials I order every year but this year has been especially exciting because of all the work the teachers did over the summer to prepare for piloting units of the Investigations curriculum. The Investigations program engages kids in meaningful work. It encourages the teachers to be just as engaged in learning mathematics and in observing their students. In this way, the program itself is a professional development tool.
The teachers have been working collaboratively and talking more about the math going on in their classrooms. They are telling stories describing how the children are grappling with mathematical concepts and developing number sense. They are noticing new things and asking themselves different questions. Some of the teachers who spent years perfecting their teaching of Everyday Math are finding themselves completely invigorated by trying the approach taken in the Investigations Program. This commitment and energy for learning “ fuels” our school and it is exciting to be a part of it.
We are fortunate to have the latest technological equipment (the SmartBoard) as well. The SmartBoard is incredible for giving workshops and presentations. And now (as you saw) we have new digital video equipment too! Jenni Voorhees is a technological wizard, and we barely can keep up with her. She has had years of classroom experience teaching math and now she is keeping us on the cutting edge of technology. It is exciting to work with her.
Together we have been exploring the ways we can pose problems to encourage the children to explain their thinking. Many of the posted problems require the students to do some “paper work” on the side. Often diagrams need to be drawn or tables created in order to find the solution. We are trying to promote thoughtful and thorough work. Mathematicians are known for their perseverance: lifetimes spent on one problem!
The children love to create their own problems and they learn a great deal in the process. Furthermore, they get to enjoy their notoriety when they are published on the web. (It is empowering when all your classmates have to solve your problem!) Jenni and I are constantly looking for new math sites and resources available on the web, and she is constantly updating the LS Math Adventures website with new hotlinks.
Huge strides in technology, including implementation of the LS Math Adventures website, two substantial Venture Grant Projects, one involving the entire LS faculty and the piloting of several units Investigations have all made an impact on the LS math program. I also think parents are more interested and involved in math education because it has received so much attention nationwide.
What a wonderful observation and question! I have had to relearn mathematics through the eyes of my students. That is perhaps the most thrilling thing about being a teacher. It is very exciting to see kids explore algebraic concepts from an intuitive base. It was ‘freeing’ to discover that I didn’t need my “high school tools,” and even more gratifying to realize I could reinvent all the mathematics I had ever learned without any textbook. My students have taught me well and working three consecutive summers as a graduate student of mathematics in a constructivist environment confirmed my trust in the whole process.
Please feel free to look at your children’s schedules if you missed them at Back to School Night. Most teachers have posted the times each week they devote to math. The Everyday Learning Journals (Chicago math workbooks) are also available for viewing and perusing in your child’s classroom or in the math lab. There are no workbooks in the TERC (Investigations) program; however, there are unit guides and again, parents are welcome to look examples over in the math lab. Please feel free to arrange a meeting time with me (adelfiom@sidwell.edu) to explore these materials.
All of us are adjusting to working in a web-based environment and children need support to learn how. The math problems on this site require reading, some pencil and paper work, and a typed response. This interaction with the computer is different than those they might have on a game site, or when they are playing in a paint program. We are asking them to work with the screen in the way they work with a sheet of paper - no bells and whistles and no instant gratification in the form of a pop-up response. In addition, the problems presented often require several steps. Help your child solve the first few and experience success. This will build into a more independent, enthusiastic approach to taking on the challenges.
As a teacher, I always wish I had a magic wand that could make all my students want to learn and to challenge themselves. I try to inspire my students (all teachers do) but clearly I’m not always successful. Sometimes children are growing in ways we can’t see; they might not be challenging themselves in math, but they are busy learning in other arenas. This is something to discuss with your child’s teacher but also to think about in terms of the broader picture, i.e., your child’s life outside of school as well.
There are reference books that are a part of the Everyday Math program, which we keep in our classrooms. These books include essays on mathematical topics, descriptions of strategies such as partial differences (used for solving subtraction problems), column addition, partial products, lattice multiplication, and partial quotients, as well as sets of tables and charts that summarize information taught in third and fourth grades. I know we have some extras that we can loan to parents. Please contact your child’s teacher about borrowing one if you are interested.
We will continue to add resource materials to the list on our Math Resources page, which is linked to the Parent's Page. Check there for additional references as we post them.
If I had a full-time job to develop math at the Lower School, I would focus on professional development for the teachers. Last summer, Carol Borut and I organized a Venture Grant for Lower School teachers that focused on our math curriculum and our approaches to teaching mathematics. Almost all of the Lower School teachers spent 4 days working with their grade level colleagues reviewing and evaluating their math programs. Many were inspired to pilot units from the Investigations in Number, Data and Space program. However, summer work is just part of equation necessary for continuing professional development. Teachers need structures for ongoing work and reflection during the school year as well. There are many models for ongoing work: teachers can pursue learning through collaborative focus groups, discussing readings gathered from relevant books, mathematics education magazine and research articles. Teachers can attend local workshops and national conferences. Teachers can take graduate level courses, participate in online seminars, arrange to work with a mentor, attend “User’s Conferences” organized by the programs, visit other schools using NSF-funded curricula, commit to watching a professional development video series, read case studies and much more. There are many web-based learning sites for teachers and now the availability of E-conferences, following the reflective “lesson study” model, have added a whole new dimension of possibilities. Jenni Voorhees and I participated in an E-conference recently and are hoping to share our experience with our colleagues soon. Jenni and I work closely together to plan faculty workshops and stay current with all the web-based learning sites available for both our colleagues and our students.
It is important to realize that Lower School teachers must consider their professional development in many areas simultaneously. Mathematics is just one of the subjects they teach in addition to reading, writing, social studies and much more. Many of our colleagues simply feel that they don’t have enough time to pursue their own learning. The day-to-day demands of running a classroom are grueling. I cannot overstate this reality, but perhaps a quick glance at most of our classroom schedules could serve as convincing evidence. Furthermore, learning about mathematics might not be on the top of some of my colleagues’ professional development desire lists, even if they were given time to study. In this way, my working in a classroom as well as serving as the math coordinator has been an asset. I am not removed from the teachers’ concerns. I experience the “problems” they experience; I understand the frustrations they face. And I don’t have time to plan more workshops than they would ever have time to attend!
It is not my understanding that the countries that rated the highest according to TIMSS introduce drills before concepts. Data for the TIMSS (Third International Mathematics and Science Study) were gathered in grades 4, 8 and 12 from 500,000 students and from teachers. The seven countries that had significantly higher scores than the United States were Singapore, Korea, Japan, Hong Kong, Netherlands, Czech Republic, and Austria. The TIMSS curriculum analysis showed that most US curricula were unfocused; they attempt to cover many more topics and they involve much more repetition than is found in most countries. In other words, “we attempt to do everything and as a consequence rarely do it in depth, making re-teaching all too common.” (Van de Walle, 2004) The curricula and instructional approaches in the US were found to be “less in line with cross-national commonalities than are the demanding curricula and classroom practices found in many high-achieving countries” (Babcock, 1998). For more information on the TIMSS see the Eisenhower National Clearinghouse (ENC) website: www.enc.org.
The TIMSS findings do not support a number of currently popular demands or the “back to basics” movement. Despite poor results, students in the United States do more homework and spend more time in class receiving instruction than the students in high achieving countries such as Germany and Japan. Tracking, common in the United States, is not practiced in the high ranking TIMSS nations either. The fundamental differences in the instructional practices showed stark contrasts between the countries. “The typical goal of a US eighth-grade mathematics teacher is to teach students how to do something. The typical goal of a Japanese teacher is to help students understand mathematical concepts.” (US Department of Education, 1997)
The National Science Foundation (NSF)-funded, “reform” curricula, which we are using, Everyday Math and Investigations in Number, Space, and Data programs, are more closely aligned with the practices in the high-ranking countries. These NSF-funded curricula were developed to help students understand mathematics and become confident in their abilities to do mathematics and solve problems. Students in reform programs perform much better on problem-solving measures and at least as well on traditional skills as students in traditional programs (Van De Walle, 2004; ARC Center, 2002; Bell, 1998; Boaler, 1998; Fuscon, Carroll, & Drueck, 2000; Reys, Robinson, Sconiers, & Mark, 1999; Riordin & Noyce, 2001; Stein, Grover, & Henningsen, 1996;Stein & Lane, 1996; Wood & Sellers, 1996, 1997.)
Research on how children learn also suggests that children must be active participants in the development of their own understanding. The teaching of mathematics is shifting from “preoccupation with inculcating routine skills to developing broad-based mathematical power. Mathematical power requires that students be able to discern relations, reason logically, and use a broad spectrum of mathematical methods to solve a wide variety of non-routine problems.” (National Research Council, 1989) The repertoire of skills needed to succeed in today’s economy has expanded and changed radically. We must educate children for their future, not in our pasts. A great deal of research shows that in fact the way we learned math wasn’t successful in preparing us, as so many adults are math phobic. For more information on changes in the teaching approaches used for mathematics learning today I recommend the following readings:
Teaching the New Basic Skills: Principles for Educating Children to Thrive in a Changing Economy by Murname and Levy (1996) Chapter 2.
Math: Facing an American Phobia by Burns (1998)
Everybody Counts. National Research Council (1989)
Setting the Record Straight About Changes in Mathematics Education (NCTM)
Before It’s Too Late. The Glenn Commission 2000
Beyond Arithmetic by Mokros, Russell, & Economopoulos (1995)
The Mathematical Miseducation of America’s Youth: Ignoring Research and Scientific Study in Education by Battista (1999) February issue Phi Delta Kappan.
If a youngster is in a traditional program in 2nd grade, he or she might appear to be more advanced if the evaluation was simply assessing fact proficiency. Traditional programs focus more on memorization and use more drill techniques. A great deal of time and emphasis is on basic operations and computation. Therefore, they might perform better on a “mad minute,” a timed test on basic fact recall. However, in this process, students are not necessarily developing the number sense they will need in the future. Furthermore, all the other content strands of mathematics- geometry, algebra, measurement, data analysis and probability and the process strands- problem-solving, proof and reasoning, communication, and connections aren’t getting the attention they deserve. The students in an Everyday Math or Investigations Program are getting more mathematics. There is a huge gap in their favor.
There are curricular materials and ideas for providing challenge and enrichment; however, the teachers in our setting must come up with more options for their students. A large part of my job as the math coordinator has been providing teachers with extra materials and activities for enrichment and challenge. Every year I spend most of my budget on new books, lesson ideas, and materials for this purpose. It takes time for the teachers to familiarize themselves with all these “new” activities and to adjust to the range in abilities in their classrooms. With only teachers in the room, it is difficult to manage four levels of activities, and yet, when the children aren’t quite old enough to be sent off independently, this is the dilemma the teachers face.
Mathematics isn’t an isolated subject only related to computations. Learning about mathematics develops our thinking and enables us to appreciate our universe. There are mathematical concepts entwined in the structure of living cells. Mathematics is everywhere and thinking of it as mere proficiency in arithmetic does not do it justice. A parallel view would be to think learning the alphabet was all there was involved in learning to read! Most people appreciate the difference between reciting their ABCs and being able to read and discuss Tolstoy – do those people really think mathematics is only about arithmetic? As Pappas describes in her books so eloquently, mathematics is so much more than doing calculations, solving equations, proving theorems, doing algebra, geometry or calculus. Mathematics is even more than a way of thinking. “Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider’s web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe.” (from More Joy of Mathematics by Theoni Pappas, 1991)
“Over thousands of years, mathematics has revealed its influence in art, mysticism of numbers, commerce and trade, architecture, the sciences….Mathematics is so subtle, pervasive, and necessary in our daily lives, we often are oblivious to its presence. Yet, with each passing day, mathematics is expanding its realm, placing its mark in more and more areas. Today, science could not function without mathematical tools, nor could we bank, build, travel, be entertained, enter the realm of electronics, invent, or explore the universe.” (Theoni Pappas from Mathematical Footprints, 1999)
Every child deserves all the glimpses into the universe he or she can experience!