Back in December, U.S. schools learned the results of the 2003 Program for International Student Assessment (PISA), a test that focuses on 15-year-olds’ capabilities in reading, math, and science literacy. The results echoed a familiar refrain in recent years: U.S. students performed lower on average than their counterparts in the participating countries in both math literacy and problem solving.
Worse still, the PISA results highlighted an achievement gap that continues to plague U.S. students, with white and Asian students outperforming Hispanic and black students. The United States also demonstrated the strongest relationship of any country between socioeconomic status and student performance. Less affluent U.S. students were outperformed by their wealthier peers, especially in problem solving.
“The PISA report is a clarion call for U.S. policy makers and educators to marshal forces to provide an equitable education for all students,” said Cathy Seeley, president of the National Council of Teachers of Mathematics (NCTM). “These results demonstrate the urgent need to look at what kinds of math programs we are offering to children in less affluent schools. Are we limiting the math experiences of our children most in need?”
The recent PISA assessment concentrated on the math literacy and problem-solving skills of 15-year-olds to measure their knowledge in the context of everyday situations. The information gathered from PISA 2003 supports the idea that students who are literate in mathematics and who understand how it can be applied are better able to solve problems they have not encountered previously but will face in real-life situations outside the classroom. The PISA results clearly point to the need for better math fluency in U.S. schools and a stronger connection between problem solving and real-life situations, NCTM said.
Despite the disappointing results on a national scale, there are many schools where students are learning math in the context of real-world situations and succeeding. And technology is playing a role in this success in a number of locations. In this Special Feature, we’ll examine some of the ways that software providers and school systems are stepping up to meet the need for greater math literacy and higher achievement.
Learning on the go
Recognizing that some students will need all the practice time they can squeeze into their busy schedules to attain math fluency, Valiant Technology has answered the call for anytime, anywhere learning for students with its MathAmigo PDA-enabled program. MathAmigo supports the teaching of mathematics from counting to pre-algebra and can be used anywhere a student goes. From the classroom to the bus, or from the playground to the home, students can use one of three kinds of handheld devices to practice math skills at their own pace, whenever they have a spare minute or two for study.
The software offers individualized activities to meet the needs of each student. It provides students with instant feedback on their work and tutorials to help develop an understanding of mathematical concepts and methods. It also tracks student mastery of concepts and adjusts the difficulty of practice questions on the fly, as appropriate. Besides giving students more opportunity to practice their skills, this handheld-powered software also could help close the socioeconomic gap highlighted by the PISA results, as it runs on lower-cost devices and therefore offers a potentially more affordable solution for schools.
For teachers, MathAmigo automatically grades students’ work to save valuable planning time. When the system is used on a network, it can be set up to send Teacher Alerts drawing attention to those students who need help. Analysis features also help teachers decide how best to help at-risk students, Valiant says. The program’s desktop management application helps teachers keep track of students and assignments, and it aids in lesson planning. A report production feature lets teachers generate formative assessment reports that are color-coded to allow educators to spot weaknesses among students at a glance; teachers can quickly create reports for individual students or groups, the company says, and the software also has a feature that permits teachers to export summative assessment data into other reporting systems.
MathAmigo is available for three different hardware designs. The classic version of the software comes installed on a proprietary hardware device that is designed with younger children in mind. The rugged design is made from thick plastic. MathAmigo also is available for any Palm OS device running version 4 or better, or it can be purchased for use with AlphaSmart’s Dana handheld device. The Dana has both a keyboard and a touch-sensitive screen. Its wide-screen display increases the size of MathAmigo activities.
Tracy Rich, K-12 technology coordinator for the Hart Independent School District in Hart, Texas, said his district uses the MathAmigo program on Palm PDAs for students in grades 3-6. Because the handheld devices are similar to the size of a Nintendo Gameboy, Rich said, the kids “think they’re playing.”
“The MathAmigo program is versatile enough to cover [several grade levels],” he said. “Teachers have the opportunity to pick the [skill] level.” Rich said his district’s use of the software doesn’t replace any of the district’s regular curriculum. “Instead of handing out worksheets and asking [students] to practice math that way, they are doing the math in a more entertaining way,” he said.
Rich added that Valiant Technology actively works with teachers to develop the MathAmigo content.
“I really feel like the software … is in its infancy right now,” he said. “[The company is] really interested in what the teachers and tech coordinators have to say, and they’ll build additional things into the program.”
Math in the context of music
An innovative program that combines math and music instruction for students in grades K-5 is catching on in schools nationwide.
Developed by the nonprofit MIND Institute, the Math + Music program combines non-language based computer math games with specialized piano training to teach K-5 math standards and enhance problem-solving skills. Evidence suggests that the use of the program has led to dramatic increases in math scores on standardized tests by enabling students to use multiple intelligences to learn the material.
The Math + Music program consists of two areas of specialization. Students attend a lab two days a week for 45 minutes each day.
The first lab consists of music training on piano keyboards. “Because music is mathematical in its architecture, the MIND Institute’s Math + Music program teaches children to recognize musical patterns and symmetries, which in turn helps them to also understand fractions, proportions, and ratios,” said International Music Products Association President and CEO Joe Lamond. The music curriculum uses and develops the brain’s innate ability to recognize patterns and use symmetry, he added.
The second lab uses math video games that take a spatial-temporal approach to math concepts. For students to progress to a new level while playing the video games, they must learn the math in order to solve the problems. As the students progress through the games, the software tracks their progress and scores are transmitted to the MIND Institute for analysis. Feedback then is aggregated and sent to teachers and used to help identify areas where improvement is necessary.
The computer games (or lessons) are aligned with the California and Texas math standards for grades K-5. The curriculum is designed to be followed by the entire class over the course of a full school year. Participants in the Math + Music program reportedly outscored non-participants at the same schools by 20 percentile points on the math portion of the 2003 California Achievement Test.
Sarah Smith, a teacher in Queens, N.Y., who has used the program in her classroom, said, “Math + Music lightens my teaching load by making math easier for me and my students. The students recognize concepts they’ve learned in Math + Music and then breeze through the standard curriculum.”
Because the program does not rely on language skills for teaching math concepts, “it levels the playing field for diverse children whose first language might not be English,” said Andrew Coulson, president of the MIND Institute’s education division. “At the same time, its emphasis on critical thinking skills and creative problem solving has also attracted magnet and private schools. And because we provide ongoing evaluation of students’ progress, we are able to assist teachers with accountability.”
Larson Learning provides a comprehensive suite of math courseware that spans elementary-level concepts through algebra. Developed by the well-known Ron Larson, who implemented a successful instructional methodology throughout his math textbooks, the software is designed to prepare students for success at the higher levels of mathematics by strengthening students’ conceptual understanding of math and preparing them for problem solving.
The Larson Math series of software includes a diagnostic-prescriptive assessment feature that places each student into the instructional sequence where he or she has the highest probability of success, according to the company. The software helps build skills and self-confidence by breaking the material down into simple, easy-to-understand components. The entire series incorporates multiple representations, inverse operations, interactive models and simulations, and verbal connections in order to address multiple learning styles.
The series places a heavy emphasis on problem-solving mastery. At each level, the series teaches 14 different problem-solving strategies, attaching meaning and methods to the use of mathematics. Regular assessment checkpoints are included to ensure students’ mastery of key concepts and their ability to succeed at higher levels. Cumulative assessments in every chapter continue to draw upon previously mastered materials, so that students internalize the concepts and are prepared to move on to higher levels.
For teachers and administrators, the program design is consistent, predictable, and straightforward, so it is easy to integrate into lesson plans, Larson says. Student and class reporting features allow teachers to document successes and set a course for continued learning. Plus, extended-use materials add depth and flexibility to accommodate the needs of teachers, students, and instructional programs. These materials include open-ended journal prompts, extra practice pages for skills and concepts, topic quizzes, module tests, and family activity letters.
Larson’s Pre-algebra and Algebra 1 were recently selected as two of the seven applications to be used in a national study on the effectiveness of educational technology interventions. In addition, the company recently signed a two-year contract with the Los Angeles Unified School District, the nation’s second-largest school district, to provide a two-year, district-wide Algebra Intervention Program.
Virginia’s Henrico County Public Schools has been using Larson Learning programs in its high schools since 1999. Steven Lapinski, mathematics coordinator for the district, said that students and teachers in his district love the Larson programs.
“It truly individualizes,” Lapinski said of the software. “One teacher with 25 students in the classroom and six classes each day can’t do that with paper and pencil. What I tell students is that you cannot fail. If the Larson program realizes there’s no way you can pass a test, it stops the test and tells you to go back and review certain topics before continuing.”
Interactive explorations of concepts
Count David Lieberman, mathematics department chair for North Hills High School in Pittsburgh, Pa., among the many fans of SAS in School’s Curriculum Pathways math software programs. Lieberman particularly likes the interactive nature of the web-based contentsuch as its hands-on simulations and other exploratory features.
“I’ve used other kinds of software, but none of them let kids control what they were doing” to the extent of SAS inSchool’s solutions, Lieberman said. He said he was thrilled to see how enthusiastic his students were about exploring mathematics using the Curriculum Pathways software.
“In essence, I had 20 little classrooms going at the same time,” he said. “And they were all doing things that I just can’t do on the chalkboard.” Curriculum Pathways’ web-based math programs target students in grades 8-14. SAS inSchool says the materials employ research-validated instructional techniques and acknowledged best practices. All lessons, learning activities, tools, and simulations integrate standards-based content.
In the introductory algebra courseware, students practice simplifying, solving, and graphing. Resources stress investigation and discovery as well as skill and mastery. This varied approach helps students develop both computational and conceptual understanding, the company says.
SAS inSchool offers 17 Simplifier, Solver, and Grapher tools for introductory algebra, each of which comes with topic-specific recommendations. The program features practice problems with access to line-by-line feedback. Users can view multiple representations of mathematical concepts instantaneously and are also permitted to view problem solutions and submit quizzes to instructors.
The Geometry InterActivities Curriculum Pathway is designed for teachers and students of Euclidean geometry. Each InterActivity contains a multimedia tool that clarifies key elements of the topic. With a teacher’s guidance, students use the interactive features and animations to explore conjectures such as “what if” scenarios. Students can resize and manipulate graphs and figures, rotate three-dimensional models, view dynamically changing measurements, access a calculator, practice a skill, and obtain immediate feedback.
Rhonda Wessman, technology coordinator at Benjamin N. Cardozo High School in Bayside, N.Y., said that SAS inSchool has set the proper priorities for teachers to use instructional technology to benefit their students.
“Curriculum Pathways addresses the higher-order needs of both students and teachers. Students are presented with a wealth of resources and activities,” Weissman said. “More importantly, they are challenged to analyze, synthesize, and evaluate information.”
All materials are correlated with NCTM standards. SAS also has mathematics programs available for intermediate algebra, advanced algebra, data analysis, and trigonometry.
Two brand-new, promising programs
The Scholastic company Tom Snyder Productions is set to release two new math software programs that directly address the deficiencies identified in the latest PISA results: FASTT (Fluency and Automaticity through Systematic Teaching with Technology) Math, for students in grades 2 and up, and GO! Solve Word Problems, for students in grades 3-6.
FASTT Math is designed to help struggling students develop fluency in basic math facts such as addition, subtraction, multiplication, and division. The company says the program helps students achieve computational fluency with basic math facts, freeing mental resources to engage in higher-order math exercises like advanced computation, problem solving, and algebra. The software continuously adapts to student skill levels to increase math fluency by having students take part in 10-minute daily sessions. FASTT Math also includes prescribed worksheets for additional practice in basic math facts or multi-digit arithmetic for those students who need more instruction.
For instructors, a teacher’s guide with assessments and lesson plans is included in the package. The Fact Fluency Foundation Guide includes age-appropriate activities for counting, number concepts, and computation strategies. Twelve detailed performance and usage reports at the student, class, grade, and school levels help teachers and administrators evaluate their students’ performance against state and national standards.
The creators of FASTT Math, Ted Hasselbring and Laura Goin, began collaborating on the research, design, and development of instructional software at the Learning Technology Center of Vanderbilt University in 1984. Their research and development have focused on the use of technology for enhancing learning in students with mild disabilities and those who are at risk of school failure.
For students who are ready to take on real-world, problem-solving challenges, GO! Solve Word Problems can help. The program gives students step-by-step instruction in using graphic organizers to map out mathematical situations for addition, subtraction, multiplication, and division. Research has shown that graphic organizers help students organize information to aid in reading comprehension. Tom Snyder Productions says they can be used in the same way to help students comprehend word problems, too.
Students master mathematical concepts by practicing word problems at their own pace, and the GO! Solve program automatically adapts to the progress of each student. The software includes hints for students and a calculator, notebook, mathematical glossary, and text-to-speech translator. And the text is customizable; students and teachers can add their own names, places, and things to the problems to make them more familiar and fun. The program contains more than 1,000 practice word problems at varying levels of complexity.
Both FASTT Math and GO! Solve Word Problems will be available to educators in June.
Teacher professional development
A new professional development service for math teachers uses streaming video on demand to demonstrate exemplary teaching practices. The Connecticut-based Math Channel offers more than 120 hours of recorded lessons taught by math guru Rachel McAnallen, also known as Ms. Math, who has spent more than 45 years as a teacher, high school administrator, school board member, consultant, and author.
McAnallen’s presentations are geared to K-12 instruction according to age-appropriate skill levels as defined by NCTM. Her pedagogical strategy develops from the belief that parents, guardians, and preschool through first-grade teachers are the most important math teachers a child will ever have.
McAnallen reminds educators that students must use their senses to understand numbersthat a good understanding of mathematics develops at the intersection of real-world, concrete examples and the symbolic numerical value. Her lessons focus on teaching mathematics in concrete, real-world scenarios, reminding teachers that nothing is to be taken for granted when teaching children the basics of math.
One simple lesson reminds teachers that numbers are used as adjectives to better describe qualities of the physical world. For example, McAnallen says that, when counting on their fingers, young children will immediately begin from two when asked to begin from the middle finger. Two, then, has been perceived in the child’s mind as the adjective that distinguishes the finger by name, not as a value that describes the finger as the second in a series. Her lessons often begin from these points of simple confusion in developmental thinking and help teachers sort them out in the minds of their students.
Gary Petersen, math and technology specialist for the Charter Oak Academy in West Hartford Public Schools, Conn., said using the professional development lessons of Ms. Math has given him “a deeper understanding of mathematics and a much deeper understanding of how children perceive mathematics.”
“I use the K-5 lessons, and I have found that a lot of the videos are applicable to multiple grades,” Petersen said.
Petersen gave the example of fractions, which teachers generally start teaching in the third grade. “Using Rachel’s lessons, you can bring the basic concepts down to kindergarten and have [the students] understand them because it’s logical,” he said. “It’s eye-opening to see what [students are] capable of doing versus what we’re allowing them to do.”
See these related links:
National Council of Teachers of Mathematics
Tom Snyder Productions
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