New York Institute of Technology The Graduate School of Education and Psychology EDMA- 625 Mathematics, Science and Technology in Education 1 Daniel Stein, Adjunct Professor - http://nyitweb.webhop.org ♦ NYC Dept of Mathematics ♦ Everyday Math ♦ Impact Math ♦ Dept. of Science ♦ Science Standards ♦
Lynne M. Bailey Monday, October 09, 2006
Impact Math Overview NYCDOE uses Impact Math in the middle schools. Please watch the Powerpoint Presentation on Impact Math , and review the details of the approach. Summarize the approach, and comment on 5 important concepts.
Impact Math is an integrated program with four content strands: 1) Number and Operations, 2) Algebra, 3) Geometry and 4) Data and Probability. The integrated approach
Impact Math Content Strands: • Number and Operations • Algebra • Geometry • Data & Probability
provides both “traditional” and “reform” methods encompassing direct instruction, skill practice, student investigations and problem solving. In addition to text book materials, there is an extensive web site with supplemental instruction materials and assessment tools, available to both students and teachers. This comprehensive program is designed for middle school students, grade 6 – 8, and includes the equivalent of a full year of algebra by the end of the eighth grade. Three principals guiding the development of the program were 1) respect for the experience and knowledge of teachers, 2) recognition of the competency and energy of typical middle school students, and 3) development of materials that are both inclusive and intellectually challenging. There is a three part sequence to instructional flow: Introduce, Develop and Assess, and a three-part sequence to the learning flow: Expose, Develop and Apply. Teacher materials seem quite good, and rather than spell out hourly lesson plans, sections are presented as modules which may encompass several class sessions, but are designed to fit into single and double 45-minute learning periods. The five important concepts that are promoted by the publisher are:
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Informal-to-formal concept development Focus on mathematical thinking, and developing such thinking in varied contexts Algebraic thread in a comprehensive program, completing one year of Algebra by the end of the 8th grade (and 3 year course) Balanced Instructional Approach (Introduce, Develop and Assess) Strong Teacher Support
Other significant principles that flow throughout the curriculum include: • • • • • • • • • • • • •
Theoretical understandings partnered with basic skills Concept development through student investigations (Explore and Think & Discuss) Use of manipulatives and calculators Organization by content Logical reasoning that supports abstract thinking Consistent Practice Problem-solving within engaging context Consistent, ongoing practice with ample review Connections between all strands of mathematics Online instructional support Narrative text with grade level reading Illustrated cartoon characters explain, and demonstrate different concepts and ideas, and problem solving strategies Communication and representation through Share & Summarize activities and homework writing assignments, In Your Own Words
Gradual Concept Development, from Informal-to-Formal Mathematical concepts are presented throughout the course, building from informal observation to formal concept codification. The spiral design is integral to the three-year course. All four strands are woven together and evolve throughout the course. The algebra strand, for example, unfolds gradually, from fundamental ideas about patterns, function, and proportional reasoning to grow into algebraic structure, allowing the mastery of skills to build and strengthen over time. The authors based the algebraic strand on a highly successful program, Access to Algebra, created by Curriculum Corporation in Australia. This program presents materials relevant to middle school students, including interests and experiences they can relate to.
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Informal-to-formal concept development
Developmentally Developmentally appropriateconcept concept appropriate development development
Solving Equations
All courses use variables, multiple expressions of algebraic relationships, and solving equations. Algebra is the notation used for ‘generic’ math. Students learn early to use variables and create equations. To see how the curriculum spirals, consider the study of functions and their graphical representation. The first course requires only that students sketch and interpret graphs portraying real situations. They discover how a change in one quantity affects another quantity. In Course 2, linear equations are explored in depth and students learn that there is a relationship between equations and graphs. They can group these relationships by the shape of the graphs and associate them with the different types of equations. In the third year, the topic is revisited. This time, though, students make a more formal analysis of function families. Using technology, such as graphing calculators, they investigate how constants and coefficients affect graphs, and learn to discern the type of function from its equation or graph. They can express functions using variables and graph these equations. This conceptual development repeats itself in the other content strands. It allows students to perceive relationships between the strands. Skills and familiarity with these ideas evolve throughout the curriculum. The geometry strand evolves from the two-
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dimensional in Course 1, to the three-dimensional in Course 2 and on to symmetries and transformations in Course 3. Algebra 1 by the End of 8th Grade Basic numerical literacy, also known as ‘numeracy,’ includes algebraic thinking, and is just as important to successful participation in the workplace as reading and writing skills. As previously detailed, algebra is one thread that builds throughout the three year curriculum and by the end of the three year curriculum students have finished the equivalent of Algebra I. It is felt that student experience in early algebra influences their general view of mathematics. This program seeks to extend natural logical thinking processes by providing enjoyable and engaging activities students can relate to real world situations, fostering further interest in continued studies. In today’s technological society we all need math skills, regardless of what careers we pursue, together with the ability to gather and evaluate a myriad of information to make informed decisions. Impact Math contributes significant opportunities for students to gain confidence in these abilities and strengthen their skills. By completing this course of study, middle-school students gain a competitive edge, and are enriched with a better skill set for making appropriate choices. Narrative Writing Style, Cartoon Depictions Throughout the text the authors use a direct, narrative writing style with realistic contexts. Cartoons depict students explaining how they approach problems, and provide opportunities for students to choose or create their own problems. I believe this is a very important component to the success of student learning and achievement. Here are some examples:
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This style of presentation is, on its own, an engaging element of the course, and helps to explain and demystify mathematics. Concept Development via Direct Investigations; Use of Manipulatives Conceptual understanding is vitally important. It’s far easier to remember mathematical rules and procedures when you comprehend why you are doing them. Likewise, the mastery of computational skills enables students to look at the big picture, and not spend endless time completing abstract problems without reason. Students want to know why they’re doing something. By discovering the ideas through direct investigation, they discover the big picture, see how these ideas relate to each other, and then can practice skills that use those ideas. The excerpts below depict a section on adding and subtracting fractions. There is a Think & Discuss introduction, and a direct investigation activity that uses manipulatives.
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Interactive, direct investigations, is one of the truly engaging components of Impact Math. These investigations provide students with direct modeling using the cartoons, several worked-out examples and Problem Sets to apply the investigation. Investigations are designed to last for two class periods when beginning a section, and the others for one class section. As many (most?) middle-schools now have double math periods, these investigations are tailor made to fit easily into the school schedule. These constructivist exercises help students to discover different ways of thinking numerically and build their comprehension of mathematical concepts. Students are naturally drawn to hands-on activities, and Impact Math provides many opportunities for this. Another example of the use of manipulatives, this one, an introductory Explore activity, Block Patterns, is excerpted below.
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In the example above, blocks are used to explore different patterns. Impact Math provides pre-packaged Classroom Manipulative Kits for each course, for classrooms of up to 28 students. These kits include 500-2cm colored, wooden cubes, 400 color tiles, 36 pairs of numeral cubes, 14 GeoMirrors, 14-1cm overhead grids, 600 two-color counters, and six sets of Linkage Strips. These objects totally support content learning through the investigative activities. Calculator usage is also introduced, but only after students have proven they can perform tasks without them. Indeed, paper, pencil, graph paper, rulers, and protractors are manipulatives as well, and students should be able to use them appropriately as well. Technology Options with Online /CD Materials Included with Impact Math are several items that use more recent technologies. There is a StudentWorks™ CD-ROM that contains the entire text book. With this
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students can study from any computer with a CD drive and Acrobat reader, no Internet access required. There are interactive assessments on the CD and links to the Internet for further exploration. With an Internet connection, students can also find these activities on-line. The web site for the book has short movies, called Brain Pops that teach and review different concepts. On-line quizzes and tests are available as well. The publisher provides videos for teacher to use in class, What’s Math Got To Do With It?, which connects mathematics to the real world. Another CD gives students in-depth interactive activities that once again, use real world contexts to promote understanding of primary mathematical concepts. In Conclusion Essentially, Impact Math is a repetitive, spiraling program that engages the middle-school student with relevant hooks. It promotes conceptual understanding, linking mathematics to material issues, and does not dwell on repetitive drills. It bridges the abstract to the physical with realistic problem solving and manipulatives. According to 7th grade teacher, Donna Vogel, “I think before, we kept thinking numbers, numbers, numbers, and Impact Mathematics has shown that math can apply to many different areas in life. Also, it’s given us a more hands-on approach, which I’m learning to enjoy.” Another teacher at Butler Junior HS, in Bulter, Pennsylvania, Amy Bodamer, remembers when Impact Mathematics was first chosen [for their school]: “When we first saw it, I have to say I was nervous. At first, the textbooks seemed to be more ‘fluff’ than content, but we quickly realized that Impact emphasized even more algebraic skills than the traditional texts we’d been using. The major difference from the traditional texts we were using before is the de-emphasis on drill and more emphasis on learning concepts. There’s a lot more reading and comprehension; it’s not just one example and then 30 problems done exactly the same way.” 1 With a curriculum that encompasses both traditional skills and “reform” strategies, it is no surprise that Impact Math has found a place in so many schools. 1
p. 9, Results with Impact Math (http://www.glencoe.com/sites/common_assets/mathematics/rb_portfolio/Impact_combined.pdf)
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