Focus on Mathematics By David and Pam Woodbury With the school year starting, we are finding strong interest in our math offerings. So we'd like to share our thoughts and recommendations with you for assembling a solid math program and approach in your homeschool. What we offer has been used in our home by our three children with very good results. In the past year, our 16-year-old daughter scored a perfect 800 on the SAT math section, and received an A in AP Calculus. Our 12-year-old daughter won first place in a 7th and 8th grade math competition, sponsored by a local community college, as well as first place in the Math-Physics section of our regional science fair, where our 10-year-old son also won first place for his age group in the Math-Physics section. The next few sections lay out our thoughts on what to cover in the area of math. The grade levels shown are meant to serve as a guideline for you, and you'll see that the plan we present ends at tenth grade to allow two additional years of advanced study for students that show a strong interest and aptitude for math. It's also reasonable to stretch out the plan we present 1 or 2 years to allow for additional focus in specific areas.

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Birth through Pre-Kindergarten Well, we're kidding on the birth part, but you can start introducing numerical and mathematical concepts at a very early age, say as early as 3 to 6 months old. With very young children, you can play number games, which are great fun and encourage bonding between you and your child. Parents naturally teach their toddlers about their bodies and surroundings. For example, dad places little Jonny on his lap, he touches his nose and says, "That's my nose! Can you show me your nose?" Then, and this is the key, he says, "One nose." "One nose." How radical is that? You are not supposed to introduce numbers at 6 months, that's what they do in kindergarten! But, young children catch on, and they catch on fast. Soon, you've graduated to "Two ears, but just one nose. I wonder what it would be like to have one ear and two noses!" Can you imagine the fun that you and your child would have doing this? And, at the same time, your child will be learning concepts such as quantity and numerical comparison. They will also be learning that you are comfortable with math, and that math is fun. As your babies get older, encourage the use of comparison words such as big, bigger, biggest, and small, smaller, smallest. Encourage playing with shapes. And help you child discover math in their surroundings. For example, we have a traditional sofa/loveseat arrangement in our family room. But, we don't refer to them by their traditional names. We call them the "two couch" and the "three couch" after the number of cushions each has. Take these ideas and make them yours by coming up with games and activities that make sense for your family. And by doing so, you'll be well on your way towards a solid math program for your children.

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Kindergarten through Third Grade If you've skipped ahead to this section, because maybe you no longer have pre-kindergarten aged children, go back and read that section anyway. Even with older children, and especially if you need to create a math friendly environment in your home and life, you'll want to adopt some of those ideas and others such as kitchen math, where you use cooking and recipes to introduce concepts such as fractions (1/4 cup of flour) and division (make half as much as the recipe calls for). Use the "wrong" size measuring cups to make proportional relationships real. For example, if a recipe calls for one cup of flour, you could say, "We need one cup of flour, but we only have the quarter cup measure clean. How many quarter cups will we need to make one full cup?" Then, as they get older, the question could be, "How many three-quarter cups do we need to make 3 cups," etc. By this age, your child is ready for manipulatives such as Pattern Blocks and Cuisenaire Rods. Pattern Blocks include yellow hexagons, orange squares, green triangles, red trapezoids, blue rhombuses and white rhombuses. With all these shapes and colors, your child can make an endless variety of designs, and get a head start on geometry. And, he will likely enjoy his pattern blocks for many years. Our kids often play with pattern blocks during read-aloud time. They keep their fingers busy, create wonderful designs, and hear a good book all at the same time! You can see a couple of terrific examples of pattern block creations, in Pam's "Math Beyond the Basics" presentation. In addition to the use of games, manipulatives, and life experiences, this is the time that you'll want to begin a formal math program, supplemented by logic and problem solving learning opportunities. For the first grade through third grade years, we recommend Miquon Math along with a set of Cuisenaire Rods, which you'll need for the Miquon exercises, as your core curriculum. Mind Benders and Figure-it-Out booklets provide excellent opportunities to stretch their logical reasoning and other problem solving skills. Miquon Math introduces many advanced concepts, presenting them within a context that young children of many different learning styles can appreciate. However, Miquon Math does not provide a lot of drill in their program, so you will need to supplement with a drill program, and for that we recommend the CalcuLadder workbooks, which, as the name implies, consist of drills for developing calculating proficiency. By the way, if you have a computer and printer, the CalcuLadder MasterPaks are our best value. The MasterPaks include a CD-ROM with 3 levels of workbooks for $21.00, whereas the workbooks are $16.00 each.

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Fourth Grade through Sixth Grade By this age, your child has probably finished all of the Miquon Math and is ready to move on. So, what comes next? Well, after years of trying many different curricula and approaches, we've settled on Singapore Math. Our children love Singapore Math, and you may have heard that Singapore Math is frequently given credit for the high, standardized math test scores achieved by school children in the country of Singapore. We recommend a three-year sequence for Singapore Math. You start with levels 4A and 4B for the first year, levels 5A and 5B for the second year, and finally, levels 6A and 6B for the third year. We carry the home instructor's guide for each level or you can purchase a single comprehensive answer key that covers all three levels. With Singapore Math, concepts are taught clearly and comprehensively. For example, Singapore Math presents multiple techniques for solving specific types of problems. Learning multiple approaches for the same problem helps a child appreciate that "there's more than one way to skin a cat." Giving the students a chance to use all of their math and thinking skills, Singapore's word problems are especially thought provoking. In addition, Singapore Math introduces advanced concepts, such as geometry, which help you to continue to build a solid foundation for the middle-school and high-school years. If you are switching to Singapore Math from another math curriculum, and are unsure where to start, visit their website where they have useful placement tests and a forum where ideas and questions are shared. No matter how good a curriculum is, any student can benefit, and get much enjoyment, from working through problems that require more than just your basic math skills to solve. Again, for logic and other problem solving techniques, we recommend Mind Benders and Figure-it-Out booklets. While you may have started using these items before, they continue on with harder problems for the older children.

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Seventh Grade through Tenth Grade Once the basics of mathematics have been taught, understood, and made second nature, it is time to move on to higher-level mathematics such as algebra and geometry. While some students are ready for this step earlier than others, most students are ready for algebra by the ninth grade. Before beginning your child in algebra, try to ensure that they have a strong working knowledge of fractions, decimals, percents, and ratios, and that they are capable of understanding abstract ideas. Our favorite texts for higher-level math are by Harold Jacobs. Jacobs does a wonderful job of presenting the material with interesting and often times comical applications. He then goes on to help the student understand the whys behind mathematics and not just the hows. In this way, the student is better prepared for continued success in future mathematic studies. We love the Harold Jacob's math books for their explanations. The kids like them for their cartoons. By using cartoons to introduce new concepts, Jacobs engages the reader and quickly shows the student how the mathematical concept being taught applies to the real world. Jacob's Algebra Each chapter has four problem-set sections. Section I is review, sections II and III are basically the same and cover the new material, and section IV is typically a one-problem mind stretcher. The answers for section II problems are found in the back of the book, while section I, III, and IV answers are found in the teacher's guide. We would typically have my daughter work problem sets I, III, and IV and then use problem set II if extra practice was needed. We particularly like the fact that there are many problems available for practicing the new techniques and concepts learned. This gives the student the opportunity to really understand what he is doing before moving on to the next chapter. Especially when our oldest was having difficulties with distance-rate-time problems, the additional practice helped her cement the concepts into her mind. Since math skills build on each other, it can be detrimental to the student to not really learn one idea before moving on to the next. Since Jacob's Algebra also covers Pre-Algebra, many children are ready for this book immediately after completing Singapore Math 6B. Other students may need additional work in key areas, and for this, we recommend the Key Curriculum Press "Keys To" series. Also, while it is not a complete Algebra II course, many people use this book to teach both Algebra I and Algebra II. So, you can feel free to work at your child's comprehension speed and take one to two years to complete the book. In our case, after completing both this book and Jacob's Geometry course, our oldest was able to successfully complete a Pre-Calculus course and then enter into AP Calculus the following year. Others may benefit from completing another year of Algebra II before continuing on. Jacob's Geometry In the tradition of the Jacob's Algebra book, this book uses comics and other real-world illustrations to grab the reader's attention. Jacob's goal is to engage the student and enable him to discover ideas. He does this through carefully sequenced and structured exercises that lead the student not only to learn the ideas but also to internalize them and make them his own. While proceeding through this book, the student will learn the difference between statements that appear true based on an illustration or because of a few measurements and statements that have been proven formally. Each chapter has three problem sets. Sets I and II are both related to the chapter concepts, with Set I being the easiest, while Set III is a more puzzle-type thinking-skills problem. Set I can be used for illustration purposes or to prepare for the Set II problems. Answers to selected problems appear in the back of the book, while the others are contained in the teacher's guide. After Geometry What to do after geometry really depends upon the student and his interests. If your child is interested in attending college and pursuing a degree in science, math, or engineering, then a Pre-Calculus/Trigonometry class, followed by Calculus, is typical. Many students will attempt one of the Advanced Placement (AP) exams for Calculus. (There are two levels of AP Calculus that are recognized.) For other students, Jacob's Mathematics: A Human Endeavor, an exciting survey of many different math topics such as number theory, statistics and topology, is a good option. We also recommend this book, if the student has time to fit it in, to those who are going on to higher levels of math in their academic careers.

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Final Thoughts Our presentation would not be complete if we did not encourage you to involve your children in math clubs and co-ops such as those that participate in Math Counts (for 6th thru 8th graders) and Math Olympiads (for 4th thru 6th graders). Also, computer science is closely related to math, and can support you in developing students that can think logically and algorithmically. We hope you've found this presentation of our approach towards math instruction useful towards helping you construct your math program. Please let us know your thoughts, ideas and any questions you have regarding this wonderful subject by sending us an e-mail to: support@woodburyschool.com.

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