Mathematics

To Handle The Daily Math Problems in Life—And To Sharpen Thinking Skills

On one level, the purpose and value of math seems obvious. Students need math to function as adults: to pay bills, make change, negotiate salaries, and so on. Virtually every career today requires some mathematical skill, and specialized careers usually involve advanced mathematical knowledge.

But the importance of math goes much deeper than these obvious benefits. Mathematical training is important because it develops a student's ability to think logically and precisely. A proper math education gives an individual the capacity to approach all areas of his thought and life with rigor and discipline. When Abraham Lincoln wanted to sharpen his general reasoning skills at age 40, he did so by working through Euclid's Elements. A successful thinker must be able to structure and organize his ideas, and bring order to his mind. Mastery of math develops this mental capacity in students.

LePort students love numbers! A teacher wanted to show her 4th grade class a documentary. Because of a special event, however, the class has to end 15 minutes early. The teacher commented to the class that unfortunately, she didn't think there would be enough time to watch it. The students, eager to see the film, protested. "How long is the documentary? Maybe there is time?!" One student, looking at the box, saw that it was 32 minutes long. Immediately another student said, "OK, if the class is 50 minutes, and we have to end 15 minutes early, and the movie is 32 minutes, do we have enough time?" The class quickly surmised that there was enough time, with three minutes to spare, and sat down to enjoy the film. The problem was simple—just an easy arithmetic calculation. What was profound was their enthusiastic willingness to use their mathematical knowledge to solve the problem. That's the enthusiasm that real understanding makes possible.

Facts Practice builds a trust in one's ability to compute! Imagine yourself ordering a Latte at a coffee shop—let's say it costs $3.25. You hand the cashier a $5 bill, and he rings up the order. Then you realize you have a quarter, and you give it to her. He stares back, like a deer in the headlights. He has already rung in the order, and so a mental calculation is required. What's wrong here? Certainly there are cases where the cashier simply cannot figure out how to proceed. But in most cases it's not that—rather, it's an underlying phobia of numbers, a distrust of the process of calculation, a process that has been rendered fuzzy through schooling that did not place a premium on precision. LePort's facts practice program accomplishes the opposite; it creates in the student a general trust in his ability to calculate, and a willingness to engage in the process. When, in daily life, he unexpectedly encounters a need to compute an answer—say to figure out if $20 is enough to buy something that costs $18.50 with 8% tax—the computation is not approached with trepidation. Numbers are not to be feared!

Skill Mastery + Conceptual Understanding Through Sequential Instruction

LePorts' Dual Emphasis: Skill Mastery and Conceptual Understanding

Some schools emphasize rigor, fact-memorization, and hard skills in math. Other schools focus on developing conceptual understanding and creative mathematical thinking. At LePort, we know that neither of these approaches works without the other, so we emphasize their unity.

At our school, math education is about more than just memorizing and applying the rules. Students must understand what the rules mean and why they work—including any memorizing necessary to apply that knowledge. A successful math education involves both understanding the material conceptually, and learning how to apply it accurately in practice.

The following three components of our program work together to provide such an education:

• The facts practice component helps students build computational speed and retain basic facts such as mental multiplication (9 x 12 = 108), fraction-percentage equivalents (1/8 = 12.5%) and measurement conversions (1 mi = 1760 yds). Students start each class with a timed facts practice quiz that encourages them to improve their scores day after day. When someone says, "Baa, Baa, Black Sheep, Have you any _______", the word wool immediately jumps to mind. No effort is required. The goal of our facts practice program is to enable our students to similarly access a range of math facts without effort—so that they can instead use that effort towards analyzing a complex problem.
  Students are motivated to compete against themselves in facts practice. They chart their performance daily, set goals, and see if they can improve on their previous day's score, without focusing on their peers' scores. This enables each student to focus on achieving his personal best.
  
  

At LePort, they care enough and are able to tailor the curriculum so that every child can succeed. For example, they pulled my son out of math class, and tailored a curriculum specifically for him. They gave him small group attention, to work through math at a level and speed that he could master. In a normal school, he'd be falling more behind; here, they tailored their approach to him, recognizing that he was stellar in reading in writing, but needed help in math."

Kevin G., LePort Parent, Mission Viejo
• With the daily lesson and review, students learn a new mathematical principle or method in each class. In this part of the class, which forms the substantive portion of the LePort Curriculum, the instructor uses diagrams, examples, and physical materials to help students grasp mathematical concepts. Following the class lesson, students work through exercises in order to practice the concepts they have been learning. For more advanced students, the problem set is supplemented with extensions and elaborations that deepen their grasp of the lesson.
  
  In contrast to many schools, methods of solving problems are always made explicit. Many students can often "intuitively" answer problems, such as finding the common denominator of 1/12th and 1/8th, without being introduced to a formal method. However, without a formal method these same students often struggle later when the questions become more difficult—say finding a common denominator between 1/25th and 1/29th—or when the format of the questions change—e.g. if the student encounters the same type of question as a word problem, rather than in the form of numerals. At LePort, we ensure that students retain the method explicitly, so that when they later encounter more sophisticated versions of the same type of problem, they're able to answer it. For instance, students learn a technique to find the least common multiple of two or more numbers, a technique they can then employ whenever they need a common denominator. Students learn to do math somehow, that is, with an explicit method, which not only leads to success, but eliminates the feeling that math is something that you're either good at or not.
  
  In teaching these daily lessons, LePort Schools draws heavily on the Singapore Math Primary and Elementary Math series of workbooks and textbooks. Singapore Math employs a unique pedagogical approach of moving from pictorial to concrete to abstract levels of understanding. The program has gained recognition among homeschoolers, academics, and recently even some public schools for its integrated and sequential presentation of topics.
  
  
• Word problems train students to bring their skills to bear in an applied way, and help us ensure students understand the concepts, rather than memorizing process steps by rote. Our word problems usually do not introduce new mathematical concepts. Rather, they are designed to make students apply known concepts in contextually sophisticated ways. They tend to involve multiple steps, and to require a critical thinking approach. Students are given guidance and training in how to employ this approach. The purpose of the word problems is to develop a student's ability to analyze a mathematical situation and figure out which method to use to solve the problem.

At LePort, Even Wrong Answers Result in Learning!

Consider this 4th grade question: There are 100 people on a plane. There are 10 more women than men on the plane. How many women are on the plane? In answering this question, many fourth graders think as follows, "Half of 100 is 50, but there are 10 more women so there must be 60 women". This common and natural mistake provides a wonderful learning opportunity—if a teacher has students think through why this is a mistaken approach. A LePort teacher would not simply tell a student his answer was wrong. He would ask: "Well, then, if there are 60 women, how many men are there?" When the student answers "40", the teacher points out that the difference between 40 and 60 is 20. This immediately raises an interesting question in the child's mind—and fuels the investigation that leads to the discovery of his error. In contrast to a student who is simply told that he got the wrong answer and is either given the correct answer, or an explanation that he cannot properly internalize, our approach develops his capacity and motivation to do math problems, because he finds the process of discovery tremendously satisfying.

LePort's word problem program recognizes that math is not only a science, but is also an art—as a skill to be tried, tested, modeled, and acquired through practice. As students grapple with and solve problems, they are guided to make their understanding explicit. The result is a satisfying awareness of their own thinking process, and the experience of math as a rewarding process worth engaging in.

Individualization

An important aspect of the LePort method of teaching math is that our programs are individualized. Students who are advanced are challenged to push forward at their own pace, while remedial students are afforded an opportunity to receive extra assistance. LePort achieves this individualization as follows:

• Individualization within assignments. For each lesson, teachers have exercises that require varying depths of understanding and analysis. While all students must learn the basic principle, the depth at which a student is expected to grasp and apply that principle depends on his ability level.
  
  
• Individualization through enrichment. In addition to their core assignments, students are often placed in enrichment programs based on their ability level. For instance, a mathematically strong student may only be required to complete the odd numbered questions of his class assignments, and then in the remaining time work on a series of exercises on geometric constructions. Enrichment work is based on the conjunction of a student's abilities, needs, and interests—some students enjoy working with their hands (using compasses to draw circles and prove theorems), while others enjoy learning about a subject's history (researching a famous mathematician and explaining his achievement). All students master the core academic program—and then each is encouraged to pursue enrichment work as appropriate.
  
  

 

• Individualization through STAS pull-outs. There are, on rare occasions, cases when other forms of individualization within class are not sufficient to meet remedial needs of a child in a particular class. (This occurs in particular when students enter LePort mid-way through the year.) In such cases, rather than have the remedial child lose confidence while struggling through material he is not prepared to understand, LePort recommends "pull-out" classes for such students under its Supplemental Tutoring and Academic Support (STAS) program. Instead of working within a classroom setting, STAS students work individually or in small groups with a different teacher. The goal of such programs is to re-integrate the child into the standard classroom when he is prepared to make the transition.

Joyful mastery of an essential tool for understanding the world

At LePort, students come to see math as the very practical science of measurement, not just a game in which the goal is to learn the rules, and score well on tests. Students see, across classes, that every skill in math is a tool that helps real people solve real-life problems. In history, they see figures such as Aristarchus and Galileo use math to transform our understanding of the physical world. In science, they see that observable phenomena like velocity and momentum can be quantified into mathematical formulas.

LePort students do not merely acquire the ability to exercise mechanical steps. They acquire a tool of measurement. When a fourth grader learns how to calculate area in square feet, she applies her knowledge through exercises such as estimating the size of various rooms at school and at home, and habituates the ability to assess the square footage of a given area. As a result, when later in life she hears that a house is 2,000 square feet and another is 4,000 square feet, that information will be meaningful to her. When an eighth grader learns about the Pythagorean theorem, he doesn't just memorize a formula—he discovers triangles everywhere, in roads and buildings and bridges. He looks around and sees the world as the engineer he has started acquiring the capacity to become.