AMATH Systems, Inc.

Have you ever joined a group of people talking about a seemingly complex and interesting topic? After listening for a while you just can't participate because you missed the first half of the conversation. Nothing makes sense. Even after someone tries to quickly bring you up to speed, you just can't get to the same point as everyone else. You feel intimidated and anxious so you leave the group and feel like you missed out. This is how people feel every day as they get left behind in math education.

Millions and millions of people around the world feel like this as they progress through math education. One fails to master a prerequisite topic for one reason or another and the group moves on. Then they fail to grasp the next topic because it built on the topic they just missed and they get left further and further behind. When someone finally tries to get that student caught up, that effort often fails because they do not spend enough time making sure that all of the prerequisites are met.

Foundations works regardless of students' skill level, pace, or lack of prerequisite mastery. Using AMATH Foundations is like having a guide that ensures that you master each concept without missing any prerequisites. Most students begin in math facts and progress at their own pace until they finally master high school level math. Foundations is efficient because when students encounter a topic they have already mastered, the program allows them to quickly demonstrate mastery and move on.

This page provides a very detailed tour of AMATH Foundations. As you read each section, try to picture how the features will apply to your individual needs and then choose the path to success that best fits your situation.

Approximate Audience	Method of Use - Path to success
High School Class College Class Adult Literacy Class Vocational Class Summer School Class Detention Center or Prison Literacy Effort	Comprehensive Literacy Effort/Class This is the primary use of Foundations
Middle school and advanced elementary school Class Students seeking to master or remediate only selected topics Students who have been prescribed areas of study based on diagnostics (not recommended)	Selective Remediation and Reinforcement
Corporate employees seeking advancement Individuals looking to increase their score on the ACT or SAT Individuals preparing for high school graduation, the GED, GRE, or math placement test.	Self Study

After logging into AMATH Foundations, with a unique user number and password, students are directed to the Main Menu or Navigation Screen. AMATH Foundations' content is organized into eight folders. Five of the folders represent a Strand or a Subscale. One folder is dedicated to Problem Solving, and the remaining two folders contain testing/evaluation modules and instructions. The contents of each folder are further divided into topics. Within each topic are a set of related Study Modules.

Notice the folders and topics are numbered and the Study Modules are arranged in a top down design. This highlights the suggested path of study. The strands, topics, and study modules have been very carefully organized. A student following the numbered sequence will never encounter a topic for which they have not achieved prerequisite mastery. We do, however, recognize the demand for accelerated progress and for selective remediation and reinforcement. This is why AMATH is not in lock-step, meaning students are not prevented from accessing a module out of the recommended order.

AMATH Foundations is graphically pleasing, easy to use, and intuitively organized.

Foundations organization promotes step-by-step mastery and helps students and teachers to immediately identify the appropriate area of study.

The small certificate next to the Math Facts/ Addition Study Module lets the student know that module is complete and that he should progress to Subtraction.

Reward, Feedback, and Reporting are an important part of Foundations that is continuous throughout the entire program

Figure 1: AMATH Foundations Main Menu/Navigation Screen

Often, before beginning a math class, students are tested. This helps teachers determine the appropriate level to begin teaching the class. After examining the test results, the curriculum is planned. Because skill levels and progress rates vary, inevitably some of the class will be left behind and others will not be challenged.

In a similar situation, diagnostic tests pinpoint the areas students need math remediation. These diagnostic tests are frequently misleading. Most diagnostic attempts fail to recognize students’ troubles in computation, arithmetic, and even the multiplication table. Diagnostics are inefficient because they fail to identify important, early holes in math literacy.

Foundations makes the formidable challenge of teaching a wide range of students easy. It is the solution to question of where to start an intervention class or remediation effort. AMATH works best when all students start with the very first study module in integer arithmetic. As you will earn in the next section, AMATH’s unique Streak Grading method allows students to complete study modules for which they already have mastery in only a minute or two and then progress at their own pace through the remainder of the program. It works like an insurance policy and a comprehensive review all in one.

The Foundations software handles the tasks of problem generation. It generates the problems, the path, grading, a solution, testing, and documentation of progress. This means teachers will have more time to focus on individual student’s needs. Their most important responsibility when using Foundations becomes motivating the students to progress and providing assistance when needed. AMATH Software handles the rest!

In many cases, often with working adults, AMATH Foundations serves as a complete self-study course. In this case, the student also becomes the teacher and must motivate himself or herself. In schools and for home users, AMATH Foundations functions as a complete, stand-alone, developmental math course. Students can view a solution to every problem that AMATH Foundations generates, and there are billions of potential problems in the AMATH System. In any given study module, a student can learn, and teach himself or herself, from a virtually unlimited bank of example problems.

One of the most common skills students lack is multiplication table mastery. Frequently, diagnostics fail to identify this shortcoming. In standardized tests, where a calculator is not allowed, just eliminating simple arithmetic mistakes often enough will often raise scores to a passing level. While AMATH Foundations starts with integer addition, these factors make the Math Facts/Multiplication Study Module ideal for introducing the Foundations Working Environment.

Take a look at the sample screen in Figure 2. There are six areas common to the work environment in all AMATH Foundation Study Modules. Common areas are dedicated to the Problem Presentation, Feedback, Progress, Answer Entry, Correct Solution, and a Menu Bar provides additional tools.

The Problem Area takes the form of a Chalkboard. This is where all of the problems are displayed. Frequently additional instructions are supplied in this area as well. All of the problems in AMATH Foundations are randomly generated. This prevents students working near one another from copying. Competitor’s software often uses problem banks. These allow students to enter an incorrect answer for all of the problems, record the correct answers, then retry the problems and enter the answers. They never have to do the problems to get the correct answer. In fact, we recommend that students using Foundations do all of their work with on pencil and paper, in a spiral notebook. This way teachers can see all of the students work as they progress, helping them to provide better assistance when problems arise.

The Problem Entry and Feedback Area is simple. This is where the student enters their solution to the problem. After grading the problem, Foundations either provides the correct answer or the words “Good Job”. Whether the problem is solved correctly or not, the student is given the option to click on a “Show Me” button to see a solution.

In many modules, students can chose to enter their answers with the onscreen keypad or the computer keypad. As the problems become more complex, students often place the computer keyboard out of the way to make room for their math journal. They can then use the mouse to navigate the entire program and enter any answer. In modules where speed is important and the answers are not complex, the computer students may prefer the computer keyboard.

Notice in Figure 2 that the solution highlights the importance of the multiplication table as well as the concept of how multiplication comes from adding sets. As you read on more sample screens of more advanced problems are presented. Pay careful attention to the solutions. They show exactly how a student should solve the problem using a pencil and paper.

Additional reference tools are available to the student while working in AMATH including language tools, formulas, and constants. This area will be addressed in detail further down the page. In Figure 2, note the menu bar has an option called Choose Range. In integer arithmetic, students choose to practice problems from a given range. For example, some students may need additional work with the 11s and 12s in the multiplication table. With Foundations, students can get that specific practice. Even though practice ranges are available, students will only be given credit for completing a module once they have demonstrated mastery of the entire range.

AMATH further differentiates itself by not using a percentage based grading system for evaluation. Rather, we use the streak concept. A streak is a set numbers of problems in a row performed without error. Instead of asking a student to do sets of problems until he/she can perform at a certain percentage, we ask that a student work continuously until they can get a set number in a row correct without error. The progress area of the work environment lets students know how much work they have done, and what they need for mastery.

Streak grading ensures mastery. It does not penalize them for the wrong answer. Quite the opposite, it creates more math problems until mastery is achieved. Streak grading gets the student to care about each problem they solve, because if they solve one incorrectly, their streak is reset. For example, if a student has to solve 15 problems in a row correctly and they have already solved 13, how important do you think those next 2 problems are to that student? In comparison, a student may not care about getting any of a 15 question homework assignment correct, rather they just want to get something on paper to get credit for the attempt. Streak grading allows you to reinforce efficiently a concept while constantly evaluating for mastery. In AMATH, mastery is achieved when one’s streak is completed. They then move on to the next module. If a student has already mastered a topic, they can finish a streak quickly and move on. This is how AMATH efficiently takes advantage of previous learning.

As students progress through the study modules, the content gets more advanced, the required streak changes, and the solutions become more detailed. In math facts, the required streak is 15. In the whole number multiplication study module pictured below, the required streak is 5. There are two reasons for this. First, when a 3-digit by 3-digit multiplication problem is broken down, it takes 36 correct steps performed in a row to get one correct answer. When a student has done 5 in a row correct, then they have successfully performed 180 arithmetic calculations without error. Let us say for example purposes that a student is 98 percent proficient in math facts. The probability of that student correctly performing 180 operations in a row without error is only 2.6%. The second reason the streak number is reduced to 5 is because students are encouraged to complete their work in AMATH Foundations with a pencil and paper. Manual problem solving takes significantly more time than using a calculator.

In standardized testing, performance is critical. Performing at only 98% in simple arithmetic will dramatically reduce students' test scores. This is why AMATH’s streak grading is so valuable.

Textbooks provide only few solved problems in each section. With Foundations, students can learn from an unlimited number of solutions. Notice the detailed solution in Figure 3 above. The solutions show step-by-step exactly how a student could have solved the problem using pencil and paper.

AMATH’s grading is very meticulous. Each answer must be formatted with correct punctuation and units of measure including the comma for the thousands separator. If correct notation is not used, but the value is still correct, then it will be graded wrong. In the case of an expression or equation, the variable must always be isolated to the left. There are many reasons for this. First, it develops very good math writing habits that will help in testing, and in learning more complicated math and science. Second, demanding specific notation makes for more opportunities to miss your streak. The type of notation required is most frequently stated in the problem’s directions, and it may change from time to time. This forces the student to read each problem, and read it for content. Reinforcing this skill dramatically increases reading comprehension and attention to detail. Teachers will notice that students who have used Foundations check their answer to almost every problem with one common exception. The first couple of problems a student tries in most study modules, they usually do not enter any answer at all. What they are doing is just seeing how the computer formats the correct answer and how to do it. With AMATH, students automatically start Self Teaching, even if it is just teaching themselves the correct way to format an answer. The ability to self teach, and the confidence that it provides, stays with students long after they have finished AMATH.

When you examine the contents of Foundations, you’ll notice that it even covers some abstract topics like reading a gas meter. Not only is reading a gas meter a valuable skill for a homeowner or a tenant, but the procedure learned for reading a gas meter directly translates into the same procedure for reading many more complicated measurement instruments like the vernier caliper. The measurement and conversion skills learned in AMATH easily apply to a current or future vocation that students may choose and directly influence performance and learning in science.

In Foundations, students learn to measure and convert between both English and Metric systems. Notice in Figure 5 above that when the correct answer uses the metric system. This means that the answer must be in the form of a Decimal number. When the system of measurement is English, the solution must be in fractional format.

Throughout Foundations, reading the question thoroughly is reinforced. In the case of measurement, students must read the problem and identify the correct system of measurement and the correct number of significant digits. If the answer does not use the correct unit of measure or if it is not formatted correctly, the problem is graded wrong and the streak resets to zero. Skills like whole number arithmetic, fractions, decimals, rounding, and working with units of measure are all reinforced in this study module, but will have been mastered by completing previous AMATH modules.

The Study Module on Inequalities shown below helps to further highlight the scope of Foundations and the quality of the feedback provided to the student in form of a solution. In this Module, students solve simple, single variable, linear inequalities by any combination of Addition, Subtraction, Multiplication, or Division operations. Note the step-by-step solution and how it is exactly what you could have typed on paper.

At this point, you have very good understanding of the layout, organization, grading, and features of AMATH. Let’s address one more very important concern. How does AMATH Deal with word problems? Almost all of the questions on standardized tests are presented in the form of word problems. We address that very ardently. In Foundations, students master computation, then they master math as a language, before intimidating problems are presented. In addition to ensuring that students are not missing the skills needed to solve word problems, we also integrate simple word problems into early topics. This builds confidence. Look at the Navigation Screen shown in Figure 1 at the top of this page. Notice that associated with each of the major sub strands is a Study Module called “Simple Word Problems”. These serve as an introduction to word problems through exercising the concepts that they just mastered. Once students progress into the more advanced strands, almost all of the problems are introduced as a simple or even moderate word problem. Throughout the program mastery in computation, reading for content, and confidence in tackling word problems is increased. Not only does AMATH make sure that students have all the skills and confidence gradually built for approaching the much-feared “Advanced Word Problems”, but we dedicate an entire Folder to learning and practicing those word problems.

Figure 7 above shows the folder on Basic Problem Solving Skills. If completed in order, by the time a student reaches this folder he/she will already have all of computational tools needed to the word problems. To make learning word problems easier and build confidence, we start by building their language and reading skills. Think back to the ability to identify important words and look up a definition. That was put there to help build math vocabulary and to prepare students for this folder. We help students translate terms, phrases, and then sentences into operators, expressions, and equations. Next, we work on identifying the question, making assumptions, and isolating relevant data. Finally, we practice these skills with multi step problems that build on themselves and with very wordy problems. Students then apply their skills to the most common types of complex word problems, in increasing order of difficulty. They enter their answer here using short-answer format, so they must fully master the problem, and its relevant computation. Note, they can also try these types of problems in multiple choice through the AMATH Testing folder.

Evaluate Performance with Comprehensive Reviews, Quizzes, and Sample Graduation Tests

Earlier we touched on the question—where does the student start? AMATH works best when a student starts with Math Facts and precedes one Module at a time throughout the whole program. Now that you understand streak grading, you can see that for students who may have mastered a given study module, they very quickly move through that module reinforcing that skill and building confidence. It serves as in insurance policy that they will have the prerequisites needed to succeed in later modules. Therefore, as is, the streak grading helps to speed up progress by taking advantage of previous learning, but with the added advantage of reinforcement and insurance that nothing was missed.

While it is best to complete AMATH in its entirety, Foundations does include tools that can be used for diagnostics. For each topic in Foundations, students can take a randomly generated 10 question test. Teachers occasionally choose to have the students take each 10-question test. This may take a couple hours. Then have the students complete the sub strands for which they did not score an 80% or above. As an added feature, you may choose to time the tests and have them take the tests as a class. You may also choose to modify the score required to “opt out” of a study module.

Comprehensive reviews are provided at the end of each topic. They are yet another tool teachers can utilize in a hurried literacy effort. For example, teachers prescribe that students do only the comprehensive review contained in each folder, without using any pre-testing. The students complete the comprehensive reviews in order until they have trouble, or are unable to finish the comprehensive review in a timely manner. When they reach a spot that they have trouble, then they should start completing each Study Module in the Sub strand.

The Foundations Testing Folder is most frequently used for evaluation and practice, rather than for diagnostics. Schools using AMATH as a comprehensive literacy effort may choose to have students complete the corresponding 10-question test after completing each Study module. The teacher can then assign a performance-based score in combination with or in place of progress passed score. For practice and evaluation of progress, we have provided a section that generates practice High School Graduation Tests for students to take on the computer. These tests can be taken in two sections, or all at once. The tests were divided into two sections to better accommodate the time allotted in a traditional class period.

Figure 8 shows a sample, interactive High School Graduation Test. The features of the practice High School Graduation tests are many. First, they look just like a test, right down to the blank first page that allows teachers to get everyone to the same point before starting a timer. Once they start the test, they page through the test answering the questions on a simulated bubble sheet. The computer checks to see that the student marks his/her answer in the correct area and warns them if they do not. The software also warns of stray marks on the Scantron sheet and tabulates both stray marks and attempts at answering on the wrong line. The student can skip problems and come back to them, just like with a traditional test booklet. When finished, AMATH grades the test, records the score, and provides the student with the opportunity to page throughout the simulated test booklet, but this time the test booklet contains solutions, correct answers, and study suggestions.

Accountability - Reports on Individual and Class Progress are available in clicks a click of the mouse!

One of the most important features of AMATH is that it records all of the students work and displays it in a useful manner, either on-screen, on paper, or in a format compatible with spreadsheet software. Teachers do not have to spend any time grading, reporting, creating problems, or a curriculum. All they have to do is get the students to work, interpret their progress/performance data, and provide assistance as needed.

Evaluating student progress is easy! The reporting function helps teachers identify, at a quick glance, specific areas that students may need additional help, but are afraid to ask. It also helps teachers to establish a mean performance and progress standard for each class so that they may provide additional motivation/assistance to slower students and additional praise to students who motivate themselves achieve more.

AMATH Administration software helps teachers to create classes for summary reporting and easy location of student data. It also serves as the management hub for allocating student login numbers and passwords.

While AMATH Foundations is a tremendous and efficient program, it does not work like a smart pill. Students who use AMATH Foundations are often patching holes in 8 or more years of math education, and that does not happen overnight. It takes continued work and dedication. Through examination of our usage data, we estimate that an average at risk high school student will work for approximately 43 hours to complete AMATH Foundations. We define "at risk" as testing at the 3rd to 5th grade level in math.

In our study of over 200 at risk urban high school students, we found that after an average of 20 hours of work in AMATH, half of the students raised their tested grade level by 3 or more years.

We believe that students who complete AMATH Foundations test higher than 80% of the United State population in math literacy!