A Diagnostic is an area of study. Think of a smokestack. All portions of Fractions - Addition and Subtraction are parts of the FAS Diagnostic.
Since FAS is actually taught in schools in what we refer to as teaching segments over a multiple year/grade period, there is no way to associate the way that we are covering the Diagnostic to specific grade levels.
When a Diagnostic is taught in school the Diagnostic is generally broken down into what we call Probes. A Probe is a a specific series of rules and transitions that apply to a specific problem type to decompose a problem from a higher level to a simpler level. Thus there are simpler problems for FAS that deal with addition and subtraction where the result is a positive value, and then a more complex Probe that deals with addition and subtraction where the result is a negative value, a more complex Probe to deal with addition and subtraction with different denominators, and then a more complex Probe that deals with addition and subtraction where the fractions result is an improper fraction, and so on.
As we transverse a Diagnostic in learn mode we transverse up the Probe tree moving from simple to greater complexity within the Diagnostic.
The way current educational systems work is to look at one area at a time. To focus on a single misconception. For instance, in Fractions Addition and Subtraction (FAS) the first thing that would be looked at is whether a student can do simple addition, so a problem like 1/3 + 1/3 would be presented to see if they could get to 2/3. Then 1/3 + 2/3 to see if they could get to 1. Then 2/3 + 2/3 to see if they could get to 1 1/3 instead of 4/3. Finally they would get to working with different denominators and work with 1/4 + 1/3, 3/4 + 2/3 etc.
We have taken a different approach since we look at the transitions between complex and simple problems. We start with the assumption that the students are already competent as simple math (can add 1+1) and that they have been given some rudimentary introductions to the simple concept of fractions in the lower grades when they were working with pizzas (fraction circles). Thus they understand the 1/3 + 1/3 and even the 1/3 + 2/3. They have been introduced at some level to the 2/3 + 2/3 even. We make these assumptions because we are starting at 4th grade level at the lowest and by 4th grade FAS has been taught in picture form (fraction circles) time form (1/4 of an hour = 15 minutes) and money form (1/4 of a dollar = $0.25). Thus, starting at "Add the Fractions and Reduce" looks to see if they can cover a grouping of basic lower level elements that cover the ability to perform the simple math, but also understand the core concept of fractions that the LCD must be used before you can manipulate a fraction. For that is really what differentiates a fraction from a whole number. The transitions necessary to make unlike items like so that they can be compared, added, etc. In addition, at the same time, we also cover the other difference between fractions and whole numbers, the concept of reduction.Thus you get what you were presented:
4/5 + 4/6 = 1 7/15
Which covers the two key areas of elementary fractions transitions LCD and Reduction, as well as all the basics of how to do the simple math. In fact, there are 4 steps (if you were to go through the step-by-step). They are:
- LCD transition.
- "do the simple math addition"
- Reduction
- Normalization if improper fraction to mixed
Once the student has been able to complete this without seeing a step-by-step, we move up to the next type of problem. Here we cover Subtraction with mixed numbers and reduction which allows us to explore the concept of borrowing outside of base-10 and also permits us to present the proper way (via the step-by-step) of doing that borrowing (ie: only borrowing from the whole what you need, not the entire number to an improper fraction), etc.
I understand where you may think that your class is not "ready" to work on these things, but the reality is that they most likely already know and understand these concepts by 6th grade, they have just not had them presented in this way.
You state (in a follow up e-mail) "the first subtraction problem was a negative fraction minus a negative fraction. It progresses too fast for my 6th graders." If they understand how to subtract a negative from a negative in whole numbers, then they shouldn't get confused by that. The balance of the problem is fully explained in the SBS in how to borrow correctly from a mixed number. The rest they should have covered in the first problem (level 1 of FAS) they were presented with.
The key thing to remember here is that we aren't a drill system (although that will be added shortly), but the power of our system is the Step-By-Step which details the transitions (new areas of learning) that the problem undertakes. And while there is some remedial explanations provided in the areas where they work the math not covered in the transitions, that is not the goal of the system, to remediate basic math.
I hope this more detailed explanation helps you understand a bit more on how we approach math and how our tool is designed. We chose the term "tutor" for what we do for a reason. We are NOT a "teacher" where the task if to teach, but a tutor, where the task if to help the student overcome a specific series of issues that could be core to misunderstanding primitive concepts that are needed to continue on. We can not teach a student to do 5+4, and we won't even try to do that. What we can tutor a student in is how to determine the LCD and transition the fractions. How to reduce a fraction. The concept and the mechanics via demonstration. We are a tool, not a replacement to a teacher such as yourself.
