Developing Fluency in Math-Delayed Children
The acquisition of math facts generally progresses from a deliberate, procedural, and error-prone calculation to one that is fast, efficient, and accurate (Ashcraft, 1992; Fuson, 1982, 1988; Siegler, 1988). For many children, at any point in time from preschool through at least the fourth grade, they will have some facts that can be retrieved from memory with little effort and some that need to be calculated using some counting strategy. From the fourth grade through adulthood, answers to basic math facts are recalled from memory with a continued strengthening of relationships between problems and answers that results in further increases in fluency (Ashcraft, 1985).
In a typical developmental path in addition, for instance, students begin adding using a strategy called “counting all” that gives way to a “counting on” strategy, which in turn gives way to linking new facts to known facts (Garnett, 1992). In multiplication, a student might employ repeated addition or skip counting as initial procedures for calculating the facts (Siegler, 1988). With repeated exposures, most normally developing students establish a memory relationship with each fact. Instead of calculating it, they recall it automatically.
In contrast, most math-delayed children, along with those who have never received systematic math fact instruction, show a serious problem with respect to the retrieval of elementary number facts. Learning-disabled children are substantially less proficient than their non-disabled peers in retrieving the answers to basic math facts in addition and subtraction. Although information is still emerging about the particular difficulties experienced by these children in the retrieval of this information, the evidence that does exist suggests that these children do not suffer from a conceptual deficit, but rather from some sort of disruption to normal development of their network of relationships between facts and answers. That is, these students often have well-developed number sense and procedural knowledge — they can figure out the answer to any fact given enough time. But because they have poorly developed declarative knowledge, they have minimal ability to recall anything but the most basic facts from memory.
What this suggests is that there are huge differences in the amount of instruction individual children need to become fluent at retrieving answers to basic math facts. By age seven, non math-delayed students can recall more facts from memory than their math-delayed peers, and this discrepancy increases as age increases. As math-delayed students get older, they fall farther and farther behind their non math-delayed peers in the ability to recall basic math facts from memory (Hasselbring et al., 1988).
Drill and practice programs have demonstrated a positive effect on improving the retrieval speed for facts already being recalled from memory. However, drill and practice had no effect on developing automaticity for non-recalled facts (Hasselbring, Goin, & Sherwood, 1986). To facilitate the automatic recall of all facts, instruction must be focused on non-automatized facts while practice and review are given on facts that are already being recalled from memory. Thus identifying and separating fluent from non-fluent facts is important.
Effectiveness of the FASTT Math Approach
The FASTT Math software program begins with a computer-based assessment that presents basic facts and records the amount of time that the child takes to answer each fact correctly. By measuring the latencies of student responses the program can accurately determine the facts that are being recalled from memory and those that are solved using a counting strategy.
Following this initial placement quiz, FASTT Math constructs a fact grid. The grid allows the student (and teacher) to visually see the fluent (‘Fast’) facts and those that the student answered slowly or incorrectly (‘Study’ facts). The program expands the student’s declarative knowledge network by building on existing knowledge. As a general rule, the program selects facts to be automatized based upon the size of the minimum addend. For example, once all facts with a minimum addend of 1 have been automatized, FASTT Math begins to select facts with a minimum addend of 2, and so on, until all the “2s” have been automatized.
The research suggests that it is best to work on developing this declarative knowledge by focusing on a very small set of new target facts at any one time — no more than two facts and their reversals. Instruction on this target set continues until the student can retrieve the answers to the two new facts consistently and without using counting strategies.
Once a problem/answer relationship is established, FASTT Math uses controlled response times to reinforce the memory connection and inhibit the use of counting or other non-automatic strategies. A controlled response time is the amount of time allotted to retrieve and provide the answer to the fact. FASTT Math begins with a controlled response time of 1.25 seconds, forcing students to abandon inefficient strategies and to retrieve answers rapidly from the declarative knowledge network. If the controlled response time lapses before the child can respond, or if the student answers incorrectly, the program provides corrective feedback by presenting the problem/answer relationship again. This continues until the child gives the correct answer within the controlled response time.
FASTT Math develops a declarative knowledge network by interspersing the two new “target” facts with other already automatized facts in a pre-specified, expanding order. Each time the target fact is presented, another automatized fact is added as a “spacer” so that the amount of time between presentations of the target fact is expanded. This “expanding recall” model requires the student to retrieve the correct answers to the target facts over long and longer periods. Only after a student is consistently able to retrieve the answer to a target fact within the controlled response time is that fact added to the child’s set of drill and practice facts.
The principles embodied in FASTT Math have been validated over several years of research with more than 400 students. Generally, the findings show that when used daily, for about ten minutes, most math-delayed children can develop fluency with all basic facts in a single operation after approximately 100 sessions. The key to success appears to lie in the consistent use of the program. As expected, students who use the program regularly do much better than students who are only occasional users.
In a study conducted by Hasselbring and Goin (1988), three groups of students were matched for age, sex, and race. Two of the groups consisted of math-delayed students and the remaining group consisted of non math-delayed students. In the experiment, one of the math-delayed groups (Math-Delayed Experimental) received an average of 54 ten-minute sessions on the software program for addition, the other two groups (Non Math-Delayed and Math-Delayed Contrast) received only traditional fluency instruction delivered by their classroom teachers. The math-delayed students receiving instruction with the FASTT Math approach gained, on the average, 19 new fluent facts while their math-delayed peers receiving traditional instruction gained no new facts and their non math-delayed peers gained only 7 new facts. Perhaps more impressive are the maintenance data. When the experimental students were tested four months after the post-test following summer vacation, the students regressed by only 6 facts, indicating that once facts become fluent through this method, they are retained at a high level.
The results of this experiment have been replicated several times across all four operations. In all cases, when used consistently, the FASTT Math approach has a positive effect on developing mathematical fluency in both math-delayed and non math-delayed students. Although FASTT Math is effective for all students needing assistance with developing fact fluency, it appears to be especially effective for students labeled as at-risk and learning disabled.
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