MATH: WHAT WORKS WELL AT HOME

Howard B. Richman
Pennsylvania Homeschoolers Research Project
RD 2 Box 117
Kittanning, Pennsylvania 16201

William Girten
Pennsylvania Homeschoolers Research Project

Jay Snyder
Pennsylvania Homeschoolers Research Project

Keywords: Homeschooling, home schooling, home education, academic achievement, math, mathematics.

While studies of home education achievement have reassured homeschooling parents that their students are succeeding on achievement tests, they have not provided guidance to parents as to how they could improve their students’ achievement.
The purpose of this study is to provide information for parents about what works well at home so that parents can adapt their own programs in more successful directions. We chose mathematics as our subject area to study because previous studies have shown it to be one of the weaker areas for home education students on standardized achievement tests (Richman et. al., 1990; Wartes, 1990a).

Method

Our method was to correlate students’ scores on a standardized achievement test with their parents’ answers to a questionnaire. The 346 achievement tests were administered in group settings, usually in the Sunday school rooms of churches at locations sponsored by homeschooling support groups in every region of Pennsylvania in the fall of 1990. Cost to the parents was $20 for each test administered. Many parents enrolled their children in order to meet the testing requirement of the Pennsylvania home education law. All test administration was closely supervised by two Pennsylvania certified teachers, including one of the authors of this study. Confidentiality of answers and students’ achievement test scores were protected and all statistical analyses were performed using files from which names had been removed.
Of the 346 tests, 224 of the parents returned the questionnaires. Nine questionnaires could not be matched with tests because of errors with numbering the tests and questionnaires. This left 215 questionnaires for further analysis.
Only math and reading testing are required in Pennsylvania. Science and social studies tests were also administered on a voluntary basis to some of the participants, but were not included in the data analysis. Overall, the mean scores for the 346 students tested, if converted from normal curve equivalents to percentile rank, were the 86th national percentile for the Total Reading score, and the 73rd national percentile for the Total Mathematics score. These are precisely the same national percentile levels that were found in our previous year’s testing (Richman et. al., 1990). The mean scores for the 215 students who were matched with questionnaires were almost exactly the same as the scores for those without the questionnaires. The mean scores for the 215 with questionnaires, if converted to percentile ranks, were the 86th national percentile in reading and the 74th national percentile in math. As noted in our previous study which was performed under similar conditions (Richman et. al., 1990), these scores can not be considered as representative of
the overall achievement of homeschoolers in Pennsylvania, but may be representative of home educated students in a loosely defined Pennsylvania Homeschoolers support network.
Testing is only required in Pennsylvania in third, fifth, and eighth grades. Of the 346 students whose scores were included in the study, 276 were in the mandatory testing grades.
Individual normal curve equivalents were tabulated from the raw test scores based upon tables provided in the Fall Norms Book by CTB McGraw Hill. The normal curve equivalent scores for Total Reading and Total Mathematics were each found by averaging the scale scores from subtests (math subtests were Mathematics Computation and Mathematics Concepts and Applications, reading subtests were Vocabulary and Comprehension) and then looking up the average scale score in the normal curve equivalent table. When the average scale score was half‑way between two integers (i.e. 545.5) the correct normal curve equivalent was obtained by first rounding upwards and rounding downwards, then looking up both scale scores, and then averaging the normal curve equivalents that were obtained. All data analysis was performed using these normal curve equivalents. When national percentiles were reported, they were computed after calculating the mean normal curve equivalent and then converting the normal curve equivalent to percentile rank using the conversion table published by Wartes (1990b, Appendix B).
Before data analysis was performed, the questionnaire results were used to further refine the sample. The 24 children who had not been homeschooled the previous year were eliminated as were an additional 28 whose parent reported that they had a degree of handicap or disability that might affect their scores in mathematics. The following data analysis is based upon the 163 questionnaires which remained and their corresponding achievement test scores.
All statistical procedures were performed using Systat: The System for Statistics for the PC.

Findings

The results that were obtained were quite diverse.

Older Students Score Higher
The math achievement levels of home educated students were related to their grade levels. Those in higher grades tended to score higher than those in lower grades. When all of the 346 students who participated in the test were considered, not just those whose parents returned questionnaires, the average Total Mathematics score for third graders was the 64th national percentile and the average score for fifth graders was the 80th national percentile. When just the 163 students were considered whose parents returned the questionnaires and where the student was not handicapped and had been homeschooled the previous year, the third graders averaged at the 61st national percentile, and the fifth graders averaged at the 86th national percentile.
For these 163 students, the correlation between Total Math scores and grade level was modest but significant (r = .27, p < .001). The correlations between grade level and the Math Computation subtest (r = .23, p < .01), and between grade level and the Math Concepts and Applications subtest (r = .24, p < .01) were also significant. Figure 1 shows the percentile ranks for each of the grades from third through eighth when the average normal curve equivalents are converted to percentile ranks. There were 68 students in third grade, 10 students in fourth grade, 46 students in fifth grade, 10 students in sixth grade, 7 students in seventh grade, and 18 students in eighth grade. The one student in second grade, the two students in ninth grade, and the one student in tenth grade were not included in Figure 1 because of the small sample size for each of those grade levels.

Math Seminars Help
Parents were asked, “How many talks, presentations or seminars on helping your child learn mathematics did you attend in the last year? (Don’t include today!)”  Parents were asked not to include seminars that would be attended that same day because the support groups which sponsored the tests usually sponsored workshops during the afternoon or evening after testing and some of those workshops were about how to teach math at home.
Of the 163 questionnaires, 112 reported attending no math talks, 32 reported attending one talk, 12 reported attending two talks, 3 reported attending three talks, 3 reported attending five talks, and one did not answer the question. The more talks that the parents reported attending, the
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Figure 1. Total Math percentile rank for each grade.

higher the student scored with math. The correlation between the number of talks attended by the parents and the child’s Total Mathematics score was weak but statistically significant (r = .17, p < .05). The correlation was also weak but significant between the number of talks attended by the parents and the child’s Mathematics Concepts and Applications subtest score (r = .18, p < .05). The relationship between number of talks attended and Total Mathematics achievement is even higher if only those students who are fifth grade and up are considered (r = .23).

Correcting Mistakes
Parents were asked, “When this child does math exercises, problems or tests, do you check his or her work and make him or her correct any mistakes?”   Parents answered on a scale of 1 to 7 where 1 was “never,” and 7 was “always.”
The more that parents said they checked the student’s work and made him or her correct mistakes, the worse the student scored. A weak but statistically significant negative relationship was found between correcting mistakes and Total Mathematics score (r = ‑.19, p < .05), as well as the Mathematics Concepts and Applications subtest (r = ‑.16, p < .05). If only students fifth grade and over are considered, however, this correlation almost disappears (r = ‑.04) and is no longer statistically significant.
There is a low but significant correlation between correcting mistakes and the student’s grade (r = .18, p < .05). The higher a student’s grade level, the less likely the parents were to check his or her work and make the student correct his or her mistakes.

Parents Like Programs that Work
Question: Parents were asked, “Did you like the math program that you used last year?”  Parents rated on a scale of 1 to 7 where 1 was “disliked that math program,” and 7 was “liked that math program.”
Result: The more that parents said they liked the math program, the better the students did on achievement tests. The relationship between the parents’ attitude and Total Mathematics scores was weak but statistically significant (r = .16, p < .05). The differences with each of the two subtests, which when combined make up the Total Mathematics score, were not statistically significant. The correlation is lower (r = .10) and is no longer statistically significant if only students fifth grade and up are considered.
At the same time, there is a weak but statistically significantly relationship between grade level and parents’ attitude toward the curriculum (r = .17, p < .05). Parents are more pleased with their math curriculums at higher grade levels.

Comparison Between Programs
Question: Parents were asked, “What was the main math program that this child used last year (during the 89‑90 school year)?”
Result: Only five categories had enough data to be computed, books from Saxon Publishers, books published by A Beka Books, books published by Bob Jones University Press, textbooks borrowed from the local school district, and all of the other programs in a category together. The Total Math percentile rank of those who used the Saxon textbooks was 91, school district’s textbooks 80, A Beka 77, Bob Jones 77, and other programs 65 as shown in Figure 2. The number of students in each program were 25 in Saxon, 18 in school district’s textbooks, 38 in A Beka, 28 in Bob Jones, and 54 in the other category.
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Figure 2. Total Math percentile rank for each program. (All grades combined.)

There appears to be a wide disparity between textbooks with Saxon far in the lead, however, it is partly due to the fact that the Saxon textbook series did not span all of the grade levels. At the time of our study it included high school texts and upper elementary school texts, but not lower elementary school texts. On the other hand, the A Beka and the Bob Jones University textbook series covered the full range of grades.
A comparison of students’ total math scores with these textbooks showed much less differentiation if only students who were 5th grade or higher were considered. At these grade levels there were 24 students using the Saxon textbooks, 10 using textbooks borrowed from the school district, 18 using A Beka textbooks, 8 using Bob Jones textbooks, and 24 students using other programs. The students using the Saxon textbooks scored at 90th percentile rank, school district’s textbooks also scored at the 90th percentile rank, A Beka textbooks at the 87th percentile rank, Bob Jones textbooks at the 84th percentile rank and all other programs at the 79th percentile rank as shown in Figure 3.

Which Programs do Parents Like?
Question: Parents were asked, “Did you like that math program (which you just circled)?”  Parents rated the program that they had used on a scale of 1 to 7 where 1 was “disliked” and 7 was “liked.”
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Figure 3. Total Math percentile rank for each program for fifth grade and higher.

Result: Only five categories had enough data to be computed. Parents who used the programs rated Saxon at 6.6, Bob Jones at 5.6, other programs averaged at 5.5, A Beka at 5.1, and textbooks borrowed from the school district at 4.1 as shown in Figure 4.
In order to make sure that this question did not just tap into the large disparity between third and fifth grade homeschool math achievement we did a further data analysis which just included students in fifth grade and higher. At these grade levels the parents rated the Saxon textbooks at 6.5, Bob Jones at 6.5, other programs at 5.3, A Beka at 4.7, and textbooks borrowed from the school district at 4.7 as shown in Figure 5.

Geometry Exposure is Important
Question: Parents were asked, “Was your child exposed to geometric concepts over the past year.”
Result: 91% of the parents answered “yes.”  The 149 students whose parents answered “yes”
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Figure 4. Parent rating of programs (all grades).

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Figure 5. Parent rating of programs (for 5th grade and higher).

scored much higher than the 14 students whose parents answered “no.”  The students of the parents who said “yes” scored at the 79th percentile rank, while the students of the parents who said “no” scored at the 50th percentile rank in Total Mathematics. The difference between means was found to be statistically significant using a two‑tailed t‑test (t = 2.65, p < .01, df = 161). The same students also scored better on the Mathematics Computation subtest (t = 2.32, p < .05, df = 161) and the Mathematics Concepts and Applications subtest (t = 2.52, p < .05, df = 161).

Effective Supplementary Activities
Question: Parents were asked, “Did you supplement your math book with other activities?  Please check any supplemental math activities or textbooks you used with your child last year?  You may check more than one.”
Result: Only two supplementary activities were related to achievement test scores to a statistically significant extent: (1) Mortensen Math (V. J. Mortensen Co, PO Box 98, Hayden Lake ID 83835), and (2) Computer programming by child.
Mortensen Math is a math program for elementary school students which uses math manipulatives to introduce algebra, arithmetic, problem solving, calculus and measurement. Use of Mortensen Math as a supplementary activity was related to higher achievement with the Mathematics Concepts and Applications subtest but not the Mathematics Computation subtest. The 17 students who used Mortensen Math as a supplementary activity scored at the 94th percentile rank on the Mathematics Concepts and Applications subtest compared to the 74th percentile rank for 146 students who did not. The difference between these means was statistically significant using a two‑tailed t‑test to compare the two means (t = 2.87, p < .01, df = 161). On the other hand, students who used Mortensen Math did not score any higher on the Mathematics Computation subtest. The Mortensen students scored at the 68th percentile while the non‑Mortensen students scored at the 72nd percentile. Differences between means on the Mathematics Computation subtest and the Total Mathematics score were not statistically significant.       Computer programming as a supplementary activity was related to higher achievement although using computer programs for math drills was not. The 16 students who programmed computers scored at the 95th percentile with the Total Mathematics score compared to the 73rd percentile for the 147 students who did not. The difference between these means was found to be statistically significant using a two‑tailed t‑test (t = 3.72, p < .001, df = 161). The students who programmed computers also scored higher on each of the two subtests. The difference between the means of the two groups was significant in each case according to a two‑tailed t‑test. For the Mathematics Concepts and Applications subtest, the students who program computers scored at the 95th national percentile compared to the 73rd national percentile for those who did not (t = 3.96, p < .001, df = 161). For the Mathematics Computation subtest, the students who program computers scored at the 92nd national percentile compared to the 69th national percentile for those who did not (t = 2.83, p < .01, df = 161).
On the other hand, computer programs for math drill did not have much of an effect. The 45 students who used computer programs scored at the 79th percentile in Total Mathematics, while the 118 who did not scored at the 75th percentile in Total Mathematics. This small difference between the means was not statistically significant.

Non‑significant Results
Correlations between several other variables and achievement, as measured by the Total Mathematics scores did not achieve statistical significance.
Question: Parents were asked, “About how many hours per week does this student spend doing formal schooling (structured lessons that were preplanned by either the parent or the provider of educational materials).”   The correlation between hours per week spent in formal schooling and math achievement was very weak (r = .05).
Question: Parents were asked “”About how many hours per week does this student spend doing formal mathematics lessons (structured lessons that were preplanned by either the parent or the provider of educational materials).”   The correlation between hours spent doing formal math lessons and math achievement was very weak (r = .03).
Question: Parents were asked, “What is the highest number of years of “formal education” completed by the parents who is supervisor of the home education program?”  The correlation with math achievement was very weak (r = .08).
Question: Parents were asked, “What is the current, before tax, annual income of this family?”  Parents indicated the income as being in one of 11 brackets. The correlation between the family’s income and math achievement was very weak (r = .01).
Question: Parents were asked, “When you were in school did you (the parent) like mathematics (as compared to other subjects)?”  Parents rated math on a scale of 1 to 7 where 1 = least favorite subject and 7 = most favorite subject. The correlation between the degree that the parent liked math and math achievement was very weak (r = .01).
Question: Parents were asked, “How well do you think this child likes math (as compared to other subjects)?”  Parents rated the degree that the child likes math on a scale of 1 to 7 where 1 was “least favorite subject” and 7 was “most favorite subject.”  The correlation between student’s attitude and math achievement was very weak (r = .14).
When parents were asked how often, per month, they used tests and quizzes to assess their child’s progress. The correlation between number of tests used and math achievement was very weak (r = .02).
Question: Parents were asked to what extent did you use the results of diagnostic tests to help you decide what to teach?”  Parents rated on a scale of 1 to 7 where 1 was “did not use diagnostic tests at all,” and 7 was “diagnostic tests determined what I would teach.”  The correlation between use of diagnostic tests and math achievement was very weak (r = .06).
When parents were asked to check off activities that they used to supplement their math textbook, the correlation between the number of supplemental activities checked and math achievement was very weak (r = .04).

Discussion

Because these results are so disparate, we will discuss each result separately.

Older Students Score Higher
We found that the math scores of homeschooled students in higher grades are generally higher than those of homeschooled students in lower grades. The biggest jump appears to occur somewhere between the beginning of third grade and the beginning of fifth grade.
While it is possible that this correlation is unique to homeschoolers in Pennsylvania or the CTBS/4 achievement test, evidence from Wartes (1990b, p. 44) suggests that a similar phenomena takes place in the state of Washington using a different test measure. When Wartes compared homeschool math achievement he found five different grade levels with positive differences where students at a higher grade level out‑scored students at a lower grade level and no negative differences where students at a lower grade level significantly outscored higher grade level students. On the other hand, data released by the National Center for Home Education based upon their nationwide testing of 10,750 homeschooled students using the Stanford Achievement Test does not indicate any obvious nationwide correlation between math achievement and grade level (National Center for Home Education, 1992). These results suggest that homeschoolers in Washington and Pennsylvania tend to gain in achievement, relative to their school educated counterparts, as they progress through elementary school, although homeschoolers in the rest of the country do not.
Pennsylvania and Washington have the latest beginning compulsory education ages of any states in the country. In other states children must begin formal education  at age 5, 6, or 7, but in Pennsylvania and Washington they do not need to begin until age 8. It may be that many homeschooled students in Pennsylvania do not begin their formal math studies until the age of 8, but, once they begin, their progress, relative to their school educated counterparts, is quite dramatic. On the other hand, it is also possible that students who transfer to home education during third or fourth grade score higher in mathematics than students who have always been homeschooled.  A future questionnaire which includes a question regarding how long a student had been homeschooled or a future longitudinal study of home educated students might disambiguate whether the gains in achievement by Pennsylvania and Washington homeschoolers are due to gains in achievement by children who are being homeschooled, or the transfer into homeschooling of children who are scoring higher in mathematics.

Math Seminars Help
We found that the more the parents reported attending presentations or seminars on helping their child learn mathematics the better the child scored on math tests. There are several possible explanations of this weak correlation. It is possible that resourceful parents are more likely to attend seminars and are more likely to be resourceful when teaching math to their students. It is also possible that those parents who are most involved with their children’s education would be more likely to attend seminars, and that interest in their children’s education would cause higher math achievement. A third possibility is that parents gained understandings from attending these seminars that they were able to use to improve their students’ programs.
In our previous study (Richman et. al., 1990) we noted that students in a loosely‑defined Pennsylvania Homeschoolers support network appear to score higher in reading and math than students who receive their information about homeschooling from school districts. It may be that the higher achievement levels within the network may be partly the result of the seminars and meetings provided to parents by the support groups in the network. These support groups often have local meetings which focus upon educational topics. They also organize regional seminars and a large state‑wide conference for thousands of parents which features many helpful workshops.

Correcting Mistakes
We found that the more parents checked their students’ math work and required their students to correct all of their mistakes, the worse the students did in math. However, this relationship largely disappeared when we just considered students who were fifth grade level and higher. This question appears to tap into a general trend where homeschooled students tend to become more independent with mathematics as they move into higher grade levels. The positive correlation between achievement and independence in math may result just because both achievement and independence are related to grade level in Pennsylvania.
A small study conducted in Virginia by Jacque Williamson (1991), using the same test (the CTBS/4) and a similar questionnaire, divided our question into two questions, the question of who does the checking, and the question of whether math errors are discussed. Williamson’s questions were, (1) “After the child does math exercises, problems, or tests, are the answers checked either by you or by the child?”, and (2) “When the answers are checked, do you and the child discuss the incorrect problems?”  Williamson found no correlation between math achievement and whether the answers were checked, but found a positive correlation (the reverse direction of our correlation) between math achievement and whether errors are discussed by parent and student.  Her results contradict any interpretation of our question which would argue that math work should not be checked at all by anyone, parent or student.

Parents Like Programs that Work
We found that parents liked their math programs better when the children were succeeding. We also found that parents tended to like their programs better as their children moved into higher grade levels. It may be that parents like the programs because they know that their students are doing well with mathematics by using those programs. Alternatively, it may be that teacher enthusiasm for a program may be a factor for program success.

Comparison Between Programs
When we compared the math programs on how well the students scored, we found that the students in all five of the groups scored higher than the average child in school. Our first data analyses appeared to indicate that those who used the Saxon textbooks clearly scored higher than those who used other textbooks, however further analysis revealed that the differences were partly due to the general trend for students to score higher in upper elementary school than in lower elementary school.

Which Programs Do Parents Like?
When we compared parents ratings of the programs. Only five categories had enough students involved to be able to calculate statistics. These were Saxon, Bob Jones, A Beka, school district’s textbooks, and “other” (all of the other alternatives). Saxon was rated the highest, with Bob Jones next, other ranked third, A Beka fourth, and textbooks borrowed from the school district rated the lowest. The Saxon textbooks were the most popular of the five categories.
The Saxon textbooks (available from Saxon Publishers, 1320 West Lindsey ‑‑ Suite 100, Norman OK 73069) are based upon the incremental developmental model of mathematics instruction of John Saxon, a retired Air Force Academy Engineering instructor who developed his method while teaching first year algebra to junior college students. The hallmark of his textbooks is gradual incremental introduction of new skills, and cumulative review in each daily lesson that gradually combines the various concepts that have been taught.
Our informal discussions with parents indicate that they like Saxon because their students can work with the Saxon textbooks more independently, the examples are clear, and the students generally have success with the problem sets. Families also seem to appreciate the clear delineation into daily lessons, and perhaps this clear structure helps families work consistently and thoroughly enough to gain skill and understanding. Some complain that the Bob Jones textbooks do not include enough review. Some find that the A Beka textbooks move too quickly or do not include adequate explanations in the student text.
It is not surprising that parents prefer books that they purchase themselves to the textbooks borrowed from the school district. When parents purchase a textbook, they would be more likely to select a book that they like. When they ask the school district for a textbook, they must accept whatever the school district has available.
The preference for the Saxon textbook was similar to a result that was found in studies conducted in public high schools which compared the Saxon algebra textbook with other algebra textbooks (McBee, 1984; Johnson & Smith, 1987). In both studies, teachers reported that they preferred the Saxon textbook. In one study (Johnson & Smith, 1987) students also displayed a more positive attitude toward the Saxon book and those who used Saxon had more self‑confidence in their math abilities.

Geometry Exposure is Important
We found that the students who had not been exposed to geometric concepts in their math programs scored lower than those who had been exposed to geometric concepts. It may be that programs which involve geometry are more effective because geometry is an essential aspect of math understanding. Alternatively, it may be that those programs that do not teach geometry, do not teach several aspects of mathematics which are tested by the CTBS/4 achievement test. Programs which do not introduce geometry concepts over the year may have a narrow focus.

Effective Supplementary Activities
Only two supplementary activities were clearly paired with higher math achievement, Mortensen Math and computer programming by the student. Mortensen Math helped with the Mathematics Concepts and Applications subtest but did not help with the Mathematics Computation subtest while computer programming helped with both math subtests.
It appears that Mortensen Math is an effective supplementary program for helping students score better on Mathematics Concepts and Applications subtests. Perhaps the math manipulatives used in the Mortensen math program help students visualize math concepts and really understand the processes involved. Another factor which may have helped were the seminars for parents that were often conducted by the salesman of the Mortensen materials.
It also appears that students who program computers score better on tests of math achievement, although math drill on computers did not correlate which significantly higher achievement. Perhaps homeschooling parents can help their students learn to program computers as a way to increase their students’ math abilities. On the other hand, it may be that students who do better in math are the ones who choose to do computer programming. A study by McAlister (1985), for example, found that when students were given a choice between computer activities, those who were better at math more often chose computer programming in LOGO, while those who were not as good at math tended to choose games and computer‑assisted instruction.

Non‑Significant Results
Many other results did not produce significant statistics. Some of these non‑results repeat earlier findings that the number of hours spent per week in formal lessons, parental educational level, and parental income do not correlate with achievement at home (Richman, Girten, & Snyder, 1990; Wartes, 1990a).

Conclusion

This study is based upon the assumption that the CTBS/4 standardized achievement test is a valid measure of students’ math achievement. There are aspects of math achievement, such as becoming an independent math learner, learning to apply math in everyday life, and being able to communicate effectively about math topics to create original math problems, that may not be measured by this achievement test. Many of the newer math assessments such as the Integrated Assessment System/ Mathematics Performance Assessment designed by the Psychological Corporation address these other areas and require students to be more active than they are in the multiple choice format of the CTBS/4.
The correlations that we have found are weak correlations. Therefore there are many exceptions in each case. Certainly policy decisions that would require changes for all homeschooling students would not be supported by these findings. On the other hand, we hope that individual homeschooling families might find them useful as suggesting possible directions that they might choose to try in their own mathematics programs.
In correlational studies it is always difficult to determine cause and effect. We have suggested all of the interpretations that we have thought of with each of our findings, but it may be that we have missed the correct interpretations. We look forward to the possibility that future research may cast more light upon the causes of these correlations.
One of our clearest findings was the tendency for homeschooled students’ math achievement in Pennsylvania to be higher in upper elementary school than at the beginning of third grade. Wartes found similar gains in math achievement in the state of Washington using a different testing instrument. It appears that homeschooled students in Pennsylvania tend to improve in mathematics relative to school‑educated students, especially between the beginning of third grade and the beginning of fifth grade. Other explanations are also possible. Longitudinal studies of home educated students are necessary to disambiguate this question.
Perhaps this result indicates that there is little cause for concern when homeschooled students in Pennsylvania score lower than expected in math at the beginning of third grade. It may be that they will advance quickly in the ensuing years. There is anecdotal evidence that a similar pattern occurs with reading at home. Some homeschooled students in Pennsylvania begin reading later than most school‑educated children their age, but they advance, relative to school‑educated children their age, once they begin to read silently to themselves for their own enjoyment (Richman & Richman, 1988).
In addition, the following suggestions for homeschooling parents are indicated by some of our interpretations of the results of this study. All of these suggestions are based upon interpretations of our data which may or may not be incorrect.
1. Homeschooling parents should consider attending presentations or seminars on math instruction as a way to discover new approaches that might be more successful with their children. Students whose parents attended such presentations or seminars tended to score higher on the mathematics portion of the CTBS/4 standardized achievement test.
2. Homeschooling parents should consider switching their math program if they do not like the program that they are currently using. Where parents liked the math program they were using, students tended to score a little higher on the math sections of the CTBS/4 achievement test.
3. Parents should consider changing or supplementing their math programs if the curriculum they are using does not expose their children to geometric concepts over the course of the year. Homeschooled students who were not exposed to geometric concepts by their math programs scored lower on the mathematics portions of the CTBS/4 standardized achievement test then homeschooled students who were exposed to geometric concepts.
4. Parents should consider using the Mortensen Math program as a supplement to their regular program, perhaps because of the helpfulness of the manipulatives or of the seminars by Mortensen Math salesmen. Students who used the Mortensen Math program as a supplement tended to score higher on the Mathematics Concepts and Applications subtest of the CTBS/4 standardized achievement test.
5. Parents should consider helping their children get started with programming computers. Homeschooled Students who program computers tended to score at a high level on the mathematics portion of the CTBS/4 standardized achievement test.

References

Johnson, D. M., & Smith, B. (1987). An evaluation of Saxon’s algebra text. Journal of Educational Research, 81(2), 97‑102.
McAllister, A. (1985). Problem solving and beginning programming. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL. (ERIC Document Reproduction Service No. ED 259 032)
McBee, M. (1984). Dolciani vs. Saxon: A comparison of two Algebra I textbooks with high school students. Oklahoma City Public Schools, OK. Dept. of Planning, Research, and Evaluation. (ERIC Document Reproduction Service No. ED 241 348)
National Center for Home Education (1992). Homeschoolers score in top third of nation on achievement tests. Press release. Purcellville, VA.
Richman, H. B., Girten, W., & Snyder, J. (1990). Academic achievement and its relationship to selected variables among Pennsylvania homeschoolers, Home School Researcher, 6(4), 9‑16.
Richman, H., & Richman, S. (1988). The three R’s at home. Kittanning, PA: Pennsylvania Homeschoolers.
Wartes, J. (1990a). Recent results from the Washington homeschool research project. Home School Researcher, 6(4), 1‑7.
Wartes, J. (1990b). The relationship of selected input variables to academic achievement among Washington’s Homeschoolers. Woodinville, WA: Washington Homeschool Research Project.
Williamson, Jacque. (1991). Personal Communication. Learning Connection, Rt. 2 Box 340B, Crozet VA 22932.