Predicting phenotypic outcomes from genetic crosses is often very difficult for biology students, especially those with learning disabilities. With our mathematical concept, struggling students in inclusive biology classrooms are now better equipped to solve genetic problems and predict phenotypes, because of improved understanding of dominance and recessiveness. Furthermore, this new concept has gained popularity among our dyslexic biology students who usually have difficulty in conceptualizing alleles and genes with alphabetical letters. In addition to giving these students the option of using numerical representations, this method minimizes confusion that is often caused by alphabetical representations.

Many biology students struggle to understand key terms such as gene and allele even after classroom and laboratory instruction (Yilmaz et al., 2010). These students face difficulties in comprehending related concepts such as dominance and recessiveness (Browning & Lehmen, 1988; Freidenreich et al., 2010), thus impeding their knowledge-building process in fundamental genetics. Consequently, they tend to wrongly predict phenotypic outcomes from genetic crosses (Browning & Lehmen, 1988). Among our struggling college-level biology students, we commonly find that slow learners and those diagnosed with dyslexia express confusion in interpreting phenotypes from genetic crosses. To help struggling students in inclusive biology classrooms, we have developed a mathematical concept to predict phenotypes from genetic crosses.

## Mathematical Concept in Monohybrid & Dihybrid Crosses

From Mendel’s pea experiments, we formulated the following story problem. Rita crosses pea plants as a hobby. Rita decides to cross a pea plant that is homozygous for the purple allele with a pea plant that is homozygous for the white allele. Purple is the dominant trait for color of flowers. Predict the genotypic and phenotypic outcomes for the F1 and F2 generations for Rita.

First, students are asked to build a key table (Table 1) to list all alleles and genes involved in the monohybrid cross. Using Table 1, they proceed to build a Punnett square to show genotypic outcomes. Because the main problem for our students is in predicting the phenotype for heterozygotes (the gene Pp), the mathematical concept we developed specifically addresses this issue. We ask students to use 100 as the number to represent the dominant purple allele P and 0 to represent p, the recessive white allele. The next step is to add 100 to 0, and the resulting total can be matched to the phenotype (Table 2), which is then used to complete the Punnett square and conclude whether the genotype yields white or purple flowers (Table 3). The mathematical concept can also be extended to predict homozygous dominant traits such as purple flowers, which is represented by the numerical value of 200 because the dominant purple allele P has been given a numerical value of 100 (Table 3). Similarly, homozygous recessive traits can be numerically predicted by a value of zero (Table 3).

Table 1.

Key table for alleles and genotypes.

From QuestionLetter RepresentationDefinition
Purple is dominant Allele for purple flowers
White is recessive Allele for white flowers
Dominant homozygous purple gametes PP

Genotype for purple flowers
Recessive homozygous white gametes pp

Genotype for white flowers
From QuestionLetter RepresentationDefinition
Purple is dominant Allele for purple flowers
White is recessive Allele for white flowers
Dominant homozygous purple gametes PP

Genotype for purple flowers
Recessive homozygous white gametes pp

Genotype for white flowers
Table 2.

Mathematical concept in monohybrid tests for predicting phenotypes of flowers.

Alleles & GenotypesMathematical ConceptPhenotypic Interpretation
P
100+ 0 Purple White
Pp 100 Purple
Alleles & GenotypesMathematical ConceptPhenotypic Interpretation
P
100+ 0 Purple White
Pp 100 Purple
Table 3.

Mathematical concept in Punnett squares for predicting flower color in F1 and F2 generations.

For a dihybrid cross, we ask students to predict the F1 genetic outcomes when two homozygous pea plants with yellow round seeds and green wrinkled seeds are crossed. Following the same procedure as the monohybrid test, students build the Punnett square using the mathematical concept in Table 4.

Table 4.

Mathematical concept in dihybrid tests for predicting phenotypes of seeds.

Alleles & GenotypesMathematical ConceptPhenotypic Interpretation
Y y 100+ 0 Yellow Green
Yy 100 Yellow
R r 100+ 0 Round Wrinkled
Rr 100 Round
Alleles & GenotypesMathematical ConceptPhenotypic Interpretation
Y y 100+ 0 Yellow Green
Yy 100 Yellow
R r 100+ 0 Round Wrinkled
Rr 100 Round

## Conclusion

With our new mathematical concept, biology students can better understand that, in the case of heterozygous traits, a dominant allele determines appearance by contributing 100% to the overall phenotype, whereas the recessive allele has no observable effects on the organism and, hence, contributes 0%. Furthermore, using numerical representations of alleles in addition to alphabetical letters can be less confusing and provide a better understanding of allelic effects for dyslexic students and slow learners. We believe that this technique might be beneficial for inclusive biology classrooms as well as special-education settings.

## References

References
Browning, M.E. & Lehman, J.D. (1988). Identification of student misconceptions in genetics problem solving via computer program. Journal of Research in Science Teaching, 25, 747–761.
Freidenreich, H.B., Duncan, R.G. & Shea, N. (2011). Exploring middle school students’ understanding of three conceptual models in genetics. International Journal of Science Education, 33, 2323–2349.
Yilmaz, D., Tekkaya, C. & Sungur, S. (2011). The comparative effects of prediction/discussion-based learning cycle, conceptual change text, and traditional instructions on student understanding of genetics. International Journal of Science Education, 33, 607–628.