Michael F. Z.
Page
*,
Patrick
Escott
,
Maritza
Silva
and
Gregory A.
Barding
Jr.

Chemistry and Biochemistry Department, California State Polytechnic University, Pomona, Pomona, CA 91768, USA. E-mail: mfpage@cpp.edu; Tel: +1 (909) 869-4533

Received
21st December 2017
, Accepted 7th February 2018

First published on 7th February 2018

This case study demonstrates the ability of high school chemistry students, with varying levels of math preparation, to experience learning-gains on state and district assessments as it relates to chemical reactions, thermodynamics, and kinetics. These advances were predicated on the use of a teaching style rooted in abstract reasoning. The methodology was presented to students and modeled by the instructor over an entire school year to reinforce key proportional relationships featured in the balanced chemical equation and related topics such as acids and bases, reaction rates, equilibrium, and conservation of matter. Despite the small sample size, there was a general increase in student success, indicated by a statistically significant difference between students receiving instruction rooted in concrete reasoning and students receiving instruction rich in abstract reasoning.

In most chemical education investigations of stoichiometry, a keen focus is often on the mole concept. Recently, Ramful and Narod (2014) identified that, in addition to the mole concept, many stoichiometry problems have multiple levels of proportionality that could prove increasingly difficult to solve. Generally, there are five levels of complexity involved in stoichiometric calculations including (a) the use of a simple mole ratio; (b) unit conversion and incorporation of mole ratios; (c) proportional relationships based on volume, the number of particles, or concentration; (d) utilizing various proportional relationships expressed in multiple reaction equations such as a chemical reaction followed by a titration or excess/limiting reagents; and (e) calculations in which one quantity or identity is unknown and must be determined. Furthermore, Scott (2012) compared high school students’ ability to solve analogous chemistry and mathematical questions. This study found that the mole concept is not the reason for poor success in solving problems. When difficult chemistry and math problems were solved, the math problem solutions were homogenous and often completed in an algorithmic fashion. Conversely, the more difficult chemistry questions were solved in a variety of fashions utilizing abstract reasoning. Therefore, higher-level abstract learning could be hindered if students rely on algorithms. In the US this is further complicated because a majority of chemistry textbooks present solving stoichiometric calculations using concrete reasoning (CR) that is guided by unit cancellation (Tykodi, 1987). This method of rote memorization (Dori and Hameiri, 2003) is highlighted in solved examples in seventeen contemporary textbooks reviewed by Cook and Cook (2005).

When employing concrete reasoning, a common mistake made by students is that they often invert the conversion factors; meaning a value that should be in the numerator is improperly placed in the denominator or vice versa. Furthermore, studies have shown that students who are able to properly use mole ratio coefficients, shown in Fig. 1, do not understand the chemical concepts that the balanced equation conveys (Nurrenbern and Pickering, 1987; Sanger, 2005).

In Fig. 2, the previous question is again solved, this time using abstract reasoning (Schmidt and Jingnéus, 2003; Page et al., 2012; Struck and Yerrick, 2013). Using the balanced molecular equation, a student can reason that 1 mole of Fe_{2}O_{3} can completely react with 2 moles of Al. Then, using the molar masses of the starting materials [159.7 g mol^{−1} Fe_{2}O_{3} and 26.9 g mol^{−1} Al], a student can deduce that 159.7 g of Fe_{2}O_{3} can completely react with 53.8 g of Al (shown in the denominator). Finally, a proportional relationship can be used to solve for the unknown quantity that 28.6 grams of Al are required to completely react with iron(III) oxide. Chandrasegaran et al. (2009) found that average- and lower-achieving students actually benefit from instruction that highlights reasoning strategies rooted in the balanced equation. Furthermore, abstract reasoning can also be extended to solve problems dealing with concentrations, titrations, densities, percent masses, percent yields, solution stoichiometry, and gas laws since each of these concepts is based on a proportional relationship (Beichl, 1986; DeToma 1994; Cook and Cook, 2005). Based on the cognitive development of high school students and their need to gain expertise in abstract reasoning, it seems logical that the extent of chemical learning should be examined through the incorporation of abstract reasoning in the day-to-day science instructional delivery to students with varying math aptitudes and reasoning capabilities.

Cluster | Area of emphasis | Number of questions on state exam | Percent of state exam |
---|---|---|---|

a Content questions that can be answered using abstract reasoning. Note: areas of emphasis that are underlined represent the content area description on the state exam, while the descriptions in parentheses represent the content area description of the corresponding items on the district benchmark assessments. | |||

1 | Atomic and Molecular Structure | 8 | 13.3 |

(Atomic and Molecular Structure) | |||

(Nuclear Process) | |||

2 | Chemical Bonds, Biochemistry | 9 | 15 |

(Chemical Bonds) | |||

(Organic Chemistry and Biochemistry) | |||

3 | Kinetics and Thermodynamics | 14^{a} |
23.3^{a} |

(Gases and their Properties) | |||

(Solutions) | |||

(Chemical Thermodynamics) | |||

4 | Chemical Reactions | 13^{a} |
21.7^{a} |

(Acids and Bases) | |||

(Reaction Rates) | |||

(Chemical Equilibrium) | |||

5 | Conservation of Matter and Stoichiometry | 10^{a} |
16.7^{a} |

(Conservation of Matter and Stoichiometry) | |||

6 | Investigation and Experimentation | 6 | 10 |

(Investigation and Experimentation) |

Prior to the intervention, two years of student assessment performances were collected for each instructor and served as individual control populations. Both case study teachers had taught chemistry for a minimum of five years and had previously taught their students using CR. The teachers also agreed to form a learning community consisting of chemistry faculty members and undergraduate thesis students to support and standardize their instructional methods of AR to model solving stoichiometric problems, gas laws, pH, solutions, thermodynamics, reaction rates, and equilibrium calculations. During the intervention, the treatment students received a full year of AR instruction during their chemistry course, whereas the control students had previously received instruction rooted in CR.

In the US, students traditionally matriculate sequentially through Algebra I, Geometry, and Algebra II, and then advance to various levels of calculus or statistics courses. Therefore, based on the state math exit exam, Teacher A instructed an advanced population of students who were in Summative Math courses or Algebra II as corequisites with chemistry [see Table 2]. At the time of the intervention, the school site had an Academic Performance Index (API) of 814 and 51% of the students were classified as socioeconomically disadvantaged. Teacher B instructed a proficient and passing population of students with a greater ratio of students enrolled in Algebra II or Geometry, which are less progressive courses. This school site had an API of 670 and 95% of the students were classified as socioeconomically disadvantaged (see on the Web Materials Pomona Unified School Accountability Report Card (SARC) Reports, 2013). The API is a three-year measure of academic preparation of the student body on statewide assessments across multiple disciplines. The calculation can range from a low 200 to a high score of 1000. The expected goal (determined by the state) is for schools to achieve an API of 800.

Control population | Treatment population | |
---|---|---|

Note: [y = Advanced], [+ = Proficient], [T = Passing] (for more information, see on the Web Materials California High School Exit Examination and Understanding California's Standardized Testing and Reporting (STAR) Program). | ||

Teacher A | ||

Summative Math | n = 116 | n = 86 |

State Math Exit Exam Average | 433^{y} |
425^{y} |

Algebra II | n = 119 | n = 47 |

State Math Exit Exam Average | 415^{y} |
406^{y} |

Teacher B | ||

Summative Math | n = 49 | n = 20 |

State Math Exit Exam Average | 386^{+} |
399^{+} |

Algebra II | n = 124 | n = 54 |

State Math Exit Exam Average | 378^{T} |
369^{T} |

Geometry | n = 38 | n = 5 |

State Math Exit Exam Average | 371^{T} |
366 ^{T} |

The study populations, shown in Table 2, met the following criteria: (i) students completed both the first and second semesters of the introductory year of chemistry with the same teacher; (ii) students completed all of the district benchmark chemistry and math corequisite exams, with one exception being granted to graduating seniors who had conflicting administrative activities during the period of the very last assessment; and (iii) students completed state chemistry and math corequisite assessments. The overall state and district performances of the students were compared by teachers and analyzed using the stats package in R, a free software environment for statistical computing and graphics with analysis of variance functions (R Core Team, 2016). One- and two-way ANOVA were carried out using the one-way.test and aov functions. A one-way ANOVA was used to evaluate the likelihood that the two teaching styles had an impact on student chemistry learning outcomes, assuming unequal variance. Additionally, a two-way ANOVA was used to evaluate if the math course corequisite (i.e. Summative Math, Algebra II, or Geometry) of the students influenced their chemistry performance based on the different teaching methods used within this investigation. Because of the small, unbalanced data set, the results obtained from the two-way ANOVA (see the Appendix) were only used qualitatively and assisted in determining if an additional study on a more robust sample size with equal variances was warranted.

Control population average scores | Treatment population average scores | |
---|---|---|

Teacher A | ||

Note: values indicate questions answered correctly. Standard deviation on the district benchmarks could not be calculated since graduating students in the populations are not required to participate in the final assessment in the second semester due to administrative conflicts. | ||

State Chemistry Assessment (Total 60 Questions) | 45.3/60 (±7.6) | 45.9/60 (±7.8) |

District Chemistry Assessment (Total 138 Questions) | 102.9/138 | 102.0/138 |

The chemistry proficiencies of various subset populations were also analyzed. In Fig. 3A, chemistry students in the treatment population, who were co-enrolled in Summative Math courses, demonstrated significant learning gains over the control as it pertained to Chemical Reactions (on the State Assessment). The same population of treatment students also had significant gains in the content areas of Acid and Bases, Reaction Rates, and Chemical Equilibrium during the district assessments when compared with the associated control (see Fig. 3B). Similarly, chemistry students in the treatment group, who were enrolled in Algebra II, demonstrated significant gains compared with the associated control on the district assessments in the areas of Acid and Bases and Reaction Rates. Although chemistry gains of students co-enrolled in Algebra II were not as dramatic as those of the higher-level Summative Math students, both treatment populations experienced learning enhancements correlated with receiving instruction rich in AR. The exact p-values of each one-way ANOVA regarding the population of Teacher A can be found in Table 5 in the Appendix.

In the classroom of Teacher B it was also demonstrated that the control and treatment students answered a similar percentage of overall questions correctly on both the state and district chemistry exams. On the state exam, the control and treatment populations correctly answered an average of 26.1 and 27.0 chemistry assessment items, respectively, yielding an overall average score of ∼44% for both populations. The same groups answered an average of 66.3 and 63.1 assessment questions, respectively, netting a similar overall average score of ∼47% on the district benchmark assessments [see Table 4]. Likewise, this result also demonstrates that the control and treatment populations had similar overall performances in the course, but includes material not directly influenced by AR, and was therefore not evaluated for significance within this study.

Control population average scores | Treatment population average scores | |
---|---|---|

Teacher B | ||

Note: values indicate questions answered correctly. Standard deviation on the district benchmarks could not be calculated since graduating students in the populations are not required to participate in the final assessment in the second semester due to administrative conflicts. | ||

State Chemistry Assessment (Total 60 Questions) | 26.1/60 (±7.2) | 27.0/60 (±7.9) |

District Chemistry Assessment (Total 138 Questions) | 66.3/138 | 63.1/138 |

Chemistry students in the treatment population, who were co-enrolled in Summative Math courses, maintained or exceeded learning gains compared to the control. In Fig. 4, significant gains were highlighted for the treatment students who were co-enrolled in Summative Math in the content areas of Kinetics and Thermodynamics (state exam) and Acid and Bases (district assessment). Likewise, chemistry students in the treatment population, who were enrolled in Algebra II, demonstrated a similar trend when compared with the associated control on the state test in the area of Chemical Reactions. Although gains of students, co-enrolled in Algebra II, again were not as overwhelming as the Summative Math students, both treatment populations did still experience learning enhancements. The exact p-values of each one-way ANOVA regarding the population of Teacher B can be found in Table 6 in the Appendix.

The increased capacity of chemistry content by the students was collaborated in the transcribed reflections of both teachers during the final program evaluation.

[Teacher A] I was apprehensive of teaching the new AR method, but once I saw how it worked, I found it superior to CR because it shows the process of forming relationships. This was key because it did not encourage a blind focus on the units by the students. I also felt supported based on the work of the learning community. What I have found over the past few years is that all my students learn stoichiometry now. This problem-solving strategy remains relevant as we cover molarity, gas laws, and equilibrium in the second semester. Over the years, the vast majority of my Advance Placement chemistry students (not included in this intervention) switch to AR once they gain confidence, especially when solving limiting and excess reagent calculations. There are still times when a concrete approach makes more sense, for example when there is a multi-step conversion, but I have students who will still find a way to AR since it is extremely logical.

[Teacher B] By solving stoichiometry calculations using AR, this method makes the chemistry curriculum more accessible to students of diverse backgrounds and varying levels of math preparedness. The success of the method can be attributed to the visual nature of the relationships. The relationships between mass, moles, and equation coefficients are a pivotal part of introducing the strategy that aids all students to identify the imbedded ratios and proportionality of the balanced chemical equation. Ultimately, students rely on valid relationships in order to solve calculations.

Although advanced math students demonstrated the highest levels of chemistry competencies, this classroom intervention allowed greater opportunity for all students to (i) improve their abstract reasoning skills through the construction of valid proportions and (ii) solve a range of problems building on valid proportions as it relates to chemical reactions, pH, thermodynamics, kinetics, and equilibrium. This work is timely as the US is currently modifying its framework of science education to align with the Next Generation Science Standards (National Research Council, 2012, p. 75). This work suggests that slight gains can be realized through the use of AR in chemical instruction. In the US, this represents an innovation that may not be realistic immediately on a grand scale. However, greater long-term understanding on the part of the student could be further analyzed if this pilot study longitudinally extends to explore if the gains in these high school students, utilizing a small sample of teachers in diverse socioeconomic environments, can be replicated across an entire school district and even to the university-level curriculum.

Teacher A state assessment results (Summative Math) | Teacher A state assessment results (Algebra II) | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Conservation of Matter and Stoichiometry | Chemical Reactions | Kinetics and Thermodynamics | Conservation of Matter and Stoichiometry | Chemical Reactions | Kinetics and Thermodynamics | |

(10 Questions) | (13 Questions) | (14 Questions) | (10 Questions) | (13 Questions) | (14 Questions) | |

[p-Value < 0.05 statistically relevant] Summative Math Control n = 116; Summative Math Treatment n = 86; Algebra II Control n = 119; Algebra II Treatment n = 47. | ||||||

Control population (aver. score) | 8.0 | 9.5 | 10.9 | 7.0 | 8.7 | 9.8 |

Treatment population (aver. score) | 7.9 | 10.2 | 11.2 | 6.3 | 8.6 | 10.0 |

One-way ANOVA | F(1, 186) = 0.28, p = 0.60 | F(1, 191) = 5.48, p = 0.02 | F(1, 189) = 1.42, p = 0.24 | F(1, 78) = 3.09, p = 0.08 | F(1, 97) = 0.13, p = 0.71 | F(1, 84) = 0.21, p = 0.65 |

Teacher A district assessment results (Summative Math) | Teacher A district assessment results (Algebra II) | |||||||
---|---|---|---|---|---|---|---|---|

7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |

Conservation of Matter and Stoichiometry | Acids and Bases | Reaction Rates | Chemical Equilibrium | Conservation of Matter and Stoichiometry | Acids and Bases | Reaction Rates | Chemical Equilibrium | |

(15 Questions) | (12 Questions) | (9 Questions) | (6 Questions) | (15 Questions) | (12 Questions) | (9 Questions) | (6 Questions) | |

Control population (aver. score) | 12.0 | 9.3 | 6.7 | 4.2 | 9.9 | 8.6 | 6.2 | 3.5 |

Treatment population (aver. score) | 12.2 | 10.4 | 7.3 | 4.6 | 10.1 | 9.4 | 6.8 | 3.9 |

One-way ANOVA | F(1, 196) = 0.32, p = 0.57 |
F(1, 188) = 23.7, p = 2.4 × 10^{−6} |
F(1, 182) = 8.60, p = 3.8 × 10^{−3} |
F(1, 191) = 4.23, p = 0.04 | F(1, 100) = 0.24, p = 0.63 | F(1, 83) = 6.66, p = 0.01 | F(1, 87) = 5.01, p = 0.03 | F(1, 79) = 2.62, p = 0.11 |

Teacher B state assessment results (Summative Math) | Teacher B state assessment results (Algebra II) | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Conservation of Matter and Stoichiometry | Chemical Reactions | Kinetics and Thermodynamics | Conservation of Matter and Stoichiometry | Chemical Reactions | Kinetics and Thermodynamics | |

(10 Questions) | (13 Questions) | (14 Questions) | (10 Questions) | (13 Questions) | (14 Questions) | |

[p-Value < 0.05 statically relevant] Summative Math Control n = 49; Summative Math Treatment n = 20; Algebra II Control n = 124; Algebra II Treatment n = 54. | ||||||

Control population (aver. score) | 4.3 | 5.2 | 6.4 | 3.4 | 4.5 | 6.1 |

Treatment population (aver. score) | 4.3 | 6.0 | 8.1 | 3.0 | 5.5 | 6.3 |

One-way ANOVA | F(1, 35) = 0.02, p = 0.90 | F(1, 32) = 1.59, p = 0.22 | F(1, 33) = 5.70, p = 0.02 | F(1, 85) = 2.32, p = 0.13 |
F(1, 94) = 8.22, p = 5.1 × 10^{−3} |
F(1, 109) = 0.41, p = 0.52 |

Teacher B district assessment results (Summative Math) | Teacher B district assessment results (Algebra II) | |||||||
---|---|---|---|---|---|---|---|---|

7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | |

Conservation of Matter and Stoichiometry | Acids and Bases | Reaction Rates | Chemical Equilibrium | Conservation of Matter & Stoichiometry | Acids and Bases | Reaction Rates | Chemical Equilibrium | |

(15 Ques.) | (12 Ques.) | (9 Ques.) | (6 Ques.) | (15 Ques.) | (12 Ques.) | (9 Ques.) | (6 Ques.) | |

Control population (aver. score) | 7.0 | 6.2 | 4.5 | 2.8 | 6.2 | 5.7 | 3.8 | 2.4 |

Treatment population (aver. score) | 6.9 | 7.4 | 5.3 | 2.5 | 6.3 | 6.1 | 4.2 | 2.2 |

One-way ANOVA |
F(1, 38) = 6.0 × 10^{−3}, p = 0.94 |
F(1, 30) = 4.22, p = 0.05 | F(1, 33) = 2.12, p = 0.15 | F(1, 53) = 1.43, p = 0.24 | F(1, 86) = 0.06, p = 0.81 | F(1, 77) = 1.10, p = 0.30 | F(1, 87) = 1.26, p = 0.26 | F(1, 100) = 0.93, p = 0.34 |

The interaction effects below describe the effect of the students’ math level on the treatment compared to the control population presented in Table 5:

Column #1 versus #4: F(1, 1) = 4.33, p = 0.04 Column #2 versus #5: F(1, 1) = 2.78, p = 1.00

Column #3 versus #6: F(1, 1) = 1.46, p = 0.23

Column #7 versus #11: F(1, 1) = 5.3 × 10^{−3}, p = 0.94 Column #8 versus #12: F(1, 1) = 0.61, p = 0.43

Column #9 versus #13: F(1, 1) = 2.8 × 10^{−3}, p = 0.96 Column #10 versus #14: F(1, 1) = 7.6 × 10^{−3}, p = 0.93

The interaction effects below describe the effect of the students’ math level on the treatment compared to the control population presented in Table 6:

Column #1 versus #4: F(1, 1) = 0.56, p = 0.45 Column #2 versus #5: F(1, 1) = 3.76, p = 0.05

Column #3 versus #6: F(1, 1) = 1.56, p = 0.21

Column #7 versus #11: F(1, 1) = 5.4 × 10^{−3}, p = 0.82 Column #8 versus #12: F(1, 1) = 1.89, p = 0.17

Column #9 versus #13: F(1, 1) = 0.68, p = 0.40 Column #10 versus #14: F(1, 1) = 0.21, p = 0.65

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