Abstract
Teacher-directed and self-directed learning have been compared across various contexts. Depending on the settings and the presentation of material, mixed benefits are found; the specific circumstances under which either condition is advantageous are unclear. We combined and reanalyzed data from two experimental studies investigating the effects of contrasted versus sequential presentation of materials on learning the principles of algebraic addition and multiplication in sixth-grade classrooms. In both studies, students were presented the same structured materials that differed only in whether the principles were explained by the teacher (n = 154) or inferred by the students (n = 91). We found short- and medium-term advantages of combining teacher-direction with contrasted presentation of algebra principles. An examination of aptitude-treatment interactions shows that particularly students with lower reasoning abilities benefited from teacher-direction in the demanding contrasted condition. Based on these findings, we discuss the particular circumstances under which teacher-directed instruction reveals its advantages.
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Acknowledgments
We are grateful to Sara Ziegler for her assistance in the implementation of the Study in the school classes, and Sara Ziegler and Adrienne Suter for their assistance in the evaluation of the test materials.
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Esther Ziegler. Institute for Learning Sciences and Higher Education, ETH Zurich, Switzerland
Current themes of research:
Learning through comparison and contrasting. Explicit and implicit learning. Desirable difficulties and productive failure in learning environments.
Most relevant publications in the field of Psychology of Education:
Edelsbrunner, P. A., Dablander, F. (2019). The Psychometric Modeling of Scientific Reasoning: A Review and Recommendations for Future Avenues. Educational Psychology Review, 31, 1–34. https://doi.org/10.1007/s10648-018-9455-5.
Edelsbrunner, P. A., Schalk, L., Schumacher, R., & Stern, E. (2018). Variable control and conceptual Change: A large-scale quantitative study in primary school. Learning and Individual Differences, 66. https://doi.org/10.1016/j.lindif.2018.02.003.
Hofer, S. I, Schumacher, R., Rubin, H., & Stern, E. (2018). Enhancing physics learning with cognitively activating instruction: A quasi-experimental classroom intervention study. Journal of Educational Psychology, 110(8), 1175–1191. https://doi.org/10.1037/edu0000266.
Schalk, L., Edelsbrunner, P. A., Deiglmayr, A., Schumacher, R., & Stern, E (2019). Improved application of the control-of-variables strategy as a collateral benefit of inquiry-based physics education in elementary school. Learning and Instruction, 59, 34–45. https://doi.org/10.1016/j.learninstruc.2018.09.006.
Ziegler, E., Edelsbrunner, P. A. & Stern, E. (2018). The relative merits of explicit and implicit learning of contrasted algebra principles. Educational Psychology Review. 30(2), 531–558. https://doi.org/10.1007/s10648-017-9424-4.
Ziegler, E., Edelsbrunner, P., & Star, J. R. (2019). Preventing interference: Reordering complexity in the learning of new concepts. Journal of Educational Psychology. https://doi.org/10.1037/edu0000347.
Ziegler, E. & Stern, E. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41(1), 41–51. https://doi.org/10.1016/j.learninstruc.2015.09.006.
Ziegler, E. & Stern, E. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33(1), 131–146. https://doi.org/10.1016/j.learninstruc.2014.04.006.
Peter A. Edelsbrunner. Institute for Research on Learning and Instruction, ETH Zurich, Switzerland
Current themes of research:.
Mixture modeling in educational and developmental research. Scientific thinking throughout the lifespan. Statistical and psychometric practices in educational and developmental research.
Most relevant publications in the field of Psychology of Education:
Edelsbrunner, P. A., Dablander, F. (2019). The Psychometric Modeling of Scientific Reasoning: A Review and Recommendations for Future Avenues. Educational Psychology Review, 31, 1–34. https://doi.org/10.1007/s10648-018-9455-5.
Edelsbrunner, P. A., Schalk, L., Schumacher, R., & Stern, E. (2018). Variable control and conceptual Change: A large-scale quantitative study in primary school. Learning and Individual Differences, 66. https://doi.org/10.1016/j.lindif.2018.02.003.
Schalk, L., Edelsbrunner, P. A., Deiglmayr, A., Schumacher, R., & Stern, E (2019). Improved application of the control-of-variables strategy as a collateral benefit of inquiry-based physics education in elementary school. Learning and Instruction, 59, 34–45. https://doi.org/10.1016/j.learninstruc.2018.09.006.
Ziegler, E., Edelsbrunner, P. A. & Stern, E. (2018). The relative merits of explicit and implicit learning of contrasted algebra principles. Educational Psychology Review. 30(2), 531–558. https://doi.org/10.1007/s10648-017-9424-4.
Ziegler, E., Edelsbrunner, P., & Star, J. R. (2019). Preventing interference: Reordering complexity in the learning of new concepts. Journal of Educational Psychology. https://doi.org/10.1037/edu0000347.
Ziegler, E. & Stern, E. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41(1), 41–51. https://doi.org/10.1016/j.learninstruc.2015.09.006.
Ziegler, E. & Stern, E. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33(1), 131–146. https://doi.org/10.1016/j.learninstruc.2014.04.006.
Elsbeth Stern. Institute for Research on Learning and Instruction, ETH Zurich, Switzerland
Current themes of research:
Academic learning in science and mathematics. Knowledge transfer by the use of visual-spatial cognitive tools.
Most relevant publications in the field of Psychology of Education:
Edelsbrunner, P. A., Dablander, F. (2019). The Psychometric Modeling of Scientific Reasoning: A Review and Recommendations for Future Avenues. Educational Psychology Review, 31, 1–34. https://doi.org/10.1007/s10648-018-9455-5.
Edelsbrunner, P. A., Schalk, L., Schumacher, R., & Stern, E. (2018). Variable control and conceptual Change: A large-scale quantitative study in primary school. Learning and Individual Differences, 66. https://doi.org/10.1016/j.lindif.2018.02.003.
Schalk, L., Edelsbrunner, P. A., Deiglmayr, A., Schumacher, R., & Stern, E (2019). Improved application of the control-of-variables strategy as a collateral benefit of inquiry-based physics education in elementary school. Learning and Instruction, 59, 34–45. https://doi.org/10.1016/j.learninstruc.2018.09.006.
Ziegler, E., Edelsbrunner, P. A. & Stern, E. (2018). The relative merits of explicit and implicit learning of contrasted algebra principles. Educational Psychology Review. 30(2), 531–558. https://doi.org/10.1007/s10648-017-9424-4.
Ziegler, E., Edelsbrunner, P., & Star, J. R. (2019). Preventing interference: Reordering complexity in the learning of new concepts. Journal of Educational Psychology. https://doi.org/10.1037/edu0000347.
Ziegler, E. & Stern, E. (2016). Consistent advantages of contrasted comparisons: Algebra learning under direct instruction. Learning and Instruction, 41(1), 41–51. https://doi.org/10.1016/j.learninstruc.2015.09.006.
Ziegler, E. & Stern, E. (2014). Delayed benefits of learning elementary algebraic transformations through contrasted comparisons. Learning and Instruction, 33(1), 131–146. https://doi.org/10.1016/j.learninstruc.2014.04.006.
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Appendices
Appendix 1. Script excerpts of the blackboard instruction
Under both instructions: 1. T writes an addition problem on the blackboard without any comment, e.g., “xy + xy + xy =”. 2. T reads the problem aloud. 3. T asks S: “How could one solve this problem? Does anyone have a good idea? Who can guess the solution?” 4. After a correct response, “3xy”, T confirms the solution and demonstrates how to proceed step-by-step (6a–b). 5. After an incorrect response, “3x3y”, T writes the incorrect response on the blackboard and crosses through it. 6. After an incorrect or no response: T demonstrates how to proceed step-by-step (A). a. One counts the number of summands, i.e., 3 times xy; thus, one can write “3 ∙ xy”. b. The multiplication sign can be excluded; thus, the result is summarized as “3xy”. 7. T then continues with the second problem. | |
Under the contrast condition: 8. After solving the first addition, T presents the first multiplication on the right side of the blackboard. 9. T writes the multiplication problem on the blackboard, e.g., “xy ∙ xy ∙ xy =”. 10. T states: “You now know how to solve an identical problem using addition. Does anyone have a good idea of how to proceed with multiplication? Who can help? Who can guess the solution?” 11. After a correct response, “x3y3”, T confirms the solution and demonstrates how to proceed step-by-step (A): a. In contrast to addition (T points to the left side of the blackboard); in multiplication, double letters are split into single letters separated by multiplication signs, “x ∙ y ∙ x ∙ y ∙ x ∙ y =”. b. T asks S: “What’s the next step? Who can help?” c. Then, identical letters are placed together, “x ∙ x ∙ x ∙ y ∙ y ∙ y =”. d. T asks S to help. e. Then, the number of letters is counted and written as an exponent after the corresponding letter, “x3 ∙ y3”. f. The result is subsequently written without the multiplication sign or intervening spaces, “x3y3”. 12. T steps back from the blackboard: “Let us have a look at both problems, which includes addition on the left side and multiplication on the right side”. 13. “What can you say? What are the differences? Can you compare the two?” 14. S provide responses (…). 15. T confirms or corrects S responses. 16. Then, T provides a summary by comparing both sides: “In addition, (…). However, in multiplication, (…)” (A). 17. T continues with the 2nd addition on the left side, followed by the 2nd multiplication on the right side. 18. The sequence is repeated. | |
Under the sequential condition: 8. After solving the first addition problem, T presents the second addition problem below the first problem on the left side of the blackboard. 9. T writes the addition problem on the blackboard, e.g., “2b + 2b + 2b + 2b =”. 10. T states: “You previously solved an addition problem. Does anyone have a good idea of how to proceed with this next addition problem? Who can help? Who can guess the solution?” 11. After a correct response, “8b”, T confirms the solution and demonstrates how to proceed step-by-step (A): a. “Similar to the first addition problem (T points to the first example), we can count the number of summands, in this case, 2b. This is “4 times 2b”; thus, we can write “4 ∙ 2b =”.” b. T asks S: “What’s the next step? Who can help?” c. Then, 4 times 2 = 8, which gives “8b”. 12. T steps back from the blackboard: “Let us have a look at both addition problems and their solutions”. 13. “What can you say? Can you describe how to solve addition problems?” 14. S provide responses (…). 15. T confirms or corrects S responses. 16. Then, T provides a summary by describing: “In this addition (…). In that addition (…). In addition in general” (A). 17. T then continues with the 3nd addition, followed by the 4th addition (when the left side of the blackboard is full, T continues on the right side). 18. The sequence is repeated. |
Appendix 2. Coding scheme for the follow-up test “Algebraic transformation explanations”
For every one of the following features mentioned in the answer, one point was scored. The features were divided into defining features and secondary features. The defining features were the principles considered central and essential for distinguishing algebraic addition from algebraic multiplication. The secondary features were external features and conventions, e.g., reporting the result. Points were given equally if: (a) a key element was explicitly mentioned or (b) a key element became visible in the solution or intermediate steps of the example. If an element was described incorrectly, it was counted as an error.1 | |
Sub-analysis of addition: Defining features/principles (total of 6 points) (d = defining feature, s = secondary feature): d1: Sorting by summands (summands are the letter endings (often with exponents: a, ab, a2, …) ➔ only ½ point: if only single letters were mentioned or used (a, b, c, …) ➔ 2 points: if sufficient detail was provided in a way that examples became redundant or dispensable d2: Summands are not split d3: Summands do not change in the result, exponents remain (a3 + a3 = 2a3), no points are given if only single letters d4: Letters with different exponents are not summarized, merged, or unified (x + x2) d5: Single letters are not summarized, merged, or unified with double letters (a + ab) d6: Letters and numbers are not summarized, merged, or unified (x + 4) Secondary features/principles (total of 4 points): s1: In the result, there are groups with different summands (2bc 3b 2c) s2: In the result, the different summands are connected by a plus sign (2a + 3b) s3: The number of letters stays directly in front of the letter (5a) s4: A single letter counts as 1 (c = 1c) To summarize, an exemplary instruction of the concept of algebraic addition: 1. Sorts according to identical summands. Pays attention to double letter variables and variables with exponents that are added unchanged, i.e., cx and c2 are not separated. This underscores the identical combinations of letters. 2. Each type of unit (cx, c2, c, 2) is added separately by adding the numbers of summands and appending the unit unchanged. Detached numbers are added separately. 3. The result consists of the number of each different unit connected with a plus sign. | |
Sub-analysis of multiplication : Defining features/principles (total of 5 points) (d = defining feature, s = secondary feature): d1: All factors are split ➔ only ½ point: if only single letters are mentioned or used (2x ∙ 3y ∙ 3x) ➔ 2 points: if a sufficiently detailed, full description is provided (separate units in the single components or mention important rules twice) d2: Letters with exponents are separated into single letters (x2 = x ∙ x) d3: Double letters are separated (ab = a ∙ b) d4: Numbers and letters are separated (2z = 2 ∙ z) d5: Exponents are added (c2 ∙ c2 ∙ c2 = c6) Secondary features/principles (total of 4 points): s1: In the result, everything is merged into one expression (16a2b4) s2: In the result, the numbers are first s3: The number of equal letters is noted as an exponent behind and above the letter (a3) s4: The numbers are multiplied In summary, an exemplary instruction of the concept of algebraic multiplication: 1. Splits the term into single numbers and letters. Pays attention to split units, such as ab, b2, and 2a, into their components. 2. Combines numbers and always combines identical letters 3. Multiplies all numbers and writes the amount at the first position. Then, counts the number of each letter and writes their number as an exponent after the corresponding letter. Writes the letters one after the other without any intervening spaces. |
Appendix 3. Treatment fidelity check for the self-study materials
To ensure comparable instructions for the contrast and sequential groups, the accuracy of generated examples was included as a treatment fidelity check. The accuracy was measured by the solution rates of correctly generated examples and can be considered an indicator of equality of the self-study parts for both conditions. There was no difference between conditions in the solution rates of generated examples, (contrast condition: M = 94.4%, SD = 5.9; sequential condition: M = 95.0%, SD = 4.8), F(1, 152) = 0.29, p = 0.589. The examples were generated by imitating the worked examples presented on the worksheet and were an indicator of processing accuracy. Both groups had comparable solution rates for the generated examples, indicating that they processed the worked examples equally. This suggests comparable commitment and engagement of the learners under both types of instruction. | |
Treatment fidelity check for the teacher-instructions To ensure that comparable instructions were given to the contrast and sequential groups, the lessons were videotaped to allow a treatment fidelity check. The instructor followed detailed scripts, which were exactly observed. Therefore, the videos were not analyzed for content but instead were rated for student participation. Six different single problems in the contrast and sequential versions were chosen for comparisons. The main measure was student participation, which had two sub-measures: (1) participation, as measured by the number of student statements or student answers during blackboard instruction, and (2) disturbances, as measured by the number of rebukes by the teacher (more detailed information in, Ziegler & Stern 2016). The files were rated by an observer who was not informed about the experimental variation with regard to the instruction types. Twenty percent of the files were rated by a second observer, and there was an exact agreement of 100%. A multivariate analysis of variance (MANOVA) revealed no difference between conditions on the measure student participation, F(2, 7) = 0.36, p = .707, η2p = .09. None of the separate univariate ANOVA tests revealed an effect for participation (contrast condition: M = 155.8 statements, SD = 26.0; sequential condition: M = 140.2 statements, SD = 29.7), F(1, 8) = 0.78, p = .403, η2p = .09, or for disturbances (contrast condition: M = 12.8 interruptions, SD = 4.8; sequential condition: M = 13.6 interruptions, SD = 12.4), F(1, 8) = 0.02, p = .897, η2p = .00, indicating that the contrast condition and sequential condition did not differ in students’ commitment and engagement. |
Appendix 4. List with the distribution of the principles for every learning step
Addition learning steps | Multiplication learning steps | Principles |
|---|---|---|
A1/A2 | M1/M2 | Add/multiply like terms with single variables |
A3 | M3 | Add/multiply like terms with multiple variables |
A4 | M4 | Add/multiply like terms with variables with exponents |
A5 | M5 | Add/multiply like terms by adding their coefficients |
A6 | M6 | Add/multiply like and unlike terms with single variables and numbers |
A7 | M7 | Add/multiply like and unlike terms with single variables and multiple variables or variables with exponents |
A8 | M8 | Add/multiply like and unlike terms with coefficients with single variables and numbers |
A9 | M9 | Add/multiply like and unlike terms with coefficients with single variables, multiple variables, variables with exponents and numbers |
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Ziegler, E., Edelsbrunner, P.A. & Stern, E. The benefit of combining teacher-direction with contrasted presentation of algebra principles. Eur J Psychol Educ 36, 187–218 (2021). https://doi.org/10.1007/s10212-020-00468-3
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DOI: https://doi.org/10.1007/s10212-020-00468-3