The Relationship of Goal-Setting in the Teaching of Mathematics with its Technological Support
Abstract
Technological support of teaching mathematics depends on what methodological and pedagogical goals are put for learning. Achieving or failing to achieve these goals is connected with the used type of feedback or in other words, the method of assessing the educational activities of students. In this work, two types of assessment are contrasted: a test form of knowledge testing (implemented by a system of mid-term and final exams) and a formative assessment (determined by the teacher’s informal reaction to the student’s productive activities and the way these activities are organized). It is shown that the first type of assessment corresponds to the consideration of the curriculum as a learning goal, the second — as a learning tool. In the first case, the purpose of training is the acquisition of specific knowledge and skills, and in the second, the mastery of the general mechanisms of educational activity inherent in a given subject area (mathematics). For the first goal, it is effective to use template tasks including generated exercises and simulators, for the second — to use various tools that support constructive and research activities.
The article shows how “non-invasive monitoring” is used to achieve the second goal, when the teacher and the student are not on opposite sides of the academic barrier (the student answers — the teacher sets a mark), but on the same side and jointly perform actions to create conditions for the most effective mastery of the course material by each student. The basis of non-invasive monitoring is modeling the presentation of the results of this activities to the scientific community, including all intermediate stages of such activities. Instead of testing knowledge and issuing formal marks, feedback is used, various approaches and ways to solve the problem are discussed together, and monitoring is limited to students’ self-esteem, which is not necessarily communicated to the teacher. At the same time, the discussion process itself is open, and the teacher can always evaluate the problems of students, without turning them into an instrument of formal pressure on the student by third parties.
References
W. He, A. Holton, G. Farkas, and M. Warschauer, “The effects of flipped instruction on out-of-class study time, exam performance, and student perceptions,” Learning and Instruction, no. 45, pp. 61–71, 2016; doi: 10.1016/j.learninstruc.2016.07.001
S. Vanslambrouck, C. Zhu, B. Pynoo, V. Thomas, K. Lombaerts, and J. Tondeur, “An in-depth analysis of adult students in blended environments: Do they regulate their learning in an “old school” way?” Computers & Education, vol. 128, pp. 75–87, 2019; doi: 10.1016/j.compedu.2018.09.008
J. Broadbent and W. L. Poon, “Self-regulated learning strategies & academic achievement in online higher education learning environments: A systematic review,” Internet and Higher Education, vol. 27, pp. 1–13, 2015; doi: 10.1016/j.iheduc.2015.04.007
J. Xia, J. Fielder, and L. Siragusa, “Achieving better peer interaction in online discussion forums: A reflective practitioner case study,” Issues in Educational Research, vol. 23, no. 1, pp. 97–113, 2013.
C. M. Mueller and C. S. Dweck, “Praise for intelligence can undermine children’s motivation and performance,” Journal of Personality and Social Psychology, vol. 75, no. 1, pp. 33–52, 1998; doi: 10.1037/0022-3514.75.1.33
A. A. Holland, “Effective principles of informal online learning design: A theory-building metasynthesis of qualitative research,” Computers & Education, vol. 128, pp. 214–226, 2019; doi: 10.1016/j.compedu.2018.09.026
M. Ally, M. Cleveland-Innes, N. Boskic, and S. Larwill, “Learners’ use of learning objects,” Journal of Distance Education, vol. 21, no. 2, pp. 44–57, 2006.
K. Clark, “Serving underserved communities with instructional technologies: Giving them what they need, not what you want,” Urban Education, vol. 40, pp. 430–445, 2005; doi: 10.1177/0042085905276388
G. Liu, “Understanding tree: a tool for estimating an individual’s understanding of conceptual knowledge,” arXiv preprint arXiv:1708.00335v.2, 2018.
T. Crooks, “The Validity of Formative Assessments,” in Proc. British Educational Research Association Annual Conference, University of Leeds, pp. 13–15, 2001.
A. Huhta, “Diagnostic and Formative Assessment,” B. Spolsky and F. M. Hult, eds., The Handbook of Educational Linguistics, Oxford, UK: Blackwell, pp. 469–482, 2010.
L. A. Shepard, “Formative assessment: Caveat emptor,” in ETS Invitational Conference The Future of Assessment: Shaping Teaching and Learning, New York, Oct. 10–11, 2005.
P. I. Laina, “Rezul’tativnost’ obucheniya matematike v shkole” [The effectiveness of teaching mathematics at school], Ph.D. dissertation, Leningrad, USSR, 1991 (in Russian).
A. Amigud and T. Lancaster, “246 reasons to cheat: An analysis of students’ reasons for seeking to outsource academic work,” Computers & Education, vol. 134, pp. 98–103, 2019; doi: 10.1016/j.compedu.2019.01.017
V. F. Shatalov, Kuda i kak ischezli troiki [Where and how did the triples disappear], Moscow: Pedagogika, 1980 (in Russian).
L. M. Fridman, “Analiziruem poiski, nakhodki uchitelei” [We analyze the searches, finds of teachers], Voprosy psikhologii, no. 3, pp. 146–151, 1981 (in Russian).
M. Wertheimer, Produktivnoe myshlenie [Productive thinking], Moscow: Progress, 1987 (in Russian).
M. I. Bashmakov, Teoriya i praktika produktivnogo obucheniya [Theory and practice of productive learning], Moscow: Narodnoe obrazovanie, 2000 (in Russian).
I. A. Zimnyaya, “Klyuchevye kompetentsii — novaya paradigma rezul’tatov obrazovaniya” [Key competencies — a new paradigm of education outcomes], Vysshee obrazovanie segodnya, no. 5, pp. 34–42, 2003 (in Russian).
“Kontseptsiya modernizatsii rossiiskogo obrazovaniya na period do 2010 goda. Prilozhenie k prikazu Minobrazovaniya Rossii ot 11.02.2002 №393” [The concept of modernization of Russian education for the period until 2010: Appendix to Order of the Ministry of Education of Russia dated 11.02.2002 No. 393], Moscow, 2002 (in Russian).
O. E. Lebedev, “Kompetentnostnyi podkhod v obrazovanii” [Competency-based approach in education], Shkol’nye tekhnologii, no. 5, pp. 3–12, 2004 (in Russian).
L. B. Ershtein, “Vliyanie kompetentnostnogo podkhoda v obrazovanii na razvitie obshchestva” [The impact of the competency-based approach in education on the development of society], Nauchnometodicheskii elektronnyi zhurnal ’Kontsept’, vol. 13, pp. 751–755, 2015 (in Russian).
M. L. Minsky, Freimy dlya predstavleniya znanii [A Framework for Representing Knowledge], Moscow: Energiya, 1979.
D. Poiya, Kak reshat’ zadachu: Posobie dlya uchitelei [How to solve it], Yu. M. Gaiduka ed., Moscow: GIZ MP RSFSR, 1961 (in Russian).
D. Poiya, Matematicheskoe otkrytie [Mathematical discovery], Moscow: Nauka, 1970 (in Russian).
D. Poiya, Matematika i pravdopodobnye rassuzhdeniya [Mathematics and Plausible Reasoning], S. A. Yanovskoi ed., Moscow: Nauka, 1975 (in Russian).
M. Minsky, The Society of Mind, New York: Simon & Schuster, 1986.
R. Deng, P. Benckendorff, and D. Gannaway, “Progress and new directions for teaching and learning in MOOCs,” Computers & Education, vol. 129, pp. 48–60, 2018; doi: 10.1016/j.compedu.2018.10.019
C. M. Mueller and C. S. Dweck, “Praise for intelligence can undermine children’s motivation and performance,” Journal of Personality and Social Psychology, vol. 75, no. 1, 33–52, 1998; doi: 10.1037/0022-3514.75.1.33
J. Sweller, “Cognitive load during problem solving: Effects on learning,” Cognitive Science, vol. 12, no. 2, pp. 257–285, 1988; doi: 10.1207/s15516709cog1202_4
B. Dahl, “What is the problem in problem-based learning in higher education mathematics,” European Journal of Engineering Education, vol. 43, no. 1, pp. 112–125, 2017; doi: 10.1080/03043797.2017.1320354
Ya. Khinchina, Pedagogicheskie stat’i [Pedagogical articles], B. V. Gnedenko ed., Moscow: Izdatel’stvo Akademii pedagogicheskikh nauk RSFSR, 1963 (in Russian).
S. I. Shapiro, Ot algoritmov — k suzhdeniyam: Eksperimenty po obucheniyu elementam matematicheskogo myshleniya [From Algorithms to Judgments: Experiments in teaching elements of mathematical thinking], Moscow: Sov. radio, 1973 (in Russian).
E. Dubinsky and M. A. Mcdonald, “APOS: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research,” The Teaching and Learning of Mathematics at University Level (New ICMI Study Series), D. Holton at al. eds., Dordrecht, Netherlands: Springer, 2001, pp. 275–282; doi: 10.1007/0-306-47231-7_25
D. O. Tall, “Reflections on APOS theory in elementary and advanced mathematical thinking,” in Proc. of the 23rd Conf. of the Int. Group for the Psychology of Mathematics Education, Haifa, Israel, 1999, pp. 111–118.
S. N. Pozdnyakov, “Sistema komp’yuternoi algebry kak pedagogicheskaya zadacha” [Computer Algebra System as a Pedagogical Task], Computers tools in education, no. 2, pp. 25–41, 2017 (in Russian).
V. A. Rokhlin, “Lektsiya o prepodavanii matematiki nematematikam” [Lecture on teaching mathematics to non-mathematicians], Matem. prosv., vol. 3, no. 8, pp. 21–36, 2004 (in Russian).
This work is licensed under a Creative Commons Attribution 4.0 International License.
