探討學生數學學習中意動與認知、情意間的交響協作：以奠基活動為例

Abstract

在學生進行數學學習的過程中，認知與情意同時都扮演了相當重要的角色，兩者之間會進行交互作用影響學生的數學學習，因此在研究與實務面向上同時希望兩者都能夠正向發展，然而在國際評比中卻有高成就、低興趣這樣一個現象，這個現象需要且有證據顯示可以被解決，而在臺灣，國立臺灣師範大學數學教育中心在教育部的補助下執行「就是要學好數學(Just Do Math)」計畫，在該計畫中產出許多數學奠基模組以期望學生在該模組的學習過程中能夠對於數學知識有感，且對於數學學習感覺到有趣，而研究中也的確顯示出這樣的結果，除此之外，Yang等人(2022)在覺動理論的理念與思考實驗下也提出任務設計的三個進程。然而較少文獻探討在這樣同時兼顧學生認知與情意正向發展的學習活動中，其學習的推動力為何，而在心理學領域針對認知與情意的交互作用進行探討的過程中早已融入了推進力的觀點，心理學以意動來稱呼這樣的推進力，也因此本研究著重於探討在數學奠基模組這樣同時兼顧學生認知與情意正向發展的數學學習活動中，學生的意動與認知、情意之間是如何進行交響協作，進而協助學生認知與情意的正向發展。在此研究方向上，研究者根據相關理論發展研究架構，包括引起意動的來源分類架構、意動與認知、情意間交響協作結果的架構，接著利用實徵研究法歸納交響協作的來源、型態及結果。意動的來源可以依據「引起意動的脈絡」與「學而不足的狀態」這兩個維度進行分類，「引起意動的脈絡」包含學生本身、任務以及社會互動三個面向；「學而不足的狀態」則包含偏認知方面的懸缺性、不確定性與不一致性以及偏情意方面的情意狀態與情意特質這幾個面向。當來源引起學生意動與認知、情意的交響協作之後，根據覺動理論，學生的學習是目標導向的，據此可以發現到當交響協作被引起之後，交響協作結果的部分則可以依據「目標」與「型態」這兩個維度進行分類，「目標」包含求理解的目標、求表現的目標以及社會性的目標；「型態」包含沒有認知、情緒或行為產生；純粹產生行為；透過表徵的操弄與形成產生行為；產生控制與評估產生行為或是產生新的意動來源；產生控制、評估與直接調整，調整後產生行為這五種。在此便可以呈現出本研究在探討意動與認知、情意之間的交響協作時，會以意動的來源、型態與結果作為對象進行探討，進而呈現出學生在這樣的環境下對數學學習感覺到有趣以及對於數學知識有感的原因及其機制。本研究的研究對象為六位國小四年級學生，在這六位學生中分別有高、中、低成就表現的學生各兩位。本研究採用質性研究法，希望能夠透過分析學生在進行數學奠基模組的學習過程，以了解學生意動與認知、情意之間的交響協作關係。本研究的資料是來自學生在進行長方形數與三角形密碼這兩個數學奠基模組的過程中所進行的課室錄音錄影，搭配學生的學習單、學生的筆記與紀錄、認知問卷、認識論下的情緒問卷與課後訪談，透過這些資料的搜集與互相校正，進一步呈現出學生在進行學習過程中意動與認知、情意之間的交響協作關係。研究結果的部分首先呈現出學生在數學奠基模組的學習歷程中，其意動與認知、情意間的交響協作共有四十種來源，接著指出學生在學習過程中意動與認知、情意間交響協作共有五種型態，這五種交響協作型態之間會有串連與並聯兩種互動模式。最後，對於型態間的互動進行歸因，進一步形成了學生在數學奠基模組學習過程中，意動與認知、情意間交響協作的機制。本研究在理論上的貢獻是透過連結本研究的實證研究結果與Yang等人針對數學奠基模組所提出的活動設計理論，進一步呈現出在學生進行學習的過程中，學生之所以會對於欲學習的數學知識有感是因為透過具體物的操作以及表徵間的轉譯，進而讓學生在學習過程中覺察不足，並且持續讓學生進行隱喻推理與體現思考，藉此協助學生連結源域與目標域，讓學生對於欲學習的數學知識有感；而學生之所以會對於數學學習感覺到有趣是因為學生在此過程中因成功所產生正向情緒的累積、滿足了學生好奇與想探索的態度或是因為學生贏得遊戲所經歷的正向情緒等，因此讓學生感覺到有趣。透過本研究的實證研究結果與Yang等人所提出的活動設計理論進行連結，進一步呈現出數學奠基模組學習環境下如何讓學生的認知與情意正向發展。

Cognition and affect play important role during students’ mathematics learning. Both of them will interact with each other to influence students’ mathematical learning. Hence, both of them are expected to be positively developing. However, the international assessment, Trends in International Mathematics and Science Study, shows that there is a phenomenon, high achievement with low interest, needed to be addressed, and there is evidence which shows that the phenomena can really be addressed. In Taiwan, ShiDa institute of Mathematics Education launched “Just Do Math” program in order to address the problem, and mathematics grounding activities (MGAs) are the products under the program. Some researches show that MGAs can really lead students to make sense of mathematics knowledge and fully engage in learning mathematics. Yang et al. (2022) also proposed three key design processes of MGAs based on the thought experiments under the rationale of enactivism. However, very few articles investigate the driving force of the cognition-affect-positively-developing learning. In order to investigate the driving force, the researcher searches for the literature and finds that in psychology, conation is the term used to investigate the driving force of interaction between cognition and affect. Hence, the purpose of the study is to investigate the orchestration between conation, cognition and affect under the context of MGA.In order to investigate the orchestration, the researcher proposes the framework of the study based on the related theories. The research framework includes the classification framework for the resources triggering conation and the framework of the products of the orchestration. Then, the researcher carries out the empirical study to provide empirical evidence. Based on the theory and the empirical evidence, the resources of the orchestration can be classified based on two dimensions, the context triggering the conation and the state of the awareness of insufficiency. The context triggering the conation includes students’ own self, task and social interaction; the state of the awareness of insufficiency includes cognitive missing, cognitive uncertainty, cognitive difference, affective state and affective trait. When the resource triggers students’ orchestration, students’ learning happens. Under the rationale of enactivism, students’ learning is goal oriented. This shows that the results of the orchestration can be classified based on two dimensions, the setting goal and the type of the orchestration. The setting goal includes learning goal, performance goal and social goal; the type of the orchestration includes no any behavior, pure behavior only, the manipulation or formation of representations, control and self-evaluation without directly adaptation, and control and self-evaluation with directly adaptation. This shows that the study focuses on the resources, the types and the results of the orchestration during students’ learning under the context of MGA in order to investigate the reasons and the mechanism for students’ sense-making and fully engagement.The participants are six fourth-grade students, two high achievers; two medium achievers, and two low achievers. The data are the video and audio records of the MGA lesson; students’ worksheets and notes; cognitive questionnaires; epistemic emotional questionnaire, and post-lesson interview.The research results first show that there are forty resources of the orchestration, and then, based on the resources and the results of the orchestration, the research results show that there are five types of the orchestration and two interaction modes of the types. Based on the induction of the resources of the interaction between the five orchestration types, the study proposes the mechanism of the orchestration between conation, cognition and affect under the context of MGAs.Finally, the study combines the research results and the theory of designing MGAs proposed by Yang et al. (2022) to show that the reasons for students to make sense of the mathematics knowledge are that the manipulation of the manipulable object and the conversion between representations can lead students to aware the insufficiencies, and these insufficiencies will trigger students’ conation to lead students to cyclically doing metaphorical inference and embodied thinking in order to connect the source domain and target domain. The reasons for students to feel interesting during mathematics learning are the accumulation of the positive epistemic emotion, the satisfaction of the attitude of curiosity and inquiry or the positive emotion when students win the game, etc. Based on the aforementioned, students can positively develop their cognition and affect under the context of MGAs.