A Study On Learning Trajectories Of Pupils For The Equal Sign

Pupils' understanding of the equal sign is related to the cultivation of many aspects of their abilities in the field of number and algebra,so it is necessary to cultivate their understanding of the equal sign.At present,researchers have begun to clarify the meaning of the equal sign and the relation between the equal sign and algebraic thinking,but there is a lack of researches on learning trajectories which is beneficial to the primary school students to understand the concept of the equal sign.The study was therefore carried out.This study focuses on four issues: How is the learning trajectories for the concept of equal in the textbook designed? What is the learning trajectories of the equal sign after optimized? How to verify that the learning trajectories of the equal sign concept has been optimized? What is the understanding of the concept of equal sign among other pupils of different grades?This study is based on the action method.First of all,the literatures on the understanding and learning trajectories of the concept of primary school students in China and abroad is reviewed,and the framework and methodology guidance are provided for the research.Through a great quantity of literature analysis and the characteristics of the teaching material situating of the concept of the equal sign,the relevant learning trajectories are sorted out,and the hypothesis learning trajectories are designed as a basis for the teaching in Class A.Secondly,combining measured changes in students' performance(pre and post)and discussion group recommendations,optimize the learning trajectories,and then carry out teaching in class B.Third,through the change of thinking level and performance in class,verify the optimization of learning trajectories,and combined with after-class discussion to form a complete learning trajectories.Finally,to carry out the delayed post-test on first-year students who have attended the class,as well as third and fifth grade,thus comparing understanding of equal sign concept among different grades,and provide some suggestions for first-line teachers regarding the layout of teaching materials.Based on the research process,main conclusions are obtained:(1)The optimizedlearning trajectories of equal sign tend to be more appropriate in the use of context.In the big situation of "Forest Games",a series of small situation strings are arranged,including "tug of war","weightlifting competition","football match",etc.,which connect the whole learning process from introduction to three learning tasks,and then to two exercises.(2)The optimized learning trajectories attache importance to intuitive operation.The introduction of "why doesn't the tug of war competition start" allows students to generate comparative needs in the conflict of questions.Combined with the operation of hand-made circle,the introduction of "one-to-one correspondence" makes it clear that there are different animals on the two sides.In task 1,add two on one side to make both sides equal,and lead to 9?7+2.Students are used to thinking arithmetically.The teacher guide students to observe the process of patch,think with relational thinking,and transform from the intuitive diagram to the abstract equation.(3)The optimized learning trajectories give students more opportunities to think and express.The teacher guides the students to observe the equation and think deeply.In the process of communication,the teacher not only changes the students' stereotype of the equation,but also guides the students to get the balance between the equal signs by focusing on the relationship between numbers.In task 2 and task 3,students experience the process of grasping the relationship between sum and difference.With sufficient guidance,students have enough time to communicate with classmates and the teacher,which not only gives students the confidence to participate actively,but also creates opportunities for teachers to deal with students' generated opinions.(4)The optimized learning trajectories' delay effect is better.The results of Class A students in the delayed posttest are lower than those in the post test,while the results of Class B students in the delayed posttest maintain a level,or even slightly improved.Class B students experience the process from reversing the misunderstanding of the operational significance of the equal sign to being able to rely on the relationship between the two sides of the equation and each of the complex structures.They constantly improve the level of relational thinking.They're more powerful in the aspect of equal sign understanding and have more development potential for other related content in the future.(5)With the increase ofgrades,students' thinking level is not necessarily higher.According to the survey,not the senior students perform better in the understanding of the concept of equal sign,and the fifth grade students have no advantage in the level of relational thinking on the of equal sign compared with the first grade students after class.Students of all grades need to have a rich understanding of the equal sign,and a better understanding of the mathematical relationship structure.In primary school,teachers need to pay more attention to the diversity of students' thinking.They shouldn't use a fixed problem-solving model to tie students down.Grade three is an important year node.The study made the following recommendations for textbooks,teachers and students: The layout of textbooks should take note of the choice and creation of the context,and add more content that embodies the relationship between mathematics and the structure of mathematics;Teachers should fully understand the importance of the concept of equal sign,pay attention to the current situation of students' thinking,grasp the key tasks in the learning trajectories,strengthen guidance,encourage independent knowledge construction,reduce rules and constraints on students' problem solving process,and increase flexibility in students' thinking;Students should look at the equal sign with a more open-minded view and develop their own relational thinking on the basis of mastering arithmetic thinking.