The results indicate that mathematical conjecturing has the necessity of group collaboration, especially in the first two stages (construction stage and formulation stage). In the construction stage, students first create 1-2 individual cases, and then collect into 8-10 cases in a small group, and finally collect about 48-100 cases in a whole class, which becomes data with observable and stable patterns, as the evidence for the argumentation . The second stage is to formulate conjectures. First, individuals make conjectures, then they are compiled into group-conjectures, and finally they are compiled into whole class-conjectures. It is possible that the individual conjectures are incorrect or not based on the data in the first stage. The purpose is to develop the habit of students to speak something relied on evidence. The group collaboration at this stage is to check the correctness of conjectures. Students first make conjectures based on the data, they confirm that they are evidence-based conjectures, and then integrate them into group conjectures.

The communication in the classroom is achieved through the collective argumentation in the mathematics conjecturing teaching. The quality of conversations in the conjecturing teaching is not only for explanations, but more about for argumentation. Argumentation is different from explanation. Argumentation is from the perspective of the listener, and explanation is from the perspective of the speaker. The purpose of explanation is not to effectively persuade the others to believe, but the purpose of argumentation is to communicate effectively and persuade the listener to believe their own proposition. The speaker of the argumentation must not only make the listener hear clearly, but also persuade the listener.

There are four types of argumentation elements in elementary school mathematics classrooms: (1) Support: using examples or mathematical ideas to support one's own or others’ views; (2) refute: using examples or mathematical ideas to refute others’ ideas; (3) Premise: turn non-true conjectures for all cases into true conjectures with restrictions or exceptions; (4) Universal quantifiers: use universal quantifiers to describe true conjectures, and use conjunctions (and, and, or) to describe a complete in conclusion. Both support and refutation are able to enhance students' critical ability, and premise and universal quantifier are the content of criticism. Explanation, explanation, and argument are three modes of criticism.

The publications relevant with the study include 17 papers and journals, 23 seminar papers, three books on mathematical conjecturing tasks and practice, a book on mathematical conjecturing teaching norms, and a book on the theory and practice of competence-oriented mathematical guesswork teaching model.