Asking Concept-specific Questions to Promote Meaning-making
Asking deep questions in mathematics lessons is not a new idea. Research on teachers’ questioning has been going on for over a century (e.g., Chi et al., 2018; Gall, 1970; Stevens, 1912). However, asking effective questions in mathematics classrooms still presents a great challenge to many teachers. The IES recommendations (Pashler et ah, 2007) have suggested example questions with key stems such as why, how, what if, what if not. While such guidance is helpful, I have found that teachers who asked these questions (e.g., “Can you tell me why?” “How did you get that answer?”) did not necessarily elicit deep explanations from their students. One of the reasons may be that the nature of these questions (e.g., “Why is that true?”) are too broad to cue relevant knowledge for explanation. On many occasions, it seems critical to ask questions (or the follow-ups) that draw students’ attention to specifics that may help elicit explanations about targeted concepts. I call such questions “concept-specific questions.” In my project, I have identified two types of opportunities to ask concept-specific questions: during the use of semi-concrete representations and during the use of numerical solutions.
When semi-concrete representations are used for modeling, a teacher may ask questions that draw students’ attention to the relational aspect of a model. This is different from focusing on an individual part of a model (e.g., “What does this cube represent?” “What does this tape represent?”). Recall that when teaching additive comparisons, a Chinese G2 teacher asked “Why is there a dot on the second tape?” This question specifically targeted the relational aspect of the tape diagram. In that specific case, the dot marked out the section on a second tape that was the same size as the first tape. It also indicated that the second tape could be separated into the “same as part” and the “more than part.” (see Section 2.4). As such, asking students a question about the “dot” on the tape may help draw students’ attention to quantitative relationships represented by the tape diagram. Consider a second situation: when teaching multiplicative comparisons, one Chinese G3 teacher asked her students how they could manipulate the stieks to clearly show “five times.” (see Section 3.5). Similar questions were observed in a U.S. G4 classroom, as well (see Section 3.5). These questions also focused students’ attention on the relational aspects of their respective models. An analysis of these aspects likely contributes to students' understanding of the concepts and quantitative relationships.
Another type of concept-specific question focuses on the meaning of numerical solutions. This is different from explaining meanings of individual numbers or computational strategies; those types of questions are relatively common in the U.S. classrooms. Rather, the focus of this type of question is on why certain operations worked. The conceptual understanding of an operation serves as a foundation for further exploration of the inverse relations and basic properties that use these operations. During my project, I found that after a numerical solution was generated, Chinese teachers almost always asked students to explain their chosen operation (e.g., “Why did you use addition?” “Why did you multiply?”). Such a question demands that students understand the quantitative relationships that led to the operation (e.g., To combine two quantities, we need to use “addition”). Without such understanding, students may depend on the key words or make random guesses about how to generate numerical solutions (e.g., “more” = add; “times” = multiply). In addition, Chinese teachers often stressed the meaning of operations in the context of comparison (e.g., why one problem is solved with addition but the other subtraction). I will discuss more about comparisons in the next section. Questions about the meaning of operations may help make sense of the basic properties as well. When teaching the CP of multiplication, some U.S. teachers asked students to explain why an instance like 3 x 4 = 4 x 3 made sense using the meaning of multiplication (e.g., the same array can be viewed as either 3 rows of 4 or 4 rows of 3). The above questions all focus on the meaning of operations and quantitative interactions (as opposed to individual quantities) in ways that would likely result in deep learning (Chi & VanLehn, 2012).
Asking Comparison Questions to Promote Connection-making
In addition to concept-specific questions, my project lessons demonstrate that comparison questions are powerful tools for facilitating connectionmaking and promoting generalization. The literature in cognitive science and educational research has long stressed the importance of comparison activities in enhancing learning (e.g., Kotovsky & Centner, 1996; Star & Rittle-Johnson, 2009). When teaching an early algebra lesson, a teacher can ask comparison questions either within a worked example or between examples (and practice tasks).
Within a worked example, teachers can ask questions that prompt students to compare different models and solutions. In my project, a U.S. G3 teacher asked students to model a division problem using both counter manipulatives and tape diagrams. She then asked students to explicitly compare both models for similarities and differences (see Seetion 3.3). This type of comparison can enable students to grasp the structural information embedded in the models. Additionally, teachers can ask students to compare multiple solutions to the same task. For instance, when teaching the “making a ten” lesson that integrated the AP, Chinese G1 teachers asked students which solution method they liked the most and which seemed to be the most effective and why (see Section 4.4). When teaching fact families, after students came up with two addition and two subtraction number sentences, the Chinese teachers commonly asked students to explain how these number sentences were similar and different and how they were related to each other (see Section 2.3). Likewise, when formally teaching the basie properties, Chinese teachers always asked students to solve a word problem in two (or more) ways, which were then compared to generate an instance of the targeted property (see Sections 4.3,4.4, 5.3, 5.4, and 5.5). The above explicit comparisons occurred much more frequently in Chinese lessons, although some U.S. teachers did encourage students to solve a problem in multiple ways (e.g., breaking apart an array or a factor differently to multiply). Multiple solutions provide opportunities to incorporate comparison questions in ways that promote connection-making, even if these opportunities are not always taken advantage of.
Teachers can also ask comparison questions about different, but related, examples to promote a search for connections. As demonstrated in the Chinese lesson, a G2 teacher taught inverse relations by presenting two sub-tasks under the worked example (see Seetion 2.4), modeling each with a tape diagram and then solving numerically. The teacher then asked students to compare the pair of diagrams (e.g., why was the second tape longer in the first diagram but shorter in the second diagram?). She also asked students why the first task was solved with addition but the second task was solved with subtraction. These comparison questions are likely to contribute to students’ understanding of the conditions under which addition and subtraction are used and how these two operations are related to each other. Moreover, when teaching the distributive property', a Chinese G4 teacher invited the elass to compare across five tasks and associated instances2 to identify the common features of the DP (see Section 5.5). In fact, some Chinese teachers also linked the practice tasks back to the worked example of the lesson to encourage explicit comparisons (see Section 2.4). The above technique of asking comparison questions can be widely incorporated into U.S. classrooms to promote explicit learning about early algebraic ideas such as inverse relations and the basic properties of operations.