Side view of a open book.

Key Takeaways for Educators

Why use Modeling and manipulatives?  

We created this curriculum to scaffold learning different topics in math with plenty of opportunities to give help. It also emphasizes using hands-on learning and modeling to teach topics related to ratios, proportions, linear equations, and functions. 

Modeling is a method of guided inquiry that allows students to construct their own explanations and defend their conclusions with evidence. It has been shown to build conceptual knowledge better than similar traditional methods. Modeling may take somewhat more time than standard lectures in the beginning as students are learning the process.

Modeling curricula encourage collaborative interactions, develop learners’ sense of community, and engage students with math practices such as explanation and critique of data interpretations. Scaffolded learning provides more engaging and exciting opportunities for students to grow and gain a deeper understanding of concepts that students can apply to future topics, allowing for less need for repeated reteaching with each new context. See our linked one-pager on modeling for more information. 

The modeling curriculum developed as part of this study also used many digital and physical manipulatives to scaffold learning. Manipulatives offer hands-on labs and experiences to learn math concepts at any age group. 

Reflections on In-person and Digital Learning

Over the course of a multi-year study, we used this curriculum in digital and in-person learning settings, combining several modes of instruction and collaboration across digital and physical classrooms. Here are some lessons we learned from running through our study and curriculum in many different contexts: 

One of our key observations was that student-to-student discourse is extremely challenging for some students in different modalities. Some students who are hesitant to speak up during a class whiteboard meeting can feel comfortable commenting when they are in a synchronous or asynchronous digital format. It can be important for students, whether you are in-person or digital, to use multiple ways for students to express ideas and organize their thinking before a whole class discussion. This is why the gallery walk, or other commenting time, is important for students to see other possibilities, question their own thought processes, and construct ideas in a way they can be expressed to others.

Encouraging Help-Giving Behaviors

This curriculum is also created to encourage students to collaborate and give help to each other. There is an introductory lesson built-into the curriculum which aims to: 

  1. help students share, expand, and clarify their own thinking
  2. help students become better listeners
  3. help students deepen their reasoning 
  4. help students think with others

In addition to learning strategies to help each other, help-giving behaviors also contribute to the co-construction of knowledge, which help students learn and transfer their knowledge in multiple contexts.

We compared how students collaborate in different types of digital settings, learned more about how students give help to each other, and what their motivations are for collaborating and giving help. We also aimed to understand if and how students giving help to each other contribute to how well they learn new content, and what kind of giving help contributes to learning most. 

One of the major takeaways was that help-giving is potentially more helpful for content learning when students are given the space to elaborate their thoughts and make mistakes. In the study, we observe help giving in a few contexts. When we looked at help-giving in a chat platform where students had to think in real time and talk between classmates, we found that students gave higher quality help than on a platform where responses were public and more polished (Khan Academy). As an educator, this might mean finding more opportunities to encourage giving help that isn’t perfect and allowing room for mistakes as well as elaboration. 

We also observed that, somewhat intuitively, different students have varied preferences for types of participation and collaboration. Just because they’re not participating in one mode doesn’t mean that will be the case in another.