Neuroscientific Foundations of InsightMath

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Relying on decades of neuroscience research, InsightMath shifts the way the field commonly does mathematics teaching and learning by changing the fundamental question underlying instruction. Instead of asking “How do we teach?” InsightMath asks, “How do we learn?”

 

Perception–Action Cycle

Human learning is a remarkable process that we both intuitively engage in and yet is routinely poorly supported in classroom instruction.  Often, mathematics is taught in the hopes that students walk away from the lesson being able to reliably produce right answers. However, the human brain is not a repository to fill up with information.  Our brains are instead continually undergoing a process of building on top of a highly structured and interconnected framework. This adaptive thinking is fundamental to human-learning.

Our brains are wired to learn by making sense of each success or failure.  By trying something, we are able to build and edit our perception of the concept or task (say mathematics, basketball, or walking). As we shift our thinking, we in turn shift our action. By engaging in a cycle of actions and perceptions refining one another, we learn by adapting our perceptions and actions. The term for this “Perception–Action Cycle” originated from neuroscientific research . 

The cycle, represented in Figure 1, describes learning as a feedback loop that consists of 

  1. a prediction guided by existing knowledge and the perception of the environment, which is followed by
  2. an action through the learner, which results in
  3. an outcome that is
  4. perceived by the learner.
     

Through this cycle, the learner compares the perceived outcome with the initial prediction and, based on that comparison, the knowledge base of the learner changes. It is through the constant interaction of an individual with the environment (i.e., InsightMath and ST Math, MIND Education’s first and flagship supplemental product and an important component of InsightMath) that learning happens: through the integration of previous experiences and novel sensory information (i.e., the feedback, conversation), the learner creates an expectation (through creative reasoning, discussed below) and compares it with the result of a performed action (i.e., the response of a learner) which then modifies and refines existing knowledge. 

 


 

This Perception–Action Cycle is continuous, resulting in creations and modifications of schemas. For example, a young student interacting with base-10 blocks might predict how many blocks will be left in an initial stack after a certain number of blocks is removed from it. This experience is either in line or not with the student’s expectations and will either reinforce or adjust the existing schema for subtraction, which then leads to a modified prediction in a next attempt, and so on. 

In colloquial terms, under this model, learning builds as we “get the Perception–Action Cycle spinning.” By this we mean that learning is not a single pass through these steps . Instead, learning happens as we continually cycle through them, engaging in creative reasoning to refine our perceptions and actions. MIND has tested these principles through the use of ST Math with millions of students, consistently showing measurable gains. 

InsightMath builds on this foundation to expand from supplemental puzzles into a full core curriculum. In InsightMath, students are presented with a problem, for which they may only have a crude understanding initially. The student will make a prediction of what the solution could be and then receive feedback (e.g., through ST Math puzzles or through social interactions and conversations) about the result of the student’s response. This allows students to refine their perceptions and iterate in a next attempt and so on. Going through such cycles, the student can refine and improve existing math-related schemas. However, in order to be able to improve at all, it is of crucial importance what kind of outcome the student perceives. 

In InsightMath, students’ perceived outcome is the formative feedback students receive from their work in the digital platform, classroom conversations with teachers and peers, and their own reasoning as they consider if their outcomes make sense. This understanding of and reliance on the Perception–Action Cycle represents a pivotal cornerstone of InsightMath.

 

Doing-First Approach with Embedded Practice

InsightMath is unique in that it does not implement a ‘teaching by telling’ approach, but instead facilitates a ‘learning by doing’ approach. By ‘learning by doing’ we most specifically refer to students engaging with the Perception–Action Cycle as early and as often as possible during the math lesson.

Unlike traditional approaches that focus on some variation of  “I Do, We Do, You Do,” InsightMath is built on students actively tackling problems before they have the solution path mapped out, allowing students to engage in discovering a solution path using the Perception–Action Cycle.  This is evident not only in the integration of ST Math into InsightMath lessons, but also in the amount of practice and discussion within a lesson itself.  With InsightMath, students are the busiest ones in the classroom.
 

Visual-First Pedagogy and Intentional Vocabulary

Learning is most often associated with language. It’s popular that students with specific language needs (e.g., multilingual students learning English, students identified as needing Special Education language support) are held back in math because “learning requires words”. At MIND, our foundational neuroscience and years of experience turning ST Math into a world class visual-first tool has shown that spatiotemporal thinking (thinking in a sequence of pictures) is the foundation of all thought.  Barbara Tversky, in her book Mind in Motion: How Action Shapes Thought, echos the same:

“… spatial thinking, rooted in perception of space and action in it, 

is the foundation for all thought.”

Being visual first not only removes language barriers, opening up mathematical rigor to students traditionally excluded due language-based learning needs (e.g., students in need of Special Education services related to language, multilingual students learning English), it ties in with our inherent way of thinking. For students learning and processing new mathematical ideas, having an existing schema (some kind of useful prior knowledge) can dramatically impact and improve memory retention and sense-making of these new ideas. This can be quickly and effectively implemented by using visuals to give context to new ideas. InsightMath consistently implements this schema-based approach by using visuals, and interactive visual tools and models to engage students thinking and creative reasoning about a mathematical topic before developing in detail, the relevant computational and procedural skills.

In both ST Math and InsightMath, our technique consists of language-independent representations of math concepts presented through a series of puzzles that make learning math fun and engaging. This allows students to bring their own language to the math concepts before formal teaching of vocabulary.  This way, students describe what they are noticing and learning in their own words before vocabulary is introduced in an intentional manner, reinforcing student understanding of that vocabulary.  Our approach introduces vocabulary as a tool to help explain a complex and familiar topic (say multiplication) without language being the primary gatekeeper to gain that familiarity.
 

Working Memory Focused on the Mathematical Content

Many digital tools take a “kitchen sink” approach, offering all digital possibilities at all times.  This often becomes distracting, generating unnecessary cognitive load.  InsightMath, instead, offers the technological tools just in time so that the students’ effort is always maximally focused on mathematics with only the tools needed for the task at hand.

When designing InsightMath, our curriculum developers were intentional about which learning modality would be most appropriate for any given learning opportunity. As a result, you will find fully digital, fully off-line, and hybrid lessons for students depending on the mathematical content and activity, ensuring students experience the mathematical content in the most advantageous way possible. 

Additionally, multiple-choice and quiz-like user interfaces can often lead to random guessing. Instead of simply selecting a value of a set of choices, the digital components of the InsightMath system, including ST Math, use an educationally-rich user interface with multiple means of interaction. This minimizes the tendency to randomly guess and ensures that student interaction is thoughtful, purposeful, and in the service of learning mathematical content. 
 

Formative Feedback with Intentional Discourse

Immediate and instructive feedback is a necessary component of effective learning. The type of feedback students receive is crucial for learning. Broadly speaking, feedback can be described as being confirmatory or elaborative. Whereas confirmatory feedback entails limited information mainly explaining whether a response was correct or not and sometimes also includes the correct answer, elaborative feedback provides more detailed information why a response was correct or not, adding an important instructive element to the response the learner receives. 

InsightMath calls this “formative feedback” because it’s elaborative and instructive with another critical element—digestibility.  Feedback that is lengthy or too difficult to decode stops the Perception–Action Cycle.  By offering digestible feedback, the Perception–Action Cycle continues to spin and students continue to learn.  This formative feedback has a particularly crucial outcome in that it allows students to adapt their solution strategies as true problems-solvers. For example, the user interface and animation with an ST Math game treats correct and incorrect answers the same, removing any stigma around mistakes, while at the same time providing a visual explanation of the mathematical validity of the student’s response. 

InsightMath builds on this foundation to intentionally shift the origin of the feedback—from the one-on-one computer experience of ST Math to a shared conversation through the lesson platform.  When students have problems that capture their attention and for which they engage in interactive and iterative thinking, they want to talk about it.  The InsightMath platform offers students multi-modal ways to “get their ideas out” including the pen tool, speech to text, text entry, and visual manipulatives to drag and drop. Additionally, it offers teachers many one-click ways to make student-thinking visible including sharing up to four bits of student work at a time, useful for comparing student strategies, or to share a student’s screen in real-time to enable a student to share their thinking at the speed of thought. Using technology during a lesson to enhance the social components of learning (especially discourse) is an intentional design component of InsightMath.

 

Embedded Formative Assessment

Formative feedback is also one essential component of formative assessment. Formative assessments, in contrast to summative assessments that aim to measure student learning at the end of an instructional unit, focus on improving learning during the course of instruction.  InsightMath takes a holistic approach to formative assessment with the goal of providing, embedded real-time assessment. By this we mean that formative assessment does not have to be a separate event from the process of learning. In fact, any time a student speaks, uses body language, acts, writes, draws, models, or in any way communicates about mathematics, is an embedded opportunity to assess that student’s understanding.  

The design of InsightMath includes specific, identified formative assessment opportunities within each cluster of content. These tasks are tied to cluster outcomes and called out in the Teaching Guide. They act as particularly relevant “checks for understanding” in which teachers take stock of student understanding of intended cluster outcomes. The inclusion of these explicit opportunities, however, does not imply that other moments of student work time are not also formative assessment opportunities. As students work, particularly as they discuss mathematics together while engaged in tasks, teachers receive formative feedback on how well students understand the material and make instructional decisions about which discourse questions to ask or which student work to discuss next in order to move student thinking forward.

Additionally, the InsightMath platform provides teachers with a steady stream of student work that teachers use to drive meaningful conversation or to edit the pacing or flow of a lesson. For example, by sharing student work, the teacher makes a real-time, formative decision to reinforce or extend a particular concept within the lesson block itself. Such student-work is also saved so that teachers revisit outside of the time constraints of a lesson. The platform also offers a Paper Scan, indicated by an icon and the word “camera” in lessons. This allows students to upload their physical work into their lesson in the same ways that submitting digital work is used and stored. Notably, formative feedback in our embedded ST Math games, used in many InsightMath lessons, offer student-facing formative assessment by giving students opportunities in real-time to adapt their thinking. 

Every activity in InsightMath becomes an embedded opportunity for formative assessment. As students engage in meaningful tasks and receive formative feedback, teachers are able to gain insight into student thinking and adjust instruction to highlight key understandings, insightful strategies, challenges or misunderstandings for discussion. 
 

Creative Reasoning

The Perception–Action Cycle offers a clear learning mechanic with vast implications.  For example, since learning is an adaptive process of shifting perceptions and strategies, learning is inherently a creative endeavor and not the "absorption endeavor” common in traditional approaches. InsightMath was built for students to develop their capacity as, and desire to be, creative reasoners.

The dominant teaching method in the United States is to present students with a problem and provide an adequate solution method for it, which is then followed-up by extensive practice that makes use of this method. Relying on such an approach is understandable and seems appropriate because teaching of pre-defined solution methods (or algorithms) saves time, prevents miscalculations, and a substantial majority of the tasks in textbooks can be solved with them.

Although it is beneficial to know about specific mathematical techniques, merely presenting predefined solution methods does not necessarily encourage the development of deeper understanding of math principles and students are at risk to use such solutions in an unreflected and superficial way without any conceptual understanding of them141517. Therefore, algorithm-based teaching approaches are showing themselves to have a limited impact on students' long-term math abilities development . 

Further confirming InsightMath’s learning method, a set of recent experiments demonstrated the advantage of creative reasoning over algorithm-based approaches. For example, Jonsson et al. (2014) compared a group of students either practicing with a creative reasoning approach or an algorithm-based approach. Their results showed that one week after practice, the creative reasoning group outperformed the other group on tasks that required the re-construction of the practiced solutions. The authors assumed that the struggle the creative reasoning group had to go through resulted in a deeper memory trace that students could benefit from and that the self-generation of solution methods facilitated conceptual understanding of the specific task solving methods. These results were confirmed in a follow-up study that also investigated brain activations. In addition to the replicated math results, the lower activity in certain brain areas in the creative reasoning group (i.e., an area called the left angular gyrus) suggested easier memory access to the solution method.

A further study emphasized that it is mainly the effortful struggle that seems to drive the superior performance in favor of the creative reasoning group17. Finally, a study by Norqvist (2018) considered the fact that teachers and textbooks sometimes provide conceptual explanations along with algorithmic solution approaches and compared such a group with a group that practiced using a creative reasoning approach. Again, the results revealed that creative reasoning outperforms algorithmic approaches regardless of whether they were accompanied with conceptual explanations or not.

 

Teaching the Way the Brain Learns

InsightMath is built on the premise that declining test scores and elevated anxiety towards mathematics are not something that can be changed with better explanations or illustrations.  Instead, as a society, we can unlock the immense potential of all students to learn, problem-solve, and grow by first looking at the kinds of experiences and activities that lead to the deepest learning.  By putting those principles into practice, as we’ve demonstrated with great success in ST Math, we believe that low performance and high anxiety don’t have to be the final chapter in a student’s mathematical journey. Instead, students can use their understanding of mathematics to positively impact their own lives, their community, the economy, and the world.

 

Endnotes

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