Fostering Mathematical Engagement and Love for Mathematics- Pt. 1 of 4
Building Concrete Foundations and Focus on Sense Making
“Teaching mathematics is not simply telling students what to do. It is about cultivating curiosity and supporting the development of meaning.”
-Deborah Ball, Mathematics Education Researcher
Hi, this is the GenWise team- we bring out this newsletter to help parents and educators to complement the work of formal schools and associated systems. We are also a founder-member of Gifted World- if you are interested in issues related to gifted education and talent development, and are looking for resources for students, do become a member of the Gifted World Community (membership is free).
Check out our upcoming courses for educators, parents and students in the ‘Upcoming Events’ section below.
This is the first of a 4-part series about the dreaded topic of math learning which will be published weekly. In this series, Sowmya Jatesan, Head of Teacher Development at GenWise, shares her insights from years of teaching math herself and mentoring math teachers. The 4 parts in this series are-
Building Concrete Foundations and Focus on Sense Making
Learning That Sticks – From Concepts to Connections
Make it Real – Engagement Through Contexts, Games and Projects
The Math Teacher as a Guide – Mindset, Assessment and Level-Appropriate Challenge
Sowmya and her team are offering an online course for math teachers called ‘My Misconception Mentor’ starting June 7, 2025. The course is also open to interested parents. For details, check the Upcoming Courses section.
Building Concrete Foundations and Focus on Sense Making
Mathematics trains the brain to break down complex problems into smaller, manageable components and solve them through logical reasoning - a skill that forms the foundation of all scientific thinking and technological innovation. Beyond the specific mathematical concepts themselves, learning math develops crucial abilities like pattern recognition, abstract thinking, and systematic problem-solving - skills that lie at the heart of Science and Technology and help us understand everything from how living things work to how computers operate.
However, many consider Math to be a difficult subject (this is a global phenomenon) and there’s a belief that one is a ‘math person’ or not. GenWise does not agree with this view and strongly believes that all children can have a strong foundation in Math and enjoy it.
In order to develop a love for Math, students need to, across many ideas of Mathematics, understand core concepts, deepen understanding and build connections through guided explorations. Building sufficient procedural fluency is absolutely necessary but this must come only after a basic conceptual understanding and at least some explorations. Students then need rich tasks to strengthen their learning, nurture their innate creativity and come up with their own solution paths to problems. It is through this kind of problem solving that students develop the confidence needed to engage with unseen problems in unfamiliar contexts. As they repeatedly experience the delight of working their way through a problem, they internalize the identity of being a “Math-person.”
Through our work with both students and teachers, we find that the role of the teacher here is central. What a teacher essentially does is to engineer an effective learning environment for all students. The foundation for this lies in the teacher’s deep knowledge of both subject and pedagogy. Atop this foundation, teachers need to be able to introduce new concepts (using concrete materials as needed) and make connections to known ideas - both in Math and in other subjects. They present relevant rich tasks to deepen student engagement and understanding and provide necessary and sufficient input for the different students to take ownership of the learning process. As part of the class, teachers must be able to assess student learning through deliberate questions and seize teaching moments in class. An excellent teacher is an adaptive expert who can both zoom out to inspire and zoom in to clarify.
When teachers are not able to address these aspects effectively, students are not engaged and may develop anxiety, fear, or disinterest in mathematics. It is through understanding these aspects that good Math teachers create effective learning environments, so students experience the delight of working with Math over and over and construct their identity as Math lovers.
It is also to be kept in mind that research in education shows that success in learning is a significant motivator for students. Thus, it is important that ‘fun in learning’ is not taken as a goal in itself- students ultimately need to learn well and doing well will spur motivation to do ‘boring work’ that leads to success.
The teacher should aim to set up a virtuous cycle between motivation and success where one feeds the other.
We now look at some key curricular and teaching aspects that foster math engagement, love for the subject, and effective learning.
Adequate Foundation with Concrete materials before Abstraction
Particularly in the primary grades, when students are pushed to abstract/symbolic representations (e.g. learning oral counting up to 100 without understanding the idea of Place value) before understanding underlying concepts and are made to proceed without hands-on experiences to make sense of the new mathematical ideas, then their conceptual understanding suffers because of little or no use of visual and physical models to bridge concrete and abstract thinking.
At the primary stage, students need to move and use their limbs to work with and explore manipulatives to create Mathematical understanding. The significant topics in the Primary - Place Value, Operations, Fractions, Shapes, Measurements - all lend themselves to this approach.
E.g. Consider this question for the age group 6-8 yrs,
If 238 is expressed as tens and units, how many tens would that be?
Students who have worked with concrete material and have changed 10 units to one ten, 10 tens to one hundred and vice versa, will be able to see the approach needed here. Students who have not, are stuck and do not understand how to think about this and may feel anxious about such problems. On the other hand, students who have first engaged with such concepts using concrete materials will answer such problems with ease and joy.
This engagement with the concrete material is initially braided with the representation (pictorial/semi-concrete) and the abstract (numerical) representation and as students begin to hold the concepts and work on them intellectually, they should be encouraged to let go of the material.
Here’s an extract (p 133-134) from the ninth edition of Elementary and Middle School Mathematics, by John A. Van de Walle, Karn S. Karp and Jennifer M.Bay-Williams.
The CSA (concrete, semi-concrete, abstract) intervention has been used in mathematics education in a variety of forms for years (Heddens, 1964; Witzel, 2005). Based on Bruner and Kennedy’s stages of representation (1965), this model reflects concrete representations that encourage learning through movement or action (enactive stage) to semi-concrete representations of drawings or pictures (iconic) and learning through abstract symbols (symbolic). Built into this approach is the return to visual models and concrete representations as needed or as students begin to explore new concepts or extensions of concepts previously learned. As students share reasoning that shows they are beginning to understand the mathematical concept, there can be a shift to semi-concrete representations. This is not to say that this is a rigid approach that only moves to abstraction after the other phases. Instead, it is essential that there is parallel modeling of number symbols throughout this approach to explicitly relate the concrete models and visual representations to the corresponding numerals and equations.
Sense making of Procedures for Understanding
Effective mathematics instruction requires development of conceptual understanding before procedures and explicit scaffolding for students to make sense of the connections between concepts and procedures.
For example, a student whose teacher has taught him multiplication shortcuts is quick at multiplying a number by 10 or multiples of 10 because he ‘remembers that he must add the same number of zeroes’ (e.g. 73 x 100= 7300). But if he is not familiar with the concept of place value and where decimals are placed on the numberline, he may give answers such as 0.73 x 100 = 0.7300, when working with decimals.
In contrast, a student whose teacher has taught him the core principles and connected the concepts, realises that 0.73 is a number close to 1 on the number line, and when made ‘100 times larger’ (multiplied by 100) has to be a number near 100 on the number line. Thus, even if this student initially makes a mistake by applying a rule blindly and writes 0.7300, he checks the reasonableness of his answer and corrects it. In contrast, the student who knows only shortcuts and rules, does not even think in terms of whether something is reasonable or not.
When students are helped to make sense of Math, they develop confidence and enjoy the subject. We will never forget the student who said
I love Math because I have to remember less!
on being asked what her favourite subject was.
Share your thoughts in the comments section of this post or write to sowmya@genwise.in. Next week, we will look at the importance of connected learning, scaffolding key transitions (e.g., Arithmetic → Algebra) and the use of visualization and Tech Tools (e.g., number lines, Desmos, Geogebra).
Upcoming Courses
My Misconception Mentor (Math Series- Primary and Middle)- for educators and parents
Early Bird Registration ends by May 20, 2025; Course starts June 7 and ends in Sep 2025
Online on Saturdays- 1 to 3 PM IST (Primary); 330 to 530 PM IST (Middle School)
‘My Misconception Mentor’ is an 8-session online course based on the data of top misconceptions identified by Ei ASSET, the leading diagnostic assessment in India and the UAE. ASSET data have been bringing student misconceptions and common errors to light for over 2 decades. By understanding the roots of these misconceptions, we prioritize the most pressing learning gaps. Focusing on addressing well-documented misconceptions, we prevent cascading learning gaps, build stronger foundations for later concepts, and improve learning outcomes. There are 2 series of Math courses- one covering grades 1-5 (primary) and another covering grades 6-8 (middle).
While the course is targeted at teachers, parents are welcome to participate. Parents who participated in our ‘Mathtastic Teaching’ and ‘GenAI for Educators’ courses found great value.
The course is composed of 2 parts
Misconception Decoder- The first two sessions directly focus on misconceptions and the most common errors as seen in national data from the ASSET assessment.
Deep Dive: Content & Connections- The remaining 6 sessions focus on Subject Knowledge, Pedagogical Content knowledge and teaching strategies in the context of specific topics.
More details about the courses are available in this document and the links therein.
Participants also have an option to register only for the Misconception Decoder part of the course.
Register for My Misconception Mentor (Early Bird Fee of INR 16,000- GST included)
Register for Misconception Decoder (Early Bird Fee of INR 4,900- GST included)
An example of a student misconception from ASSET data and how the course would help teachers address such learning issues is discussed below.
This grade 8 question was administered to 6710 students nationally.
Only 30% of students estimated the answer correctly (slightly less than 16). 27% said it would be slightly more than 4!
Symptomatic approaches to fixing this misconception would involve teaching tricks like "move the decimal point to make it a whole number" or giving rules like "when dividing by a decimal less than 1, the answer will be bigger than the number you started with".
Such symptomatic approaches are insufficient because
They don't address the fundamental misunderstanding (division and decimals here)
Students can't explain why their answers make sense
The learning doesn't transfer to new situations
Students struggle when numbers or contexts change
The misconception often resurfaces in more complex problems
Our approach helps the teacher go deep below the surface; sometimes she may even have to work with teachers in other grades to tackle the roots of such learning problems. It is a slower and more painstaking process that takes time as the teacher builds on her subject and pedagogical content knowledge and also how students think. However, it is also motivating and rewarding when you start seeing students make sense of their learning and come up with their own innovative methods. Most importantly, this approach results in a sustained change in the teacher’s mindset, knowledge and skills.
Misconceptions are just the tip of the iceberg- we don’t want to address just the tip but the root causes that form the huge underwater mass of the iceberg.
Thus, for the decimal division example shared above, teachers would, over a period of time, focus on building number sense beyond whole numbers, connecting division and fraction concepts etc.
Gifted Online Summer Program- July 2025- for students
Early Bird Registration ends by May 31, 2025
This program provides young school students a front-row seat to mind-expanding courses, brilliant mentors, and a tribe of gifted thinkers. Courses range from Astrophysics, Storytelling, Physics of Ancient Arms, Algebraic Patterns to GenAI and Scientific Inquiry (please visit this page for more details including the eligibility criteria for the program).
✅ 2 - 4 weeks of fun learning
✅ Each Learning Quest is 12+ hours of live, small-group sessions with world class mentors.
✅ Resources and assignments from each session help expand and consolidate your learning.
✅ Saturday Circles with inspiring guest speakers and alumni
✅ Quizzes, challenges, and chances to win recognition
✅ A fun and supportive online community where the child finds ‘their tribe’
✅ Certificate and written feedback at the end of the course
Dates: July 7 – Aug 2, 2025
Location: 100% Online | Serving Gifted Students in India & the UAE
Grades: 5–9 (2025–26 Academic Year)
Eligibility: Ei ATS top performers for advanced courses; Other courses open to all Gifted World members. Check details here.