Mathematics is a particular way of studying phenomena; a way we, as humans, seek to solve the puzzle of our place in this world and understand relationships between states of natural elements, including predicting rates and magnitudes of action and reaction. It is at once, a language, a blueprint, a discipline, a pathway for reasoning, and a tool that improves the way each of us navigates life. This world needs mathematicians.
As early childhood and elementary educators, we need to embrace the mathematician within in order to empower young learners as mathematical inventors and explorers of their unfolding world. In this short series of articles, we will highlight three central themes that impact achievement in mathematics: 1) the human propensity toward number sense (and how to keep that alive), 2) the importance of spatial perception skills for mathematical acumen, and 3) the role of language in mathematical conceptualization.
This is the first article in the series considering the question – what is the human propensity toward number sense and how can we build on the early intuitive relationship with numbers, while interrupting the very real and crippling effect of anxiety driven math-avoidance?
The truth is that time spent diving into mathematics could and should be one of the most engaging parts of a learner’s day. Unlike the act of reading, early mathematics – including seeing patterns, numerical magnitude, and changing amount – is hardwired into our developing brains. Park and Brennen (2014) refer to the nonverbal primitive number sense, also known as the Approximate Number System (ANS) (Eliot, 2019), which is evident in very young infants who are able to recognize differences in magnitude without any verbal or formal framework (Xu, 2007). When toddlers are encouraged to use this skill often, they build fluency recognizing the total of a number of objects without counting. Early childhood educators call this subitizing and recognize the benefits of intentional practice with this skill to lay a strong conceptual relationship with numbers.
The question then becomes, how do we, as early math teachers, leverage this propensity?
For answers, we at the CTTL, turn to the body of research, historical and emerging, as more and more academics study what works in the mathematics classroom.
- Design opportunities to practice, extend, and build arithmetic accuracy with the ANS early in a learner’s journey, even before verbal and written use of the numerical system is introduced.
- Use knowledge of the developmental steps in cultivating mathematical sophistication to skillfully build capacity for reasoning and logic. For this, we must increase our own understanding of mathematics and remember that children will move through these developmental benchmarks at different rates!
- Never assume; probe a child’s mathematical awareness and skill through frequent formative assessments using a variety of modalities. Hone your questions and activities to target and surface hidden capacity. Always attach new knowledge to already established concepts.
- Design mathematical explorations using concrete artifacts chosen wisely to highlight the concept or objective, and watch out for any seductive distraction they may cause (fingers are magical manipulatives for emerging mathematicians of every age).
- While the human propensity toward an approximate number system and pattern-seeking is evident early in life (and therefore can be thought of as innate), a belief that math ability is set at birth changes the way you design lessons and interact with individual students. This perspective has been shown to impact the motivation and outcomes for low-achieving students.
- Understand the role that metacognition plays in math achievement. Actively incorporate reflection and analysis early on, so that young mathematicians become explicitly cognizant of useful strategies and have opportunities to practice choosing the right strategy for any given problem. Trial and error learning has a place in building awareness of this repertoire of skills.
Teachers have a great impact on learner academic and lifelong outcomes (Nye, 2004). This matters in three ways in the context of this article. A) is through the implementation of an inquiry-based, concept-rich and coherent approach to math instruction, shown to have a positive impact on math outcomes. B) is in the clarity with which teachers explain mathematical content, and by extension, the reduction of incorrect presentation of content in the early years (Blazer, 2015). C) is in the active interruption of confidence-destroying anxiety connected to mathematics.
- Knowing that math concepts are built one upon another over a number of years, the idea of coherence becomes very important. Teachers of earlier grades need to be very familiar with the direction in which the ideas they are teaching are heading. To think about how an understanding of later math can inform stronger math instruction in elementary grades, let’s consider the concept of decimals.
Over the last two decades, I have had the good fortune to teach in every grade, Kindergarten through fourth. In that process, it became very evident how early exposure to coins, in particular counting pennies and writing the total as dollars and cents, laid the foundation for a later relationship with the value scale that digits hold beyond the decimal place. After 6 years away from first grade, I returned for one more year with six-year-olds, only to find that the curriculum developers had a new edition of the program our school used and had eliminated coin exposure. The reason given was that hand-held money was becoming obsolete. This may be true, but it misses part of the reason coin and note inquiry was included in the first place. Changes in scale can be very difficult for a young mind to visualize, and the concrete experience of exchanging 100 pennies for a $1 note and sometimes having pennies left that get written after the decimal place made this change in scale tangible for young learners. As you can imagine, I was mortified and diverged from the curriculum every now and then to provide spaced practice counting and writing the notation for money. While it is true that these tools are not the only way to lay the foundation for decimals, if you remove them, you need to replace them with a different concrete tool, relevant to the children, that explores the same conceptual change in scale. This is an example of horizontal decision-making in math, when what is needed is vertical understanding, in order to ensure conceptual foundations are anchored early for the more complex math concepts of higher grades.
- This kind of vertical mental clarity helps us articulate concepts and methodology in a more precise way, including the why that is connected to broader goals of mathematics. A practical strategy I’ve used in the past when preparing to teach with clarity a process I was accustomed to doing differently, is to create a short tutorial for my students using an app like Explain Everything Whiteboard. This serves two purposes; refining my language while noticing any mistakes in my explanation through review of my work and providing a solid follow-up resource for students to use after the lesson if I missed anything in the moment as I taught the concept.
- Recognize math anxiety when you see it. Parents pass their anxiety around performance in math to their children, but only if they regularly support their child with math homework. This is true for both parents with limited math knowledge and parents with more robust math knowledge (Maloney, 2015). In addition, If you identify as a female teacher, you pass your math anxiety on to learners who also identify as female in your classroom (Beilock, 2009). To combat this, be fully prepared for every math lesson and exude confidence. Anxiety of this kind can undermine a child’s ability to show you everything they are capable of, so it is critical that we address factors that contribute to math anxiety in young learners to build mistake resilience and grow a thirst for difficult problem-solving.
Humans are natural mathematicians. Believing this becomes important as we seek to strengthen our approach to math instruction and incorporate pedagogy shown to positively impact outcomes for ALL our students. Believing in the unrealized high potential of all the learners in our mathematics classroom is essential as we design, plan, and prepare high quality instruction every day.
Beilock, S., Gunderson, E., Rameriz, G., & Levine, S. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences of the United States of America, 107(5).
Elliott, L., Feigenson, L., Halberda, J., & Libertus, M. (2019). Bidirectional, Longitudinal Associations Between Math Ability and Approximate Number System Precision in Childhood, Journal of Cognition and Development, 20(1), 56-74.
Heyder, A., Weidinger, A. F., Cimpian, A., & Steinmayr, R. (2020). Teachers’ belief that math requires innate ability predicts lower intrinsic motivation among low-achieving students. Learning and Instruction, 65, .
Izard, V., Sann, C., Spelke, E., & Streri, A. (2009). Newborn infants perceive abstract numbers. Proceedings of the National Academy of Sciences of the United States of America, 106(25). 10382-5. 10.1073/pnas.0812142106.
Maloney, E. et.al. (2015). Intergenerational effects of parents’ math anxiety on children’s math achievement and anxiety, Psychological Science, 26(9)
Nye, B., et.al. (2004). How large are teacher effects? Educational Evaluation and Policy Analysis, 26(9) 1480-1488. Sage publishing.
Park, J. & Brannon, EM. (2014). Improving arithmetic performance with number sense training: an investigation of underlying mechanism. Cognition. 133(1),188-200.
Xu, F., & Arriaga, R. (2007). Number discrimination in 10-month-old infants. British Journal of Development Psychology, 25, 103-108.