“Surprisingly deep down we were all born to be mathematicians!” (Woo, 2018)
Number Sense
“Animal research on number pointed to an age-old competence for processing approximate quantities. This ‘number sense’, which is also present in infants, gave humans the intuition of number. Cultural inventions, such as the abacus or Arabic numerals, then transformed it into our fully-fledged capacity for symbolic mathematics.” (Dehaene, 2011) The Number Sense discusses the role of numbers in our daily lives, history and communication. Dehaene is among many authors to discuss the innate role of number sense in human beings and their development. “Number sense builds on students’ natural insights and convinces them that mathematics makes sense, that it is not just a collection of rules to be applied,” (Howden, 1989) number sense is innate in all humans, we are born with a drive to understand quantities as part of our survival. Infants as young as six months have been observed to distinguish between small quantities. Humans and many animal species have an internal approximate number system, this is our first mathematical competence and paves the way for all other mathematics learning. Our innate approximate number system “can represent number sense without language.” (Clements & Sarama, 2014)
Humans have been using numbers since our earliest days in caves, with number systems occurring in every civilization around the globe. “Some of the earliest written records of ancient cultures include numerical symbols in reference to counting people and animals. The development of trade and agriculture brought with it needs for numbers,” (Chapin & Johnson, 2006) ancient peoples tallied crops, calculated the area of parcels of land, calculated astronomic tables and explained their world using mathematics building the foundation of the number systems and mathematics we continue to use and to teach. In ancient Egypt officials calculated the area of parcels of land and calculated how much was lost due to the annual flooding of the Nile. Mathematics was essential for the development of civilization for communication, trade and taxation. As long as we have lived in groups humans have relied on mathematics to communicate ideas and information.
“Number sense is the primary focus of Elementary Mathematics, one of the goals of instruction in arithmetic is for students to become numerically powerful and proficient.” (Chapin & Johnson, 2006) students who are numerically powerful are able to interact with numbers flexibly adapting their strategies, and justifying their thinking. Number sense is essential to students’ ability to access the mathematics curriculum and to participate in many day-to-day activities. Strong number sense is an indicator of over all numeracy skills as well as student success with the mathematics curriculum. To demonstrate number sense students, need “a good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden, 1989)
“The discipline of mathematics comprises three worlds: the actual quantities that exist in space and time; the counting numbers in the spoken language; and formal symbols, such as written numerals and operation signs. Number sense requires the construction of a rich set of relationships among these worlds. Students must first link the real quantities with the counting numbers. Only then can students connect this integrated knowledge to the world of formal symbols and gain an understanding of their meaning.” (ASCD, 2004)
Definitions
Many authors who address the area of number sense have their unique way of breaking number sense down into constituent skills. The table below represents the definitions of number sense presented by five authors, the skills represented in the table help to inform the structure of Number Agility and which skills have been chosen to be part of the assessment.
| Cunningham (Cunningham, n.d.) | Small (Small, 2021) | Van De Wall (Van de Wall, Karp, Bay-Williams, McFarvey, & Folk, 2015) | Chapin and Johnson (Chapin & Johnson, 2006) | Clements and Sarama (Clements & Sarama, 2014) |
| – Understanding quantities Grasping concepts like more and less, and larger and smaller – Understanding the order of numbers in a list: 1st, 2nd, 3rd, etc. – Understanding symbols that represent quantities (7 means the same thing as seven) – Making number comparisons (12 is greater than 10) – Recognizing relationships between single items and groups of items (seven means one group of seven items)understanding quantities – Grasping concepts like more and less, and larger and smaller – Understanding the order of numbers in a list: 1st, 2nd, 3rd, etc. – Understanding symbols that represent quantities (7 means the same thing as seven) – Making number comparisons (12 is greater than 10) – Recognizing relationships between single items and groups of items (seven means one group of seven items) | – Comparing and relating numbers – Comparison language – Spatial comparison – Numbers to 10 – The teen numbers – Multiple representation – Reading and writing numerals – Counting | – Quantity, counting and knowing how many – Subitizing – Numeral writing and recognition – Comparing sets: more than, less then, equal to – Relationships for numbers 1-10 – Part-part-whole relationships – Estimation and measurement – Data collection and analysis | – Classifying numbers – Numbers represent quantities – Numbers represent relative position. – Numbers can be identifiers (jersey numbers, addresses, phone numbers) – Numbers are parts of sets. – Place value. – Number lines. – Number relationships – Part and whole reasoning | – Counting skills – Subitizing – Comparing and ordering numbers – Cardinality – Conservation of number – Composition and decomposition of numbers – Fraction concepts |