Co- funded by the European Union Knowledge and Pratice Standards Number Sense WG Corin Mathews, Lise Westeway, Zain Davis
Knowledge and Practice Standards (KPS) 9 to 17 KPS 12 : KPS 13 : KPS 16 : KPS 10 : Additive Multiplicative KPS 14 : Common KPS 17 : Early KPS 9 : Pre Number KPS 11 : Place KPS 15 : Relations with Relations with Rational Fractions and Algebraic Number Systems and Value Integers whole whole Numbers Proportional Reasoning Theory numbers numbers reasoning
Sub-Standards of the Number sense Knowledge and Practice Standards Foundational Concepts Sub- Standards Curriculum Content Sub- Standards Pedagogical Sub- Standards
Sub-Standards of the Number sense Knowledge and Practice Standards Foundational Concepts Sub- Standards Curriculum Content Sub- Standards Pedagogical Sub- Standards
Curriculum Content Sub- Standards – Refers to the teaching curriculum content for Foundation and Intermediate Phase Understand counting as a fundamental mediating resource used to relate operations on aggregates to the basic arithmetic operations. Understand the use of additive relations in the teaching of addition and subtraction of natural numbers. Teachers should know addition and subtraction problem types and the range of representations, calculation strategies and tasks used to solve addition and subtraction problems.
Sub-Standards of the Number sense Knowledge and Practice Standards Foundational Concepts Sub- Standards Curriculum Content Sub- Standards Pedagogical Sub- Standards
Pedagogical Sub-Standards Representations Fluencies Strategies Refer to Refers to the Referred to as contexts used ‘instant and strategies or for posing tasks, accurate’ methods for supporting manner of learners’ responding to a reasoning about mathematical a task and for calculation or classroom procedure discussions.
Pedagogical Sub-Standards Instructional Discourse SAY Do Write Encompasses This details the This unpacks the This outlines the natural language, nature of teacher demonstrable micro-content actions and talk to be used in actions teachers related to inscriptions the class which should be teachers’ needs to cohere engaged in while inscriptions with the other teaching the during teaching elements of KPS’s. of the KPS’s. ‘instructional discourse’
Pedagogical Sub-Standards Representations Fluencies and Instructional Discourse Strategies • objects Fluencies SAY DO WRITE • diagrams number facts • words (less and more) adding and subtracting 10 to/from • • • speak about addition as a demonstrate addition as record addition as ‘forward’ • number track any number combination of two sets to ‘forward’ jumps on a number jumps on a number track or • beadstring adding and subtracting tens and produce a combined set, and track or number line and number line and subtraction as • number line hundreds also as increasing a starting subtraction as ‘backward’ ‘backward’ jumps • number chart adding a single digit to a decuple • quantity by a certain amount jumps calculations can be recorded • (e.g. 30+7= ☐ ) p-p-whole diagram • • refer to subtraction as take present different types of bare using number sentences (or • Dienes blocks adding up to a decuple away and difference: starting calculations and word the column algorithm) in • (e.g. 34+ ☐ =40) base-5 and base-10 reps (5- with a collection, taking away problems: join, separate, part- symbolic form – these can also frame, 10-frame, abacus, subtracting a single digit from a from it and being left with a part-whole and compare (the be recorded as jumps on a arithmetic rack) decuple smaller collection, and also as names for the different number line (e.g. 50 – 6 = ☐ ) • making a comparison between problem types do not have to flow diagrams and arrow subtracting to a decuple two amounts be mentioned or explained to notation can also be used to (e.g. 78 – ☐ =70) • additive property – the learners) represent additive reasoning • quantity represented by the vary the position of the Strategies whole numeral is the sum of ‘unknown’, that is, start counting the values of the individual unknown, change unknown using known facts (bonds) digits (e.g. 36 is the sum of 30 and result unknown bridge-through-ten and 6) jump strategies (N10) • additive reasoning speaks split strategies (1010) about how quantities are compensation related in terms of how much column algorithm more or less. doubling • talk about the inverse relation near doubles between addition and halving subtraction (which means that one action
12 Foundational Concepts Sub-Standards Number systems: Knowledge of construction of Relations and numeration Structure Functions: Counting: systems and preservation and Knowledge of Understand number representations: relations and Equivalence counting as a systems Operations Knowledge of functions: in and order Aggregates: form of (natural as functions: structure order to relations: Knowledge of quantification Parts and numbers, Knowledge of preservation as a Sets and set understand Knowledge of mathematical and wholes: rational relations and basis for Discrete and relations: the nature of equivalence notions of distinguish Knowledge of numbers, functions Computation understanding of continuous Measure: Knowledge of mathematical relations, collections representations, between lists parthood, integers and (including al structures: quantity: Understand sets, set operations as equivalence and the of number part-whole real numbers) domain, Knowledge of particularly, part- Distinguish measure as a membership, well as the classes, relations words and relations, in base 10 as codomain, the whole operations between equality and form of set relations, relations between lists of part-whole well as other range), properties of and operations on discrete and quantification set between the equations. collections, numerals, operations, number bases operations as computationa numbers, to set continuous . operations, mathematical FP/IP sequences of and part- number in order to functions and l structures. operations and quantity. and set objects teachers also words, number whole develop a the basic operations on partitions. belonging to require numerals and words and partitions. deeper arithmetic numbers, and the the knowledge of numbers. sequences of understandin operations as use of computationa order numerals, g of the base functions. manipulatives and l domains relations. 10 number models in relation and counting they are proper. system. to basic arithmetic required to Understand operations. work with. the computationa l features of number systems.
Foundational Concepts Sub-Standards • Aggregates, counting, parts and wholes, sets and set relations, number system, relations and fractions, equivalence and order relations, operations as function, computational structures, structure preservation and representations.
Foundational Concepts Sub-Standards Parts and wholes: Knowledge of parthood, part-whole relations, part- whole operations, and part-whole partitions. Whole Part Part 10 8 2 8+2=10 10=8+2 2+8=10 10=2+8 10-2=8 8=10-2 10-8=2 2=10-8
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