Common Misunderstandings - Level 2.2 Efficient Counting Tool

Level 2: Place-value

Materials

56 stackable counters
A container to hold counters
13 bundles of ten icy pole sticks and 16 single sticks/straws
Paper and a pencil/pen.

Instructions

Bold type indicates what should be said.

Place container of counters in front of the student and say, “I’m going to tip all these counters out and I would like you to count them as quickly as possible … Ready?… Tip out counters. “Go!”

Note how student counts and how he/she organises the count (eg, counts by ones, counts by twos, systematically moves counters to avoid recounting or groups to make count easier, …).

If student counts quickly and accurately by whatever method, ask “How would you write that number? ... Why would you write it like that? … If I counted that collection, would I get the same number?” Note student’s response.

If student appears to lack confidence, counts relatively slowly by ones and/or counts inaccurately. Stop them about half-way and say, “That’s going to take a long time isn’t it? Let’s put that amount over here. Let’s see if we can count what’s left in a quicker way.” Model counting by twos to 14, saying “If I do this, will this work? …” Then ask student to continue from 14, saying, “Can you keep on counting like this? … Note accuracy and speed. Stop and proceed to next task if, after one attempt to self correct, student still counts incorrectly. Otherwise, ask: “How would you write that number? ... Why would you write it like that? … If I counted that collection, would I get the same number?”

Note student’s strategy and written response.

Place all the icy-pole sticks in front of the student . Pick up one bundle of ten and ask, “How many sticks are in this bundle?” If the student guesses or appears uncertain, unbundle and ask him/her to count before proceeding. If student says “ten” fairly confidently, say: “Okay, can you count these for me and tell me how many please? … How would you write that? … Why would you write it like that?”

Note student’s strategy and written response.

2.2 Advice rubric

This tool has two different elements, the stackable counters task and a bundling sticks task. The observations and advice associated with each task are presented in turn below.

Stackable counters

Student responses to this task indicate the extent to which they can use grouping methods to count a large collection efficiently (eg, by twos, fives or tens). This also provides an indication of the extent to which students trust the count of the particular group chosen and see the group as a countable unit in its own right which is an important pre-requisite for working with multiplication later on. Consider the student’s first response and whether or not they can adopt a more sophisticated strategy after the modelling prompt.

Observed response	Interpretation/Suggested teaching response
Persists with counting by ones despite prompt	May not trust the count for twos or fives and/or may not be familiar enough with counting word sequences involved (eg, 2, 4, 6, 8 … or 5, 10, 15, …) Use 0-99 Number Chart to practice counting word sequences for twos, fives and tens Model and practice counting large collections. Point out the inefficiency and inaccuracy of counting by ones. Establish the value of counting by twos and fives. Chicken Scramble is a purposeful counting activity which involves placing a large amount of counters (enough for about 40-50 per student, the ‘chicken feed’) in the middle of a group of students (the ‘chickens’). Students are advised not to be greedy chickens then, on the word, “go”, students collect their share of the food. Before counting, students are asked if they think anyone has been greedy (amounts can be moved around accordingly), then the ‘chickens’ are asked to count their ‘food’ to see how fair their share was.
Starts counting by ones but after prompt counts by twos, fives or tens, or starts counting by twos, fives or tens but experiences some difficulty with counting word sequence and/or with any remaining counters	May not completely trust the count for the group size involved or recognise that counting by a given group still tells how many. This is suggested where students get ‘locked in’ to the counting sequence to the point where they cannot deal with the 1 remaining (if counting by fives) or the 6 remaining (if counting by tens) Model and practice counting large collections (see above) Use a 0-99 Number Chart to count by twos, fives and tens, stopping periodically to model counting on from, eg, “… 55, and 1 more?”, or “… 60 and 7 more?”
Counts fluently by fives or tens and adds on any remaining ones	Able to deal with composite units, and trusts the result of the count, ready to move to formal place-value Introduce 2-digit place-value by making and counting tens as countable units, eg, use bundling materials such as icy-pole sticks or straws to make tens and count as 1 ten, 2 tens, 3 tens etc For remaining steps see Booker et al (2003).

Observed response

Interpretation/Suggested teaching response

Persists with counting by ones despite prompt

May not trust the count for twos or fives and/or may not be familiar enough with counting word sequences involved (eg, 2, 4, 6, 8 … or 5, 10, 15, …)

Use 0-99 Number Chart to practice counting word sequences for twos, fives and tens
Model and practice counting large collections. Point out the inefficiency and inaccuracy of counting by ones. Establish the value of counting by twos and fives. Chicken Scramble is a purposeful counting activity which involves placing a large amount of counters (enough for about 40-50 per student, the ‘chicken feed’) in the middle of a group of students (the ‘chickens’). Students are advised not to be greedy chickens then, on the word, “go”, students collect their share of the food. Before counting, students are asked if they think anyone has been greedy (amounts can be moved around accordingly), then the ‘chickens’ are asked to count their ‘food’ to see how fair their share was.

Starts counting by ones but after prompt counts by twos, fives or tens, or starts counting by twos, fives or tens but experiences some difficulty with counting word sequence and/or with any remaining counters

May not completely trust the count for the group size involved or recognise that counting by a given group still tells how many. This is suggested where students get ‘locked in’ to the counting sequence to the point where they cannot deal with the 1 remaining (if counting by fives) or the 6 remaining (if counting by tens)

Model and practice counting large collections (see above)
Use a 0-99 Number Chart to count by twos, fives and tens, stopping periodically to model counting on from, eg, “… 55, and 1 more?”, or “… 60 and 7 more?”

Counts fluently by fives or tens and adds on any remaining ones

Able to deal with composite units, and trusts the result of the count, ready to move to formal place-value

Introduce 2-digit place-value by making and counting tens as countable units, eg, use bundling materials such as icy-pole sticks or straws to make tens and count as 1 ten, 2 tens, 3 tens etc
For remaining steps see Booker et al (2003).

Count bundling sticks

Student responses to this task indicate the extent to which they can treat ten as a countable unit, count in tens and ones and rename in terms of place-value parts.

Observed response	Interpretation/Suggested teaching response
Treats bundles of tens as ones and counts all as ones despite prompt	Can count using one-to-one correspondence but does not trust the count of 10 Work on subitising and part-part-whole knowledge for numbers 1-5 then use this to build knowledge of numbers 6-10 as mental objects (evident when these numbers can be recognised using subitising and part-part-whole knowledge) Once part-part-whole knowledge established for numbers 1-10, use Ten-frames and Open Number Lines (see Level 1) to build a sense of numbers beyond 10 Introduce 2-digit place-value by making and counting tens as countable units, eg, use bundling materials such as icy-pole sticks or straws to make tens and count as 1 ten, 2 tens, 3 tens etc For remaining steps see Booker et al (2003)
Recognises bundles as tens, partially counts by tens (eg, may stop at 100, then proceed by ones), may experience some difficulty with counting word sequence	May not be familiar enough with counting word sequence for larger numbers, may not realise that a count of tens can go beyond 100 Use extended Number chart (eg, 60-159) to practice counting forwards and backwards in ones and tens to strengthen number words and make patterns explicit.
Says and records 146 by counting tens and ones systematically	Appears to understand how tens and ones are represented, counted, named and recorded. Consolidate 2-digit place-value by comparing 2 numbers (materials, words and symbols), ordering/sequencing (by ordering 5 or more 2-digit numbers or placing in sequence on a rope from 0 to 100), counting forwards and backwards in place-value parts starting anywhere (eg, 27, 37, 47 (clap), 46, 45, 44, 43, …), and by renaming (eg, 45 is 4 tens and 5 ones or 45 ones)

Observed response

Interpretation/Suggested teaching response

Treats bundles of tens as ones and counts all as ones despite prompt

Can count using one-to-one correspondence but does not trust the count of 10

Work on subitising and part-part-whole knowledge for numbers 1-5 then use this to build knowledge of numbers 6-10 as mental objects (evident when these numbers can be recognised using subitising and part-part-whole knowledge)
Once part-part-whole knowledge established for numbers 1-10, use Ten-frames and Open Number Lines (see Level 1) to build a sense of numbers beyond 10
Introduce 2-digit place-value by making and counting tens as countable units, eg, use bundling materials such as icy-pole sticks or straws to make tens and count as 1 ten, 2 tens, 3 tens etc
For remaining steps see Booker et al (2003)

Recognises bundles as tens, partially counts by tens (eg, may stop at 100, then proceed by ones), may experience some difficulty with counting word sequence

May not be familiar enough with counting word sequence for larger numbers, may not realise that a count of tens can go beyond 100

Use extended Number chart (eg, 60-159) to practice counting forwards and backwards in ones and tens to strengthen number words and make patterns explicit.

Says and records 146 by counting tens and ones systematically

Appears to understand how tens and ones are represented, counted, named and recorded.

Consolidate 2-digit place-value by comparing 2 numbers (materials, words and symbols), ordering/sequencing (by ordering 5 or more 2-digit numbers or placing in sequence on a rope from 0 to 100), counting forwards and backwards in place-value parts starting anywhere (eg, 27, 37, 47 (clap), 46, 45, 44, 43, …), and by renaming (eg, 45 is 4 tens and 5 ones or 45 ones)