Common Misunderstandings - Level 2.1 Number Naming Tool

Level 2: Place-value

Materials

26 counters in a suitable jar or container
7 bundles of ten icy pole sticks or straws and 22 single sticks or straws
A 0-99 Number Chart and a Masking Card (pdf - 27.6kb)

Instructions

Bold type indicates what should be said.

Empty container of counters in front of student and ask: “Can you count these as quickly as possible and write down the number please?” Note how the count is organised and what is recorded.

If not 26, ask, “Are you sure about that? How could you check?”

Once student has recorded 26, circle the 6 in 26 and ask, “Does this (point to the 6) have anything to do with how many counters you have there?” Indicate the collection. Note student’s response.

Circle the 2 in 26 and repeat the question. Note student’s response. Place counters back in the container.

Place bundles and sticks in front of the student and ask, “Can you make 34 using these materials please?” Note student’s response. If student asks or moves to unbundle a ten, say, “Before you do that, is there any way you could use these (pointing to the bundles of ten) to make 34?” Note student’s response. Remove sticks.

Tip out the container of 26 counters and ask student to count these again and record the number. Note response, then ask, “Can you put these into groups of four please?” Once this is completed, point to the 26 that has been recorded and circle the 6. Ask: “Does this have anything to do with how many counters you have?” Circle the 2 in 26 and repeat the question. Note student responses.

If counter task handled reasonably well, place the 0-99 Number Chart in front of the student. Cover the numbers 33 to 66 with the Masking Card and ask, “Can you count on by ones from 41 please?” If done easily, point to 57 and ask, “Can you count back by ones from here please?” Note student’s response.

Run your finger down the column headed by 5 and ask, “What are we counting on by now?” If student says “fives”, remove mask and try to find out what he/she is thinking by asking, “Do you still think we are counting by fives … Why?” Proceed to next question. If student answers “ten or tens” to initial question, leave mask in place and ask , “Can you count on from 15 by tens until I say ‘stop’?” If student stops at 95, encourage him/her to continue. Say “stop” when student no longer continues or at 135 (whichever comes sooner). Note response and whether or not student can proceed beyond 95.

2.1 Advice rubric

This tool has four different elements, the kidney bean task, the bundling sticks task, the regrouping task, and the number chart task. The observations and advice associated with each task are presented in turn below.

Kidney beans

Student responses to this task indicate the meanings they attach to 2-digit numerals. A version of this task was originally employed by Ross (1989) who identified five stages in the development of a sound understanding of place-value, each of which appears in some form in the advice below.

Observed response	Interpretation/Suggested teaching response
Little/no response	May not understand task Repeat at a later date
Response given but not indicative of strong place-value knowledge, eg, refers to 6 ones or physical arrangement such as “2 groups of 3” for circled 6, and “twenty” for circled 2.	Suggests 26 is understood in terms of ones, or 20 (ones) and 6 ones, may not trust the count of 10 or see 2 as a count of tens Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 ) Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place
Says 6 ones and 2 tens fairly quickly	Appears to understand the basis on which 2-digit numbers are 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) Consider introducing 3-digit place-value

Observed response

Interpretation/Suggested teaching response

Little/no response

May not understand task

Repeat at a later date

Response given but not indicative of strong place-value knowledge, eg, refers to 6 ones or physical arrangement such as “2 groups of 3” for circled 6, and “twenty” for circled 2.

Suggests 26 is understood in terms of ones, or 20 (ones) and 6 ones, may not trust the count of 10 or see 2 as a count of tens

Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 )
Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place

Says 6 ones and 2 tens fairly quickly

Appears to understand the basis on which 2-digit numbers are 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)
Consider introducing 3-digit place-value

Bundling sticks

Student responses to this task indicate their understanding of place-value and the extent to which they trust the count of 10, that is, they can treat 10 as a countable unit.

Observed response	Interpretation/Suggested teaching response
Little/no response, incorrect or insists on using/counting by ones only	May not understand task, does not trust the count of 10 Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 ) and review subitising and part-part-whole ideas for 10 (see Level 1 ) Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place
Counts ones, attempts or asks to unbundle tens in order to continue count of ones, after prompt may check count of bundle then uses either 2 tens and 14 ones or 3 tens and 4 ones	Suggests place-value ideas not well established, may not trust the count of 10 if student checks the number in a bundle Check trust the count, review subitising and part-part-whole ideas for 10 and making, naming and recording tens and ones (see Level 1 Advice ) 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)
Uses 3 tens and 4 ones to make 34 without any hesitation	Suggests sound understanding of place-value, particularly if combined with ability to recognise 6 as 6 ones and 2 as 2 tens in 26. 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) Consider introducing 3-digit place-value

Observed response

Interpretation/Suggested teaching response

Little/no response, incorrect or insists on using/counting by ones only

May not understand task, does not trust the count of 10

Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 ) and review subitising and part-part-whole ideas for 10 (see Level 1 )
Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place

Counts ones, attempts or asks to unbundle tens in order to continue count of ones, after prompt may check count of bundle then uses either 2 tens and 14 ones or 3 tens and 4 ones

Suggests place-value ideas not well established, may not trust the count of 10 if student checks the number in a bundle

Check trust the count, review subitising and part-part-whole ideas for 10 and making, naming and recording tens and ones (see Level 1 Advice )
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)

Uses 3 tens and 4 ones to make 34 without any hesitation

Suggests sound understanding of place-value, particularly if combined with ability to recognise 6 as 6 ones and 2 as 2 tens in 26.

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)
Consider introducing 3-digit place-value

Grouping task

Student responses to this task indicate the strength of their understanding of place-value by exploring the extent to which they can be distracted by the regrouping and the perceptual image it presents (6 groups of 4 and 2 ones remaining). Interestingly, some students who referred to the 2 in 26 as “twenty” in the first instance are prompted to refer to the 2 in 26 as “2 tens” after the grouping exercise.

Observed response	Interpretation/Suggested teaching response
Little/no response or refers to 6 as the number of groups of 4 and 2 as the 2 remaining ones	Distracted by the visual arrangement to override whatever else they may know about what ‘26’ means, suggests little/no place-value knowledge. May not understand task, does not trust the count of 10 Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 ) and review subitising and part-part-whole ideas for 10 (see Level 1 ) Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place
Is not distracted by visual image or regrouping, but refers to 2 as “twenty”	Suggests place-value ideas not well established, may not trust the count of 10 Check trust the count, review subitising and part-part-whole ideas for 10 and making, naming and recording tens and ones (see above) 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)
Says 6 ones and 2 tens fairly quickly	Appears to understand the basis on which 2-digit numbers are 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) Consider introducing 3-digit place-value

Observed response

Interpretation/Suggested teaching response

Little/no response or refers to 6 as the number of groups of 4 and 2 as the 2 remaining ones

Distracted by the visual arrangement to override whatever else they may know about what ‘26’ means, suggests little/no place-value knowledge. May not understand task, does not trust the count of 10

Check extent to which child trusts the count for 10 by counting large collections (see Tool 2.2 ) and review subitising and part-part-whole ideas for 10 (see Level 1 )
Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place

Is not distracted by visual image or regrouping, but refers to 2 as “twenty”

Suggests place-value ideas not well established, may not trust the count of 10

Check trust the count, review subitising and part-part-whole ideas for 10 and making, naming and recording tens and ones (see above)

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)

Says 6 ones and 2 tens fairly quickly

Appears to understand the basis on which 2-digit numbers are 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)
Consider introducing 3-digit place-value

Number chart task

Student responses to this task indicate their understanding of the place-value pattern and the extent to which they recognise the count of 10 and treat 10 as a countable unit.

Observed response	Interpretation/Suggested teaching response
Little/no response or hesitant to count hidden numbers on from 41 or back from 57	Unlikely response but may not understand task, needs to see the numbers to generate the count, and/or is unsure about the number naming sequence Check extent to which child can accurately count a physical collection (see Tool 2.2 ) Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place Draw attention to the patterns inherent in the 0-99 Number Chart by talking about the 1 ten family, 2 ten family, 3 ten family etc and how each number is represented, eg, 34 is in the 3 ten family, it is made up of 3 tens and 4 ones. Discuss what remains the same and what changes in each row (tens remain the same, ones change). Rehearse oral counting sequence and counting larger collections efficiently (see Tool 2.2 )
Manages to count on/back by ones fairly easily but says “fives” when asked to identify column count. May count on by tens once mask removed.	May not understand task but more likely to be distracted by visual perception suggesting understanding of place-value pattern not very robust, may only understand count of tens in terms of multiplies of ten and ones (ie, 10, 20, 30, 40, …) Practice making, naming and recording tens and ones, emphasising the count of tens in the tens place and the count of ones in the ones place Draw attention to the patterns inherent in the 0-99 Number Chart (see above). Discuss what remains the same and what changes in each row (tens remain the same, ones change) as well as each column (ones remain the same, tens change). Practice counting on/back in place-value parts starting from anywhere using Number Charts that extend beyond 100
Recognises count of tens and can count on by tens to 95 but hesitates or counts on by ones or fives to arrive at 105	May not appreciate that tens can be counted as countable units in the same way that any other ‘object’ might be counted, or that numbers can be renamed in terms of their place-value parts Practice renaming hundreds, tens and ones, using MAB materials (eg, 124 can be shown as 1 hundred 2 tens and 4 ones, as 12 tens and 4 ones, or as 124 ones). Record in words as in example given. Discuss which is the easiest way to remember or think about these numbers Practice counting on/back in place-value parts starting from anywhere using Number Charts that extend beyond 100
Recognises count of tens and counts to 135 with little difficulty	Suggests a sound understanding of the basis on which 2-digit numbers are recorded Introduce 3-digit place-value Use MAB and Number Expanders to consolidate 3-digit place-value by comparing 2 numbers (materials, words and symbols), ordering/sequencing (by ordering 5 or more 3-digit numbers or placing in sequence on a rope from 0 to 1000), counting forwards and backwards in place-value parts starting anywhere (eg, 327, 337, 347 (clap), 346, 345, 344, 343, …), and by renaming (eg, 845 is 8 hundreds 4 tens and 5 ones, 84 tens and 5 ones, or 845 ones) Use Number Chart activities(pdf - 27.6kb) to reinforce counting patterns