Common Misunderstandings - Level 1.2 Mental Object Tool

Level 1: Trusting the count



Place 5 counters and the container with 4 counters in front of the student.

Say: “There are 5 counters here and 4 more in this container.” As this is said, pick up the container and rattle it to indicate that the counters are there. “Without opening the container, can you tell me how many counters there are altogether? … How did you work that out?”

Note student’s response.

If done relatively quickly and easily, place the unfolded Mental Objects Card in front of the student, and pointing to each in turn, say: “There are 7 dots here and 9 dots here.” Quickly cover the 9 dots with the flap created by cutting the card as indicated (secure if necessary) and say, “Without unfolding the card, can you tell me how many dots there are altogether? … How did you work that out?”

Note student’s response.

1.2 Advice Rubric

Student responses to this task indicate the sophistication of their counting strategies and capacity to deal with an unseen number. This task was originally employed by Steffe and his colleagues at Georgia in the early 1980s to identify the steps involved in learning to count fluently and confidently but it also provides a valuable insight into the extent to which students have developed mental objects for numbers, hence the extension to larger numbers here. While the original research identified 5 different steps in the counting process, these are generally collapsed to three for the purposes of informing teaching,

  • perceptual counters , where child can only count what they see and cannot deal with the hidden collection (this suggests little/no access to a range of mental objects for the hidden number);
  • figural counters , where the child can provide some sort of visual and/or auditory cue to assist with the count of the hidden collection, eg, fingers or taps to help keep track of the count from 5 to 9 (this suggests access to some mental objects); and
  • abstract or conceptual counters, where the child immediately says “9” on the basis that they “just know 9”, or number fact knowledge, eg, “it has to be 9, because 5 and 5 is 10” (this suggests access to a range of mental objects for the hidden number).
Observed response Interpretation/Suggested teaching response

Little or no response, may count what they see to 5, but nothing further (perceptual counters)

May not understand the task or recognise “five” without counting to confirm

  • Practice counting collections and oral counting to establish the number naming sequence
  • Check and consolidate the link between collections, number words and numerals (make, name and record numbers to 10)
  • Practice counting on from 1, 2, or 3 using a conventional 6 –sided dot dice and another dice with 1-3 in dots and 1-3 as numerals. Toss dice, ask students to read numbers, cover 1, 2 or 3, then count on the dots on the other dice

Counts the 5 that can be seen and makes some attempt to count the hidden collection by counting on or counting all but unable to complete or incorrect

May not trust the count for 4, possibly still relying on perception

  • Use Subitising Cards (pdf - 49.64kb) to encourage students to recognise small numbers without counting (subitising) and build part-part-whole ideas for numbers 1-5 (eg, 4 is 1 and 3, 2 and 2, 1 less than 5 etc). This helps establish a trust of the count by developing mental objects for these numbers which support childrens’ efforts to count on
  • Practice counting on from given, eg, use a set of numeral cards and a 6- or 10-sided dice, say the number and count on dots displayed on dice
  • Model counting on 2, 3 or 4 by starting from given and clapping as you count, eg, 5 …6 (clap), 7 (clap), 8 (clap), 9 (clap). Repeat with different starting numbers and fingers or taps instead of clapping. Taps can mirror familiar pattern, eg, if counting on 5, taps could be spatially located to represent 5 pattern on a dice

Correctly counts on to 9 using fingers, taps or other ways of keeping track of the count (figural counters). Unable to deal with 7 counters task.

Indicates access to mental object for numbers less than 5 (trusts the count for these numbers). Needs to consolidate part-part-whole ideas for numbers 1-10 and number fact knowledge

  • Use ten-frames and Subitising Cards to promote subitising and the development of part-whole ideas for the numbers 5-10 (that is, that 7 is 1 more than 6, a 5 and 2, or a 3 and 4
  • Make this knowledge explicit by asking students to say what they know about a given number, eg, “6 is double 3”, “it’s 2 more than 4, 1 less than 7, 4 less than 10” and so on. Record on posters and display, review regularly

Immediately correct on the basis that “I just know” or the use of number fact knowledge, eg, “I thought of 5 and 5 and 1 less made 9” (abstract or conceptual counters). May attempt task with 7 counters but unable to complete or incorrect or counts on all by ones

Indicates sound knowledge of numbers 1-10 in terms of part-part-whole and trusting the count, needs to work on mental strategies to deal with larger numbers

  • Begin to develop mental strategies for addition (see 1.1 Subitising Tool Advice ) commencing with count on from larger (eg, 2 and 7, think: 7 … 8, 9)
  • Once this is established proceed to the doubles and near doubles mental strategy (eg, 6 and 7, think: double 6 is 12 and 1 more, 13)see 1.1 Subitising Tool Advice )
  • Use Ten-frames and Open number Lines (as indicated in Subitising Probe Task Advice) to scaffold the make-to-ten mental strategy (eg, for 6 and 8, think: 8 … 2 more to 10 and 4 more … 14)

Answers both tasks correctly on the basis of number fact knowledge or the use of an appropriate strategy such as make-to ten

Indicates sound number fact knowledge and/or access to efficient mental strategies. Ready to proceed to further work in addition and subtraction

  • Consolidate mental strategies through practice and making strategies explicit
  • Extend strategies to solve basic subtraction problems mentally, eg, for 7 take-away 3, use part-part-whole knowledge or count back 3, for 12 take-away 5 use make-back-to-ten and part-part-whole knowledge for 5 and 10, and for 16 take-away 9 use place-value knowledge and/or halving.