Mathematics Tasks

Factual knowledge/Speeded recall task: 

This task assessed participantsÕ knowledge of number facts. On each trial an arithmetic problem was presented on screen for three seconds and participants were asked to retrieve the result without mental calculation. The participants were instructed to give their answer verbally, at which point the experimenter pressed a key and inputted by the answer. Participants were instructed to say ÒI donÕt knowÓ if they could not retrieve the answer.  

Participants completed four practice trials and then 12 experimental trials in random order. An additional four easy Ômotivational trialsÕ were intermixed with the experimental trials. To ensure that performance was not at floor or ceiling level in any group we selected a different set of items for each age group (8-9 years, 11-12 years, 13-14 years, young adults). Following pilot testing, the problems given to the primary school students were composed of single-digit addition operations only, those given to the secondary school students also included subtraction operations. The problems for the 11-12 year olds involved single-digit numbers, and the problems for the 13-14 year olds were composed of one single-digit number and one double-digit number. The problems given to the young adults involved addition, subtraction, multiplication and division operations composed of one single-digit and one double-digit number. 

Procedural skills/Strategy task:

This task assessed the strategy choice and efficiency with which participants could accurately perform arithmetic procedures. On each trial an arithmetic problem was presented on screen and participants were instructed to solve it using any mental method they preferred. Prior to starting the task participants were shown pictures representing different strategies (i.e.. counting in your head, counting on fingers, decomposition, and retrieval) to ensure that younger participants understood that any strategy was acceptable in this task. The experimenter described the strategies and told participants that any of these strategies, or others, could be used to solve the task. 

Participants were given four practice trials and then 10 (8-9 years and 11-12 years) or 12 (13-14 years, young adults) experimental trials. The operations were designed to be age appropriate, and of a difficulty level where retrieval would be unlikely. The problems for all age groups involved a mix of single and double digit numbers, with a greater proportion of double digit numbers for the older groups. The trials given to 8-9-year-olds and 11-12-year-olds were composed of addition and subtraction operations and the trials given to 13-14-year-olds and young adults were composed of addition, subtraction, multiplication and division operations. The items in each version were presented in one of two orders counterbalanced across participants.

The participants were instructed to give their answer verbally, at which point the experimenter pressed a key and inputted the answer. After each answer, the participants were questioned on their strategy use. All trials were audio recorded.

Conceptual knowledge task:

This task assessed participantsÕ understanding of conceptual principles underlying arithmetic. The operations were designed to be difficult to solve mentally, to discourage the participants from attempting to do so. A different set of problems were used for each age group. The 8-9-year-olds watched a puppet solve a double-digit addition or subtraction problem using counters and were shown the example problem (including the answer) written in a booklet (e.g. 23 + 24 = 47). They were then shown four probe problems which were presented without answers and asked whether the puppet could use the example (completed) problem to solve each probe problem, or if he would need to use the counters to solve it. Of the four probe problems, one of the related problems was identical (e.g. 23 + 24 = ), one was related by commutativity (e.g. 24 + 23 = ), one was related by inversion (e.g. 47 Ð 23 = ) and one was unrelated (e.g. 32 + 24 = ). The participants were first asked to decide whether or not the example problem could help the puppet solve each probe problem, and asked to explain how. The participants completed two practice example problems, with feedback, followed by 24 experimental trials (six example problems each with four probe problems). The items were presented in one of two orders counterbalanced across participants.

The conceptual task for the 11-12-year-olds, 13-14-year-olds and young adults was presented on a computer. On each trial an arithmetic problem with the correct answer was presented on the screen. Once this was read, the experimenter pressed  ÔreturnÕ and a second, unsolved operation appeared below the first problem. The participants were asked to state whether or not the first problem could help solve the second problem, and then were asked to explain how. Participants were given four practice trials and thirty experimental trials. Eighteen of the thirty problem pairs were related. The pairs of problems were related by the subtraction-complement principle (e.g. 113 Ð 59 = 54 and 113 Ð 54 = ), inverse operations (e.g. 74 + 57 = 131 and 131 Ð 74 = ), and associative operations (e.g. 87 Ð 54 = 33 and 87 Ð 34 Ð 20 = ). The trials given to the 11-12-year-olds were composed of addition and subtraction problems involving two operands of two and three digit numbers. The trials given to the 13-14-year-olds were composed of addition and subtraction problems involving two or three operands of double-digit numbers, as well as some multiplication and division problems involving single and double-digit numbers. The trials for the young adults were composed similarly but they also included some division problems including two double-digit numbers. The items in each task version were presented in one of two orders counterbalanced across participants.
All participants gave their response verbally and the experimenter recorded this. Accuracy measures were calculated for how many relationships were correctly identified, and for how many accurate explanations each participant provided. All trials were audio recorded.

Dot comparison task:

On each trial the participants were shown two sets of white dots on a black screen and were instructed to identify which set had the highest number of dots. The dots were created using an adapted version of the matlab script provided by Gebius and Reynvoet (2011). Participants were required to ignore the size of the dots and the array on the screen and to respond based on the number of dots only. The number of dots in each array ranged from 5 to 28 and the ratio between the number of dots ranged from 0.5 to 0.8. Participants completed 6 practice trials and 80 experimental trials in random order. They were given breaks during the task as needed. 

Executive function tasks

Verbal short-term memory:

Verbal short-term memory was assessed via a word span task. Participants heard a list of single syllable words and were asked to recall them in order. There were three lists at each span length, beginning with lists of two words, and the participants continued to the next list length if they responded correctly to at least one of the trials at each list length. 

Verbal working memory:

Verbal working memory was assessed via a sentence span task. Participants heard a sentence with the final word missing and had to provide the appropriate word. After a set of sentences they were asked to recall the final word of each sentence in the set, in the correct order. Participants first completed an initial practice item with two sentences, and therefore two words to remember. They then continued to the test trials where they received three trials at each span test length. Provided they recalled at least one trial correctly, the sequence length was increased by a single item and three further trials were administered.

Verbal processing:

ParticipantsÕ performance on the processing task was assessed separately in two blocks (one before the sentence span task and one after) of 20 trials each. In these blocks they only had to provide the final word of the sentence, without the need to recall the words. Response times were measured.

Visuospatial short-term memory:

In the visuospatial short-term memory task participants saw a 3 x 3 grid on the screen. They watched as a frog jumped around the grid and after he had finished the sequence they had to point to the squares he jumped on in the correct order, which was recorded by the participant using the mouse. There were three trials at each sequence length, beginning with sequences of two jumps, and participants continued to the next sequence length if they responded correctly to at least one of the sequences at each length.	

Visuospatial working memory:

Visuospatial working memory was assessed via a complex span task. Participants saw a series of 3 x 3 grids each containing three symbols and they had to point to the symbol that differed from the other two. After a set of grids participants were asked to recall the position of the odd-one-out on each grid, in the correct order. There were three sets at each span length, beginning with sets of two grids, and participants continued to the next span length if they responded correctly to at least one of the sets at each span length. 

Visuospatial processing:

ParticipantsÕ performance on the processing task was also assessed separately in two blocks (one before the complex span task and one after) of 20 trials each. In these blocks participants only had to identify the location of the odd-one-out, without the need to recall the position. Response times were measured.

Animal Stroop/Non-numerical inhibition task:

On each trial two animal pictures were presented on the screen. One animal was selected from a set of large animals (e.g. a bear, gorilla, and giraffe) and the other animal was selected from a set of small animals (e.g. an ant, rabbit, and mouse). The participantsÕ task was to identify which animal was the larger in real life. On each trial, one animal image was presented with an area on screen four times larger than the other image. On congruent trials the animal that was larger in real life was also the larger image on the screen, and on incongruent trials the animal that was smaller in real life was the larger image on the screen. Participants were required to ignore the size of the images on the screen and to respond based on the size in real life only. On each trial, the images were presented on screen and participants responded as quickly as possible by pressing one of two buttons on the keyboard that corresponded to the side of the screen with the larger animal.

Participants completed four experimental blocks each containing 48 trials in random order. The time taken to complete each block was recorded and presented to participants at the end of each block to encourage them to respond quickly. In the first two experimental blocks 75% of the trials were incongruent and 25% were congruent and in the second two experimental blocks 75% of the trials were congruent and 25% were incongruent. Prior to commencing the task participants were shown each of the animal images in one size and asked whether the animal was large or small in real life to ensure they had the necessary real-world knowledge to perform the task. 

NEPSY Animal Sorting/Card Sorting task:

To assess participantsÕ ability to formulate basic concepts and shift from one concept to another we used the Animal Sorting subtest from the NEPSY-II. The task requires participants to sort eight cards into two groups of four using self-initiated sorting criteria. Following a teaching example the participants were given 360 seconds of cumulative sort time to sort the cards in as many different ways as they could. The test was discontinued before 360 seconds if the participant stated they had finished, or if 2 minutes elapsed without a response. In addition to the scores calculated as per the NEPSY manual we also recorded the time participants took to complete each sort. 

Visual search task: 
 In this task participants saw a screen full of simple drawings of small animals (elephants, donkeys and bears). In the first trial all the animals faced right and participants were asked to select the elephants. In the second trial half of the animals faced left and half faced right; participants were asked to select all of the elephants that faced left. In the final trial again half of the animals looked left and half right, this time participants were asked to point to all of the bears that looked right. Participants selected an animal by clicking on the animal with the mouse and the image of that animal shrunk in size. During each trial the experimenter did not remind participants which target they were looking for, or encourage them to look for any more. When participants had finished identifying they pressed the space bar to move onto the next trial. There was no time limit. The number of targets correctly identified, the number of distractors selected, the time between item selection and the distance between item selections were all recorded. 
Maths Anxiety: 

Maths anxiety in Year 4 was measured with the Child Maths Anxiety Questionnaire, or CMAQ (Ramirez, Gunderson, Levine & Beilock, 2013), and in the year 7, 9 and adult groups with the Mathematics Anxiety Scale-Revised, or MAS-R (Bai et al., 2009).    

Behavior Rating Invetory of Executive Function (BRIEF):
For participants in Year 4 teachers completed the BRIEF-teacher form. Participants in Year 7 and Year 9 completed the BRIEF-SR (self-report adolescent form). Adults completed the BRIEF-A (self-report adult versions).