Rule 1 Rule 2 Rule 3 Rule 4. Balance Weight

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107 Tble 1 Predicted Success (Percentge of Correct Responses) on Different Blnce-Scle Problems for Individuls Using Siegler s (1976) Four Rules Problem Type Level of Performnce Rule 1 Rule 2 Rule 3 Rule 4 Blnce 100 100 100 100 Weight 100 100 100 100 Distnce 0 100 100 100 Conflict-Weight 100 100 33 c 100 Conflict-Distnce 0 b 0 b 33 c 100 Conflict-Blnce 0 b 0 b 33 c 100 Both-Greter d 100 100 100 100 Incorrectly predict tht the scle will blnce. b Incorrectly predict tht the side with the greter weight will go down. c Chnce responding is predicted. d Problem type not discussed by Siegler (1976, 1981). One side hs the greter weight nd the greter distnce.

108 Tble 2 Chrcteristics of the Clusters for Network 4120 Men Hidden Unit Activity Men Torque Cluster n H0 H1 H2 H3 Difference b 1 (left) 117.84 (.12) 0 0 0-11.47 (5.07) 7 (left) 121.38 (.11).08 (.13).03 (.02) 0-5.97 (4.06) 6 (left) 50.16 (.03).80 (.16).08 (.03).02 (.03) -2.20 (1.40) 3 (blnce) 49.09 (.01).84 (.04).14 (.01).54 (.03) 0 2 (right) 127 0 0.87 (.11) 0 10.88 (5.22) 5 (right) 129.02 (.01).02 (.03).43 (.12).03 (.06) 5.66 (4.17) 4 (right) 32.06 (.01).31 (.09).21 (.02).71 (.14) 1.94 (1.22) Totl 625.26 (.32).17 (.30).30 (.33).09 (.21) 0 (8.95) Note. Stndrd devitions re shown in brckets. When men of zero is shown without brckets, the stndrd devition is lso zero. Clusters hve been reordered by output stte of the network (shown in brckets). b Torque difference = (right weight x right distnce) - (left weight x left distnce).

109 Tble 3 Chrcteristics of the 52 Problems Not Solvble by n Additive Heuristic Problem Type Torque predicts: Additive predicts: Frequency Conflict Blnce blnce tip (> weight) 2 Conflict Blnce blnce tip (> distnce) 2 b Conflict Weight tip blnce 20 c Conflict Weight tip tip opposite side 4 d Conflict Distnce tip blnce 20 e Conflict Distnce tip tip opposite side 4 f Note. All bsolute torque difference vlues for conflict-weight nd conflict-distnce problems rnge between 1-4. Conflict blnce sum weight (CBSW). b Conflict blnce sum distnce (CBSD). c Conflict weight sum blnce (CWSB). d Conflict weight sum distnce (CWSD). e Conflict distnce sum blnce (CDSB). f Conflict distnce sum weight (CDSW). These lbels were suggested by Ferretti et l. (1985).

110 Tble 4 Predictions Mde by Three Different Strtegies for Six Types of No-Mtch Problems nd Three Types of Mtch Problems Strtegy Type No-Mtch Type Torque Additive Weight Only Frequency CBSW BAL >WT >WT 2 CDSW >DIS >WT >WT 4 CWSB >WT BAL >WT 20 CWSD >WT >DIS >WT 4 CBSD BAL >DIS >WT 2 CDSB >DIS BAL >WT 20 Mtch Type CB BAL BAL >WT 20 CD >DIS >DIS >WT 64 CW >WT >WT >WT 64 Note. CBSD = conflict-blnce sum-distnce, CBSW = conflict-blnce sum-weight CDSB = conflict-distnce sum-blnce, CDSW = conflict-distnce sum-weight CWSB = conflict-weight sum-blnce, CWSD = conflict-weight sum-distnce. BAL = scle blnces, >DIS = tips to side with greter distnce, >WT = tips to side with greter weight. Out of 625, for five-peg, five-weight version of the tsk.

111 Tble 5 Number of Problems in the Test Set Used by McClellnd (1989) nd Schmidt nd Shultz (1991) by Torque Difference (TD) nd Problem Type Torque Problem Type Difference CD CW DIS WT Totl 1 1 0 0 0 1 2 1 1 1 0 3 3 0 1 0 0 1 4 2 1 1 1 5 5 0 1 0 0 1 6 0 0 1 0 1 7 0 0 0 0 0 8 0 0 0 3 3 9 0 0 0 0 0 10 0 0 1 0 1 Totl 4 4 4 4 16 Note. Absolute vlue of torque difference = (RW x RD) - (LW x LD). CD = conflict-distnce, CW = conflict-weight, DIS = distnce, WT = weight. No-mtch problem.

112 Tble 6 Number of Problems in the Test Set Used by Jnsen nd vn der Ms (1997) by Torque Difference (TD) nd Problem Type Torque Problem Type Difference CD CW b DIS WT Totl 1 2 3 1 0 6 2 3 2 1 3 9 3 0 0 1 1 2 4 0 0 1 0 1 5 0 0 0 0 0 6 0 0 1 0 1 7 0 0 0 0 0 8 0 0 0 1 1 Totl 5 5 5 5 20 Note. Absolute vlue of torque difference = (RW + RD) - (LW + LD). CD = conflict-distnce, CW = conflict-weight, DIS = distnce, WT = weight. Four of the five were no-mtch problems. b Three were no-mtch problems.

113 Tble 7 Number of Problems in the Test Set Used by Schmidt nd Shultz (1991) for Two Levels of Torque Difference (TD) by Problem Type Torque Problem Type Difference CD Mtch CD No-Mtch CW Mtch CW No-Mtch Level 1 3 1 2 2 Level 2 2 2 4 0 Note. Level 1 problems were TD = 1; level 2 problems were TD = 3.

114 Tble 8 Bnd Membership for H1 s Function of the Additive Eqution for Network 2929 Additive Hidden Unit 1 Bnd b Difference A1 A2 B1 B2 C Totl -8 1 0 0 0 0 1-7 4 0 0 0 0 4-6 10 0 0 0 0 10-5 20 0 0 0 0 20-4 35 0 0 0 0 35-3 0 52 0 0 0 52-2 0 0 0 68 0 68-1 0 0 0 0 74 74 0 0 0 45 0 0 45 1 0 74 0 0 0 74 2 68 0 0 0 0 68 3 52 0 0 0 0 52 4 35 0 0 0 0 35 5 20 0 0 0 0 20 6 10 0 0 0 0 10 7 4 0 0 0 0 4 8 1 0 0 0 0 1 Note. There re totl of 573 ptterns (mtch problems only). Additive difference = (RW + RD) - (LW + LD). b Bnds A1, A2, B1, nd B2 refer to the microbnds in hidden unit 1 (see Figure 25).

115 Tble 9 Accurcy Dt (Percent Correct) for Ech Prticipnt for Conflict Problems with Torque Difference # 4 (Mtch nd No-Mtch Problems) Prticipnt Problem Type CB CBSD CBSW b Mtch c CDSB d CDSW e CWSB f CWSD g Mtch h 1 100 50 50 65 25 80 50 78.3 2 0 0 55 30 25 40 25 80.0 3 0 0 10 20 0 60 75 81.7 4 0 0 80 15 0 25 25 90.0 5 0 0 20 0 0 45 25 68.3 6 0 50 70 15 25 45 50 46.7 7 0 0 0 55 25 45 25 81.7 8 0 100 50 60 50 0 0 75.0 Men 12.5 25.0 41.9 32.5 18.8 42.5 34.4 75.2 Note. The frequency of no-mtch problems (out of 625) is s follows: CBSD (n = 2), CBSW (n = 2), CDSB (n = 20), CDSW (n = 4), CWSB (n = 20), CWSD (n = 4). Conflict blnce sum distnce (CBSD). b Conflict blnce sum weight (CBSW). c Mtch conflict-blnce problems (n = 20). d Conflict distnce sum blnce (CDSB). e Conflict distnce sum weight (CDSW). f Conflict weight sum blnce (CWSB). g Conflict weight sum distnce (CWSD). h Mtch problems with torque difference # 4 (n = 60).

116 Tble 10 Accurcy s Function of Torque Difference for Simple Problem Types Torque Difference Totl Correct Both- Greter Weight Distnce 1 98.4 100 96.9 2 97.8 99.1 96.4 3 96.2 93.8 95.8 96.9 4 95.8 90.6 96.1 96.9 5 96.6 96.3 98.4 95.3 6 95.3 89.6 98.8 98.8 7 95.8 95.8 8 97.1 96.4 100 95.3 9 99.4 99.1 100 100 10 99.5 100 100 97.9 11 99.2 99.2 12 98.3 100 97.9 95.8 13 99.2 99.2 14 98.4 98.4 15 100 100 100 100 16 100 100 100 100 17 97.9 97.9 18 100 100 19 100 100 20 100 100 100 21-24 100 100 Note. Percentge correct for simple blnce problems ws 93%. Empty cells indicte tht problem type does not hve instnces t tht level of TD.

117 Tble 11 Accurcy for Conflict Problems s Function of Torque Difference Level Torque Totl CW CW CD CD Difference Correct Mtch No-Mtch Mtch No-Mtch 1 54.4 85 33.8 68.8 30 2 54.3 75 42.2 65.6 34.4 3 63.1 81.3 50 79.2 25 4 65.5 87.5 56.3 60.4 25 5 81.9 6 79.9 7 89.8 8 90.6 10 89.1 11 95.3 15 93.8 Note. The totl percent correct for conflict-blnce problems ws 38.0%. Percent correct for mtch nd no-mtch ws 41.8% nd 18.7%, respectively. The rnge of torque difference for no-mtch problems is 0-4.

118 Tble 12 Accurcy Dt (Percent Correct) for Ech Prticipnt s Function of Problem Type Problem Type Both Conflict- Conflict- Conflict- Prticipnt Greter Blnce Weight Distnce Weight Distnce Blnce 1 100 88 100 100 86.4 78 54.2 2 96.5 96 99 95 79.5 64.8 45.8 3 98 92 100 98 89.8 61.4 8.3 4 99 100 97 100 77.3 65.9 66.7 5 97.5 92 97 93 73.9 42 16.7 6 93.5 100 97 95 71.6 30.7 62.5 7 100 80 98 97 69.3 80.7 0 8 98.5 96 99 99 50 83 50 totl 97.9 93 98.4 97.3 74.7 63.4 38

119 Tble 13 Rule Dignosis for Ech Prticipnt for Eight Different Test Sets Test Set Ss McC b level 1 c S & S S & S level 2 d S & S level 3 e S & S level 4 f ll mtch g nomtch h high TD i 1 4 4 4 4 4 2 4 4 2 3 3 3 4 4 4 3 4 3 2 3 3 4 4 4 2 4 4 3 4 4 4 4 4 3 4 5 0 0 2 3 2 2 2 4 6 2 2 3 2 4 4 3 4 7 3 3 3 3 3 3 0 3 8 3 3 3 3 4 4 3 4 Note. Rule dignosis ws performed s outlined by Siegler (1976). A dignosis of 0 indictes unclssifible. Prticipnt s identifiction number. b McClellnd s (1989) test set published in Shultz nd Schmidt (1991). c Shultz & Schmidt (1991) torque difference level 1 (TD = 1). d Shultz & Schmidt (1991) torque difference level 2 (TD = 3). e Shultz & Schmidt (1991) torque difference level 3 (TD = 12 for simple problems; TD = 5 for conflict problems). f Shultz & Schmidt (1991) torque difference level 4 (TD = 15-20 for simple problems; TD = 10-15 for conflict problems). g A smple of problems with ll conflict problems clssified s mtch problems. h A smple of problems with ll conflict problems clssified s no-mtch problems. i A smple of problems with high torque difference for non-blnce problems.

120 Tble 14 Rule Dignosis for Ech Prticipnt for Three Test Sets nd Three Different Assessment Orders Test Set McClellnd No Mtch b All Mtch c Ss d 4321 e 1234 f 4231 g 4321 e 1234 f 4231 g 4321 e 1234 f 4231 g 1 4 3 4 4 3 4 2 2 2 2 3 3 3 3 2 2 4 3 4 3 2 2 2 2 2 2 4 2 4 4 3 3 3 3 3 3 4 3 4 5 0 0 0 2 2 2 2 2 2 6 2 2 2 3 2 2 4 3 4 7 3 2 2 0 0 0 3 2 2 8 3 3 3 3 3 3 4 3 4 Note. Rule dignosis ws performed s outlined by Siegler (1976). A dignosis of 0 indictes unclssifible. Test set used by McClellnd (1989) nd Shultz nd Schmidt (1991). b A smple of problems with ll conflict problems clssified s no-mtch problems. c A smple of problems with ll conflict problems clssified s mtch problems. d Prticipnt s identifiction number. e Scoring priority: Rules 4, 3, 2, 1. f Scoring priority: Rules 1, 2, 3, 4. g Scoring priority: Rules 4, 2, 3, 1.

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