Number Concept

Introduction

The question of how we come to develop the number concept over years of training in mathematics education has been hotly discussed and tested in various methods over the decades. The number concept refers to the ability of humans to associate quantities and magnitudes of concrete objects with abstract numbers, and the ability to perceive relationships and differences between numbers. Jean Piaget first popularized the idea of the “number concept” when he discovered that preschool children often fail to develop “one-to-one correspondence”; this is the ability to count and associate numbers with quantities of objects, recognizing that numbers are symbols to represent quantities. Over time, this number concept becomes increasingly advanced as one begins to recognize abstract relationships between numbers, such as addition, evenness and oddness, and factors and multiples.

To try to assess what features of numbers people of different ages have understood and internalized, several psychological studies administered surveys that ask participants to rate the similarities between pairs of numbers. Based on these similarity ratings, the researchers used methods such as Multidimensional Scaling (MDS) (Shepard, Kilpatric, & Cunningham, 1975; Miller & Gelman, 1983) and Individual Differences Clustering (INDCLUS) (Miller & Stigler, 1991) to find both visualizable and quantifiable methods of representing the implicit features and methods that these participants may have used in judging how similar certain numbers are to others.

Our goal is to find out whether we can achieve consistent results from using both MDS and INDCLUS in analyzing individual survey results of number similarities. To do so, we gathered similarity ratings from a control group of participants, who did not undergo any interventions during the process of rating similarities between numbers. Proving there is a consistency between the results from both analysis methods will allow us to present more compelling evidence towards understanding the number concept of people of different ages. Going further, we plan to run experiments in which subjects judge the similarities between one group of numbers, undergo a certain educational intervention, and then judge the similarities between another group of numbers, to see if there was a difference in similarity judgments. Using the analysis methods of MDS and INDCLUS, we can then explore how certain educational interventions, such as math pedagogy or math-related activities, influence the number concept in humans.

Methodology

The data were collected from an online Qualtrics survey (https://ucbpsych.qualtrics.com/jfe/form/SV_0cc37YgiYXUpx3f) that was designed by Rachel Jansen of the Computational Cognitive Science Lab at UC Berkeley. The survey randomly chose 20 numbers from 2 to 30 that were divided into two groups of ten numbers each. (The division of numbers was designed so that each group has the same number of even/odd numbers, prime/composite numbers, etc. In future studies, one group would be used before intervention and the other after the intervention.) Subjects were asked to choose the most and least similar pairs of numbers among the 20 numbers, and then rated the similarities between all possible pairs of numbers within the set from a scale of 0 to 100, yielding 190 similarity ratings for each participant. (Some pairs contained two numbers from a single group, while others contained one number from each group of ten. While the similarity ratings for pairs of numbers within a single group were technically more useful for our assessment, we also included pairs of numbers between groups so that we could compare the clusters emerging from the set of all 20 numbers with the clusters from each group of ten numbers.) Subjects then were asked a series of questions regarding features of numbers, such as “Which numbers are even?” or “Which numbers are multiples of 3?”, to assess whether they understood various numerical characteristics. Ultimately, however, only the similarity ratings are used in our analysis. A total of 84 participants’ data were deemed eligible for analysis.

We cleaned the data and converted it into a series of similarity matrices. These were generated from the average results of all 84 participants as well as each individual participant’s results. In addition, there were matrices for all 20 numbers and pairs of matrices, one for each group of 10 numbers. Using these matrices, we ran the MDS algorithm to generate 2D Cartesian Coordinate representations of the relationships between the numbers. (Generally in this spatial representation, the closer numbers are to each other, the more related they seem to be according to the participant’s judgment; the inverse is also true.) From the plots of all 84 participants, we found certain plots that seemed to highlight significant characteristics and clustering between numbers, such as a clear divide between even and odd numbers or even a close cluster of prime numbers. Thus, we decided to use the INDCLUS model to find overlapping clusters of the numbers.

Although we tried to use INDCLUS to run analyses based on these results, the documentation regarding the algorithm was sparse. The best code we could obtain for the algorithm was limited to an antiquated Fortran document, which we did not have the ability to operate. Therefore, we resorted to a similar algorithm called ADCLUS, or Additive Clustering, to analyze our results. ADCLUS is a slight simplification of INDCLUS, as it takes in only one similarity matrix and therefore does not account for all differences between individuals as INDCLUS does. In other words, it is a form of INDCLUS in which there is only one subject (n = 1). To get around that issue, we ran a function named ADCLUS2 for the matrices of certain participants whose MDS plots seemed to have significant clusters (Used from Michael Lee’s website: http://faculty.sites.uci.edu/mdlee/similarity-data/). We did this to determine whether these clusters from the MDS plots were consistent with the clusters generated from the ADCLUS algorithm.

The ADCLUS2 function is a modification of ADCLUS that uses additive clustering combined with stochastic hill-climb to get a better result than ADCLUS. It takes in an NxN symmetric matrix of pairwise similarities with a number of clusters, which specifies the number of clusters to use in the clustering process. After optimizing the clusters with respect to variance, ADCLUS2 eventually returns the best cluster membership matrix and its associated weights for each cluster. The weights represent how significant each cluster is compared to the others. For example, if the cluster with the highest weights happens to contain even numbers, then we can speculate that the participant emphasized even/oddness more than other numerical features when judging similarities between numbers.

Results

1. Average Results of 84 Participants

We first ran MDS and ADCLUS2 on the average results of all 84 participants. We tested different numbers of clusters, and it seemed like 4 clusters gave the optimal clustering of numbers. From the results, it seems like both algorithms recognize that most of the clusters are based off of magnitude. There are clusters of “small” numbers (below 10), “medium” numbers (between 10 and 20), and “large” numbers (over 20), as well as a cluster of even numbers.

MDS Analysis of Average Results


ADCLUS2 Analysis of Average Results
4 clusters:
● 14, 16, 8, 18, 12, 28, 6, 4 (even)
● 29, 20, 21, 25, 28, 27 (>=20)
● 14, 16, 9, 11, 20, 18, 21, 12, 17, 13, 10 (>=10 <=20)
● 5, 9, 8, 7, 6, 4, 10 (<10)

2. Case Study Results

These are the MDS and ADCLUS2 results for participants who seemed to have significant clusterings of numbers based on their MDS plots.

(A) Participant 5

MDS Analysis of 20 Numbers

*The bolded weight and list of numbers belong to the cluster with the highest weight, meaning it seemed to be the feature that the participant emphasized the most.

ADCLUS2 Analysis of 20 Numbers
Weights = 24.2764 56.5536 35.9281 23.7502 ● 5, 8, 20, 7, 25, 6
● 14, 16, 8, 20, 18, 12, 28, 6, 4, 10 (even)
● 14, 21, 7, 12
● 5, 9, 11, 12, 13, 10

MDS Analysis of Two Groups of 10 Numbers


ADCLUS2 Analysis of Group 1
Weights = 61.8217 64.4453 66.8217 20.5881
● 5, 11
● 14, 16, 8, 20, 18 (even)
● 5, 20 (multiples of 5)
● 9, 8, 20, 18, 21

ADCLUS2 Analysis of Group 2
Weights = 41.6386 29.1386 24.4404 65.5946
● 12, 13
● 7, 25, 28, 27
● 7, 12, 6
● 12, 28, 6, 4, 10 (even)

Here, we see that the algorithms seem detect significant clusters for even numbers, for both the set of 20 numbers and the two groups of 10 numbers.

(B) Participant 32

MDS Analysis of 20 Numbers


ADCLUS2 Analysis of 20 Numbers
Weights = 29.8831 41.3112 32.0941 40.8001
● 9, 21, 7, 25, 6, 13, 27
● 14, 16, 8, 20, 18, 12, 28, 6, 4, 10 (even)
● 5, 9, 8, 20, 25, 10
● 16, 8, 25

*11 and 17 are not in any clusters. As could be seen from the diagram, those numbers are distant from the rest.

MDS Analysis of Two Groups of 10 Numbers
ADCLUS2 Analysis of Group 1
Weights = 36.4607 37.2306 47.5376 51.3812
● 9, 29, 21
● 9, 8, 20, 21
● 5, 9, 18
● 14, 16, 18, 20, 18 (even)

ADCLUS2 Analysis of Group 2
Weights = 29.4314 34.0314 46.2419 31.0314
● 7, 25, 27, 10
● 7, 17, 27
● 12, 28, 6, 4, 10 (even)
● 25, 6, 13, 27

(C) Participant 51

MDS Analysis of 20 Numbers


ADCLUS2 Analysis of 20 Numbers
Weights = 26.0371 26.2686 17.2405 38.7405
● 14, 16, 9, 8, 20, 18, 21, 25, 12, 28, 6, 4, 27, 10
● 16, 8, 20, 18, 12, 28, 6, 4, 10
● 5, 14, 0, 18, 21, 7, 25, 12, 6, 27
● 5, 11, 7, 13 (prime)

MDS Analysis of Two Groups of 10 Numbers


ADCLUS2 Analysis of Group 1
Weights = 12.9116 23.6286 31.9450 26.7850
● 9, 8, 11, 18
● 16, 9, 8, 20, 18
● 14, 16, 8, 20, 18, 21
● 14, 9, 18, 21

ADCLUS2 Analysis of Group 2
Weights = 26.5000 74.9545 53.9545 26.5545
● 25, 12, 28, 6, 4, 27, 10
● 7, 13 (prime)
● 17, 13
● 12, 28, 6, 4, 10

Here, we see that overall for all 20 numbers, this participant’s data yielded a more significant cluster of prime numbers. When the numbers are broken apart into two groups, we see that in the first group, 5 and 11 don’t seem to form a cluster, but 7 and 13 from the second group form a heavily weighted cluster.

(D) Participant 80

MDS Analysis of 20 Numbers

*We used seven clusters for the analysis of this participant’s data, because this participant’s results had more features observed in the MDS plot.

ADCLUS2 Analysis of 20 Numbers
Weights = 29.6526 48.1872 19.4657 71.0419 58.3805 35.8621 59.4179
● 16, 8, 20, 12, 4 (multiples of 4)
● 14, 16, 8, 20, 18, 12, 28, 6, 4, 10 (even numbers)
● 14, 16, 18, 21, 7, 28, 10
● 5, 20, 25, 10 (multiples of 5)
● 9, 18, 27, 21, 12, 6 (multiples of 3)
● 21, 7, 25, 28
● 14, 7, 17 (numbers related to 7)

MDS Analysis of Two Groups of 10 Numbers


ADCLUS2 Analysis of Group 1
Weights = 82.0266 64.8810 23.4562 65.5412
● 5, 20 (multiple of 5)
● 14, 16, 8, 20, 18 (even, multiples of 4)
● 14, 11, 18, 21
● 9, 18, 21

ADCLUS2 Analysis of Group 2
Weights = 47.1127 75.5277 37.2448 38.5719
● 25, 28, 10
● 7, 28 (multiple of 7)
● 12, 28, 6, 4, 10 (even, multiples of 4)
● 12, 28, 6, 27

Here, we see that while the analysis of all 20 numbers yields many significant clusters, the analyses of the two groups of 10 numbers do not seem to yield as many clusters. Group 1 and the set of 20 numbers both contain a significant cluster of multiples of 5, but Group 2 does not. Also, both Group 1 and Group 2 contain clusters of even numbers, but Group 1’s most signficant cluster contains multiples of 5, while Group 2’s contains multiples of 7.

Discussion and Conclusion

From our results, we can generally conclude that there was some consistency between the MDS and ADCLUS analyses of similariy data. Clusters that were apparent visually in the MDS plots were generally detected by the ADCLUS algorithm and given significant weights. Clusters that were apparent when analyzing all 20 numbers together were generally also detected when analyzing the two groups of 10 numbers. However, there were exceptions, such as the results of the data from Participant #80, in which clusters that were significant when aggregating the 20 numbers were lost when splitting the numbers apart into two groups. Nevertheless, the analyses from the 20 numbers were just frames of reference. The main point of the study was to examine whether there was consistency between the analyses of the two groups of numbers, based on the data of our control group. (Without any educational or cognitive interventions, there should be no difference in how participants judge the similarities between numbers of each group, so the clusters of the two groups should arguably be similar.) Regarding that aspect, the results were fairly positive; generally, the clusters significant in Group 1 also showed up in Group 2, although there were still exceptions.

These issues of discrepancies between clustering analyses of different groups may have been alleviated if there were more numbers being compared in the study, but that would lead to an exorbitant number of comparisons between numbers that participants would have to make (with only 20 numbers, there were already 190 comparisons to be made). Nevertheless, the relatively small quantity of numbers we included in our study was a limitation to our analysis.

However, we do believe the overall results are positive enough for us to use both MDS and ADCLUS to assist our analyses of future studies. We plan to design studies in which a group of participants rates similarities between a group of 10 numbers, undergoes a certain intervention pertaining to mathematics (such as a lesson, a problem-solving task, or an activity), and rates a different set of 10 numbers (with roughly the same numerical features as those of the first group). Our analyses from the data of these future studies hopefully shed light on how certain interventions affect how one perceives relationships between numbers.