K12 students often rely on testing examples to explore and determine the truth of mathematical conjectures. However, little is known about how K12 students choose examples and what elements are important when considering example choice. In other domains, experts give explicit consideration to the typicality of examples – how representative a given item is of a general class. In a pilot study, we interviewed 20 middle school students who classified examples as typical or unusual and justified their classification. We then gave middle school students and mathematicians a survey where they rated the typicality of mathematical objects in two contexts – an everyday context (commonness in everyday life) and a mathematical context (how likely conjectures that hold for the object are to hold for other objects). Mathematicians had distinct notions of everyday and mathematical typicality – they recognized that the objects often seen in everyday life can have mathematical properties that can limit inductive generalization. Middle school students largely did not differentiate between everyday and mathematical typicality – they did not view special mathematical properties as limiting generalization, and rated items similarly regardless of context. These results suggest directions for learning mathematical argumentation and represent an important step towards understanding the nature of typicality in math.
Educated adults are more likely to judge that an equilateral triangle is a triangle than that a scalene triangle is. Such adults also report that 400 is a “better” even number than is 798 (
The general explanation for why equilateral triangles are more often judged to be triangles than scalenes, and why 400 is said to be a better even number than 798, is that the former member of each pair is understood to be more
One of the most interesting features of mathematical typicality is not that people judge some numbers or shapes to be better instances of their classes than others, but how they do so. Mathematically, an equilateral triangle is atypical. Equilateral triangles have many distinctive properties, and facts about equilateral triangles may not generalize. For example, AngleSideSide congruence holds for equilaterals, but not for all triangles. Similarly, given our base 10 system of counting, 400 is a relatively special quantity. Notably, many calculations are much easier with 400 than with 497 (e.g., squaring). However, equilateral triangles and numbers that are multiples of 100 are likely more familiar and frequently encountered in daily life (
We examine middle school students' and mathematicians' judgments of typicality in two contexts  how often numbers and shapes are seen in everyday life, and how generalizable numbers and shapes are in terms of their mathematical properties. These judgments have important implications for teaching K12 students about the strategic use of examples in mathematical arguments, which aligns with current standards in mathematics education.
We begin our review in the area in mathematics education that we have been exploring issues of typicality – mathematical justification, generalization, and proof. We then discuss typicality relations in general, before moving to discuss their significance in mathematics.
Justification and proof are important activities in mathematics education (
Research suggests K12 students often struggle to learn how to construct viable and convincing mathematical arguments and provide valid generalizations (
Deductive reasoning is, however, fundamental to human reasoning generally, and mathematical reasoning specifically (
Examples are culturallymediating tools that connect learners and mathematical ideas – they provide a specific and familiar context for exploration and understanding of nuances and constraints (
Both K12 students (
The role examples play in the justification activities of K12 students is quite different. K12 students can be overly reliant on examples and mistakenly use examples alone as a form of proof (
Research in other domains suggests that experts have a set of heuristics for judging which relations are important in their domain of expertise (
While deductive inference appears to be a lateemerging skill, requiring explicit practice and instruction, inductive inference seems to be a cognitive primitive (
By 2 years of age, children understand that objects can belong to the same category, even if they are perceptually dissimilar, and use this to reason successfully about properties of those objects (
Although typicality relations are clearly important in domains like biology, little is known about what features are relevant within mathematics and whether mathematicians and K12 students demonstrate different knowledge about mathematical features and relationships. For a mathematician, zero may be an extremely atypical (and important) number in the mathematical context of testing examples. However, people also have ideas about typicality relations in everyday life. Zero may be a fairly typical number as it is commonly encountered in stores (e.g., $0 down payment), on devices (0% downloaded), etc.
The features of objects that matter for judgments of typicality in mathematical contexts are an open question, but guesses are available. For geometric shapes in particular, there is a clear hierarchy of important mathematical properties that may matter when testing conjectures, such as having congruent sides and angles, right angles, etc. The mathematical properties of numbers that may matter for typicality are less clear and may be dependent on characteristics of particular conjectures – although numbers like zero and one clearly have special properties. We next discuss how mathematicians consider examples when exploring conjectures.
Examining how mathematicians think about the examples they test when exploring conjectures offers insight into important elements of strategic thinking about examples. In an online survey distributed to 220 mathematicians at 27 universities around the country (
Mathematicians would often cite that they used simple examples (e.g., “Easy ones! Start with toy cases and slowly build up the complexity.”) and complex examples (e.g., “I try to find examples that include all of the (foreseen) barriers to a proof. The "hardest" examples in the sense of what I'm trying to prove.”), as well as examples that made computation easy. Mathematicians also reported choosing counterexamples, examples with specific properties (e.g., “For numberbased conjectures, I choose 0, numbers close to 0  both positive and negative  very large and very small numbers.”), and examples that represent a boundary case (e.g., “I will next try to test some strange or pathological examples, to really push the boundaries of what might be possible in this situation.”). Some mathematicians also reported using generic examples (e.g., “Try the most general example which is still practical to test”), examples that are familiar, unusual examples (e.g., “Next, I try something slightly more obscure.”) and common examples (e.g., “Ones that are not special, ones that I judge to be typical.”), as well as random, exhaustive, and dissimilar examples.
It also is important to consider mathematicians’ use of examples when exploring specific conjectures, particularly conjectures that would not be trivial to them with their expertise. In a followup study (
Middle school students may have less welldeveloped notions about choosing examples. In a recent study, middle school students and STEM doctoral students were engaged in cardsorting tasks where they constructed sets of numbers, triangles, and parallelograms (
Although this study provided useful ideas about properties of mathematical objects that are salient to different groups, we wanted to more specifically explore ideas about
For our pilot interview study, our research question was: What reasons do middle school students give for classifying numbers and shapes as typical or unusual?
Oneonone interviews with 20 middle school students (11 female, 9 male; 8 sixthgrade, 7 seventhgrade, and 5 eighth grade or higher math courses) from a large suburban school district in a Midwestern state were conducted. Parental consent forms were distributed to students in 6 middle schools and students who returned consent forms were selected on a firstcomefirstserve basis. The district used a reformoriented curriculum,
Specific Conjectures 

Generic Conjectures 
We focus on a series of questions the interviewer asked each participant
The examples that middle school students generated for specific and generic conjectures were coded based on (1) whether they deemed the example to be “typical” or “unusual” and (2) the reasons the middle school students gave for
Results showed that for the two specific conjectures related to numbers (Eric and Amy in
For the two specific conjectures related to geometric objects (Lewis and Bob), the most common reasons given for shapes being unusual were: a shape having unequal sides (33% of instances), shapes that were elongated or in nonstandard orientations (14%), shapes that were not commonly encountered in life (9%), and shapes with right angles (9%). The most common reasons given for shapes being typical were: a shape being commonly encountered in everyday life (22% of instances), a shape having equal sides (14%), a shape being labelled as a square or equilateral triangle (12%), and a shape being labelled as an acute triangle (8%). One student discussed the triangles he generated, saying “I guess I think of the typical ones are all equal. And I guess I just think of triangles that are typical that are all in just I don't know…. equilateral I guess. And then these that are all different lengths, I think they're just different. I don't think of them as typical.” Another student said a parallelogram was unusual because “Probably because it's so like thin and long. Cause when I think of a parallelogram I think of like a house or a door.”
Being encountered in everyday life was often mentioned as a reason for being typical, so the patterns for these justifications were examined more closely. Most often (37% of typical cases coded as “everyday life”), the reason would describe how frequently the object was encountered or used. Other common reasons within this category related to how the object was seen when working with money, packaging, school books or tasks, and when examining architecture or other real world structures/objects. Thus both the idea of frequent real world encounters as well as specific, salient real world connections seemed important for typicality.
The results for general conjectures were similar. The most common reasons for a number being typical were being even (45% of instances), often encountered in everyday life (44% of instances), BaseTen properties (22%), divisibility properties other than 10s/2s (20%), or being small (11%) or an “origin” number (like 1; 11% of cases). The most common reasons for a number being atypical were not being encountered in everyday life (47% of instances), being odd (35%), being prime (22%), being large (16%), and being difficult computationally (16%). The most common reasons given for a shape being typical were being encountered in everyday life (62% of instances), being an equilateral triangle or square (21%), having equal sides (18%), or being symmetrical or not elongated (15%). The most common reasons for a shape being atypical were not being encountered in everyday life (32% of instances), having unequal sides (32%), and being elongated or in nonstandard orientation (12%).
This study suggested that middle school students were able to decide whether the examples they tested were “typical” or “unusual,” and give reasons for this classification based on mathematical and nonmathematical properties. There were characteristics of both number and shape that were consistently cited as either relating to typicality or atypicality, with frequency in everyday life being a major consideration for both. The ways in which these typicality relations interact with middle school students’ ideas about mathematical generalization is less clear, as are the ways in which their judgments relate to the judgments of mathematicians, who are more experienced with mathematical justification. We explore these questions next.
We frame our next study by discussing two aims that came out of prior research. Taken together, the card sort study (
Our first aim was to explore which particular items mathematicians and middle school students consider typical or atypical,
Relating to our second aim, recall that in other domains, like biology, an awareness of typicality relations means understanding deep features, those that will impact your ability to generalize from examples, while disregarding surface features that do not impact important domain relations. In our prior studies, middle school students and mathematicians brought up a variety of mathematical properties and other characteristics that contributed to an object being deemed “typical” or “unusual” (e.g., being a multiple of 10, a small number, or an equilateral triangle). Our second aim was to compile a list of
Domain  Property  Justification for Inclusion 

Number  
Prime  Salient to middle school students in pilot interview study 

Perfect Square  Salient to STEM doctoral students in card sort study  
The number 0 or 1 (identity element)  Salient to middle school students in pilot interview study 

Multiple of 5/10 
Salient to middle school students in pilot interview study  
Small/Large/ 
Salient to middle school students in pilot interview study, card sort study 

Odd Parity  Salient to middle school students in pilot interview study 

Parallelogram  
Square, rectangle, or rhombus  Salient to middle school students in pilot interview study 

Size (small, large)  Salient to middle school students and STEM doctoral students in card sort study  
Orientation (nonstandard/leftleaning)  Salient to middle school students and STEM doctoral students in card sort study  
Skinny/elongated  Salient to middle school students in pilot interview study  
Triangle  
Isosceles, equilateral, right  Salient to middle school students in pilot interview study 

Scalene, Obtuse  Salient to middle school students in pilot interview study  
Size (small, large)  Salient to middle school students and STEM doctoral students in card sort study  
Skinny/elongated  Salient to middle school students in pilot interview study  
Orientation (nonstandard)  Salient to middle school students and STEM doctoral students in card sort study 
^{a}Items with this property were given to mathematicians only.
We investigate the following research questions (each related to one of our aims):
R1.
R2.
A total of 475 middle school students (46% female) took the survey with pencil and paper. These students were from a school in the pilot study. The school was 48% Caucasian, 21% African American, 14% Asian, 11% Hispanic, and 1% Native American, with 37% free/reduced lunch and 10% English Language Learners. Participating students were distributed across grades 6 (144 students), 7 (160 students), and 8 (163 students).^{i}
A total of 430 mathematicians began the webbased version of the survey; however, 104 did not supply any typicality ratings, for a final sample size of
Of the 326 mathematicians, over half (185) were working on a Ph.D. in Mathematics or Applied Mathematics, while most of the others (106) already held a Ph.D. in Mathematics or Applied Mathematics. Twelve participants were working on a Master’s degree in Mathematics or Applied Mathematics, while one was working on a Bachelor’s in this field. The remainder (
Each survey form contained questions from two of four different domains: numbers, parallelograms^{ii}, triangles, and birds (birds are omitted from this analysis). For each domain, middle school students were presented with mathematical objects or items in that domain (e.g., a small equilateral triangle or the number “6”) and asked to rate each item’s typicality on a 1 (
Example of items from survey instrument.
Mathematical objects to be placed on the survey were selected by the researchers to either cover the space of possible mathematical properties in the domain (e.g., the parallelogram in
The number items were divided into three sets; there were two sets of parallelograms and two sets of triangles. There was also an additional set where middle school students were asked to judge the similarity or dissimilarity of two items, not considered here. Each student received two of these sets on their survey, and for each set they were asked to give ratings of the items in the set in both the mathematical and everyday contexts. Whether they rated items in a mathematical context or everyday context first was randomized. The order of the items (the 9 triangles, 9 numbers, or 10 parallelograms) within each context was randomized, as was the order of the 2 sets each student received.
On Version 1 of the mathematician survey (given to 186 of the 326 participants), mathematicians would rate 7 number items in a mathematical context, and 7 number items in an everyday context. Then they would either rate 5 triangles or 5 parallelograms in a mathematical and everyday context. The instructions, prompts, and rating scales were all identical the middle school student survey to ensure strict comparability (
The items each mathematician received were randomly selected by the Qualtrics survey environment from a set of items that was twice as large (sets of 14 numbers, 10 triangles or parallelograms). A new version of the survey (Version 2) was introduced approximately halfway through the data collection period and was taken by 140 of the 326 participants. In this version the number and shape items were different, and the participants would rate a total of 8 number items in each context instead of 5. When collapsing both versions, the mathematician survey used the same number and shape items as the middle school student survey, except that one parallelogram and one triangle were left off of the mathematician survey. However, two additional number items (163452 and 3432984) were added to the mathematician survey to see if mathematicians rated the typicality of very large numbers differently than smaller numbers.
Middle school students completed the paperbased survey during their regular math class. They would first respond to a prompt about how typical it was to eat oatmeal for breakfast to familiarize themselves with the Likert rating scale and the idea of typicality. They were instructed to answer every question on the survey and were told that there were no “right” or “wrong” answers. Demographic data relating to gender, grade level, and teacher was requested.
Mathematicians were recruited by sending emails to mathematics departments at 39 universities in the United States. The email was sent to department personnel with the request that it be forwarded to mathematics faculty and doctoral students. The email invited the mathematicians to participate in an online survey in Qualtrics – there was no compensation offered and participation was voluntary. The email and the introduction to the survey stated that the purpose of this research was to better understand how middle school students use evidence in mathematics by examining mathematician data on example usage. Demographic data relating to gender, education, area of specialization within mathematics, language, and country of origin was collected from the mathematicians.
The first part of the first research question asked whether participants distinguished between two contexts: everyday and mathematical typicality. We addressed this question by calculating and visualizing the correlation between everyday context and mathematical context typicality ratings for each group (mathematicians, middle school students) in each domain (numbers, parallelograms, triangles). We computed Pearson correlations by averaging for each group their ratings of each item within a domain, for everyday and mathematical contexts. We then looked at the correlation between those two sets of item averages.
Middle school students’ everyday and mathematical typicality ratings were strongly correlated for the triangle domain,
We explored these trends further by plotting the average mathematical typicality rating for each item versus the average everyday typicality rating for each item for both middle school students and mathematicians. If mathematical typicality and everyday typicality are not distinct, we would expect all items to fall on the line
Middle school students’ and mathematicians’ average ratings of numbers (top), triangles (middle), and parallelograms (bottom), in a mathematical (yaxis) and everyday (xaxis) context, normalized.
Of interest in the number domain are a group of apparent outliers where everyday and mathematical typicality do not correspond. Mathematicians rated the additive and multiplicative identity elements (0 and 1) as less mathematically typical than other numbers, despite their high level of commonness in everyday life. For middle school students, there is one extreme outlier in the number domain – 13 – which is rated as less mathematically typical than other numbers of similar everyday typicality (e.g., 14, 11). This deviation likely reflects the status of 13 as a “superstitious” number. However, it is also possible that 13 is a salient example of the mathematical property of being a prime. A second outlier, 10,000, was rated by middle school students as more mathematically typical than other numbers, but less everyday typical. This rating may reflect the special significance of powers of 10 in our base ten counting system. Thus there was evidence that middle school students may distinguish the mathematical and everyday typicality of numbers. However, it is not clear that they are basing their judgments on mathematically significant features of the numbers. We see fewer obvious outliers in the shape domain – middle school students have a very consistent pattern where they rate shapes similarly in everyday and mathematical contexts, and mathematicians have a very consistent pattern where they rate shapes with high everyday typicality as having low mathematical typicality.
The second research question asked what specific properties make objects typical or atypical for mathematicians and for middle school students. The preceding analyses demonstrate that mathematicians and middle school students show very different judgments about mathematical typicality. Moreover, they suggest that middle school students may use everyday typicality as the basis for judging mathematical typicality. To explore this issue further, we used a list of specific properties relevant to typicality, determined from prior research (
Data were analyzed using mixedeffects linear regression models (
Predictor  Middle School Student 
Mathematician 


B  B  
(Intercept)  4.16  0.45  2.41  0.11  
Context Mathematical  (ref.)  (ref.)  
Context Everyday  0.03  0.17  0.16  .8644  0.19  0.07  2.72  .0056 
Magnitude Large  (ref.)  (ref.)  
Magnitude Small  0.76  0.15  5.07  .0001  0.02  0.12  0.18  .8178 
Ends in 5  0.96  0.25  3.81  .0016  0.09  0.20  0.43  .6662 
Power of 10  1.25  0.21  5.91  .0001  0.04  0.17  0.24  .8572 
Odd  0.55  0.16  3.39  .0036  0.03  0.13  0.26  .8640 
Identity  0.27  0.30  0.91  .3792  0.72  0.25  2.82  .0088 
Context Everyday: Magnitude Small  0.59  0.09  6.74  .0001  1.41  0.08  18.00  .0001 
Context Everyday: Odd  0.31  0.08  3.64  .0002  0.43  0.09  4.76  .0001 
Context Everyday: Identity  0.46  0.16  2.95  .0026  1.23  0.17  7.39  .0001 
Magnitude Extra Large  0.20  0.22  0.88  .4340  
Prime  0.19  0.17  1.14  .2588  
Perfect Square  0.09  0.15  0.59  .5176  
Power of 2  0.02  0.20  0.12  .8188  
Context Everyday: Magnitude Extra Large  1.04  0.15  7.11  .0001  
Context Everyday: Prime  0.55  0.12  4.69  .0001  
Context Everyday: Perfect Square  0.42  0.10  4.13  .0002  
Context Everyday: Power of 2  0.64  0.14  4.74  .0001 
Predictor  Middle School Student 
Mathematician 


B  B  
(Intercept)  4.97  0.41  3.90  0.15  
Context Mathematical  (ref.)  (ref.)  
Context Everyday  0.15  0.20  0.75  .4572  1.00  0.17  5.90  .0001 
Skinny  0.74  0.17  4.28  .0024  
Isosceles  0.63  0.17  3.64  .0042  1.21  0.11  11.30  .0001 
Equilateral  0.75  0.33  2.28  .0374  1.91  0.20  9.48  .0001 
Acute Triangle  (ref.)  (ref.)  
Obtuse Triangle  0.05  0.19  0.25  .7942  0.27  0.13  2.02  .0664 
Right Triangle  0.20  0.27  0.75  .4530  0.25  0.17  1.48  .1874 
Standard Orientation  0.59  0.18  3.19  .0096  0.06  0.13  0.44  .6762 
Context Everyday: Equilateral  0.68  0.21  3.19  .0016  2.96  0.27  10.81  .0001 
Context Everyday: Obtuse Triangle  0.16  0.10  1.56  .1242  0.36  0.18  2.00  .0562 
Context Everyday: Right Triangle  0.46  0.16  2.91  .0042  0.90  0.23  3.95  .0001 
Context Everyday: Isosceles  1.75  0.15  11.82  .0001  
Context Everyday: Standard Orientation  0.34  0.18  1.91  .0744 
Predictor  Middle School Student 
Mathematician 


B  B  
(Intercept)  2.87  0.61  4.63  0.11  
Context Mathematical  (ref.)  (ref.)  
Context Everyday  0.39  0.27  1.41  .1650  1.59  0.11  15.10  .0001 
Square  0.88  0.34  2.58  .0150  2.50  0.15  16.57  .0001 
Rectangle  0.63  0.31  1.99  .0548  1.99  0.20  9.87  .0001 
Rhombus  0.57  0.40  1.44  .1538  1.53  0.20  7.81  .0001 
Standard Orientation  0.43  0.24  1.78  .0798  
Size Large  (ref.)  
Size Small  0.19  0.55  0.34  .7104  
Leans Left  0.51  0.34  1.52  .1336  
Context Eday: Standard Orientation  0.49  0.11  4.49  .0001  
Context Eday: Size Smll  1.04  0.25  4.20  .0001  
Context Eday: Lean Left  0.52  0.15  3.48  .0002  
Context Eday: Square  0.55  0.15  3.64  .0002  4.13  0.19  21.52  .0001 
Context Eday: Rectangle  1.31  0.14  9.30  .0001  3.23  0.26  12.42  .0001 
Context Eday: Rhombus  0.55  0.18  3.14  .0014  2.21  0.26  8.60  .0001 
Golden  0.05  0.26  0.19  .9048  
Context Eday: Golden  0.77  0.34  2.28  .0246 
In general, the analyses presented in
Analyses of triangles and parallelograms showed similar patterns. Mathematicians judged Isosceles and Equilateral triangles as less mathematically typical than non Isosceles and nonEquilateral triangles (respectively). However, the effects of these dimensions reversed for everyday typicality (e.g., Isosceles more typical than nonIsosceles). Right triangles were more everyday typical than nonRight triangles for mathematicians. Middle school students judged nonSkinny, Isosceles, and Equilateral triangles to be more typical in both contexts, although equilateral triangles were significantly more typical in an everyday context, compared to a mathematical context. Standard orientation and Right contributed significantly to middle school students’ everyday typicality but not mathematical typicality judgments of triangles. Thus for triangles, mathematicians recognized that special mathematical properties (Isosceles, Equilateral) make triangles less mathematically typical, while middle school students had the opposite trend. In an everyday context, both groups recognized properties (Equilateral, Isosceles, Right) that make triangles more typical in an everyday context.
For parallelograms, mathematicians judged Squares to be less mathematically typical than nonSquares, Rectangles tend to be less mathematically typical than nonRectangles, and Rhombi less mathematically typical than nonRhombi. However, in each case these effects reversed for everyday typicality (e.g., Squares more typical than nonSquares). Parallelograms with near Goldenratio proportions were rated by mathematicians as more everyday typical than other parallelograms, but proportion had no significant effect on mathematical typicality. For middle school students, being Square and being Rectangular significantly increased mathematical typicality. These same features also increased everyday typicality, with squares being significantly more everyday typical than mathematically typical. Several other features, including Size, Orientation, and Skew, affected everyday but not mathematical typicality. For mathematicians, special mathematical properties like Rectangle and Square increased everyday typicality but decreased mathematical typicality; for middle school students, these properties increased both ratings. Middle school students and mathematicians also attended to other more everyday properties (Goldenratio, Orientation) when considering everyday typicality.
Expertise is often characterized by distinctive judgments regarding typicality relations among objects in a domain. In mathematics, it is reasonable to assume that experts see different connections among objects than others. Psychological principles of inductive inference have been explored in other domains, such as living things (
Mathematicians and middle school students both showed robust and consistent judgments about what we termed “everyday” typicality. Some numbers and shapes were consistently cited as more common in everyday life than others (e.g., small numbers and equilateral shapes). Middle school students’ reasoning often cited that these objects are encountered more often in their experience. People may have perceptual (
Both studies showed interesting contrasts between this everyday sense of typicality, and a “mathematical” sense of typicality – i.e., the degree to which one mathematical object was a reliable basis for inferences about other objects. Middle school students showed robust and consistent judgments about mathematical typicality for both number and shape that did not differ from their judgments about everyday typicality. In contrast, mathematicians made sharp distinctions between everyday and mathematical typicality; these judgments were almost perfectly negatively correlated for geometric shapes. The most typical shapes in an everyday context, like squares and right triangles, were the least typical in a mathematical context. For numbers, mathematicians either did not have consistent judgments about the mathematical typicality or did not see a relevant network of typicality relations when considering a generic conjecture in this domain. Whatever this signals about the nature of expertise in the domain of numbers, the contrast with middle school students is important. Mathematical typicality is not the same as everyday typicality for mathematicians, for numbers or shapes.
Our studies varied whether participants were given a specific mathematical conjecture to explore (like in the pilot interview study) or given a generic prompt about conjecturetesting (like in the survey study). Typicality judgments are certainly applied most productively when a specific mathematical task given. For example, when considering conjectures about additive properties, concerns relating to whether a number is positive or negative might be most paramount for generalizability. In contrast, when considering conjectures about multiplicative properties, properties relating to divisibility might matter most. The type of mathematical task may be especially important for numbers (which have many potential properties of importance) compared to shapes (which have a clearer hierarchical organization of relevant properties).
In addition, in the survey study the classes to which generalizations were invited were more specific for shapes than for numbers. Participants were asked whether facts about one number generalized to “most other numbers,” while facts shapes were generalized to "most other triangles/parallelograms” instead of the larger class of all geometric figures. There was also a difference in specificity – shapes were presented without side or angle measurements, whereas the number items were specific – “13” or “102.” Finally, the example numbers were all one special subtype of numbers: whole numbers. If respondents interpreted the task as involving generalization to all types of numbers (rational, real, imaginary) then they may have viewed all whole numbers other than 0 and 1 as roughly equally typical.
A recent publication of the National Council of Teachers of Mathematics (
If middle school students are using their everyday senses of typicality to reason about mathematical relations, a central question these studies confront is whether there is an alternative network of typicality relations in the domain of mathematics that would serve them better. For this reason, we included Research Question 2, where we examined whether there are overarching properties of numbers and shapes considered important for generalizability by mathematicians. For the domain of Euclidean geometric shapes, it appears that such a network of properties may exist – mathematicians recognize a network of special geometric properties that limit the potential of an example shape for inductive generalization. Middle school students may have not yet adopted this lens for thinking about the mathematical properties of their examples, which may limit the quality of the inductive arguments they generate, and the utility of these arguments for allowing them to understand geometric properties and discover more general proofs.
The story for whole numbers is more complex – there does not seem to be a clear network of typicality relations that is recognized, but rather localized, microlevel properties of number (like primeness or parity) that become relevant when considering particular conjectures. Much of the situated work of mathematicians involves determining which properties are important to consider in the context of a conjecture, and then taking those properties into account, when possible, when exploring the conjecture (
In other research, we have questioned middle school students on whether typicality considerations are important to take into account when they are proving conjectures (
With generalization being framed as central to mathematics education, our results suggest that classrooms would benefit from exploration of the properties of the examples K12 students choose, and discussion of how these properties interact with ideas relating to generalization in the context of the particular conjecture athand. Instruction on exploring conjectures could include discussions about the typicality of mathematical objects – i.e., which objects it makes the most sense to test as K12 students first consider the conjecture, as they try to “break” the conjecture, and as they try to find a pattern to prove the conjecture, and why. Understanding that special mathematical properties of an object should be a consideration when determining whether a conjecture is likely to generalize to all other objects could be another instructional goal relating to the nature of inductive evidence in mathematical domains. Such discussions would also include what constitutes strong versus weak evidence in inductive reasoning. It is also important to consider how reasoning inductively can support the development of deductive proofs. Testing examples can be especially useful for understanding the structure of conjectures, or how they work, in order to reveal an underlying mathematical pattern. Finally, everyday aspects of typicality could be discussed – K12 students could critically examine whether surface features like a number’s magnitude or digit patterns or a shape’s skinniness or orientation are important considerations in the context of their current mathematical exploration.
Experts understand objects and ideas in their domain of expertise as a complex interwoven network of relationships based on important and useful properties. This network of relationships allows them to strategically confront and solve novel tasks through a consideration of relevant domain principles. Typicality is one such principle that is relevant to practices of inductive inference, but this principle has mainly been identified as being useful in fields other than mathematics. Here we provide evidence that mathematicians have a distinct sense of mathematical typicality that can guide their activities related to justification and generalization.
Recent reform movements in mathematics education have pushed for mathematical argumentation and generalization in elementary and middle school. A long tradition of research in math education has demonstrated that K12 students do use empirical strategies of inductive inference to make sense of content they learn during mathematics instruction (
*These items were on the mathematician survey but not the middle school student survey.
**These items were on the middle school student survey but not the mathematician survey.
The research was supported in part by the NSF, Award DRL0814710. Any opinions, findings, and conclusions or recommendations are those of the authors and do not necessarily reflect the views of the NSF. It was also supported through funding from the U.S. Department of Education Institute of Education Sciences (Award No. R305B100007) Postdoctoral Training Program in Mathematical Thinking, Learning, and Instruction.
The authors wish to thank Eric Knuth, Amy Ellis, Elise Lockwood, Fatih Dogan, and Andrew Young, for their contributions to this work.
Portions of this data were previously presented at the 2012 the Psychology of Mathematics Education – North America (PMENA) Conference, the 2013 Annual Meeting of the American Educational Research Association, and the 2011 Annual Conference of the Cognitive Science Society.
A parallelogram was defined for the middle school students on the survey: “A parallelogram is a 4sided figure with opposite sides that are parallel.”
The authors have declared that no competing interests exist.