Design of Search Interfaces for MathematiciansAndrea Kohlhase
Jacobs University Bremen and FIZ Karlsruhe

Abstract. The availability of mathematical knowledge on the Web changes mathematical practices. In a study of professional mathematicians’
search behavior we identified ten behavioral patterns particular to mathematicians. In this paper, we present a set of design goals for mathematical search interfaces implied by these patterns and insights derived from
them.

1

Introduction

The availability of mathematical knowledge on the Web is constantly increasing. Research especially in the Mathematical Knowledge (MKM) and Digital
Mathematical Libraries (DML) communities have looked into the representational and organizational aspects of this. In the Mathematical User Interfaces(MathUI) workshop we are especially interested in the aspects of use and handling of this fast growing mathematical web knowledge. In this paper we areespecially interested in improving search for math. For that, we might look at
general resources like [10] or [2]. But even a complete redesign of e.g. the reviewing service Zentralblatt Mathematik (zbMath.org) according to the newest bestpractices didn’t result in a jump in usage levels. We suspected that the issues
lie deeper as mathematicians have specific, idiosyncratic needs and thus requiredistinct design goals.
To assess the usability of the European Digital Math Library in [4] Kitchen
conducted interviews with (professional) mathematicians — mathematiciansthat not only have degree(s) in mathematics, but also are actively pursuing
mathematical research. It provided the insight that mathematicians didn’t know
how to use the offerings of mathematical search interfaces. The reason can either
reside in the content, i.e., not offering resources interesting to mathematicians,
or in the form, i.e., offering valuable resources in not suitable ways. In [16]Zhao analyzes user-centric and math-aware requirements for math search. He
lists “general keyword search, browsing math-specific resources and personal contact ”
as main math search activities. This doesn’t yet answer the general question,
whether there is a need for a specific math-oriented design for search interfaces.
To find out what distinguishes mathematicians from non-mathematicians,
we interviewed professional mathematicians and other people interested in mathto find idiosyncratic, but typical behavioral approaches, expectations or understandings with respect to mathematical search interfaces (reported in [5]). We
elicited 10 patterns that give us insights about the special user group of professional mathematicians.

In this paper, we want to turn to the design aspect of these 10 patterns. We
will argue for some principles that don’t seem to have been taken into account incurrent designs of math software systems. Note that you should not expect a setof handy statements of the kind “Do this, but not that” as the underlying powerof design consists in the fair balance of concurrent and partially contradictive requirements. The formulation of math search specific design goals is consequently
done rather cautiously.
In Human-Computer Interaction (HCI) it is common practice to establish aset of “best practices” for the design of user interfaces, which are directed towards
a general user profile as a human being. Therefore, they hold for professional
mathematicians as well. Note that we cannot provide best practices for designers
of math UIs, as we haven’t conducted any usability study with a system that
adopted our principles yet. The presented design goals are supposed to serve asa basis for discussions about future math user interface design.

2

Ten Patterns of Mathematical Approach to mSIs

In [5] we used the set of exemplary math search interfaces (“mSI”) and established the following 10 patterns of mathematicians when approaching mathsearch interfaces:
Pattern 1 (“Familiarity”):“Mathematicians do not assess mSIs based on familiarity.”
Pattern 2 (“Community”):“Mathematicians trust human and community resources.”
Pattern 3 (“Finding”):“Finding is the primary mathematical search task.”
Pattern 4 (“Social Interaction as Tool”):“Mathematicians appreciate social interaction as a mathematical tool. In particular, it is a mathematical practice to collaborate and exchange feedback.”
Pattern 5 (“Medium”):“Mathematicians aim at adopting a search tool as a medium.”
Pattern 6 (“Function”):“Mathematicians appreciate function over form.”
Pattern 7 (“Outcome”):“Mathematicians care more for the outcome than the input.”
Pattern 8 (“Empowerment”):“Mathematicians want to be empowered in the search process.”
Pattern 9 (“Transparency”):“Mathematicians base their information search process on transparency of thesearch result.”
Pattern 10 (“Expectations”):“Mathematicians expect to find meaningful information in the search result.”2

3

Design Implications for mSIs

The ISO standard for Ergonomics of Human System Interaction ISO 9241-11(1998) sets usability to be the “extent to which a product can be used by specified users to achieve specified goals with effectiveness, efficiency and satisfaction in a
specified context of use ”. The elicited patterns didn’t even mention the keywords“effectiveness”, “efficiency” or “satisfaction” once. The underlying reason is not
the author-specific phrasing of the patterns in [5] but rather the disregard themathematicians showed for these concerns in the common form.
This might be just as well as usability has lost its status of being the criterion for a well-designed product in modern HCI. There, the new criterion isuser experience (UX), which comprises according to ISO 9241-210 (2009) “aperson’s perceptions and responses that result from the use or anticipated use of a product, system or service ”. This replacement seems much more fitting, if we recall

that the math-specific properties of the above mSIs were cooked down in [5] toevaluating the “preconditions of the search”, the “precision of the input” and the“relevance of results driven by either user or data”. The expectations concerning
the input and the anticipated meaningfulness of the output clearly indicate anholistic approach by mathematicians.
According to [1] user experience consists of four components: value, desirability, adoptability, and usability. Therefore, we will analyze the patterns in thefollowing with respect to these four components.
3.1

Value

The value aspect of UX concerns the question of usefulness.
Finding As pointed out in [5] the term “search” comprises numerous different
search processes. In particular, it entails for example:– Finding = already knowing what one is looking for– Browsing = getting an overview over a topic or an idea of a concept– Surfing = surrendering to the links, drifting from one to another– Solving/Information Gathering = creating a search plan, i.e., specifying
a sequence of actions that achieves the solution of a problem– Asking = posing a question to find an answer
In different phases (like selection, formulation and collection) a user seeking information translates a search intension into a query or series of queries optimizing
for the relevance of the final collection of search results (see [8]).
The value aimed for by mathematicians according to Pattern 3 (“Finding”)
consists in the satisfaction of the search intension. It can only be redeemed, if
the entire search environment is streamlined towards finding.
The presumptions for such a streamlining Shneiderman et al. summarizedin [14] under the formulation phase with the following distinct aspects:a.) Sources: How large is the underlying database?
b.) Fields: How structured is the underlying database?
c.) What to search for: How expressive is the input language?
3d.) Variants: How big is the error tolerance?
Additionally, in the collection phase they point out the following aspects:e.) Entropy: How much information is conveyed via the presentation of thesearch results?
f.) Correctness: How good is the precision and recall according to the searchintension?
g.) Refinement: How easy is a refinement of the search results (towards thesearch intension)?
To optimize the finding process, thus, several handles can be used:
Goal 1 (“Optimization of Resources”):“Optimization of sources, fields, query language, presentation of variants, low entropy,
and refinement features — with respect to the search intension ”

Expectations The Pattern 10 (“Expectations”) from [5] stresses the importanceof the collection phase. If the precision of the information retrieval is not high,then mathematicians need tools to turn this data into meaningful information,e.g., for aligning their subsequent search plans.
Goal 2 (“Structure”):“Maximization of structure in the search results ”

Note that mathematicians do value function over form (Pattern 6 (“Function”)),
so a complex presentation (maybe causing cognitive overload) is allowed, aslong as it provides value. In [9] Libbrecht posed an interesting question: Is
there something like a “too high precision” of (mathematical) search queries
with the effect that search results become too small? Considering Pattern 10(“Expectations”) we can assume that mathematicians didn’t plan for low recallwith their search query. Goal 2 (“Structure”) therefore should include a query
extension or finetuning mechanism like the one presented in [9]. There, a bad
result gives, e.g., hints about query variants.
Function Obviously, Pattern 6 (“Function”) strongly supports the need of mathematicians for value, as they evaluate form less than function. Even though thedifference in layout in the set of mSIs is perceived by most (notes taken in interviews in [5] confirmed that), it hasn’t found its way to the assessment schemata
of mSIs, therefore can be deemed irrelevant for mathematicians.
Empowerment But the question arises whether mathematicians appreciate functionalities independent of their search intension? In particular, is there something
like too much functionality for mathematicians, where too much value turns intoless value? Pattern 8 (“Empowerment”) suggests that this is not the case, as theywant to have all the world at their fingertips and decide autonomously what to
do with it in their search strategy. The side condition “serving their search strategy” should be considered nevertheless.
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Goal 3 (“Flexible Search Strategies”):“Addition of as many functionalities to support flexible search strategies as possible ”

Transparency The value of a mSI for mathematicians also consists in a correct
mental model of its inner workings. Mathematicians do not appreciate blackboxes, they want to evaluate themselves how to make use of a service. For that
(according to Pattern 9 (“Transparency”)) they need to be able to assess its
value for them, e.g., when working out a search plan to fulfill a search intension.
Therefore it might be helpful if math designers adopted
Goal 4 (“Semantic Transparency”):“Semantic transparency, i.e., if the user interface allows access to the relevant information and its underlying semantic relations [6] ”

In combination with Pattern 10 (“Expectations”) we might even venture to saythat only a not understood, that is, unpredictable search result is a bad one. Sucha search result breaks the original search plan as it does not allow a refinementof the search process.
Social Interaction as a Tool Mathematical literacy is an ability that computers
as of now do not possess, but mathematicians do. The more they have dealt
with mathematics the stronger their literacy becomes (see [7]). Not only reflection, but also communication within the center of the mathematical Communityof Practice (CoP) gets so much more efficient with this growing literacy. Thismight be the underlying reason why we found in [5] that mathematicians could
approach social interaction within their CoP as if it were a mere search tool
for mathematics. The value of social interaction is not about establishing social
bindings or networking, it is about efficiency. Therefore, we suggest:
Goal 5 (“No Social Media”):“No inclusion of social media in an mSI ”

Note that we refer to social media with a focus on establishing social networks.
A math search interface like mathoverflow (mathoverflow.net) shows how aservice can successfully introduce social interaction in accordance with Pattern(“Social Interaction as a Tool”).
Community mathoverflow draws on Pattern 2 (“Community”) as well.
Goal 6 (“Community Idiosyncracies”):“Integration of community aspects ”

Recall, for instance, the following mathematical community aspects: polite anddistinguished feedback, anecdotes, regular e.g. small community meetings in
Oberwolfach, etc.
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3.2

Desirability

To understand the desirability of an interactive product we ask whether it is fun
and engaging. But what exactly does this mean for mathematicians? The disgust
for playfulness shown in the study in [5] by them indicates that it is not about theengagement factor in the normal sense. Entertainment and gaming do not comeinto the fun experience of mathematicians when searching for math. But maybe
the beauty of Math does? In [15] there is even some scientific evidence thatmathematicians desire elegance. What concretely does for example the following
statement mean for design? “Mathematics, rightly viewed, possesses not only truth,
but supreme beauty a beauty cold and austere, [...], sublimely pure, and capable ofa stern perfection such as only the greatest art can show.” [13]. We can derive aninterpretation of “desirability” for mathematicians:

Goal 7 (“Beauty”):“Utmost reduction of complexity — without losing functionality ”.
One best practice of a well-known set of usability guidelines [12] points in thesame direction. It states that “any extra unit of information in a dialogue competes
with the relevant units of information ” and therefore a minimalist design is tobe preferred (see also [11]). For a search interface Google’s layout is not only
a prototypical example, it has even reached an idol status, the Google search
design factor, which ’has’ to be aimed for. In particular, Google achieves this(see [3]) based on the maximes of an ethics of simplicity, rewarding opennessand drawing on the effects of the long tail. In [5] we have explained that theredesign of zbMathNew interface towards a Google-like design does not need to
be a successful move with mathematicians. Reduction of form is good, but the
impression of loosing functionality destroys the basic idea.
Transparency Here, the Pattern 9 (“Transparency”) kicks in. For instance, inthe zbMathNew example above the new Google-like design did not disable functionality, it merely neglected to point it out to the user. That is, the redesign
was not transparently carried out. The desirability of a math search interface isimproved when the interface is transparent enough, so that search results can be
anticipated. Pattern 7 (“Outcome”) is also about the ability of looking-ahead.
Therefore, we note that a rather coy but with a lot of potential design might bedesirable for mathematicians:

Goal 8 (“Cinderella”):“Diffident design with the potential to dig deep ”

The metaphor “Cinderella” is based on Cinderella’s diffident attitude and appearance contrasting her brilliant heritage and future.
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Empowerment To be able to showcase mastery of a tool is the more impressive
the more the tool empowers one to do. Here, mathematicians are no exception,
they like to be experts. Thus, Goal 2 (“Structure”) and Goal 3 (“Flexible Search
Strategies”) support fun and engagement, that is, desirability.
Community Note that the desirability is strenghtened with Goal 6 (“Community
Idiosyncracies”) as well.
3.3

Adoptability

Familiarity It is an HCI best practice to bridge the gap between new and oldservices by extending old and therefore known interaction rules into the new
ones. In particular, the old service can usually serve as a metaphor for the useof the new service. E-readers, for instance, do use the good old book metaphor
to be handled.
Pattern 1 (“Familiarity”) breaks this best practice. It seems to suggest thatprofessional mathematicians, from here on abbreviated to “mathematician” only,
can be used as guinea pigs for whatever new designs one can come up with aslong as some basic usability guidelines are followed. But in fact, it is quite theopposite. Even though they are open to new designs in general, they have to
be convinced in the first place to adopt the new service. These criteria seem to
be much higher than with non-math services as some of the other patterns like
Pattern 6 (“Function”) or Pattern 5 (“Medium”) are not really optional.
Community One reason for adoption of an application consists in the added value offered. As the community is highly appreciated by mathematicians, Goal
6 (“Community Idiosyncracies”) supports the adoptability.
3.4

Usability

Outcome One main usability goal of applications is the efficiency of its userinterface. But for mathematicians the outcome is more important than the input
according to Pattern 7 (“Outcome”). In particular, efficiency is not a main goal
for them. Other main usability goals are learnability and memorability, which
didn’t come up in the interviews in [5] at all. Yet another usability goal consists inthe fun factor of using an application. We discussed this in regard to desirability
already and decided that fun has to be reinterpreted for mathematicians. So,
all in all, we can state that the usability criteria for mathematicians have to be
redefined. The suggested goals in this paper lead towards such a redefinition.

4

Conclusion

We have presented and argued for design implications based on the patterns
elicited in [5]. In particular, we have:7G1 (Optimization of Resources): “Optimization of sources, fields, query language, presentation of variants, low entropy, and refinement features — wrt searchintension ”
G2 (Structure): “Maximization of structure in the search results ”
G3 (Flexible Search Strategies): “Addition of as many functionalities to support flexible search strategies as possible ”
G4 (Semantic Transparency): “Semantic transparency, i.e., if the user interface allows access to the relevant information and its underlying semantic relations [6] ”
G5 (No Social Media): “No inclusion of social media in an mSI ”
G6 (Community Idiosyncracies): “Integration of community aspects ”
G7 (Beauty): “Utmost reduction of complexity — without loosing functionality ”
G8 (Cinderella): “Diffident design with the potential to dig deep ”

Future work now entails the implementation of these design suggestions in, forexample, Mathematical Knowledge Management applications. Once this is done,
the value of the goals can be verified with usability studies. If so, then we canestablish the above set of design goals as best practices for mathematical searchinterfaces.
Note that this study didn’t prove common usability goals invalid, but when
considering a fair balance of essential requirements for a good design of math(search) interfaces, the weights should be specifically assigned according to mathematicians. If those requirements are satisfied, the finetuning regarding generalbest practices concerning form but not content can begin — if there are resources
left. Mathematicians will be the happier.
Acknowledgement This work has been funded by the Leibniz association
under grant SAW-2012-FIZ.

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