in a 2009 report , the american association of medical schools ( aamc ) and howard hughes medical institute ( hhmi ) articulate goals for quantitative biology teaching that can serve as a starting point . specifically , biology students should be able to 
 demonstrate quantitative numeracy and facility with the language of mathematics , interpret data sets and communicate those interpretations using visual and other appropriate tools , make statistical inferences from data sets , extract relevant information from large data sets , make inferences about natural phenomena using mathematical models , apply algorithmic approaches and principles of logic ( including the distinction between cause / effect and association ) to problem - solving , andquantify and interpret changes in dynamical systems ( aamc and hhmi , 2009 , pp .
demonstrate quantitative numeracy and facility with the language of mathematics , interpret data sets and communicate those interpretations using visual and other appropriate tools , make statistical inferences from data sets , extract relevant information from large data sets , make inferences about natural phenomena using mathematical models , apply algorithmic approaches and principles of logic ( including the distinction between cause / effect and association ) to problem - solving , and quantify and interpret changes in dynamical systems ( aamc and hhmi , 2009 , pp .
although these quantitative skills are critical , we propose that other skills inherent to being a quantitative biologist may be equally important , such as the ability to collaborate across disciplines ( e.g. , hackett and rhoten , 2009 ; milton et  al . , 2010 ) .
we also posit that engendering positive student attitudes about quantitative work is important , because students ' interests and values , as well as their emotional responses , such as anxiety or enjoyment , can have significant effects on their willingness to engage in learning activities , persevere when they face difficulties , and persist in certain educational or career paths ( steiner and sullivan , 1984 ; glynn et  al . , 2007 ; rheinlander and wallace , 2011 ; matthews et  al . , 2013 ; poladian , 2013 ) .
finally , we anticipate that positive skill and attitudinal outcomes will likely lead to desirable behavioral outcomes , such as enrollment in additional quantitative courses , completion of more quantitative degree programs , and pursuit of further education or careers in quantitative biology .
thus , we propose that our goals for students should go beyond development of quantitative skills to include 
 more positive emotional responses to quantitative work , such as greater enjoyment or reduced anxiety;more positive beliefs about the ability to do quantitative work , such as increased confidence and self - efficacy;increased interest in quantitative work;greater sense of the centrality of mathematics , statistics , and computation to the practice of life sciences , including their relevance and importance;improved ability to work in interdisciplinary teams ; andincreased intentions to pursue or actual pursuit of further education and careers in quantitative biology . more positive emotional responses to quantitative work , such as greater enjoyment or reduced anxiety ; more positive beliefs about the ability to do quantitative work , such as increased confidence and self - efficacy ; increased interest in quantitative work ; greater sense of the centrality of mathematics , statistics , and computation to the practice of life sciences , including their relevance and importance ; improved ability to work in interdisciplinary teams ; and increased intentions to pursue or actual pursuit of further education and careers in quantitative biology .
just as data are needed to support hypotheses and conclusions in science research , data are needed to support the contention that students are benefiting from instruction .
in particular , educational data should be collected to demonstrate whether students are making progress toward achieving the intended goals .
what can students to do to convince us that they can use statistical methods to make inferences from data sets ?
how can we be sure that students understand the theory behind a mathematical model of a biological phenomenon and can use that model to make predictions ?
how will we know that students are less anxious about tackling quantitative problems or that they see math as relevant to their work as biologists ?
we must collect assessment data , such as students ' solutions to problems , explanations of results or phenomena , or analyses and models of data .
published methods for assessing students ' development of quantitative skills and other important outcomes are fairly limited .
a number of assessments of quantitative reasoning or quantitative literacy have been developed , primarily to document quantitative skills of non  science majors ( e.g. , steele and kilic - bahi , 2010 ; sundre and thelk , 2010 ) .
similarly , attitudinal assessments have been developed primarily for general audiences rather than biology majors ( reviewed in chamberlin , 2010 ) .
more tools are needed to document students ' progress toward quantitative biology  related outcomes , especially beyond introductory or non  majors biology . to this end , we encourage teams of biologists , quantitative scientists , and education specialists to collaborate in developing and testing a broader suite of assessment tools related to quantitative biology .
assessments are much more than homework problems or exams  they are data - collection tools .
it is not necessary , or even desirable , to always use published assessment tools when we teach , since these tools may not align with our teaching goals . just as in science , educators often have to develop tools or techniques to address the task at hand . we can use results gathered from these tools to inform what and how we teach  a process called formative assessment .
others have offered very useful advice on designing assessments ( angelo and cross , 1993 ; sundberg , 2002 ; tanner and allen , 2004 ) .
we will highlight just a few ideas here that are easily adapted to teaching in any domain . to identify areas of confusion
, we can periodically hand out index cards near the end of class and ask students to write one thing they learned and one thing they are still confused about . we can quickly peruse student responses and address the confusing ideas directly during the next class .
we can also give challenging problems for students to work on or data sets to analyze , either as homework or in class , alone or in groups .
we need to select these problems carefully to align with learning goals and give students practice solving the kinds of problems they will be expected to solve on exams or other high - stakes assessments .
students earn a small amount of credit ( e.g. , a few points for genuine , on - task effort ) for completing the work before we discuss it as a whole class
. we can ask students to share how they went about tackling the problem or conducting the analysis , what their results are , and where they ran into difficulties . we can then offer guidance as needed .
with goals and assessments in mind , we can choose what and how we will teach .
some resources we can adapt or adopt for use with our own students are already available .
some educators have developed entire research projects , courses , or course series ( e.g. , edelstein - keshet , 2005 ; chiel et  al .
, 2010 ; duffus and olifer , 2010 ; miller and walston , 2010 ; rheinlander and wallace , 2011 ) . for example , the symbiosis curriculum at east tennessee state university is a tightly integrated , three - course sequence that replaces the standard introductory biology , introductory statistics , and calculus i requirements .
an initial assessment of the first two cohorts demonstrated student gains in both mathematical and biological knowledge , and further assessment is planned ( depelteau et  al . , 2010 ) .
introductory courses have also been created that focus on learning the process of science , with data analysis as the main quantitative element ( e.g. , bell , 2011 ; goldey et  al . , 2012 ) .
at the upper level , math  bio courses such as bioinformatics , mathematical biology , and mathematical modeling have been developed with the aim of bringing biologists , computational scientists , and mathematicians together to learn from one another ( hydorn et  al . , 2005 ) .
others have launched collaborations between upper - level math and biology classes for students to learn from one another as they work together on a math  bio project or problem ( greer and palin , 2012 ) .
most of us do n't have the time , resources , or inclination to develop an entire course .
instead , we can teach more quantitatively by using single lessons or modules that emphasize quantitative outcomes ( e.g. , jungck et  al .
the american biology teacher contains numerous quantitative biology activities scattered throughout their issues , but lse is especially useful for finding evidence - based instructional strategies and curricula .
for example , speth and colleagues ( 2010 ) infused data - driven problems and quantitative tasks into their ecology and evolutionary biology course , which led to improvements in students ' abilities to represent data graphically .
one of these sites , mathbench , was developed by biologists at the university of maryland and includes 30 + freely available online math  bio modules that can be integrated into existing life sciences courses .
thompson and colleagues ( 2010 ) have shown that students who complete these modules report gains in their quantitative skills and increased comfort with solving quantitative problems . web and journal resources for teaching quantitative biology .
challenges of teaching quantitatively include fitting quantitative elements into an already - packed curriculum and convincing colleagues that a new version of an existing course serves students as well as what has been done in the past .
concerns about failing to cover content may be assuaged by work from hester and colleagues ( 2014 ) , who demonstrated that students in a quantitative introductory molecular and cell biology class scored equally well on biology test questions when compared with students who completed a traditional content - driven class .
they stress the importance of using a learner - centered approach to achieving such outcomes .
indeed , results from numerous meta - analyses ( e.g. , freeman et  al . ,
2014 ) demonstrate that science students perform better when taught using active - learning approaches in place of traditional lectures  in other words , when instructors stop talking and engage students in tasks that promote meaningful , intellectual engagement ( allen and tanner , 2005 ) .
to prepare future generations of life scientists to take full advantage of mathematics and computation to understand the living world , we need to change how we are teaching .
start by articulating goals for quantitative biology instruction  consider quantitative and other professional skills as well as attitudinal outcomes , as all are important for developing expertise in quantitative biology . then define the evidence needed to show that students have made progress toward achieving the goals and how this evidence will be gathered through assessments . finally , design instruction to align with goals and assessments . some quantitative biology curricula and assessments are already available , but new resources are needed and should be developed through the collective efforts of quantitative biologists and education specialists .
 more than a decade has passed since the publication of bio2010 , calling for an increased emphasis on quantitative skills in the undergraduate biology curriculum . in that time , relatively few papers have been published that describe educational innovations in quantitative biology or provide evidence of their effects on students . using a  backward design  
 framework , we lay out quantitative skill and attitude goals , assessment strategies , and teaching resources to help biologists teach more quantitatively . 
 collaborations between quantitative biologists and education researchers are necessary to develop a broader and more appropriate suite of assessment tools , and to provide much - needed evidence on how particular teaching strategies affect biology students ' quantitative skill development and attitudes toward quantitative work . 
GOALS: WHAT ARE WE AIMING FOR IN TEACHING QUANTITATIVE BIOLOGY?
ASSESSMENT: HOW DO WE KNOW QUANTITATIVE BIOLOGY TEACHING WORKS?
TEACHING STRATEGIES: HOW DO WE TEACH QUANTITATIVELY?
CONCLUSIONS