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The lessons are running once a week during the winter term 2015/16. The students will be shown (i) how to see the links between formal descriptions and graphs; (ii) how to draw schemes and calculate particular examples when reading a new, biological text with mathematical equations; (iii) to see differences between discrete and continuous world; (iv) to see that the mathematical formalism is only a part of our language, which can help with understanding to our problems in biology. Please note that the lessons are given in the Czech language only; we can, however, switch to English on request.
During the forthcoming term, we will also run a class of practical computational skills that will support the lessons. The class is not mandatory, but only recommended. However, the class will be closed for those who will not attend the lessons. Last update: Šizling Arnošt Leoš, RNDr. Mgr., Ph.D. (29.04.2015)
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Caswell, H. 1989. Matrix Population Models. Sinauer Associates, Inc. Publisher Sunderland, Massachusetts. Last update: Sacherová Veronika, RNDr., Ph.D. (29.10.2019)
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Written test. Last update: Sacherová Veronika, RNDr., Ph.D. (29.10.2019)
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Ten modules are offered to the students during the basic course of mathematics for biology. Not all of these modules are expected to run each term. As declared in the annotation, the lecturers prefer robust understanding over the quantity of information, and therefore they could skip some of the modules if the progress of the students were slow. The lecturers are, however, rather the moderators of the student’s discussion, than speakers, and so the students can ask them not to skip a module if they were interested.
Notation: lesson - how to turn biology into math equations; "grammar" and "syntax" of mathematical "formulas" ; types of queries and information that one can impose and receive, respectively when solving mathematical equations or inequalities; unit invariance.
practice - basic skills in mathematical notation; how to solve and simplify equations and inequations with respect to the question in focus.
Function 0: lesson - function as a modell of biological datasets; functions that fitt and bound datasets; graph and equation of a linear, polynomic, logarithmic, exponencial and hyperbolic function.
practice - link between formal notation and graphical pattern for selected functions.
Recurrence relations and differential equations: lesson - recurrence relations and their use in population ecology; differentiation of a recurrence relations and the most frequent mistakes; derivative of a function, meaning and definition; case studies.
practice - training in function derivatives and its graphical representation.
Integration: lesson - we introduce the idea of integral using a case study from biology; definite and indefinite integral; link between integral and addition and patch area; integral and probability; basic in statistical testing; units of an integral.
practice - training in function integration and graphical representation of integrals.
Functional equations: lesson - functional equations; differential equations; integral equations - graphic visualisation; case studies from biology.
practice - training in functional and differential equations.
Logarithm: lesson - we show that a logarithmic function is a solution of a functional equation; maening and usage of a logarithmic function; logarithmic data transformation for statistical analyses; case studies.
practice - training in logarithms; practice in the reading of the biological texts with logarithms in it.
Invariances: lesson - mathamatization using invariances; unit invariance, principle of superpozition, taxon and area invariance, scale invariance, self-similarity and fraktals; case studies.
practice - practice in the reading of the biological texts with invariance in it.
Functions 1: lesson - functions of multiple arguments; derivative of a function of multiple arguments along a curve and in a focal direction - we show all the problemes in a visual way; case studies.
practice - training in the function of multiple variables; simple calculations.
Statistical methods: lesson - probability distribution functions; we show usage of mean, median, modus and maximum likelihood in case studies from biology; basic in multivariable statistical methods (GLM) and a link between these methods and linear function of multiple arguments.
practice - practice in usage of mean, median, modus, maximum likelihood and multivariable statistical methods.
Transformations: lesson - transformation of variables; how to prepare data for statistical analyses; link between a function and data transformation; changes in functions, frequency distributions and units while we apply a transformation.
practice - transformation of a dataset, graphic representation of a function and frequency distribution; case studies.
Matrices: lesson - how to design Leslie matrix and a vector of the population in population ecology; determinant and eigenvalues of a matrix; the meaning of the determinants and eigenvalues for biology.
practice - practice in matrices calculus; determinant and eigenvalues of a matrix; how to use matrices when one solves a system of linear equations.
Last update: Šizling Arnošt Leoš, RNDr. Mgr., Ph.D. (29.04.2015)
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University education differs from that provided at secondary schools and technical lyceums. Its aim is not the memorization of "knowledge," the acquisition of "skills," and certainly not the imposition of "attitudes (an item in the official assignment)" (the university was founded and should remain an apolitical institution). The goal of university education is not to understand the world in the sense of "being able to apply and modify an already known principle" or to be competent in the sense of "being able to solve a typical problem in a standard way." The purpose of university education is to understand the world to such an extent that graduates should be able to distinguish whether a piece of knowledge or a statement is consistent with existing knowledge (in today's jargon, "critical thinking") and to uncover new principles and phenomena. These skills cannot be directly tested in individual subjects, but in final theses such as bachelor's or master's theses. For this reason, I do not focus on individual knowledge, skills, competencies, and attitudes in my teaching, but rather use discussion to show different perspectives on the relationship between mathematics and biology, what mathematical thought patterns can be used in biology, and what the pitfalls of "common," informal thinking are. During the lessons, I try to show why mathematics is a language, how to interpret a biological problem to a mathematician or mathematical software. Students complete the course with a written exam. If I try to describe the teaching using the required scheme, it could perhaps be written (albeit inadequately from the point of view of educating future scientists) as follows: Extended annotation Basic Mathematics (MB162P05) is a course for bachelor's and master's students that seeks to bridge the gap between theoretical mathematics and its application in biology. Instead of memorizing facts and routinely solving typical problems, the course focuses on understanding the world through exact forms of thinking and logically consistent argumentation. Teaching takes the form of discussions about the relationships between subjects, properties, and quantities, presenting mathematics as a universal language capable of translating biological problems into a logically consistent form. Students are shown various types of argumentation, such as proof by contradiction, complete induction, graphical proofs, the principle of superposition and transformation of state spaces (e.g., graphs), and mechanisms generating distribution functions. The aim is to equip future scientists with the ability to distinguish whether new claims are consistent with existing knowledge and enable them to discover new principles in their own research. Technical competence could perhaps be described as training in reading mathematical formulas in biological literature. This course prepares graduates for scientific work, where the synthesis of biological theory and logically consistent argumentation is a prerequisite for interpreting observations and testing hypotheses. KNOWLEDGE · Logical structure of biological argumentation: Understanding the principles of logically consistent argumentation and interpretation of basic mathematical objects in a biological context. · Mathematical schemes in biology: Knowledge of thought schemes (e.g., symmetry; superposition; unit, scale, taxon, and other invariances; conservation principles; minimum and maximum principles) commonly used in physics and which (according to the lecturer's experience) are applicable in biology. · Relationship between properties, functions, and geometric insight: Understanding the connections between properties of “animated world” and their functional and graphical expressions (logarithm, polynomial, Taylor series, logit, distribution function). · Biological interpretation of basic operations: Linearization, logarithmic transformation, limit transition, differentiation, derivation, integration, matrix multiplication and addition (if there is enough time in the term). · Distribution mechanisms: Knowledge of the mechanisms that generate different types of frequency distribution functions and understanding to the conditions under which the existence of a distribution function in biology is meaningful. SKILLS · Interpretation of biological problems: Ability to communicate the biological issues with a trained mathematician or SW (demonstrated using Wolfram Alpha). · Analysis of invariants: The ability to identify different types of invariance (unit, scale, translation, taxon). · Quantification of dynamic changes: The ability to distinguish between effective and instant values and to use the concept of limit transition for the analysis of biological processes. · Interpretation of mathematical formulas and graphs: The ability to read the physical and biological meaning from formulas and graphs in biological literature (e.g., estimating tree biomass, or population abundance using graph area (integral)). · Working with information, structure, and symmetry: The ability to argue using considerations of symmetry, structure, and information; apply principles of superposition and the laws of minimum and maximum (e.g., linear regression, the principle of least action, the principle of maximum entropy) in modeling natural phenomena. COMPETENCES · Critical scientific thinking: The ability to independently assess whether a particular statement or finding is consistent with existing biological and mathematical knowledge. · Interpretation (translation) from biology to mathematics and back: The competence to integrate biological theory with a more exact mathematical form. · Recognition of cognitive traps: The ability to identify errors in biological interpretation stemming from a misunderstanding of mathematical principles (e.g., the construction of indices (Wolterstorff index, diversity index, and others). The above is demonstrated to students, with the understanding that it is up to each student to evaluate this teaching in their master's thesis. Last update: Šizling Arnošt Leoš, RNDr. Mgr., Ph.D. (29.01.2026)
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