# Mathematics, Science, and Technology

The Mathematics, Science, and Technology (MST) curriculum integrates these three connected disciplines. Throughout the MST sequence, students learn to:

Understand how scientific and mathematical ideas have developed throughout history

Actively explore mathematical and scientific content

Identify new problems and pose original questions

Carry out original investigations in pure and applied mathematics and science

Apply their learning to new situations

MST classes harness students' natural curiosity about the natural and man-made world, the origins of our universe, and human history. Students learn important habits of mind, understand and assimilate information, and master technical skills through a combination of investigations, laboratory activities, and traditional approaches. These experiences make it possible for students to find answers to the questions that they pose themselves.

*Division One (Grades 6 and 7)*

## + Engineering with Robotics and Computer-aided Design

The essential question explored in this course is “How can we apply scientific, mathematical, and technical facts and methods to the solution of real world problems?” Beginning with the book *Catastrophe!* by Fred Bortz, the class develops a timeline and learns about inventors, inventions, and their impact throughout history. Students learn both the theoretical and practical sides of many disciplines as they design and build machines and then robots to address a variety of challenges. Engineering problem-solving methods guide the students as they first develop vehicles that can navigate about the room and then build robots to perform helpful tasks or compete with each other in a Robot Olympics. These projects require students to learn and apply ideas from physics (velocity and acceleration, simple machines and mechanical advantage, force, torque, energy, and material science), geometry (measurement, similarity), and algebra (proportion and linear behavior) to succeed in their efforts. Students write a research paper on an application of robotics (e.g., a self-driving car or surgical assistant). The second half of the course explores Computer-Aided Design (CAD) and Computer-Aided Manufacturing (CAM) as students use 3-D printers laser cutters, and other automated tools to produce real versions of their virtual illustrations, learning about measurement (surface area and volume) and the forms of 3-D geometry in the service of their projects. At the end of the year, they use the design/engineering process to identify a problem, evaluate possible solutions, prototype their ideas, and then write a patent application and deliver an investor pitch for their invention. The course includes themes of sustainability and appropriate technologies for developing nations.

## + Doing Research in Mathematics and Science

This course asks the question “How is new knowledge acquired in math and science?” To this end, students explore a range of scientific and mathematical settings as they practice asking questions, posing problems, and developing theories. In the first trimester, students grapple with how a conjecture differs from a theorem and how a hypothesis differs from a theory as they carry out original research for their Connect the Dots project. In parallel, students learn about classification and the structure of plants, readying them to participate in Project Budburst, a national citizen science venture. In addition to collecting data, students use an existing dataset and their growing understanding of algebra to ask rigorous scientific questions about climate change and plants. Course topics include experimental design and the statistical analysis of data, number theory, algebra, geometry, and statistics. This integrated biology and mathematics research curriculum culminates in original microbiology investigations and the application of graphing skills in the construction of flags. Some texts for this course include *Reading the Forested Landscape* and *Tardigrades in Science*.

*Division Two (Grade 8)*

## + Marine Science

Marine Science develops and applies biological, physical, algebraic, and geometric ideas to the study of the environment. Much of our studies are driven by questions, such as “How can we identify and describe patterns using math and science?” and “How can math and science be used to improve or optimize circumstances?” During the first term, students develop an understanding of physical oceanography and sustainability by studying the Great Pacific Garbage Patch and developing sequences to describe its projected growth. Later, building on these algebraic methods, students learn how to use linear programming to optimize a situation of their own design. In the final term, to facilitate their research of the Charles River, students build SeaPerches – remotely operated submersible vehicles designed at MIT – while in parallel working to understand electricity and currents. Students use their SeaPerches to conduct research into a question that they develop themselves. Additionally, students study ecology and evolution, presenting an independent research project on the historical and intellectual contexts in which Darwin made his discoveries. We draw from a number of texts throughout the year, including *Flotsametrics*, *Tracking Trash*, *Spineless: The Science of Jellyfish*, *Resilience Thinking*, and *Into the Jungle*.

*Division Three (Grades 9 and 10)*

## + Revolutions in Math and Science

This course compares and contrasts the relatively recent historical evolution of a truly scientific understanding of chemistry with the ancient development of a rigorous approach to mathematics and the blossoming of geometry that were reflected in Euclid's Elements. Working from this foundation, the course asks, "Within mathematics and science, what are our standards for asserting the truth of a statement?" "What types of evidence do we accept?" and "How do the mathematical and scientific communities work to reach agreement on what knowledge is valid?" Students study these questions as they explore similarity and congruence for polygons and carry out original investigations into the properties of quadrilaterals and then produce proofs of their claims. Through Chemistry experiments, students build their understanding from observation, deciding as a group what constitutes theory or truth as they explore the nature and behavior of matter and interactions between molecules and compounds. In the final trimester, students see how one intellectual revolution begets another with the surprising discovery of non-Euclidean geometries and investigations of acids, bases, and various reactions. Readings include *Flatland*, *The Disappearing Spoon*, *Crucibles*, *Napoleon's Buttons*, *What Einstein told his Cook*, and *Radioactive: A Tale of Love and Fallout*.

## + Human Biology and Decision-Making

This course compares the biology of different organisms with an
emphasis on human biology through the lens of molecular, developmental, genetic, and evolutionary concepts. In all of these contexts, we ask “How does form inform function?” Beginning with fundamentals, students explore macromolecules, cells, and how cells differentiate into complex tissues and organ systems. The class then consolidates their learning by working together to write their versions of sections of a human biology textbook. Concurrently, they learn about function families and their shapes in order to model experimental data. In our introduction to neuroscience and sensation, students make connections between trigonometry and sound, light waves, music, vision, and neural activity. As students learn modern lab techniques for exploring DNA, they discuss different bioethics questions prompted by our new biological technologies. Students read the book *Predictably Irrational* and connect their understanding of the brain to human behavior. Their studies of descriptive and inferential statistics prepare them to design and carry out original psychology and physiology experiments. Readings also include the books *Gulp* and *Your Inner Fish*.

*Division Four (Grades 11 and 12)*

## + Calculus, Physics, and Modeling

Throughout this course, students refine their abilities to model situations and see the myriad historical connections between math and science. Driven by the question “How do we describe change?” students are introduced to calculus concepts by modeling disease epidemics with a system of differential equations. Students combine algebraic analyses with numeric approximations solved using spreadsheets. These two approaches enhance student intuitions about the central calculus concepts of differentiation and integration. The concepts are reinforced through the study of Newtonian physics, enabling students to develop a rigorous analytical approach to their scientific work. Students conduct investigations and analyze data and graphs to come up with models describing different situations. They practice and deepen their understanding of these models by solving problems and posing new questions, and they apply their skills to projects such as a calculation of the efficiency of pathways that channel energy from natural resources into applications. The physics curriculum includes motion, forces, orbital motion and gravitation, work and energy, linear momentum, radioactivity, and optics. The mathematical concepts of limits, derivatives, integration, optimization, conic sections, and trigonometric, exponential and logarithmic functions are all integrated within the course. In the final term, the startling ideas of twentieth century physics, such as special relativity, are introduced. Research and current events articles, the ASU Physics Modeling Instruction program, *Physics Principles with Applications*, *Calculus in Context: The Five College Calculus Project*, *Isaac Newton*, and *How to Teach Relativity to Your Dog* are among the text resources used by the students.

## + Mathematical Modeling and Computer Science in the Social and Natural Sciences

Mathematical Modeling is the process of bringing a mathematical perspective to the study of real-world issues. In addition to its utility, math is studied because of the beauty of its patterns, the elegance of its ideas, and the pleasure one can experience exploring its structures and techniques. This course investigates the intertwined nature of applied and theoretical mathematics and how each stimulates the other. Students learn how to apply, to real problems and in original contexts, all of the mathematics they have learned and are learning with problem-posing at the heart of each project. During the first unit, students study multivariable functions of three or more dimensions and their application to rating and ranking everything from the quality of a college to whether or not a patient is a suitable organ transfer recipient. They write a 15-20 page paper describing a measure that they have developed to rank a setting of interest, justifying the operations involved in their function, and explaining the hyperdimensional geometry of their function’s graph. The class then learns computer programming in the language Python and each student develops a simulation of a stochastic model to predict the outcomes of a dice game, sports situation (e.g., the likelihood of Dimaggio’s 56-game hitting streak), or board game (e.g., expected length of a game of Chutes and Ladders). Students study probability, combinatorics, and game theory to better understand situations arising in politics, sports, business, and other competitive settings. Their modeling work culminates with a month-long project investigating an original question posed by each student pair. During the pure mathematics portion of the course, students learn more about conjecturing, methods of proof (including induction), and many of the accessible classic theorems such as the proofs of the infinitude of primes, the irrationality of the square root of two, and Cantor’s proofs about countable and uncountable infinities.

*Electives*

## + Introduction to Computer Programming

Students study how to analyze a problem and to design a structured solution to that problem in computer languages, such as Python and SNAP, which provide powerful and accessible graphical tools. Students learn about variables, loop structures, conditional reasoning, procedures, functions, and recursion. Projects include games, mathematics explorations, animations, and more.

## + Computer Science - Atoms, AND Gates, and Abstraction: How Computers Work

Students explore how computers work from the electronic level to logic gates, chips, and computer structure. Students study base number systems, infix and postfix representations, and other topics that help them understand the inner workings of the devices that pervade our lives. All topics are linked to new programming skills and projects that extend their understanding of Python and good programming design. Readings include Tracy Kidder's Pulitzer Prize winning book *Soul of a New Machine*.

## + Computer Science - Data, Algorithms, and the Ideas that Make the Internet Work

Students study how to represent data, design algorithms, and connect the two in programs. Students build a number of applications both individually and through group projects which require the class to coordinate their programming specifications. Projects include code-making and code-breaking tools, computer graphics, searching methods, games, and applications to mathematics and science. *Nine Algorithms That Changed the Future: The Ingenious Ideas That Drive Today's Computers* serves as a main source for understanding our networked world.

## + Non-hierarchical Systems, Fractals, and Chaos

This class instroduces programming in the graphically powerful and parallel processing programming language NetLogo, which makes possible explorations in real-world science and social science settings. Students explore how simple rules followed by many agents (e.g., termites, molecules, people, etc.) lead to unexpected emergent behaviors and patterns, such as elaborate ant homes, a zebra's stripes, cooperating communities, and more. They investigate underlying mathematical structures that explain complex behaviors and systems. Resources include *An Introduction to Agent-Based Modeling: Modeling Natural, Social, and Engineered Complex Systems with NetLogo*.

## + Mathematics Research Seminar

This class provides the opportunity to do sophisticated investigations in pure mathematics. Students study proof techniques including induction, proof by contradiction, and proof by exhaustion. Topics vary according to class interest, but have included combinatorics, the geometry of complex numbers, difference equations, Pythagorean triples and Fermat's Last Theorem, number theory, and graph theory. Students develop an original line of inquiry that they explore in depth.

## + Computer-Aided Design and Manufacturing

This class provides the opportunity to study Computer-aided Design (CAD) and Manufacturing (CAM) and three-dimensional geometry and algebra. Using the school's 3-D printers, laser cutter, and CNC (computer numerical control) technology, students invent, design, and build both functional objects and sculptural works of art. Students work with the powerful and professional CAD program Onshape.

## + Advanced Placement Physics

This class is offered in partnership with Boston University. Students complete some of their studies online and some on site at B.U. Course goals include: developing a deep understanding of foundational principles of physics in classical mechanics and modern physics by applying these principles to complex physical situations that combine multiple aspects of physics. Students will design and conduct inquiry-based laboratory investigations to solve problems through first-hand observations, data collection, analysis and interpretation.