t ±I. pva. mathematical reasoning mathematical modelling problem solving THEN

Σχετικά έγγραφα
Introduction to mathematical thinking. Dag Wedelin

General perspectives on modelling and problem solving. Dag Wedelin

Section 8.3 Trigonometric Equations

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

2 Composition. Invertible Mappings

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

C.S. 430 Assignment 6, Sample Solutions

EE512: Error Control Coding

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Section 7.6 Double and Half Angle Formulas

Srednicki Chapter 55

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

Matrices and Determinants

Definitions. . Proposition 1

Section 9.2 Polar Equations and Graphs

Math 6 SL Probability Distributions Practice Test Mark Scheme

Finite Field Problems: Solutions

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Concrete Mathematics Exercises from 30 September 2016

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

LESSON 14 (ΜΑΘΗΜΑ ΔΕΚΑΤΕΣΣΕΡΑ) REF : 202/057/34-ADV. 18 February 2014

Οι αδελφοί Montgolfier: Ψηφιακή αφήγηση The Montgolfier Βrothers Digital Story (προτείνεται να διδαχθεί στο Unit 4, Lesson 3, Αγγλικά Στ Δημοτικού)

Code Breaker. TEACHER s NOTES

Every set of first-order formulas is equivalent to an independent set

derivation of the Laplacian from rectangular to spherical coordinates

The Simply Typed Lambda Calculus

On a four-dimensional hyperbolic manifold with finite volume

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

Areas and Lengths in Polar Coordinates

Homework 3 Solutions

Example Sheet 3 Solutions

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

Inverse trigonometric functions & General Solution of Trigonometric Equations

ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ

ELEMENTS BOOK 3. Fundamentals of Plane Geometry Involving Circles

ST5224: Advanced Statistical Theory II

Areas and Lengths in Polar Coordinates

Lecture 2. Soundness and completeness of propositional logic

PARTIAL NOTES for 6.1 Trigonometric Identities

Advanced Subsidiary Unit 1: Understanding and Written Response

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

Homework 8 Model Solution Section

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

F-TF Sum and Difference angle

ΑΓΓΛΙΚΗ ΓΛΩΣΣΑ ΣΕ ΕΙΔΙΚΑ ΘΕΜΑΤΑ ΔΙΕΘΝΩΝ ΣΧΕΣΕΩΝ & ΟΙΚΟΝΟΜΙΑΣ

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

BECAUSE WE REALLY WANT TO KNOW WHAT YOU THINK ABOUT SCHOOL AND YOUR GARDEN. Fairly true If I decide to learn something hard, I can.

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Approximation of distance between locations on earth given by latitude and longitude

EUCLID S ELEMENTS OF GEOMETRY

Επίλυση Προβλήματος σε Προγραμματιστικό Περιβάλλον από Παιδιά Προσχολικής Ηλικίας

Example of the Baum-Welch Algorithm

the total number of electrons passing through the lamp.

( ) 2 and compare to M.

Paper Reference. Paper Reference(s) 1776/04 Edexcel GCSE Modern Greek Paper 4 Writing. Thursday 21 May 2009 Afternoon Time: 1 hour 15 minutes

ίκτυο προστασίας για τα Ελληνικά αγροτικά και οικόσιτα ζώα on.net e-foundatio // itute: toring Insti SAVE-Monit

[1] P Q. Fig. 3.1

LESSON 6 (ΜΑΘΗΜΑ ΕΞΙ) REF : 201/045/26-ADV. 10 December 2013

Συστήματα Διαχείρισης Βάσεων Δεδομένων

Fractional Colorings and Zykov Products of graphs

Partial Differential Equations in Biology The boundary element method. March 26, 2013

6.3 Forecasting ARMA processes

Statistical Inference I Locally most powerful tests

ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. Τα γνωστικά επίπεδα των επαγγελματιών υγείας Στην ανοσοποίηση κατά του ιού της γρίπης Σε δομές του νομού Λάρισας

ΣΥΓΧΡΟΝΕΣ ΤΑΣΕΙΣ ΣΤΗΝ ΕΚΤΙΜΗΣΗ ΚΑΙ ΧΑΡΤΟΓΡΑΦΗΣΗ ΤΩΝ ΚΙΝΔΥΝΩΝ

5.4 The Poisson Distribution.

LESSON 12 (ΜΑΘΗΜΑ ΔΩΔΕΚΑ) REF : 202/055/32-ADV. 4 February 2014

Μηχανική Μάθηση Hypothesis Testing

Reminders: linear functions

Right Rear Door. Let's now finish the door hinge saga with the right rear door

ΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ. ΘΕΜΑ: «ιερεύνηση της σχέσης µεταξύ φωνηµικής επίγνωσης και ορθογραφικής δεξιότητας σε παιδιά προσχολικής ηλικίας»

( y) Partial Differential Equations

Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.

CRASH COURSE IN PRECALCULUS

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

TMA4115 Matematikk 3

Chapter 2 * * * * * * * Introduction to Verbs * * * * * * *

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ

How to register an account with the Hellenic Community of Sheffield.

ΟΡΟΛΟΓΙΑ - ΞΕΝΗ ΓΛΩΣΣΑ

Block Ciphers Modes. Ramki Thurimella

7 Present PERFECT Simple. 8 Present PERFECT Continuous. 9 Past PERFECT Simple. 10 Past PERFECT Continuous. 11 Future PERFECT Simple

Numerical Analysis FMN011

Volume of a Cuboid. Volume = length x breadth x height. V = l x b x h. The formula for the volume of a cuboid is

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Math221: HW# 1 solutions

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

Εγκατάσταση λογισμικού και αναβάθμιση συσκευής Device software installation and software upgrade

Πώς μπορεί κανείς να έχει έναν διερμηνέα κατά την επίσκεψή του στον Οικογενειακό του Γιατρό στο Ίσλινγκτον Getting an interpreter when you visit your

SOLVING CUBICS AND QUARTICS BY RADICALS

ΠΑΡΟΥΣΙΑΣΗ ΙΔΕΠ ΣΥΜΒΟΥΛΕΣ ΓΙΑ ΣΩΣΤΗ ΔΙΑΧΕΙΡΙΣΗ ΕΡΓΩΝ ERASMUS+ STRATEGIC PARTNERSHIPS

Transcript:

Course summary

mif p =. THEN pva. of mathematical reasoning xf # It* m. - t ±I. mathematical modelling problem solving

mathematical reasoning

We are naturally able to imagine and reason about abstract mathematical concepts! In our mathematical thinking we can therefore connect common sense and the use of mathematics.

The importance of precision! - How much fuel do we have? - Quite a lot - So how far can we go? - Pretty far - Will it be sufficient for our trip? -... begin with clear definitions!

conjecture proof also in pure mathematics you try and guess first before you prove!

" Mathematics is the art of explanation " ( P. Lockhart )

mathematical modelling

The math is not always visible!

Discrete versus continuous mathematics The computer may be digital but the world is often not!

We can scale up the math!

Models, algorithms and software interface algorithms model

Mathematics is the science of patterns! Associates both to patterns within mathematics itself, and to how we use them to model the world! (Hardy, Steen)

problem solving

How can we make the most of our limited capacity?

If you want to move this bookshelf to another wall you need to work in small steps!

Investigate the problem for deeper understanding! Explore different paths towards a solution! Always think and reflect!

The importance of simplicity! vi kanske gjort saker och ting en aning mer invecklat än nödvändigt försöka finna ett annat sätt att tänka på som kanske inte är så komplicerat som det vi först tänkt på Begin with something simple! Continuously try to simplify your current thinking!

Working mathematically use pencil and paper use computational tools explain and communicate well familiarity with different kinds of problems and how to handle them appropriate use of mathematical knowledge

how to think more independently

ELEMENTS BOOK 1 ΓΕΒ μείζων ἐστὶ τῆς ὑπὸ ΒΑΓ. ἀλλὰ τῆς ὑπὸ ΓΕΒ μείζων (the sum of) BD and DC. ἐδείχθη ἡ ὑπὸ ΒΔΓ πολλῷ ἄρα ἡ ὑπὸ ΒΔΓ μείζων ἐστὶ Again, since in any triangle the external angle is τῆς ὑπὸ ΒΑΓ. greater than the internal and opposite (angles) [Prop. Εὰν ἄρα τριγώνου ἐπὶ μιᾶς τῶν πλευρῶν ἀπὸ τῶν 1.16], in triangle CDE the external angle BDC is thus περάτων δύο εὐθεῖαι ἐντὸς συσταθῶσιν, αἱ συσταθεῖσαι τῶν greater than CED. Accordingly, for the same (reason), λοιπῶν τοῦ τριγώνου δύο πλευρῶν ἐλάττονες μέν εἰσιν, the external angle CEB of the triangle ABE is also μείζονα δὲ γωνίαν περιέχουσιν ὅπερ ἔδει δεῖξαι. greater than BAC. But, BDC was shown (to be) greater than CEB. Thus,BDC is much greater than BAC. Thus, if two internal straight-lines are constructed on one of the sides of a triangle, from its ends, the constructed (straight-lines) are less than the two remaining sides of the triangle, but encompass a greater angle. (Which is) the very thing it was required to show. Εκ τριῶν εὐθειῶν, αἵ εἰσιν ἴσαι τρισὶ ταῖς δοθείσαις [εὐθείαις], τρίγωνον συστήσασθαι δεῖ δὲ τὰς δύο τῆς λοιπῆς μείζονας εἶναι πάντῃ μεταλαμβανομένας [διὰ τὸ καὶ παντὸς τριγώνου τὰς δύο πλευρὰς τῆς λοιπῆς μείζονας εἶναι πάντῃ μεταλαμβανομένας]. Α Β Γ. Proposition 22 To construct a triangle from three straight-lines which are equal to three given [straight-lines]. It is necessary for (the sum of) two (of the straight-lines) taken together in any (possible way) to be greater than the remaining (one), [on account of the (fact that) in any triangle (the sum of) two sides taken together in any (possible way) is greater than the remaining (one) [Prop. 1.20] ]. A B C Κ K Ζ Η Θ Ε D F G H E Λ L Εστωσαν αἱ δοθεῖσαι τρεῖς εὐθεῖαι αἱ Α, Β, Γ, ὧν αἱ Let A, B, andc be the three given straight-lines, of δύο τῆς λοιπῆς μείζονες ἔστωσαν πάντῃ μεταλαμβανόμεναι, which let (the sum of) two taken together in any (possible αἱ μὲν Α, ΒτῆςΓ, αἱ δὲ Α, ΓτῆςΒ, καὶ ἔτι αἱ Β, ΓτῆςΑ way) be greater than the remaining (one). (Thus), (the δεῖ δὴ ἐκ τῶν ἴσων ταῖς Α, Β, Γτρίγωνονσυστήσασθαι. sum of) A and B (is greater) than C, (thesumof)a and Εκκείσθω τις εὐθεῖα ἡ ΔΕ πεπερασμένη μὲν κατὰ τὸ C than B, andalso(thesumof)b and C than A. So ΔἄπειροςδὲκατὰτὸΕ, καὶ κείσθω τῇ μὲν Α ἴση ἡ ΔΖ, it is required to construct a triangle from (straight-lines) τῇ δὲ Β ἴση ἡ ΖΗ, τῇ δὲ Γ ἴση ἡ ΗΘ καὶ κέντρῳ μὲν τῷ equal to A, B, andc. Ζ, διαστήματι δὲ τῷ ΖΔ κύκλος γεγράφθω ὁ ΔΚΛ πάλιν Let some straight-line DE be set out, terminated at D, κέντρῳ μὲν τῷ Η, διαστήματι δὲ τῷ ΗΘ κύκλος γεγράφθω and infinite in the direction of E. AndletDF made equal ὁκλθ, καὶ ἐπεζεύχθωσαν αἱ ΚΖ, ΚΗ λέγω, ὅτι ἐκ τριῶν to A, andfgequal to B, andgh equal to C [Prop. 1.3]. εὐθειῶν τῶν ἴσων ταῖς Α, Β, Γτρίγωνονσυνέσταταιτὸ And let the circle DKL have been drawn with center F ΚΖΗ. and radius FD. Again, let the circle KLH have been Επεὶ γὰρ τὸ Ζ σημεῖον κέντρον ἐστὶ τοῦ ΔΚΛ κύκλου, drawn with center G and radius GH. And let KF and ἴση ἐστὶν ἡ ΖΔ τῇ ΖΚ ἀλλὰ ἡ ΖΔ τῇ Α ἐστιν ἴση. καὶ ἡ KG have been joined. I say that the triangle KFG has 25 Thinking creates knowledge!

What is needed to solve a problem? very different balance for different problems knowledge needed for solving a problem = knowledge created by own thinking + knowledge from others because of the variation you often have to add something here!

Very restrictive to always think about what somebody else wants. Instead ask what makes sense to you!

About the right track It is not important to get it right at first! It is only at the end that it needs to be right or good enough. What is important is a self-correcting way to work. (monitoring/reflecting/metacognition is key!)

Investigate and search broadly! Find a selfcorrecting way to work! non-linear (monitoring/reflecting/ metacognition is key!)

Don t take the problem for granted! Have I understood the problem? Is the problem clearly defined? Is this the right problem to solve? Can I go further? (than providing the requested answer)

Alchemy (by Arturas Slapsys) In research you often end up solving a different problem!

some ways in which we have worked towards independence in the course

solving a varied set of challenging problems AND reasoning about reasoning

Bloom s taxonomy (1956, improved by Anderson et al 2001) We try to move upwards in this hierarchy! To do this you must take control of your own thinking!

Self-evaluation and management What do you expect? Can we assume that? Are we on the right track? Is this answer correct? How well did we perform? What would be a sensible and useful answer? What are sensible assumptions that we can try? How can we ensure that we eventually get there? How can we check and evaluate the answer ourselves? Self-assessment

reflection self-check

Awareness of your abilities your limitations not always comfortable

You can develop several roles within yourself! solve! ask questions! explain! guess! evaluate!

You can develop several roles within yourself! You can supervise yourself as we supervised you! solve! ask questions! explain! guess! evaluate!

So don t be a follower! Take control of your own thinking! Trust your own judgement! Focus on problems, not on solutions! Create your own problems!

Continue to think about how knowledge is created! When you see or read about an interesting solution Try to imagine a situation where it would have been natural to look for this solution. And how could you possibly in that situation have come up with this solution?

Supplantive learning http://skills4trainers.blogspot.se/2013/05/resistance-to-learning-and-learning.html

so where are we now?

realistic problems and situations experience with a varied set of realistic problems thinking reasoning modelling problem solving functions/equations geometry optimization differential equations probability and statistics discrete math pencil and paper computing knowledge and tools

realistic problems and situations experience with a varied set of realistic problems thinking reasoning modelling problem solving functions/equations geometry optimization differential equations probability and statistics discrete math pencil and paper computing knowledge and tools

For you as a software engineer (IT, CS-programme, ) Your ability to use mathematics and mathematical thinking as a problem solving tool is an important part of your capacity as engineers! Many other IT-people do not have that.

For you as a teacher of mathematics Modelling and problem solving are at the heart of thinking and working mathematically! Thank you!