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

Transcript

1 Course summary

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

3 mathematical reasoning

4 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.

5 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!

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

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

8 mathematical modelling

9 The math is not always visible!

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

11 We can scale up the math!

12 Models, algorithms and software interface algorithms model

13 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)

14 problem solving

15 How can we make the most of our limited capacity?

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

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

18 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!

19 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

20 how to think more independently

21 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!

22 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!

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

24 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!)

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

26 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)

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

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

30 solving a varied set of challenging problems AND reasoning about reasoning

31 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!

32 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

33 reflection self-check

34 Awareness of your abilities your limitations not always comfortable

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

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

37 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!

38 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?

39 Supplantive learning

40 so where are we now?

41 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

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43 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

44 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.

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