On October 6th, it rained at my house [click on the image to see the video].

So I saw, immediately, the chance to make some questions:

- How much did it rain when rainfall is finished? Because of that, I decided to put my watering can outdoors and then mesure the water inside [can you see in the video?].
- Because there is an storm, could we prove the “Rule of Lightning and Sound”: is the distance between the storm and my house [2]?
- The day after the rainfall, there will be official records of rainfall (we have a network of weather stations). So, can I estimate the amount of rainfall from their records? By this reason, I’ve choosen near weather stations and I’ve made a interactive map with rainfalls (between 00h - 24h on October 6h) and put my home in red on the map
^{2}

Do you have the whole activity here: in english and in catalan

[1] Meyer, D. *Category: 3acts*. 2016. http://blog.mrmeyer.com/category/3acts/.

[2] WikiHow. *How to calculate the distance from lightning*. 2016. http://www.wikihow.com/Calculate-the-Distance-from-Lightning.

Recently, I discovered a analogy about mathematical activities: what kind of writing task do you do?

- You just complete the calligraphy copybooks: follow the marked line with pencil. So you are not able to write free content nop form.
- You could write a formal application to Government for example [2]. With this kind of document, you are restricted with a lot of format constraints but you could freely write the content.
- And finally, you could write a book [3]. You are not restricted to form or content.

Following *5 Practices for Orchestrating Productive Task-Based Discussions in Science* [1] these categories rise up the demanding of knowledge. And I think that students really “write a book” if they do Project-based learning.

I give you an example of this analogy for practicing fractions as operator. I want students to calculate of .

- Activity: “Calculate of ”
- Students possible response: “”

- Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second.

Divide this plot to verify this requeriment"

- Students response:
- understand what is “three times”
- calculate somehow
^{1}that one farmer has 12 squares and other 4 squares. - draw

- Activity: “Two farmers want to divide a 4 x 4 plot but the first should have three times surface than the second. What’s the
*best*way to do it? Consider costs like fencing, buying seeds, irrigation, etc. and crop benefits.”^{2} - Students responses: ?

Update: I change the book analogy from this:

Can you find three different ways to divide this plot verifying this requeriment?

What is the division which has the minimum cost? (each fencing side has a cost of $10)?

Can you compare yours with your neighbours’?

Can you find out what is the minimum cost division among all possible divisions?"

to above.

[1] Cartier, J.L., Smith, M.S., Stein, M.K., and Ross, D.K. *Practices for orchestrating productive task-based discussions in science*. National Council of Teachers of Mathematics, 2013.

[2] Govern de les Illes Balears. *Llibre d’estil*. amadip.esment, 2006.

[3] Wikipedia. *El ingenioso hidalgo don quijote de la mancha*. 2016. https://es.wikipedia.org/wiki/Don_Quijote_de_la_Mancha#/media/File:El_ingenioso_hidalgo_don_Quijote_de_la_Mancha.jpg.

More thoughts about flipped classrooms….

- I do not like the fact of
*prescribing*students to see the theory outside the classrooms (typically with videos). I do not like it because two reasons:- first, I think students have to do
*all*things*inside*the classrooms. The subject time is the classroom time, I think. Students are bussy persons (like everyone) and we (teachers) have no right to command (typically compulsory) homework. When you go home, you don’t (at least you should not) take your reports for next morning at work. This is more general critique. It also happens in direct instruction. - second, what happens if students don’t understand the theory? Do teachers have to re-explain it in classroom? So does gained time equal wasted time? I don’t know how
*flipped-teachers*manage that.

- first, I think students have to do
- Flipped classrooms are sympton that something happens with direct instruction. Something is not working on that. I think that flipped classrooms pedagogy shows that we have an overloaded curriculum. If we had reasonable curriculum, then we would have time
*in class*to teach theory*and*practice (in whatever form we chose to do that). But because we have overloaded curriculum we have two options: direct instruction (which clearly is the fastest method of teaching all the curriculum^{1}) or cutting theory and sending as homework. In my case, I have no legal obligation to cover the curriculum up with objective justification [1] but you have pressure to end it up because sometimes you have external exam students need to take (university or vocational training entering exams)^{2} - I like flipped classrooms like a warning that we need to change in what we spend time in classrooms: we need time to relate to our students. The majority of classes I have seen have been with superficial interaction between teachers and students.

[1] Cultura de les Illes Balears, C. d’Educació i. *Orientacions per a l’elaboració de la concreció curricular i de les programacions didàctiques*. http://weib.caib.es/Documentacio/orientacions_elaboracio_cc_pd/orientacions_concrecio_curricular_i_programacions_didactiques_25-02.pdf.

[2] Kirch, C. *Flipping with kirch. the ups and downs from inside my flipped classroom*. The Bretzmann group, 2016.

[3] Kirch, C. *Flipping your math classroom: More than just videos and worksheets. how i got time back in my classroom to support all learners and deepen the learning experience for students*. 2016. https://www.bigmarker.com/GlobalMathDept/Flipping-Your-Math-Classroom-More-Than-Just-Videos-and-Worksheets.

[4] Wikipedia. *Flipped classroom*. 2016. https://en.wikipedia.org/wiki/Flipped_classroom.

- 37.5 hours per week of working time.
- 9.5 hours for making work at home
^{1} - 28 hours for making activities in school (
*mandatory permanence*)- 20 hours of teaching in class
- 8 hours for library dedication, burocracy tasks, meetings
^{2}, etc.

In some cases, these 20 hours could be reduced when you are the *department chief*^{3} (3 hours), tutorials, etc. In my case, this season I teach *only* 14 hours in classrooms (37% of my working time). I do not complain about the number of hours of teaching^{4} but I complain about the efficiency of non-teaching hours. In the majority of our non-teaching hours, we spend (read waste) time discussing about irrelevant things for students. It’s sporadic to mention how to improve teaching or what specific mesures we could adopt for engage students in our meetings. We spend time discussing things like:

- Reading previous meeting acts in public
- Reminding the holidays and calendar of the season
- Our struggles with politicians for having more or less teachers
- Reminding what to say to students on the first day
- When and how to fill forms reviewing the real time spending in the class (
*memoirs*) - Where to go outside school
- What is the
*knowledge state*of our students and if they would success - Proposing teaching training activities
- Aproving fire prevention plan
- Suggesting new materials for teaching
- Aproving the school budget
- Making lectures about statistics or the studies offered by our school
- Talking about computers issues

Things which are, mostly, not really important. We spend almost 30% of our (permanence) time on improductive meetings or burocracy. And the most disappointing thing is that it is marked by law or by *tradition*. But I think we could (and should) spend (invest) this 30% of our time doing valuable things, things for students, things for improving our school, things for free us of filling forms, etc.

What are your situation? How much time do you spend to unproductive tasks?

[1] Cultura de les Illes Balears, C. d’Educació i. *Intruccions per a l’organització i el funcionament dels centres docents públics d’educació secundària obligatòria per al curs 2015-2016*. 2015. http://www.caib.es/sacmicrofront/archivopub.do?ctrl=MCRST4905ZI147029&id=147029.

In my opinion, it’s an euphemism for “doing nothing” for

*bad teachers*because these hours are not controled. Good professionals spend more than 10 hours per week at home for preparing classes↩teaching staff meetings, inter subject department (

*Comissió de Coordinació Pedagògica*in catalan), school board, etc↩you are the boss of all teachers teaching the same subject↩

Politicians use it for abusing and deteriorate working conditions many times.↩

In a certain city, gas service is paying 15 € fixed a month and 0.75 € for each cubic meter consumed.

- How much do you pay for ? And for
- Plot the function which relates consumed cubic meters and the cost of the service

When do you need the formula of the relation there? Clearly you could get (a) without a formula. And you could plot the points using value table (with previosly calculated values in (a) if you want) but you *need* the formula of the relation () for *assuring* that you could join the points with a straight line (using the theoric fact that: afine functions correspond to straight lines). Otherwise you would not know if the plot is a straight line or a curve.

My students understood more clearly the connection between plots and functions when I ask them for which step needs here the formula.

*If an athlete is running at 6.833 meters per second in 1500 meters race, at which speed she would run in 3000 meters race?*

This activity is *very* uninsteresting. It does not bring nothing to our students. How can we enrich that? In my case, I think I acomplished it incorporating as much as actions I could in the activity. In the original activity, our students just calculate. No more. In the evolved activity they do more things than that. Perhaps, the evolved activity were not interesting and it would not engage our students. I don’t know, but at least it touches more aspects of the reality. Knowledge is about connections and one way to connect is to do diffents things (or same thing in different ways).

No more preambles. Here is my activity:

It involves:

- reason
- sketch a graph
- place points in a graph
- read coordinates
- express your opinion
- and, obviously, calculate.

I hope my students would consider it interesting.

PS: Sorry for orthographic mistakes in animation. I corrected in pdf.

[1] Bordoy, X. *Athletics records*. http://somenxavier.xyz/static/blog/athletics-races.pdf.

As it is expressed in its solution this problem involves an optimization problem of a radical function, which obviously does not match the level of my students:

But I have liked the problem very much. So I decided to transform this to a more simple one: changing the metric in from usual metric to taxi metric^{1}. With this discretization, the problem is more easy (even it could be solved with *try-and-see* tactic): students *just* have to minimize an afine function. The moral of the tale is that we have to maximize the trip with the highest velocity.

Perhaps this tactic of simplifying could be useful to you with other problems.

[1] Bolt, B. *Aún más actividades matemáticas*. Labor, 1989.

Note that in the city the width and the length of the streets are not the same. So the metric is a modified taxicab: , where and are the width and lentgh of streets.↩

But it’s needless. We don’t need to calculate the cutting points to -axis because the vertex is “positive”

What?, I said. Can you explain it with more detail?

And then, we put in blackboard what we have known as Aitor’s theorem. We wrote in blackboard its hypothesis and its thesis:

- Hypothesis:
*The parabola is concave*and*the of the vertex is positive* - Thesis:
*the parabola has not cutting points of -axis*

Since then, we have applied this theorem a lot of times for saving us the time to calculate the cutting points of -axis and, more important, my students know firsthand why mathematical reasoning is.

]]>We could do applying the formula :

But we don’t need it. We surely have got our students confused. And we would introduce a mathematical arsenal that we will not need.We could do as a product: . So we could apply distributive property.

This reminds our students the “rule” seen before, but surely most of them will have a mess after their application. And surely most of them would not understand what would be doing.Why not teach simply as we can: . So let’s multiply it.

This reduces the unexpected thing to familiar and known thing: algorithm for multiplying integers.

When does we need to know the formula ? In my opinion, only when our students need to expand expressions like *many* times or if they needed in order to solve a major problem (in which it is embeded) and they need not to spend their time in calculations (like they *do* in third way).

Note: For making this article, I used *SimpleScreenRecorder* [2] for recording an specific area of my desktop while running *LibreOffice Writer* formula editor [4] and *Geogebra* [3]. For conversion from `webm`

to animated `gif`

I used `ffmpeg`

[1]

**Update**: Fred G. Harwood (@HarMath) suggested we could calculate it also visually. It gets better understanding to students and avoid typical errors.

[1] Albino, B. *How to do i convert an webm (video) to a (animated) gif on the command line?* 2014. http://askubuntu.com/questions/506670/how-to-do-i-convert-an-webm-video-to-a-animated-gif-on-the-command-line#506672.

[2] Baert, M. *SimpleScreenRecorder. maarten baert’s website*. http://www.maartenbaert.be/simplescreenrecorder/.

[3] International GeoGebra Institute. *Geogebra*. http://www.geogebra.org/.

[4] The Document Foundation. *Home. libreoffice*. https://www.libreoffice.org/.

But sometimes (many more than it should be), the rules are taught before they are needed.↩

I need it to teach it because I want it to pass from the form to the form for treating quadratic functions. In particular for representing that in cartesian plane. Yes I know we could represent directly (and more easyly) quadratic functions in the form .↩

[1] Archive, I. *Wayback machine of somenxavier.xyz*. 2016. https://web.archive.org/web/*/somenxavier.xyz.

- “Schools on Trial” of Nikhil Goyal [2] (discovered via Idzie Desmarais [1]) for knowing what I have to change
- “The Best of the Math Teacher Blogs 2015” by Lani Horn and Tina Cardone [3] for knowing what I could do
- “De los principios a la acción: Para garantizar el éxito matemático para todos” [5] (spanish edition of “Principles to Actions: Ensuring Mathematical Success for All” [4]) for knowing what USA citizens face teaching

Here is a photo of the whole set:

I don’t know if I will have time to read them before next course.

[1] Desmarais, I. *I’m unschooled. yes, i can write.* https://www.facebook.com/yesicanwrite.blog/posts/10153575355503411.

[2] Goyal, N. *Schools on trial. how freedom and creativity can fix our educational malpractice*. Doubleday, 2016.

[3] Horn, L. and Cardone, T. *The best of the math teacher blogs 2015. a collection of favorite posts.* Amazon Fullfillment, 2016.

[4] NCTM. *Principles to actions: Ensuring mathematical success for all*. NCTM, 2014.

[5] NCTM. *De los principios a la acción: Para garantizar el éxito matemático para todos*. NCTM, 2015.

Fins ara usava carpetes al meu ordinador, on tenia diverses activitats, separades per nivells. Freqüentment juntava les activitats en documents (sobretot en format ConTeXt). Però amb aquesta organització tenia diversos problemes:

- Una mateixa activitat podia servir per diversos cursos però amb el meu sistema organitzatiu només la podia desar a una carpeta
^{1}. - A més volia publicar les meves activitats, sobretot les òperes en tres actes (tal com fan altres persones [4,5])
^{2}. - Volia desar l’activitat amb múltiples formats: de vegades una activitat està bé presentar-la de diferents maneres, segons el que es vulgui aconseguir a classe [1]. Per tant, vull poder tenir la mateixa activitat en format d’òperes en tres actes [3], d’emparellament de conceptes [2], de deures clàssics, etc. Un document amb diverses activitats no serveix per fer això
^{3}.

Fa temps que vaig fer-me un *script* i vaig pujar les meves òperes en tres actes al servidor. Ara bé, això tenia limitacions: havia d’escriure les òperes en format html i només servia per a òperes i no per a altres activitats.

Després de pensar molt i molt, me n’he adonat que l’estructura de blog és ideal per això: una entrada (activitat, document, el que sigui) classificada (normalment usant etiquetes). Així tothom pot trobar fàcilment el que cerca (per exemple “qualsevol cosa relacionada amb funcions afins”). Aquesta darrera setmana m’he posat mans a l’obra i amb l’ús de `gostatic`

he creat un repositori d’activitats, documents, òperes en tres actes, etc. L’he anomentat “*theque" per allò de què és una col·lecció (“teca”) de qualsevol cosa (“*“).

Encara és una versió beta, però és totalment funcional. Tenc pendent ara passar totes les meves activitats a aquest repositori. Hi podeu accedir aquí: theque.somenxavier.xyz

[1] Bordoy, X. *Comentari 13 de l’entrada Dan Meyer’s Dissertation*. http://blog.mrmeyer.com/2015/dan-meyers-dissertation/#comment-2406667.

[2] Bordoy, X. *Activitat d’emparellament de domini, gràfica, expressió algebraica i taula de valors*. http://somenxavier.xyz/static/blog/01-matching-dominis-gràfiques.pdf.

[3] Meyer, D. *The Three Acts Of A Mathematical Story*. 2011. http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/.

[4] Meyer, D. *Dan Meyer’s Three-Act Math Tasks*. 2016. https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit?pref=2&pli=1#gid=0.

[5] Stadel, A. *Andrew Stadel 3-Act Math Tasks*. 2016. https://docs.google.com/spreadsheets/d/19sms4MpuAOO71o4qFPJyVKK-OGLnNegMgSL6WAwIdb8/edit?pref=2&pli=1#gid=0.

This is my proposition: use additive function. If you have marks, , why instead of calculating arithmetic mean just add them up. This has a lot of advantages:

- It’s more easy to calculate. In primary school, it’s easy for students to calculate a sum rather than a mean (it involves division)
- It captures better the sense of “league”, the sense of “marathon”: you could know, directly, how long you progress a day (“I add it up 5 points today”. “20 points left to get a C”)
- Students know better the weight of a score (exam for example) in relation to the whole course: if one activity worths 10 points, then if you passed, you get 10 points. But what is the weight of a 10 in an exam if you apply the mean? With additive assessment function, “you have what you get”
- There is an obvious equivalence between calculate arithmetic mean and just sum numbers: if each has a maximum score of , then you could calculate the mean or (of ).

The only disavantage I know is that you have to be aware of how many test/activities students will get. Because otherwise you could not say “If you get 50 points, you will pass the course”. If students missed tests, then you/they have to re-escale [4] it (perhaps an excuse to talk about proportions ;).

By all these reasons, the additive function will be my next assessment function next course^{1}

[1] Wikipedia. *Arithmetic mean*. 2016. https://en.wikipedia.org/wiki/Arithmetic_mean.

[2] Wikipedia. *Grading systems by country*. 2016. https://en.wikipedia.org/wiki/Grading_systems_by_country.

[3] Wikipedia. *Mean*. 2016. https://en.wikipedia.org/wiki/Mean.

[4] Wiktionary. *Rescale*. 2016. https://en.wiktionary.org/wiki/rescale.

Right now I can’t modify the existing rules.↩

L’única pega que li veig és la **legalitat**: es pot assegurar que un professor que imparteixi classe *disruptiva* no tindrà problemes legals?

- Legalment tenim llibertat de càtedra, però també hem de seguir un currículum que en la majoria de casos és tivant (diu què s’ha d’ensenyar i quan) i irreal (la quantitat de conceptes, procediments i actituds per aprendre és massa) però és la llei. Com podrem justificar que hem vist sistemes d’equacions a 1r d’ESO però que no hem vist regles de tres a 2n d’ESO perquè la pràctica
*disruptiva*ha duit a què els alumnes s’interessassin*massa*per un tema i no per un altra? O com*justificar-nos*davant les altres persones:- què avaluarà l’inspector?,
- quins comptes haurem de retre setmanalment al nostre cap de departament?,
- podrem justificar els mètodes i continguts impartits davant els pares (si és el cas) i els alumnes?
- O davant els altres companys de departament?

- Seran els continguts
*impartits*(que no apresos) els mateixos que si seguíssim l’ensenyament tradicional? Com ho podrem justificar? Fent un examen tradicional que no concordarà amb el que hem fet al llarg del curs? - Si per exemple, utilitzo les òperes en tres actes a la meva pràctica habitual, el més raonable seria avaluar-ho a un “examen de modelització”: els estudiants equipats amb tauletes intentant resoldre una òpera (cadascun una de diferent). D’aquesta manera avaluaríem la capacitat de modelització dels alumnes (que en molts casos està al currículum). Però com deixaríem constància del què s’ha fet? I de l’examen que els hi hem posat? No el podrem imprimir, perquè involucra intensivament l’ús d’un vídeo? De manera anàloga, com podrem deixar constància per escrit del què hem fet a un projecte?

En resum, quines garanties (legals) tenim que podrem anar en *contra del sistema imperant*?

[1] OCDE. *La naturaleza del aprendizaje. Investigación para inspirar la práctica*. 2012. http://www.oecd.org/edu/ceri/The%20Nature%20of%20Learning.Practitioner%20Guide.ESP.pdf.

[2] Vallory i Subirà, E. *L’actualització disruptiva de l’educació*. 2015. https://www.youtube.com/watch?v=Qg7HxXo6m5Y.

In Spain there are three types of schools:

- Public schools which are funded, only, by the Government. The management of the school (norms, expenses, principal, hiring of personel, etc.) is responsible of the Government too. There are no fees for registration.
- Private schools which are funded and managed by private companies or institutions. They have expensive fees.
- Charter schools (in spanish
*Escuelas concertadas*[7]) which are funded by a mixt of Government and private institutions. The management of the school belongs to the private organization. They have cheaper fees than private schools (around € 300).

I only know how the system of public schools works, in where I have teached for 13 years. In particular, the system of Balearic Islands [1,8].

There are two types of teachers in public schools: civil servants^{1} or temporary employees. Both kinds of teachers could receive training but they have no obligation to do that. The *motivation* to get training in both cases are a) getting points to choose the school you want to teach and b) earn more money. The points have a maximum, so it has no *sense* to get training from a certain moment. This leads to taking courses just for making points. In general, teachers choose easy courses but with no didactic interest.

The courses are offered by the Government itself and private organizations (typically labour unions and universities). When private organizations made a course, the Government has the right to accept or decline. If it’s accepted, the course is homologated. Only homologated courses are valid for making points and, even, to be recognized.

In my opinion que quality of the courses is, *at least*, arguable. The following are the main categories in which the courses offered, right now, by the Goverment [2,3] and by some unions [4,5,6] belong to (in parentesis the number of courses and the rounded percentage of the main categories):

- Technologic resources (21 - 39%): learning how to use Moodle (4), using social networks for education (2), making mobile apps (1), using Augmented reality (1), using Google products for classroom (1), Scratch (1), using scientific calculators (1), using digital boards (1), using blogs (1), etc (8).
- Language courses (6 - 11%): english (6)
- Classroom management (16 - 30%): Flipped classroom (3), Behaviour management (7), Creating activities with hook (1), Project-based class (1), Group activies (1), Other (3)
- Psicology and sociology (6 - 11%)
- Hiking
^{2}, trip and other sports (4 - 7%)

As you can see, there are not too much courses about making hooking activies (like could be 3-act operas) and making a non-traditional classroom. The research from internet, reading interesting blogs (like Mr. Meyer’s one) and finding MTBoS community is the only alternative that I have found against that. As me, very much people in my region.

I hope in other countries get training could be more useful than in mine. Please comment if you want to share how it works in your country.

[1] Cultura de les Illes Balears, C. d’Educació i. *Oferta educativa curso 2015-2016*. 2016. http://www.caib.es/govern/sac/fitxa.do?lang=es&coduo=36&codi=50606.

[2] Cultura de les Illes Balears, C. d’Educació i. *Activitats oferta cep manacor*. 2016. http://cepmanacor.caib.es/.

[3] Cultura de les Illes Balears, C. d’Educació i. *Programa fad. activitats de formació*. 2016. http://weib.caib.es/Formacio/distancia/activitats_.htm.

[4] Illes Balears, A.S.I. de les. *Llistat de cursos*. 2016. http://anpe-balears.org/ofertanew.php?Autentifica=No&Cod_categoria=25.

[5] Stei-i. *Cursos personal docent (presencial)*. 2016. http://www.steiformacio.com/formcol2.asp?idCol=6.

[6] Stei-i. *Cursos personal docent (a distància)*. 2016. http://www.steiformacio.com/formcol2.asp?idCol=28.

[7] Wikipedia. *Escola concertada*. https://ca.wikipedia.org/wiki/Escola_concertada.

[8] Wikipedia. *Balearic islands*. https://en.wikipedia.org/wiki/Balearic_Islands.

They passed a public exam which is made by Government↩

I’m not joking: there are several

*courses*just for hiking and, in theory, explain the scientific features of the route. See for example “Excursions per Palma. Les seves manifestacions artístiques” [4] which could be translated as “Trips around Palma. Its artistic manifestations”↩

- It’s more intuitive: if we want to mesure the regularity, the homogeneity of a sample, then I think that it’s more intituitive to calculate the mean of the diferences between mean and the values of the sample instead of calculate the square root of those differences squared. All my students say “Why squared?”
- We don’t need standard deviation until high courses. For examples if we teach gaussian distribution, confidence intervals or linear regression. But if we introduce statistics or if we want to analyze data samples, I think we need simple deviation parameter
^{1}

The only disadvantage I think we would have if we teached the instead of is that we could not take the most references in scientific literature, which use . I thought about it many times. And I think that the reason, the real reason, for which mathematicians use instead is because, as a function, it’s derivable and is not. So it allows getting easy bounds with known distributions (like gaussian ones).

[1] Cultura de les Illes Balears, C. d’Educació i. *Ordre de la consellera d’Educació i cultura de 22 de juliol de 2009, per la qual s’estableix el currículum de l’educació secundària per a persones adultes que condueix a l’obtenció del títol de graduat en educació secundària obligatòria a les illes balears*. 2009. http://www.caib.es/eboibfront/pdf/VisPdf?action=VisHistoric&p_any=2009&p_numero=117&p_finpag=55&p_inipag=4&idDocument=629435&lang=ca.

A range of the sample could be another one↩

I accustom to pass a (anonymous) poll to my students at the end of the course^{1}. With this, my students can *assess* me and I can see what are my good points and what have to improve in the next course. I think this time is the only oportunity they have to express their opinion about their teacher (me). I am not scared about any result, because I trust that the mean of the results and common sense compensates the optimal and pessimal opinions.

Here is my poll (translated from catalan):

You can download the pdf and its source.^{2}

Here are my results until now since I have taught in adults school:

I could say that:

- My good points are Q5, Q15, Q17, Q19 and Q20
- My week points are Q1, Q2, Q18, Q22, Q26 and Q27

So I have to improve the contents (they have to be more real life and interesting contents) and participation (the students lack of participation). For doing that, my plan is to change from traditional manner of doing class to a problem-based one. I do not know how I will do it but now, with the helping resources of MTBoS, the transition will be, surely, more smooth.

On the other hand, another important question is which items you consider important for making a teacher assessment poll? Perhaps you could comment it out…

I think it deserves an informal assessment, at least. After the MTBoS blogging initiative I got:

- An habit to blog weekly. This have allowed me to ponder what I have done in my classes. So I’m more aware now what are my better and good points.
- Discovering new ways of doing things. Following the comments from each posts of this initiative [3,4,5,6], I discovered new people and how they did things differently.
- I have received feed-back (more than I thought). My mentor, Stephen Cavadino (@srcav), commented some of my posts and spread them through twitter. And the most awesome is that I received blog comments from people I’ve never been contacted with. I received 4 comments.

Update: I decided to create a certificate which accredites that I post 4 of 4 posts (as I can prove):

The logo is made using GIMP from logo of 2016 blogging initiative [2] + Pixabay logo of certificate [7].

[1] Explore the MTBoS. *A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

[2] Explore the MTBoS. *Logo of the 2016 blogging initiative*. 2015. https://exploremtbos.files.wordpress.com/2015/12/mtbos-blogging-initiative.png?w=180&h=413.

[3] Explore the MTBoS. *Week 1 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/10/week-1-of-the-2016-blogging-initiative/.

[4] Explore the MTBoS. *Week 2 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/17/week-2-of-the-2016-blogging-initative/.

[5] Explore the MTBoS. *Week 3 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/24/week-3-of-the-2016-blogging-initiative/.

[6] Explore the MTBoS. *Week 4 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/31/week-4-of-the-2016-blogging-initiative/.

[7] Pixabay. *Certificate 98472*. 2013. https://pixabay.com/es/certificado-insignia-premio-precio-98472/.

Quadratics is the first unit I have to teach in ESPA 4 [1]. This is a course of adults education in Balearic Islands (Spain). What the students should know to do at the end of this unit are:

- Solving second grade equations
- Representing quadratic functions and getting the vertex, their orientation and the cut points with the axes.
- Solving real problems (in particular modelize situations with quadratic functions)

I have split the unit in three parts according with these aims. So there are three parts, which seem *unconnected*:

- In the first part, essentially, we solve 2nd-grade equations
- In the second part, we use 2nd-grade equations just for finding cut points with the axes
- And for solving problems we use 2nd-grade equations or finding the vertex of the parabola

For introducing this topic, I put this problem^{1}:

From this point, we make problems of finding the dimensions of a square with a fixed area.

After that, I put the analogous problem with a rectangle and we see that there are infinite solutions. So we have to restrict the problem: perhaps the height of the rectangle could be something related to the width. For example, the height could be 3 times the width. Also we trait the triangles:

But we restrict our dependency to number *times* a dimension. In this step all the 2nd-grade equations are of the form .

Then, we make a *small* change: the dependency of one dimension is a number *plus* the other dimension:

With this kind of problems, we get complete equations (of the form ). After that we practice solving 2nd-grade equations. Just equations, no problems.

In this part, I remember the cartesian plane and how we could read or write coordinates and points to/from it. And inmediately I teach how to represent quadratic functions and we just “resolve” exercises of representing them:

It is supposed that this section represents the real applications to all of the previous stuff. But I can’t achieve what is pretended. A sample of my problems is this:

(in the last two problems of this sample, the students have just to apply a formula)

I have never been able to find optimization problems which are real, easy and interesting.

- The introduction to 2nd-grade equation must be shorter. We usually spend two or three weeks.
- I need better models for introducing the second grade equations.
- I need to attach the lesson to the reality, overall the quadric functions. I have been thinking about it many times.
**The**real application of quadratic functions is the parabolic shot [4,5]. But its governing equations are very much complicated for my students. So I’ve been stucked.

*Ordre de la consellera d’Educació i cultura de 22 de juliol de 2009, per la qual s’estableix el currículum de l’educació secundària per a persones adultes que condueix a l’obtenció del títol de graduat en educació secundària obligatòria a les illes balears*. 2009. http://www.caib.es/eboibfront/pdf/VisPdf?action=VisHistoric&p_any=2009&p_numero=117&p_finpag=55&p_inipag=4&idDocument=629435&lang=ca.

[2] Explore the MTBoS. *A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

[3] Explore the MTBoS. *Week 4 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/31/week-4-of-the-2016-blogging-initiative/.

[4] Meyer, D. *Will it hit the hoop?* 2010. http://blog.mrmeyer.com/2010/wcydwt-will-it-hit-the-hoop/.

[5] Teaching Channel. *To the Moon!* 2015. https://www.teachingchannel.org/videos/paper-rocket-lesson-plan.

All the problems here are translations from their original ones in catalan.↩

“What would you need for making a shoe fabric *in your town*?”. This is the first question I ask to my students when I start the statistics unit.

They answer several things: a plot for making the building, electricity, machines for making shoes, etc. But, and the end, I say “Yes. All of this stuff it’s needed, but what would you really need for switch on the machines?” and then, if there is no answer to that, then I put a clue “What do you do, first of all, when you go to shoe shop?”. Eventually, the answer becomes “The shoe number”.

And “how is the shoe number that you would manufacture first?”. And then obviously we have to determine what is the most frequent shoe number *in our town*. So we have to make a poll? But how to do that? I introduce the concept of poll, population and sample. Assuming that our classroom is non-biased sample, we list all the shoe number of my students^{1}. This naturally introduces the frequency table (because we have a middle size sample) and the Mode.

This example of shoe number could be more useful. Depending on the course, then we represent the shoe numbers *distribution* and see what are the intervals of numbers with are the most frequent (in particular, the median).

This makes me introduce Statistics without having to be so magisterial and, with this, I realized that open questions are always good for introduce topics and engage students.

*A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

[2] Explore the MTBoS. *Week 3 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/24/week-3-of-the-2016-blogging-initiative/.

and usually their parents and siblings too. If you would make a non-inmediate poll, you could send it as a homework.↩

I work in an adults school. My official curriculum [1] stablishes that my students of the last course, ESPA 4th^{1}, have to make an scholar’s work about anything which “involves mathematical ideas and procedures” and they have to deliver a memorandum of the following process and their main conclusions. The teacher has to guide them through the process. This is my *favorite*.

When I change from teenagers education to adults school, this was my main challenge. I had not done this before. After two years in the adults school, I can *defend* myself against this. My main constraint is the time: the time for doing the work is very limited. My course is just 4-month course and I have to teach 3 lessons. So a month per lesson. And we have one month for the work. And they have to redact in a *scientific style* the work and they usually do not have any idea of how to do it. So, how do I do?

- I have 4 classes per week. In the first month, I spend one class per week to make a
*mock work*^{2}. We try to answer to the question: “Which is the most non-contaminant can in the market?” It serves to us to see the 5 phases in a scientific work:

- After this mock work, I
*push them out*for thinking their question of the work. In my opinion “Without question there is no answer to give” and so there is no possible work. After a month, if someone does not have any question, I give them a list of possible questions or topics.

- After that, I continue normal classes and I assign them a big homework: start the research and feedback with me any progress, difficulty or decission they’ve made. In this moment I use the theorical authonomy of 18+ years persons
^{3}. I encorage to share and ask me anything in any moment, in any class, via email or in the flesh. I remember their duty to deliver the memorandum, I ask them how is their progress once a week and how close is the deadline. - In the final two weeks of the course, they write the memorandum. This is the most difficult thing for them. Telling things in assay mode, reasoning each step, giving reasons of each assertion and being impartial are very difficult things for them because they usually do the contrary. I have to teach them “Linguistics” instead of Mathematics.

At the end, it is worthy: each student ends with an answer to a question. The answer could be less or more “developed” (with more or less confidence in its argument) but each of them achieve to solve the question *scientificly*^{4}. The following are the titles of some works (translated from catalan, their original language)^{5}:

- Comparison between the gregorial calendar and arabian calendar (Comparació entre el calendari gregorià i el calendari musulmà)
- Study about the increase of the muscular brute force and muscular resistance in a period of two months (Estudi de l’increment de la força bruta muscular i de la resistència muscular en un temps de dos mesos)
- Explanation and description of the numerical systems (Explicació i descripció dels sistemes de numeració)
- Working of existing calendars until the present and explanation of leap-years (Funcionament dels calendaris existents fins a l’actualitat i explicació del funcionament dels anys de traspàs)
- How much does an epidemic take to kill the population of Mallorca (Quan de temps estaria una epidèmia a acabar amb la població de Mallorca?)
- Gambling games: from which I could get more? (Jocs d’atzar: amb quin guanyaria més?)
- , division between the perimeter and diameter of a cicle (, divisió entre el perímetre i diàmetre d’un cercle)
- Study about crimes in Balearic Islands (Estudi dels delictes a les Illes Balears)
- How many handicapped people will be in 2020? (Quantes persones amb discapacitat hi haurà l’any 2020?)
- Types, calculate and elaborations of present bar codes (Tipus, càlcul i elaboració dels codis de barres actuals)
- Methods for calculating the Consumer Price Index and elaboration of a personal CPI (Métodos para calcular el IPC y elaboración de un IPC personal)
- Practical testing of the number and determination of committed error (Comprovació pràctica del valor del nombre i determinació de l’error comès)
- Study about Basa metabolic rate and types of diets (Estudi del metabolisme basal i tipus de dietes)
- How they calculate the unemployment rate (Com es mesura la taxa d’atur?)
- Study about how the protection affects the endangered animals (Estudi sobre com afecta la protecció als animals en perill d’extinsió)
- Aplications from Mathematics to diets and sport (Aplicacions de les matemàtiques en el tema de les dietes i l’esport)
- How did the countries of the world face the economical crisis? (Com varen afrontar la crisi econòmica els països del món?)
- Analysis of linear, exponential and logistic model in the growth of population (Anàlisi dels models lineal, exponencial i logístic del creixement de la població mundial)
- Study about the emissions of modern vehicles (Estudi sobre les emissions de dels vehicles moderns)
- In the EU, the widest country is the more dense? (Dins la UE el país més extens és alhora el més dens?)
- What would be the most cheap way to travel to Amsterdam? (Quina seria la trajectòria a seguir per arribar a Amsterdam de ma manera més econòmica possible?)
- Study about the fuel station in Campos (Estudi de quina benzinera és més econòmica a Campos)
- The number : history and study of several methods of approximation (El nombre : la història i l’estudi dels diferents mètodes d’aproximació)
- Relation between the growth of vegetables and the Fibonacci succession (Projecte d’investigació de la relació entre el creixement de vegetals i la successió de Fibonacci)

*Ordre de la consellera d’Educació i cultura de 22 de juliol de 2009, per la qual s’estableix el currículum de l’educació secundària per a persones adultes que condueix a l’obtenció del títol de graduat en educació secundària obligatòria a les illes balears*. 2009. http://www.caib.es/eboibfront/pdf/VisPdf?action=VisHistoric&p_any=2009&p_numero=117&p_finpag=55&p_inipag=4&idDocument=629435&lang=ca.

*A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

[3] Explore the MTBoS. *Week 2 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/17/week-2-of-the-2016-blogging-initative/.

[4] Wikipedia. *Education in the united states. educational stages*. 2016. https://en.wikipedia.org/wiki/Education_in_the_United_States#Educational_stages.

which I think it is formally equivalent to 10th grade of USA [4]↩

My students have not written a scientific work. If I want they to write it, they need at least

*one*chance to learn.↩I suppose that we have to guide the teenagers…↩

Obviosly there are students who don’t pass the work, but normally it’s due to their absence of work not due to the lack of reasoning in their argument.↩

Any mistake with the translation is mine↩

One of the aims of my official curriculum [1] is to discover regularities and patterns. Another one is to plot afine functions and knowing that the formula is associated with (straight) lines and other types of functions correspond to curves. Since two courses ago, I have combined these aims letting students discover by themselves this relation. This is the chronicle of how I have done this year (this week).

Before threating the problem of differentiating curves or lines from the algebraic formula, we need several preliminaries. The first is to know the cartesian coordinates. In the first classes, I explained what is the cartesian plane, what are the axes and I gave some terms like *origin* and *quadrants*. Then I gave exercises of reading coordinates from points and to draw points giving their coordinates (see the picture above). Since this point, we will identify the points and their coordinates.

After assuring everyone knows how to read and write points to the cartesian plane, I started to represent functions. I put a simple formula in blackboard (like ) and asked what are the points which satisfy the equation. Then we plotted these points and we got the representation of the formula. We represented several functions: for example , , or , calculating the tables of values of and . We saw the *impossibility* of knowing exactly what the graph of the function is exactly, because we represented only a finite number of points but the graph itself has infinite number of those. So we had to *guess* the form of the graph corresponding to one formula.

This is a workout exercise. After that, students know that there are a lot of possible graphs which could generate a formula and they are self-confident representing functions. Then, I asked them:

- “Is there a relation between those graphs and the formulas?”
- “Is there a way of knowing what kind of graph is generated by a formula, or not?”

For knowing that, I asked students to list a bunch of functions (we put in the blackboard) and then I asked them to **conjecture** their rule (their relation between graphs and formula). The rule could be simple or complex as they want. They choose. If they have not guessed any rule, then I suggest to represent as many function as *they need* for getting the guess. And if they got the rule, I ask them to *prove* or *falsifying*. Falsifying is always possible: we just need two graphs and two formulas which contradicts the rule. But we learnt that it’s impossible to *prove* the rule with their tools. And even we knew that there are rules more general than others and rules which implies other ones.

This week, the rules are:

- “A function of the form has a ‘U’ graph”
- “A function of the form has two curves”
- “ is a curve”
- “If the functions contains then, it is a curve because when grows the change of is greater”

Next day, we checked these rules and I informed which “proved” rules are really true or false (read mathematically proved). See graph above (in catalan). Sorry for the quality.

After that, I asked them if there is a rule to know if a functions gets a line or curve. And we deduced that we get a line when the formula is .

After that, because it’s needed only two points for defining one line, then when we would represent a function of the form (that we have learned that is a line), we will need to calculate just two points in our table of values.

I love this experience because it’s very exciting when students deduce their own rules and try to prove or disprove them. We feel like a really mathematicians. I strongly recommend you to apply in your class. Obviously you could use Geogebra or other interactive tools instead of blackboard.

*A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

*Week 1 of the 2016 blogging initiative!* 2016. https://exploremtbos.wordpress.com/2016/01/10/week-1-of-the-2016-blogging-initiative/.

On the other hand, I enabled comments in posts (via isso) for allowing feed-back with blog readers.

[1] Cavadino, S. *Cavmaths. maths, teaching and life*. https://cavmaths.wordpress.com/.

*A new exploration!* 2015. https://exploremtbos.wordpress.com/2015/10/18/a-new-exploration/.

MTBoS stands for

**M**ath**T**witter**B**log**oS**phere↩

For now, I bought “5 Practices for Orchestrating Productive Mathematics Discussions” [2] and “5 Practices for Orchestrating Productive Task-Based Discussions in Science” [1] of National Council of Teachers of Mathematics. I hope these books give me the ideas that I need.

*Practices for orchestrating productive task-based discussions in science*. National Council of Teachers of Mathematics, 2013.

[2] Stein, M.K. and Smith, M.S. *5 practices for orchestrating productive mathematics discussions*. National Council of Teachers of Mathematics, 2011.

Because I was new in my school and I had not serious experience with adults education, I had no time to incorporate these activities into my classes. However, I decided to do that in some *near* future. Meanwhile I added Dan Meyer’s site to my RSS reader. One of his posts, pointed me to Greoff Krall’s site and I used his “guide” of how transform routinary activities to interesting ones^{2} for making a matching activity among domains, graphs, algebraic expressions and tables of values [1].

It is two years since this decision and it is the time to action. I want to teach somehow different, with the minimum of magistral sessions as I can. Three-acts operas are one way of doing that (Geoff Krall sneaky activities are another). So the plan is to create and use 3-acts operas as well as other non-magistral activities for improving my classes.

Sharing is great and I appreciate the work of Dan Meyer, Geoff Krall and other teachers who share their work without any other interest but the appropiate credit and community feedback. I want to do the same. So here is my blog. I would be very glad if some material is good for you. Meanwhile, the blog could serve me as a documentary tool.

[1] Bordoy, X. *Activitat d’emparellament de domini, gràfica, expressió algebraica i taula de valors*. http://somenxavier.xyz/static/blog/01-matching-dominis-gràfiques.pdf.