FINAL POST

Hello readers!

In the last two months I have shared several posts on Didactic of Mathematics, focusing on the measurement and other fields like the statistic or the probability. Each post has had its special feature: didactic techniques and materials for the learning of the measurement (cubes, an own intervention proposal, the tangram, ICT tools - Google Maps - …) or the measurement and other sciences in real contexts (stats and sports, the Spanish deck, the acre…).

With all of these posts I have tried to show different ways to teach and learn measurement. Although there are posts which are not explicitly didactic, I have added feasible didactic applications for all the cycles that could be implemented in class in order to provide a utility for learners. In the end, I have pretended to share a wide range of posts to create an entertaining blog, avoiding the monotony.

As I wrote in the introduction, the main goal of the blog was to make readers aware of the influence and the importance of measurement in the student’s process learning and in our life, and how it is represented in some of the areas out of the school. I hope to have obtained it.

Thank you for all!

THE SPANISH 'BARAJA'

The probability in mathematics is the extent to which an event is likely to occur, measured by the ratio of the favourable cases to the whole number of possible cases. One of the contexts in which the probability is very present is in the packs of card. I want to focus on the Spanish deck.

The Spanish deck is composed by forty cards (although there are other packs which have forty-eight), and they are classified in ten groups: gold, cup, sword and clubs. In each group the cards are the ace, two, three, four, five, six, seven, jack, horse and king.

There are a wide range of games to play with the Spanish deck, but in all of them the probability has an important role to decide the winner. When we have the whole pack of cards, there are some percentages to find a concrete card. For instance, the probability to take a gold card is 10/40 (ten are the favourable cases and forty possible cases); to take an ace, 4/40 (10%); to take a jack, horse or king, 12/40 (30%); to take any card, 1/40… As readers can see, the probability could be expressed in different ways: fractions, percentages, etc.

In my view, using the Spanish deck to introduce the probability is great for Primary children for two reasons: the majority of them know at least a game to play with the pack, and the manipulative material is always interesting to attract their attention; thus it would be positive to work it with learners.

A significant activity for any cycle could be developed with the previous cases that I have detailed in the previous paragraphs. It could add more complex calculations of probability. For example, only in swords group of cards, finding the probability to take a card with less than six words (the answer would be 5/10), or to take two cards whose value is 12 (the favourable cases would be the horse -a value of 11- + the ace, the jack -a value of 10- + the 2, and the 5 + the 7; the possible cases, 45; so the result would be 3/45).

GOOGLE MAPS

Google Maps is a web mapping service developed by Google. It offers satellite imagery, aerial photography, street maps, 360° panoramic views of streets (Street View), real-time traffic conditions, and route planning for traveling by foot, car, bicycle and air (in beta), or public transportation (Google Maps, s.f).

This resource is very useful to work and improve some areas of education, such as geography and history, but for mathematics too. It involves the measurement as an important part of its utility, in terms of length and time. It is very profitable to know how much time we are going to late to go from one side to another (from home to the city centre, for example) and very easy to use it; we only have to write the place of destination and the place of destination, and the program offers us a range of ways to arrive there automatically, specifying the distance and the time. The image represents the way that Google Maps recommend us to go from Carranque to Constitution Square, in Málaga, indicating the time and diverse alternatives. 

In addition, we have the option to personalise our own way, indicating the streets and the squares for which we want to go; thus we can know the metres or kilometres of the drawn path. To carry out this option, we must click on the right side of the mouse and click again on “Medir la distancia”; then we only have to click on the preferred streets and to turn right or left. The image shows us an own route that I have designed to go from the Spanish Parliament to the Senat, in Madrid. 

In my opinion, Google Maps is excellent for learners. An activity to implement in class (second and third cycle) would be to calculate the metres they walk in a day. For that aim they should draw the route from their home to the school, the park, the academy… They would finish adding all the distances. Another activity, including the geography, would be designing our best journey. In it students ought to write the cities in Google Maps where they would go to see the kilometres and the time between places. Even they could add the prices of the planes (the program also tell us); in this way the calculations on money would be integrated too.

The benefits for children are great: they could improve their mathematical competence, with contents about measurement, such as distance and time while they use digital devices. Moreover, interdisciplinary contents, like geography, would be included, producing a more significant and deep learning.

References: 

Google Maps (s.f.). En Wikipedia. Recuperado el 29 de abril de 2019 de https://en.wikipedia.org/wiki/Google_Maps

THE SPORTS AND THE STATS

The statistics and the sport are strongly linked. In all the sporting disciplines the stats play a fundamental role in order to analyze the efficiency of the performances, and they are very useful to interpret the tactical management and to obtain conclusions. That numerical information is represented by percentages, comparison, averages, graphics… I am going to focus on two sports I like especially: football and basketball.

In football, statistics are concerned to observe both of the individual and collective performance. For example, from a group perspective, the quantity of passes and shoots indicate us if a team propose an offensive or defensive playing style, and we can compare it with the other clubs in order to make comparisons. From an individual point of view, the number of goals and assists of a player can induce us to think it is good or bad. For more advance information, the heatmaps help us to know which the most preferred position of a player is on the pitch. This image corresponds Messi’s movements on the pitch in a match in 2015.

In basketball the stats are primordial and more influential than in football: points, rebounds, assist, blocks, shot percentage, etc. These numbers are taken into account in naming the MVP of a competition. To give concrete examples, the best 2-points shooters are usually players who play near the basket (the tallest players), as well as the rebounds and the blocks are dominated by them; the best 3-points shooters are often short players. The table shows us the stats of a team in different disciplines (minutes, points, rebounds, assists…) of the play.

The most common hobby for Primary children are usualy the sports; for this reason it is a good idea to use it as a context in order to motivate them in the learning of mathematics. From my point of view, a useful activity (third cycle) could be the representation of different statistic information in graphics and finding the average. For example, the teacher would provide the number of points and rebounds of a player in ten games. Students would have to add all the points and rebounds to find the average, and then they would make a dot plot to observe the progression of their performance in points and rebounds (it would be compulsory to draw a line for each statistic discipline). To expand the activity, it could ask the best and the worst performance of the player and analyse them critically, thinking if the player is improving or getting worse observing the progression. 

THE €URO

The euro is the currency of the European Union, The are 343 million of people who live in the 19 countries of the eurozone, so  it is the second most negotiated currency in the world, after the dollar. 

The 1st of January 2002 the coins and bills were introduced in the European societies. Eight coins and seven bills were designed, and all of them are identical in the eurozone countries. The symbol (€) inspires in the Greek letter épsilon, which is the first letter of the word “Europe”, and it is crossed by two parallel lines that indicates stability.

However, they are some countries that don’t use it, such as Poland, Romania or Sweden. In addition, there are microstates that have an agreement with the EU to use the euro, like Monaco or Andorra. Finally, there are other countries, such as Montenegro or Kosovo, which have adopted the euro in a unilateral way. 

The euro has an exchange rate with the previous currencies of the eurozone countries. For instance, an euro is equal to 166 Spanish pesetas, 1’95 German marks, 1936 Italian liras or 6’56 French francs.

A feasible activity to work with third cycle students would be applying the rule of three in order to know the value of the past currencies of eurozone countries with regard to a particular amount of euros. For example, they could find how many pesetas 10 euros are, and the same with other currencies. Moreover, the teacher could plan a similar activity to compare the euro with the dollar (1 € is 1,12$ currently). These exercises are justified in the legislative framework, which has as evaluation criterion "conocer el valor y las equivalencias entre las diferentes monedas y billetes del sistema monetario de la Unión Europea".

Finally, other alternative would be to create a supermarket (second and third cycle), in which learners would sell and buy products. In this way they practice the mathematical operations mentally, boosting their mental arithmetic. To carry out this activity children could take the boxes, tetrabicks… they have at home to recycle; thus we provide a didactic utitlity to the recicled materials. Regarding the coins and bills, at the schools that kind of resources usually exists, which is given by the publishers.

MY INTERVENTION AT THE PRACTICES STAGE

In my practices stage I implemented an intervention proposal about the measurement. For carrying out it I planned several activities on units of measure, using measurement tools, ICT resources… Pupils who developed it were in the 4º year of Primary (9-10 years old).

I started with an initial evaluation to verify what they remembered on the length and the capacity. Later, I introduced the contents through different questions like “if we move towards the left, do we multiply or divide?”

I did a crossword and an alphabet soup. The crossword consisted of ten questions, two were to write the principal units of measure of length and capacity, and the rest were a series of operations that they had to answer in its respective numbers and boxes. Every number had a phrase and there was a missing word (the words that were missing were units of measure). They had to do the calculation, which was in the phrase, to find the unit of measure . The alphabet soup was easier. Learners had to look for units of measure in it. When they find six units of measure that exists in the soup, they related every unit to the object, element or living being that they thought.

Regarding measurement tools, students worked with the measuring tape. In pairs, they chose what object they were going to measure. Later, they measured that object and took notes about how many centimeters it had. Finally, they had to represent that information with another unit of measure. In this way, their relevance is higher because they must be who obtain the information of the activity, analyze it and operate with that information. It is not like in a textbook, where they only have to do calculations. The aim is that they improve their cooperative skills and that they see the whole process of the information: from its extraction up to its analysis and manipulation to obtain results.

Finally, the ICT resources were games on the Internet and Plickers. On the Internet pupils had to prove what unit of measure was most adapted for the object or living being that appeared on the screen. The final day of the intervention I used the app Plickers to review what they learned during the intervention. In pairs, they debated together and gave a response to every question between the options that I provided them. As we can observe in the image, they were questions that they knew. There were both of easy and difficult questions. 

The main goals I pretended were bosting their cooperative skills, their mathematical competence in these contents and their individual concentration. The results were very successful for me, I was so proud of the learners’ engagement and how every activity was carried out without big problems.

DIDACTIC MATERIALS TO LEARN THE CHANGE OF UNITS

The change of units of measure is one of most important learning that Primary students have to internalize when they treat the different magnitudes of measurement. From kilogrames to grames, from metres to centimetres… are several the kinds of changes that students must learn.

In the Andalusian Primary curriculum we can find that it requires this learning throughout the three cycles. For example, in the second cycle (although for the third cycle is valid too) there is a content which says the following: “3.7. Comparación y ordenación de unidades y cantidades de una misma magnitud”. For that reason teacher must get this aim. How to obtain this learning? The methods and techniques come into play in this point. Then I share with you two activities who can be very useful for children.

Firstly, the cube of units. Each face represents each unit of measure. A feasible activity would be giving a series of quantities and ask children to transform them throwing the cube. They should represent the numbers by means of the resultant units. For example, they have the quantity 1.500 litres and throw the dice. The new unit is the kilolitre, so they ought to to divide 1.500 between 1.000.

The second activity consists of pairing quantities which have the same value, represented in different units of measure. For that aim, it is neccesary to design pairs of cards. The cards are upside down and children have to turn two cards. If those two cards shape the relationship between the same quantitative value, they have to leave them face up; if it is the opposite situation, they must to come back to put them upside down. In this way, the teacher can verify if learners know to transform the units keeping the same value. Pupils group in four to carry out the activity, two of them play (they must talk together to guess the pairs) while the others are the ‘referees’ and ‘teachers’ (they have to control that those who are playing don’t cheat and guess the pairs), and viceversa.

I believe these games would be very positive for students to get these procedures in the matter of measurement. In addition, they would acquire other learning like the cooperative skills.

APPS TO LEARN MEASUREMENT

The new techonologies are changing our lifes, and the teaching is not an exception. The methodologies are being modified, and one of those causes are the technological devices. There are different IT resources to learn Mathematics in Primary Education, and the measurement concretely. Three apps to learn measurement are detailed below.

The first application is ‘Cambio unidades’. In it children can practice and improve their abilities regarding the lenght, volumen and mass. They learn all the different units of measure (metres, kilometres, litres, kilogrames, grames…) and what calculation they must do to change it (multiply by 10, divide between 100…). In addition, students can add or subtract two quantities (15 cl + 76 dl, for example). Finally, there is a challenge, in which the app ask questions to calculate mindly (how many half-metres are 8 metres? for instance).

The second has as a name ‘Matemáticas 10 años’. It is an app which treats lots of fields of Mathematics. One of those is the measurement and the change of units. The app requires eight right answers to overcome the game, and there are only ten. In other words, players only have two options to fail. At the third failure, the game is over and players must start again. The style of questions are like this one: A bus travelled 30 km in a hour. At the same speed, hoy many hours will it do in three hours (60.000 m/9.000 m/30.000 m/90.000 m). Players have to choose one of the responses.  

The last game is ‘Matemáticas 7 años’. It is produced by the same creator than the previous one, it only changes the level. There are mini-games on lenght, mass and capacity. The format is the same than ‘Matemáticas 10 años’, a question and a right answer between four possibilities, and there are options to fail (two or three, depending on the mini-game). An example would be: 9 l + 6 ½ l = _ (8 l/15 l/12 l/10 l). As the example shows, the level is minor, logically.

The new technologies provide children new ways to learn measurement at home, reinforcing their knowledge in this field of Mathematics.

THE TANGRAM

The tangram is a game whose origine is Chinese. It is formed by seven poligonal flat pieces, generally made by wood, with which it must create figures without overlaping them. In the early 19th century it was carried over to Europe, and it became very popular in the continent for a time then, and then again during World War I. It is one of the most popular dissection puzles in the world. 

The tangram provides lots of benefits: it promotes the development of intellectual and psychomotor abilities, stimulates the creativity and the formation of abstract ideas, fosters the logical-mathematical intelligence, improves the eye-hand coordination…

There are only thirteen convex shapes matched with the tangram set. Children must take into account, for example, in the equivalence  of several pieces, gathering the two small triangles, learners can build the square, the rhomboid or the medium triangle…

Primary students can play with the tangram to differentiate the area and the perimeter; in this way the would understand that a same area is not equal to the same perimeter. In my practices stage I could observe that misconception, some children believed that two polygons with the same area had the same perimeter, and I had to show them that they were wrong. I gave them an example so that they could understand in a notebook, but the tangram is the best tool in order to solve this problem, in my opinion.

An entertaining activity for any cycle would be to give freedom to learners so that they create the figures they want; thus they foster their imagination and creativity. Once they have formed it, they could measure the perimeter and the area, and they could correct the misconception I have detailed in the previous paragraph.

Playing with they tangram is very enjoyable for Primary children. They can practice and play with a didactic tool in an funny way while they are learning meaningful concepts about the measurement. Without any doubt, a great invention.  

THE ACRE

The acre is an unit to measure the area, used in the agriculture of several countries. Depending on the country and the time, it corresponds to several surfaces, generally between 0,4 and 0,5  hectares.

It is traditionally defined as the area of one chain by one furlong (66 by 660 feet), which is exactly equal to 10 square chains, ​or ¹⁄₆₄₀ of a square mile, and approximately 4.047 m².

The acre was roughly the quantity of land tillable by a yoke of oxen in a day. This explains the acre as the area of a rectangle with sides of length one chain and one furlong. A long, narrow strip of land is more efective to plough than a square plot, since the plough does not have to be turned so often. The word "furlong" itself derives from the fact that it is one furrow long (Acre, s.f.).

The acre is commonly used in a number of current and former Commonwealth countries, like United Kingdom, the United States and Canada. In addition, almost all countries of the former British Empire used it, such as Ghana, India or the Caribean islands where agricultural lands is measured in acres.

I understand that it would difficult to Primary students. Those ones who show problems to manage our system will not like the acre because it is more complex. However, it could implement a little activity (third cycle) in which they could go to the playground and measure it with their feet. They would compose small groups, and while one is measuring a side, another could measure the opposite side, and so on. At the end, they would write down the number of feet and, in the classroom, they could present their results with all their classmates. Thus they operate through estimations, not by means of an accurate system, dealing with other measurement models.

References:

Acre (s.f). En Wikipedia. Recuperado el 17 de marzo de 2019 de https://en.wikipedia.org/wiki/Acre

INTRODUCTION

Hello and welcome to my blog! I am Andrés and I am a Primary Education degree student at the University of Málaga. I pretend to learn the measurement in a didactic way; for this reason I will search information to get my aim. In addition, I want to share it with all of you.

Didactic Measure is a blog whose principal aim is to make readers conscious of the influence and the significance of the measurement in our life and to learn to teach it in a different way, through curious methods and a variety of tools and materials and by means of real contexts. The measurement, the statistic, the probability... are very present in our everyday life: in our houses, in the sport, in the society, at the school, in ourselves… In every corner of the world, the measurement and the other sciences are implied. That's the reason why learners must acquire those knowledge. In my opinion, they have to understand what it surrounds them from a mathematical perspective. For that goal, I will describe different methodologies and activities to teach measurement, didactic applications, representations of the measurement and the other sciences in real life, curiosities... 

In the following weeks I will upload several entrances related to these contents and feasible didactic applications. I hope you like it! Come on!