Marta Civil: civil@math.arizona.edu

Rosi Andrade: andrade@math.arizona.edu

The transition from literature to mathematics: I think that it is important to know about mathematics given that it is a subject that exists all around us, that day to day we share in them though we may not realize it; to be able to teach our children to coexist with them, although they believe that mathematics are only numbers when there are so many types of mathematics to discover.

For example, to arrange their bedrooms they need to use mathematics in order to know how to distribute things so that they have greater space.

We as women, even to cook or clean the house (mixing formulas and chemicals), use mathematics.

As a mother I am teaching my sons: To not fear mathematics; That they are not that complicated; That they try to play with them. And they themselves over time will realize that they can use them as a game. That they not be afraid of the teacher that tells them that mathematics must be learned as he or she says, because there are so many ways of understanding them.

Until now, as an adult person am I realizing that there is not only that one strict way of learning them. If I had realized in my own school days that I could manipulate mathematics for myself, I would have had another understanding of them.

Even though the majority of youth do not like mathematics, for me they are the most important and entertaining subject. I wanted to know more and more about mathematics each day, but sometimes in the classroom there are so many students that teachers must work slowly. Although some [students] may know things, they must continue to go over it until all have learned it. In this manner teachers lose much time and cannot focus on a single student to answer the questions of all and give explanations one by one. There are students that because they are left with the doubt, fear or shame or perhaps it is that the topic has passed that they do not ask. In this fashion, when they spoke to me of the talleres matemáticos, it appeared an excellent idea to have a teacher for a few students and one whom we could ask anything -- knowing that it would be answered. I gain a lot of satisfaction from the talleres matemáticos because I am finally coming out of a lot of doubts. I am learning many things in a different way from the one I knew. At last, I also have someone that more than a teacher is a friend and most importantly inspires me: Confianza [confidence, trust] the confianza that I in particular never had with any other teacher of mathematics. To me this is all very important for the future because I know that if one day my child or a nephew asks me something about mathematics I will know it because "lo que bien se aprende jamás se olvida [what is learned well is never forgotten]." And with these workshops everything is learned well.

With our weekly meetings in the Literature Group little by little we integrated the talleres de matemáticas as a necessity, to update ourselves with what our children are learning.

It was also something that began as a space for us as women and mothers. And it has been something useful for each one to apply in her own way in our homes and lives.

This served us as way of knowing our children’s teachers more closely in being able to speak to them and approaching the classroom with confianza and not only when there was a conference.

As the mother that I am of a family of four children, on a personal level I worry about their academic development -- mathematics are one of the most difficult subjects for them and if one cannot help them at home they are even more difficult.

We moved from Literature to Mathematics because it is another one of subjects that interests us in our personal development, we refresh our memory and learn new things that we can apply in our daily living, aside from it being a very different way of now learning Mathematics, than in my own period as a student.

Here in this group there is adequate camaraderie, confianza and we are not competing with anyone. We are all the same and we mutually support one another. Afterwards, we each practice what we have learned in our home, be it with the children or husband. By means of these talleres I have realized how it is that my children learn in School and I realize that it is the same as when I was [in school] because the tradition continues – there is no explanation as to the use of formulas or how one will use them in the future and that makes their learning them more difficult.

Literacy (reading and writing) which is what we began with, is if not the most important, one of the subjects that should be for every student, because by way of them we discover a very different world, including Mathematics.

I continue with Literature and Mathematics at the same time because they help me a great deal with my children and it helps them to develop as individuals who may know how to resolve potential problems in their future.

Talking about myself when I integrated into the grupo de las Señoras, for me the most important foundation was the confianza that each one offered me. For example, they made me feel comfortable, secure and that I could speak without any fear and that I was taken into consideration in whatever comment I made; it is for that reason that I continue in the group. I can say that all that I now know and have learned has been accomplished by means of the confianza. That is why I would like that in no matter what classroom, be it in any other group, that being well received be taken into greater consideration; instead of saying I know or know more than you and placing us to compete with one another. It is in these situations that rejection is felt and one resolves not to return.

We have to place ourselves on a level as equals, as all being human beings, to feel that we are well received – and that it be reflected in how others are received into the group. That is: project what we feel.

When the group was initiated we each brought a little bit of what we knew and in that manner the knowledge has been accumulating and we have learned one from the other. It is a way of feeling that your opinion or idea serves to develop or deepen a subject. And I suppose that arriving for the first at a group, and feeling that you are well received and that your opinions will be taken into account reinforces self-confidence and gives greater security in continuing to participate given that confianza is something very important to learning. For example, it has been awhile that the majority of us as mothers and housewives have had practice with mathematics, and feeling that you are well received motivates us to move ahead.

I believe that the confianza of the group is very important to every human. Thanks to the confianza that exists in the group we can work without problems and pose any sort of question without fear. We can also know one another and support one another mutually, and to think that all this has been achieved thanks to one simple word: confianza.

The collaboration that exists between the talleres matemáticos and the mothers is the confianza amongst both given that it is not about the levels Teacher and student. You can speak, question, express your ideas and concerns regarding what we as women would like to know about Mathematics. The collaboration exists not only here in the group, but also in our homes, given that we practice with our children what we have learned and can help in a way different from the one found in the schools. We form a group with them like those with the women that attend the talleres. We take advantage with them to increase not only the mathematics, if not also friendship and confianza between parents and children, which in these difficult times is being lost or they fear us as they do Mathematics and they do not want to speak to us, they slip from us or they complain like whenever it has to do with Mathematics.

The talleres de matemáticas are for women like a small window to the knowledge that is given to students in school (like our children or nieces and nephews) day to day. They also serve as another way of gathering the family in something enjoyable, educational and fun. It is important to promote these talleres in our community given that unfortunately when they are carried out there is not great parent participation, and we as such should create a consciousness with respect to the importance of involving ourselves in the education of all students; given that the act of teaching should be a mission for home, school, and community in conjunction. And in that way finding the best approach to help students resolve all sorts of problems, including mathematics; fostering a consciousness by means of confianza. With respect to mathematics it would mean taking into consideration the use of logic not only memorization of formulas and procedures without seeing them beyond the classroom.

In whatever angle we are taken into consideration, I believe we are the fundamental part for the development of our children (ours and one another’s), be it in the home or in the community. The world is known through our perceptions of it. As mother, they depend in great measure to the dedication that I give to the development of their abilities (call them speaking, walking, learning to interact with others, etc.).

We should show ourselves and others that woman is not only mother to care for children; wife to wash, iron and cook and comply with her husband and daughter in teaching her to respect her parents. There are those who criticize the woman who is not always at home. Those persons who think that way are not rationalizing well. A mother is someone who knows how to educate her children through her own personal and intellectual development.

December 10, 1999.

Rosi it is Alan A. that writes to you

How are you?

I really liked the mathematics.

And I am going to come again

I am back at school again.

I am happy because Christmas is coming.

What are you going to do for Christmas?

Are you going to go to your mother and father’s?

Thank you for everything.

Have a merry Christmas and New Year! Alan Alcantar wishes them for your

The experiences that I have in mathematics in school are very distinct from the mathematics that I have learned from my mom. What they teach me in school is the same as what I am learning with my mom, but the only difference is that with my mom it is more fun because we have time to talk about things that that I am doing at school just like when I was a child in kindergarten. It has been awhile since my mom and I did this because she did not understand very well what I was doing in school, but now that she is attending the talleres de matemáticas she can teach me other ways of learning mathematics. I think that they are more entertaining with my mom because we have more confianza and I can learn, I ask her questions, I ask her for explanations and I teach her things that she does not yet know or that she learned in a different manner.

I believe that one of the more important things that there should be in schools would be confianza between students and teachers, since there are many students that do not ask questions because of a lack of confianza or fear that they will be laughed at. Teachers like students are human, for that reason I believe that they could learn something from the students if they gave us an opportunity to express our ideas and feelings. The best teachers that I have had are those that have allowed me to express myself and have learned something from me – since I can communicate and have the self-confidence to express myself and not necessarily with individuals with a teacher’s title or a paper that certifies them as such; my parents are these teachers since I have learned with them not only mathematics, but also how to guide myself in life as an individual and they have given me the confidence to be myself.

The ways in which my mother was taught mathematics when she was a high school student are very similar to the way in which I am learning them now in high school; now that we talk and she tells me -- that they did not explain to her either the why of formulas and what they would be useful for in the future in her daily life and that is how I am now. Although now that she is attending these talleres she is learning in a different way, understanding the why of the formulas and where they come from and how they can be applied in her life; she shares it with the entire family and we all get involved in a mathematical reunion that is fun. We are all teachers and students at the same time, there is no difference and that there be much respect and confianza is most important.

When we began these talleres we did not choose which problems to solve. Marta brought us the first assignment: the cube problem [With cubes of 2 colors how many towers of a height of 2 can you form? How many different towers of a height of 3, 4, 5, ... can you form?], which led us to Pascal’s Triangle something that we as students had never heard mentioned.

Afterwards we ourselves asked for what we wanted to learn and we began to extend into area and perimeter for which we used different graphs wherein we discovered snippets of parabolas and hyperbolas that we had also never heard of.

After that we talked a bit about making papel picado [cut-paper art], presenting it to Marta and utilizing it in Mathematics. As we observed the papel picado we realized that there are different figures for which we could extract area and perimeter. It was at that point that we came to formulas for area, circumference, perimeter, radius, diameter, and the most important: p .

With p we investigated the why and where of it. We wanted to know the applications of p in our daily lives, and example of our explorations was:

What relationship exists between the measurement of the head (circumference) with height (stature)?

We realized that the eye can be deceiving because one would never imagine that the head is a third of our height, or if not, it approximates very closely to it or is the stature that perhaps one should have.

In children it would be like using probability given that the head develops before the body and we could calculate the height of them when they are adolescents or adults.

Tenemos un interés en saber cuales son las aplicaciones de Pi en nuestras vidas diarias.

Como ejemplo de nuestras exploraciones, estamos buscando ¿cuál es la relación entre la medida de la cabeza (circunferencia) y la estatura? A primera vista nos damos cuenta que la vista engaña -- ¿cómo puede ser posible que la cabeza sea aproximadamente 1/3 de nuestra altura?

Hemos hecho una pequeña exploración –

Figura A

Ahora estamos explorando con otros datos. Como datos estamos tomando una muestra de varias edades de menores de1-19 años de edad y nosotras (tambien incluye a maridos) – que son:

• nombre
• edad
• circunferencia de cabeza
• estatura/altura

Queremos anotar alguna otra relación a estos datos y de estos mismos datos cogemos la estatura y la multiplicamos por Pi (3.14) – una formula que desarollamos en nuestra primera exploración - visto en figura A. Queremos saber:

• ¿qué es lo que hayamos?
• ¿que relación hay?
• También queremos preguntarnos -- ¿qué se puede saber/decir con estos datos? Por ejemplo a que edad creemos que empieza ser mas real

• ¿cuál es la relación de estatura a tal (x=11, o ?) edad?
• Y por fin, queremos saber ¿que forma tendrá la gráfica de circunferencias de cabeza con estaturas reales y calculadas según Pi?