I used to have an book (in English) written by a professional which would surely have the correct terminology but this is the best I can do from memory.

The process being described is for renewing or "activating" the starter (masa madre). From what I can tell, the "pie de masa" described is the activated starter that can be used for baking.

"At the 4-hour mark, the volume in the jars had increased and it was fully active and ready to use in the next step. I forgot to take a photo of the fully active starter in the jars. I was ready to move on."

I have read the discussion entries and this is what I have found. I hope it helps!

Séptimo día. (Primer refresco)

Preparación para el refresco del iniciador.
Tomar el iniciador conseguido el día anterior.
Añadir la misma cantidad de harina de fuerza que el peso de la masa.
Incorporar el agua, la mitad del peso de la harina.

Ejemplo.

Iniciador 1000 gr.
Harina de fuerza 1000 gr.
Agua a 20º C 500 gr.
Amasar, refinar, enrollar y sumergir en agua a 20º C en invierno y a 4º C en verano, dejar el recipiente a temperatura ambiente hasta el día siguiente.
(A esta porción de masa la llamamos pie de masa madre)

http://www.cetece.net/joomla/index.php/elaboraciones-del-cet...

Pâte fermentée (pie de masa): masa de pan sin cocer de la hornada anterior que se guarda refrigerada para el día siguiente. Es masa de pan, como tal tiene sal y los demás ingredientes. Es una forma sencilla de acortar la primera fermentación sin que el pan sufra una merma de sabor.

http://www.elforodelpan.com/viewtopic.php?t=20

It is a French term that means pre-fermented dough, or as it is occasionally called, simply old dough, scrap dough or pre fermented dough, invented by Professor Calvel, at ENSMIC in Paris, France. It is done by taking small piece of fermented dough saved from one made earlier in the day or from the previous day's batch and then, adding it to the very end of mixing to today's dough. The amount to be added may vary around 15% of the weight of the total formula flour, or sometimes a bit more. Then, a portion of today's newly mixed batch is held and fermented for the next day's dough. Professor Calvel notes that "taking dough from one batch and adding it to a subsequent batch cannot be done indefinitely, or even over generations, without the danger of producing undesirable flavors."

The old dough piece should have a minimum of 3 hours fermentation after having been leavened with 1.5% yeast (based on total flour weight). It can stay at room temperature for up to 12 - 14 hours or be refrigerated for up to 18 - 20 hours.

The concept is this old dough gives bakers an easy way to use a pre-ferment without having to mix a separate preparation in advance. It provides an organic acid enrichment and improvements in both taste and shelf life. The old dough always contains all of the ingredients found in the final dough, usually in the same proportion, being flour, water, plenty of yeast and salt, which allows a reduction in the first fermentation period of bread.

http://baking911.com/learn/ingredients/yeast/pre-ferments

Para la pâte fermentée (masa blanca fermentada, pie de masa o masa vieja):
250 gramos de harina de fuerza.
175 ml. de agua.
4 gramos de sal.
5 gramos de levadura fresca.

http://unpellizcodecanela.blogspot.com.es/2013/02/pan-de-pag...

http://books.google.es/books?id=vTy0JvDgsDAC&pg=PT66&lpg=PT6...

Se elabora específicamente como masa de arranque, a diferencia de la pâte fermentée (pie de masa), que es una masa antigua que se puede obtener simplemente guardando parte de una masa ya elaborada para hacer pan.

http://www.cayetanogarcia.es/cms/la-biga-prefermento/



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Note added at 3 hrs (2014-07-20 16:48:04 GMT)
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Old dough (pâte fermentée) may be made with yeast or sourdough cultures, and in essence consists of a piece of dough reserved from a previous batch, with more flour and water added to feed the remaining flora. Because this is a piece of old dough, it has the typical ingredient of salt to distinguish it from other pre-ferments. Once old dough had rested for an additional 10 hours of age, the French named it Levain de Chef.
Sourdough starter is likely the oldest, being reliant on organisms present in the grain and local environment. In general, these starters have fairly complex microbiological makeups, the most notable including wild yeasts, lactobacillus, and acetobacteria. They are often maintained over long periods of time. The Boudin Bakery in San Francisco for example, has used the same starter dough for over 150 years. A roughly synonymous term in French baking is levain.
Mother dough often refers to a sourdough, and in this context the term starter often refers to all or a piece of mother dough; however, mother dough may also refer to a first-generation yeast sponge; so the process used in relation to the ingredients and fermentation times is important to understanding yeast versus sourdough methods. A roughly synonymous term used in French baking is Chef.

http://en.wikipedia.org/wiki/Pre-ferment

I think it refers to a piece of the original starter that is put aside each day and "refreshed " with enriched flour to to be ready for the following day's batch.