What is ZPD? Education has so many acronyms you almost need a designated webpage just to define them all. The ‘Zone of Proximal Development’ (ZPD) is a term used by psychologist Lev Vygotsky to describe the distance between what a person can do and what they cannot do on their own (Robyn’s translation). Consider the bull’s eye diagram below and the way I interpret the idea of ZPD.

The center circle is what a child can do by herself. The middle ring is called the Zone of Proximal Development. This is where a child can learn but will need some help. The outer ring is currently out of reach for this child. And if you consider moving from the inside circle towards the larger ring, one must pass a mathematical wall or any other kind of wall (discussed in the previous blog).

Ok, so what? Well, let’s say that a child is being asked to practice the subtraction algorithm with regrouping over and over. If this concept is one that this child can do by herself (in the center circle) then the practice will help this child become more adept at correctly solving these types of subtraction problems. [Does this mean the child ** understands** subtraction? This will be saved for a later post.] When a child practices what is within that center circle, the concept will eventually become automatic. Children are happy within this circle of comfort. Teachers are happy when their students are within this circle. And to be honest, teachers as students are happy when

**are within this circle. But we are not born with everything we need to know and understand in life inside that circle.**

*they*What if a different child is being asked to practice the subtraction algorithm with regrouping over and over and this concept is not inside their center circle? They are faced with a wall and have 3 choices: Stop (and wreak havoc in the classroom); go around it to just get by (which means that they probably practice all of the problems but make numerous errors and do a pretty good job of solidifying ** their** version of the algorithm inside the little circle); or they can start climbing over the wall with the help of others.

This is where the teacher role (and parent role) is so important. So the teacher is careful to not assign practice too soon; and the parent is careful to not ‘teach the way I learned it’ too soon. The teacher must know at all times what her students truly understand and what they need help with. This doesn’t mean that the help has to come from the teacher. Students can learn from each other and create knowledge together. In order for this to happen the teacher needs to provide numerous opportunities for her students to develop an understanding of what subtraction really means. (This is why the new standards have moved the subtraction algorithm to the 4^{th} grade – so they have time to hang out in the ZPD with their peers for a while until that wall is no longer a wall.)

As I mentioned in my last blog, there are many walls I would have never climbed ** by myself**. Being in that center ring is uncomfortable. We are raised in a society to be independent and do everything by ourselves. About as soon as a child begins to speak, he/she also begins to say “I do myself”. We need to teach our students (and ourselves) that it’s ok to be uncomfortable and it’s ok to need others.

Teachers seem to be afraid to make students uncomfortable – maybe for fear of students whining, or of parents calling to complain, or of simply moving out of their own comfort zone by pushing their students out of theirs. Teachers (and parents), if your children are NOT having trouble with mathematics, ** that’s** when the teacher is NOT doing what she/he should do. The only time a child should be in a comfortable place is when he/she is doing meaningful practice of a concept they already understand. They are just practicing until they can do it automatically.

And what’s in the outer ring? Probably the idea of multiplication as a way to add large groups of the same amount, or rational expressions, or quadratics equations…

Guess what happens when students spend time together in the ZPD developing understanding together. What ** was** in the ZPD for the child will move to the center circle and something that used to be completely out of reach before moves into the ZPD (this is what I have labeled as ZPD 2).

With a thoughtful and intentional teacher, mathematics students will begin to see walls as a bump, not a skyscraper. They will develop the belief that they will eventually cross that hurdle. It will probably involve working and talking with their peers. It might involve reading what someone else has written. It might involve the student asking questions. But they know that whatever they are struggling with now will eventually become comfortable.

So I will close with my question from the last post: Are you comfortable with being uncomfortable? Are you comfortable with making others uncomfortable?

As you can probably tell ‘making others uncomfortable’ has been moved from my Zone of Proximal Development to my center circle of comfort.