How can Teachers use ZPD to Help Students Climb?

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

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 4th 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.

Learning to be Comfortable with being Uncomfortable

(If you are reading this blog first, you might want to stop and read the one published before this one. Also, I have a disclaimer to add to yesterday’s blog. Although God provided for the Israelite children when they whined and complained, we know from later chapters that this was not always his response.)

My experience has been that most teachers go into education because they want to help students learn. Yet, as stated in the previous blog, we live in a society where we avoid making others uncomfortable at all costs.

The ‘best’ math class I ever had in college was Introduction to Higher Mathematics – a proof class. We had no homework, no tests and no final. As long as you participated in class adequately you got an A. We all loved it…until I discovered that I was actually supposed to learn how to mathematically prove something and I would be expected to do so in every math class that followed. I liked the easy path and did not complain about it while I was in the class. But you can bet I complained in each of the subsequent mathematics classes. I did not have the necessary foundation upon which to build future understanding.

I am quite sure that many of you can relate to “lacking the necessary foundation upon which to build future understanding” when it comes to mathematics. It is my belief that everyone will hit a wall at some point when it comes to understanding mathematics. The only question is when the wall will show up. Even full time mathematicians may spend years trying to understand a piece of mathematics that’s just beyond their reach.

When faced with the mathematical wall you have 3 choices – stop, go around it, or climb over it. Those who stop are the ones who develop self-helplessness, fear math, hate math, and/or avoid lifelong dreams because of math. Those who go around the wall are those who choose the easy way out – either intentionally or unintentionally. These are the people who seem to do well in mathematics. They are not afraid of it; as a matter of fact, they might even like it because they feel successful. I was this type of person. When I hit a wall I found success at just paying close attention to the rules that the teacher was giving. I knew that if I tried to follow the patterns just as the teacher had, then I would be able to get around that wall. The last group of people is those who choose to climb over the wall. Just like there are some people who are born with an athletic desire, there are those who are born with a desire to understand the relationships between mathematical concepts. I was not born into this group.

I believe people can learn to make a different choice. I have a family member who was the stopper. She was able to go around some walls, but always dreaded the next wall and rarely believed that she could get around it. As an adult she was faced with a dilemma – stay stopped in front of a mathematical wall called ‘The Test’ or she could choose to try to climb over the wall and follow in the career that she felt called into. She chose to climb, developed the understanding necessary to scale that wall and ‘The Test’ no longer stood between her and her dreams.

I am an example of the person who always walked around the mathematical walls until I arrived in graduate school. It was at UGA when I discovered that I would not make it through the PhD program by walking around mathematical walls. I was going to have to climb over some of them and boy were they hard to climb. I quickly learned that I couldn’t climb by myself, I needed help. We would spend hours (and sometimes hours and hours more) working together to understand a concept and guess what would happen when we found ourselves on the other side of the wall? We would rejoice! We would high five. We had worked hard, fallen, gotten scrapes, and sometimes even fights but we would always end rejoicing together. Then we would take a deep breath, start again and moan when it was time to climb another wall, but we didn’t stop. Mathematically I have stopped climbing walls but I have gotten so much better at understanding the relationships within mathematics in the early grades that I can pretty much leap over the old walls without much effort.

My cousin and I had a reason to try to conquer these mathematical walls; we had a long term goal that required us to. Young children usually do not have those long term goals to motivate them to try to climb over mathematical walls. In many cases these same children are also in classrooms of teachers who mean well but who try their best to protect their students from struggle, from discomfort, from frustration, and unknowingly from understanding.

This blog has become much longer than I thought. I have not completed my thoughts, but I need to stop. Look for the next blog where I continue my thoughts on discomfort and understanding…

Protecting Others From Discomfort – Is it Wise?

This is an idea that has danced around in my mind (and heart) for about 3 months now. It seems that everywhere I turn something causes me to contemplate the question “Is it wise to protect others from discomfort?” One thing I have determined is that this question can apply to so many areas of life – a parent protecting a child, a teacher protecting a student, God protecting his children, friend protecting a friend, benefactor protecting the recipient, etc… My blog began as a result of this question burning in my mind; yet, I couldn’t put anything in writing. It wasn’t until I typed the list of situations to which this question can apply that I realized why the words were just out of reach. I couldn’t separate my thoughts; the list above just separated them for me. (Hang with me; if you don’t already see how this idea applies to teaching mathematics to everyone, you will.)

Protecting others is a natural instinct for me. I never thought much about it, but in retrospect I think I thought it un-Christian NOT to help when I saw a need. I guess it was this summer as a participant in a mission trip that I began to question this ‘help without thinking’ reaction to every need I saw. I was required to read a book “When Helping Hurts”. I won’t go into the details but I now admit that in many instances when I thought I was ‘helping’ someone, I was actually ‘protecting’ someone from discomfort. How is someone going to learn to make the necessary adjustments in their budget if regular living expenses are being paid by other(s)? How is an alcoholic ever going to realize he/she needs help if others keep making excuses for the addiction? How is a college student to make wise choices living out on their own if they were never allowed to make their own choices while living at home? How is a student going to feel like he/she can succeed if always protected from failure? How can a person realize their need for a Savior in Christ, if we unknowingly become their savior and protect them from consequences and discomfort?

Yesterday in church the pastor continued his series in Exodus. I’ve heard this story numerous times. The children of Israel had just crossed the Red Sea on dry land and watched the Egyptians drown. They stopped on the other side of the river and sang a song of praise to God. Yet, in the very same chapter (Exodus 15), they began to complain – about bitter water, about being hungry, about having no water. And each time after their complaint God lovingly and miraculously provided for their needs (Exodus 15:22-17:7). Every time I have read/heard this story my thought was why were they so stupid? And why did God give them what they wanted each time?

Yesterday I found the answers. The Israelites were so stupid simply because they were human, just like we are. The other thing I learned was profound. My pastor said that God could have provided sweet water, abundant food, and abundant water for the Israelites and ‘protected’ them from the discomfort of drinking bitter water, being hungry, and being thirsty. Instead he allowed those discomforts to exist in order for his children to learn. They needed to learn that there is something more important than food and water and it could only come from God. He was teaching them to Hear, Trust, and Obey (Deuteronomy 8:2-3).

Think about what you have read so far and see how you can apply it – to your life, especially to teaching, and particularly to teaching mathematics. I’ll be back tomorrow. (The strategy series will continue later.)

Part 2 of ‘using strategies based on properties of operations’…

In my last post I assigned two problems for you to solve mentally and then think about the properties being used in your strategy.

Compensation or Friendly Numbers        58 + 36

To solve this problem mentally using the strategy of compensation, I chose to make one of the given addends into a friendly number to work with in my head. I know that I can add 60 in my head much easier than I can a 58. So I’m going to move 2 from the 36 over to the 58 and now my problem is 60 + 34 which is 94. Easy, huh?

Now let’s investigate why this works. First I have 58 + 36 = 58 + (2 + 34). I have simply decomposed 36 into 2 and 34, knowing that I need the 2 in order to make a friendly number. Now using the associative property of addition, I add the 2 to the 58 instead of the 34. (58 + 2) + 34 = 60 + 34 = 94.

Breaking apart into Place Value                 158 + 221

I probably don’t do this problem exactly as the strategy calls for, because to me it makes more sense to not break apart the first number. I know that the answer is 379. How? In my head I look at 221 = 200 + 20 + 1. Then I add the 200 to the 158 and get 358. Next I add the 20 to 358 and get 378 and finally I add 1 to 378 to get 379.

What properties of addition did I use? I’m not even really sure. I guess it could be called the associative property of addition. 158 + 221 = {[(158 + 200) + 20] + 1} = 379. If you have any other idea(s) please share them. I really just used what I knew about place value to solve this problem. If you look back at those Common Core mathematics standards, you’ll notice that students are supposed to solve problems based on their understanding of place value and the properties…

Let’s look at some of the subtraction strategies next. So your next assignment is the following:

335 – 219                             413-135

Remember to see if you can determine what mathematical properties, place value understandings, or relational understandings you are using to mentally solve these problems. Go back and try to solve the problems a second time using a different mental strategy. I’ll be back in a couple of days.

For Teachers’ Eyes Only

Throughout the Common Core Georgia Performance Standards for Mathematics in K-5 you read the phrase “based on place value, properties of operations, and/or the relationship between addition and subtraction; as well as multiplication and division”. What does this mean?

Let me provide a context to which we can all relate. Most all of us were taught “THE” properties in one to two days each year we were in school. There was one page in the book that described each property – commutative, associative, distributive, identity – and provided one to two examples of each. Then on the next page in the book we were given numerous examples and then we had to record what property was being presented in each example. Not only was it boring, but rarely did we see a purpose.

So when I taught 6th grade I decided to make it more ‘exciting’. I typed up a list of examples, cut them apart, handed one out to each student then had them walk to the designated part of the room for that property. Yes, students may have enjoyed it a bit more, but they did not really make any connections between what they were ‘matching’ and the other problems they did in mathematics the other 178 days of the school year.

I recently taught a workshop on how Number Talks help students develop understanding of these properties. I stumbled a bit on the first time of presenting this idea, but improved with practice and additional thought.

Doubles/Near Doubles    999+999

I honestly do not know what 999+999 is without thinking about it. But I can determine it rather easily. 999 is really close to 1000 and I automatically know that 1000 + 1000 is 2000. So now I know that 999 + 999 is 1,998. Wait a minute, how did that happen?

999 + 999 = 999 + (1 – 1) + 999 + (1 – 1)  Behind the scenes I added a ‘zero’ to each 999         without changing the value of the problem. This can be done because zero is the Identity Element of Addition. Zero does not change the identity of any number that it is added to.

999 + 999 = (999 + 1) – 1 + (999 + 1) – 1 This is where I said, “I know that 1000 and 1000 is 2000.” But what went on behind the scenes? After I added the 0 to each 999 in the form of (1-1) I then moved my parenthesis to add in a different order, using the associative property of addition.

999 + 999 = 1000 -1 + 1000 – 1 = 1000 + 1000 – (1 + 1) There are two properties that I used to write this equation string. I swapped the first -1 with the second 1000 using what is called the commutative property of addition. Then I should have been left with -1 -1. Where did            – (1 + 1) come from? I could have thought in my head “2000 minus 1 is 1999 and then minus 1 again is 1998.” But I actually thought in my head “2000 minus 2 is 1998”. I was using the distributive property of multiplication over addition. If you can remember back to algebra the negative is to be distributed to each term within the parenthesis – ( 1 + 1 ) = -1 -1.

In this one simple problem I used 4 mathematical properties ‘behind the scenes’. Why do I keep putting it like that? Teachers need to know what is ‘allowed’ and ‘not allowed’ so that they can assess on the spot whether a student is using a mathematically valid strategy. Yet, the students do not necessarily need to know the names of these properties depending on their level of understanding.

So when a student talks through the problem like I did originally, they have in fact used the CCGPS for their grade level that says students should use strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division. I have discovered that the difficulty associated with these particular standards is that teachers may not always recognize unspoken properties. I hope that this series of conversations will be helpful.

I stated that I do not believe that students need to know the names of these properties; however, when a student changes a problem by switching the placement of two numbers the teacher should ask why they did that and does it change the value of the problem, why or why not? After this ‘switching of numbers’ becomes so commonplace in the classroom then the teacher can introduce the official name of commutative property of addition (or multiplication). I know that tacking on ‘of addition’ or ‘of multiplication’ is time consuming but if you consistently use the whole name of the property once it’s introduced at some point one of your students is going to ask, “Is there a commutative property of subtraction?” What a beautiful question! Respond with, “That’s an excellent question. Talk with your neighbor about this question and see if you can determine if there is such a property and be able to explain why and why not?”

I will continue to discuss some of the Number Talk properties in following posts. But in the meantime, you have homework.

Solve the following problems mentally and record your thinking. Then go back and look at your recorded thinking and see if you can determine how you used strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; multiplication and division.

a. 58 + 36     b. 158 + 221    

I’ll assign some subtraction problems next time.