Social Math | Differentiation at Home
How do you teach all your kids at one time for math? Lately, Sonya has criticized me for not sharing my little secret. It isn’t my secret though. It is really Gattegno’s.
We need to remember the key to why we are expanding so rapidly in technology and math. It is the ease of collaboration through lightning fast communication via the web and transportation.
The key to doing math at one time with all my children is that I take advantage of collaboration to facilitate learning.
In the last online math session we had recently with Sonya from Arithmophobia No More, we dived into the manual on Teaching Fractions that we received at the most recent online Gattegno conference.
Notice and wonder is not always apparent in Gattegno’s textbook, but in the manual on Teaching Fractions, notice and wonder becomes apparent. But in the midst of the online math session, one parent brought up their own personal struggle to notice and wonder.
I can completely relate to this. When I first started with Gattegno, I too had a hard time with noticing and wondering. It was seeing Anne Fetter's notice and wonder video that I really began to understand the concept. But what really set my mind in motion was working together with my children on their first notice and wonder project, their own artwork.
In math education today, there is a need for each student to travel the path of learning on their own and prove themselves individually. It is as if sharing and building upon each other's learning would some how ruin the individual child's growth. But if society has gained much from collaboration and social learning, how much more a child?
My Experience of Social Math
Cuisenaire rods and social math activities are key to how I teach all my kids at one time. I became more aware of the benefits of using social math activities when we did our first notice and wonder activity.
Each notice and wonder of each kid fed into a notice and wonder of another kid that they began to build upon each other making amazing discoveries of patterns.
It was a rush of excitement where we often talked over each other. I am sure that not everyone gained the same awareness from the activity. Yet, each child grew in their understanding of math as a study of patterns.
Current education is adamant that each child must set out on their own individual path and prove their own individual worth through digesting and regurgitating a set of information. All of this must of course be accomplished under an overbearing constraint, the timeline. (I could rant about this timeline but let me stay focused).
The Perfect Path
Numbers have such a vast array of relationships and patterns that how could there ever be a perfect timeline by which to discover them all. You can go into so many exploratory paths and how can one path be defined as better than another?
And what person does not retrace a path that brought them delight? Yet, we ask kids to keep moving forward, rushing them down the next path. There is no lingering to play a little longer with an idea. No, that overbearing timeline is breathing down our neck pushing us forward and we must drag our kids along.
In my experience there is only one path that has the greatest value, and it is the child's own path which is filled with meaningful experiences and filled with delightful treasures special to the child. But of course a path greatly benefits when it is intertwined with other paths, and this is where social math activities can inspire and shape a child's path.
A Free Social Math Activity
The recent free activity that I posted HERE has shown me again the benefit of social math. I don’t want to talk too much of the activity because while I love social sharing, too much sharing is like ruining the ending of a great book. I don’t want to ruin this activity for you.
However, I do want to share that children of varying levels of awareness can benefit from working together in this activity. In this activity, you find all the ways to build numbers one through ten and discover the interesting patterns that reveal themselves as you record your answers.
This activity is more of a project. It is quite extensive and you should plan to just do a little each day over a couple of weeks. Children from ages 4 to 80 would benefit from this activity, so it makes a great family project.
How I differentiate Math Lessons with Gattegno
This activity is a prime example of how I differentiate a math activity so that all of us (the family) are participating in social sharing in a meaningful way.
For most math gurus, differentiation means changing the activity just slightly to accommodate for those who aren’t yet on the same awareness as everyone else. For me, the key to differentiation isn’t changing the activity, but letting the kids participate in the activity at their level of awareness.
For example, the activity of recording all the number of ways to build each rod from white to orange is based off the mat building activity Gattegno sets forth from the beginning of Textbook 1.
This means a child who is just beginning Textbook 1 can participate on some level with the older children who may be in Textbook 2 or higher. Using Number Building play mats, a young child can add their own trains to the mat and participate as they are able.
The six-year-old cannot possible find all the ways to build every number. This is exhaustive work that takes a certain amount of stamina and awareness. The young child can though make predictions in the activity and they may surprise you in their predictions.
Great questions to ask throughout the activity are
- "Based on what you have seen so far, how many 4 car trains the length of yellow can we make?"
- "Did you find all the 4 car trains?"
- "How do you know?"
- "Do you see any patterns so far?"
Appreciate the struggle of answering these questions and feel free to move on without an answer.
The Benefits of Social Math Activities
The benefit the younger child gains by participating on a small level is tremendous. They become aware that there are many more ways to build a number. They notice that each number has a finite set of ways that they can be built (a fancy term is permutation).
The older child also gains valuable experience in organizing thought. One of the greatest questions in this activity is, "Did you find all the ways to build the number?" For smaller numbers, this is an easy task, but as the number grows, the question becomes more difficult. The child learns to organize their mats to discover the answer to the question.
This is no easy task, even for an adult, and you will find yourself grappling with organizing your mats to be sure you have the exhausted discovery. This struggle is a great struggle for young and old alike. It cultivates a need for thoroughness and organization.
It should be noted that if you find the older children struggling too much with the exhaustive discovery and that their organization of thought is lacking, it is okay. I even advise that you assist them. This is a social activity of collaboration, and they will not be ruined by your assistance.
Of course, I did tell my oldest to hold back his discoveries until my second oldest made the discovery. Ever child needs different amounts of time to make discoveries, and I don't think it hurts to hold back sharing for a few minutes or even a day. Letting one child dominate an activity could effect how the collaboration plays out, and we, knowing our children, must facilitate accordingly.
Repeat, Repeat and Please Repeat
Repeat this activity yearly because overtime, I am sure they will gain the experience to need you less and less to exhaust discovery. As they repeat the activity, I imagine the rediscovery of the patterns will become more and more meaningful.
A natural by-product of our social math exploration is that the younger child’s awareness is expanded more quickly, but the older child is challenged to continue the expansion of their awareness. All that the older child discovers and shares is not always memorable to the young child. In time though, the younger child rediscovers the concept on their own terms and the understanding becomes more solid.
We know children exposed to richer vocabulary through the environment of sophisticated adult conversation have a richer learning journey . The same must be true with mathematics.
It may be seen that the younger benefits the most from social math activities and this is how life works. It is when we abandon sharing and collaboration then society fails the next generation. And maybe this is why education in America has fallen because the benefits of sharing and collaboration have been replaced with individual grades and test results.
Differentiating other Gattegno Exercises
All the basic math activities remain the same in Gattegno but the awareness of the student is expanded through variations of the activities and the wondering and noticing of patterns that reveal themselves in the activities. The last activity demonstrates how each child can participate in different levels with mat building. Let's explore other activities.
Building trains is a core activity and the many variations of this activity help children to see the many ways numbers can relate to each other. Solid color trains reveal multiples, fractions and division. Should a child see all this the very first time they do this activity? No, but hearing what each child sees, all the children will begin to develop an awareness for each concept.
For a child just starting to play with staircases, he may notice simply the difference between one stair to the next. However, an older child may notice that certain staircases reveal other interesting relationships like multiples. Even more, another child may notice squares within staircase structures. They can share in the simple activity of building staircases but each child expands their awareness on their own level.
Another activity is the substitution game. In this game, each child can substitute a number for two numbers of the same value. How the child expresses these values depends on their own comfort level and understanding of math structures. However, the child gains even more by being exposed to more sophisticated substitutions. I always participate with my children to model new and interesting equations.
What if I just have one kid?
If you have just one kid, I highly suggest you doing these activities with your child. Share what you notice and wonder and let the child participate as they are able. You will not ruin them by sharing what you see.
Let this FREE activity spark a passion within you for math and if you find your noticing and wondering lacking, find another adult to work on this activity with you. Hopefully, a curiosity for math will be stirred in you.
Such curiosity is contagious. Curiosity is a major driving force to true learning. As I have tapped into my children’s curiosity, I have found my task of teaching easier as resistance is reduced. Their energetic curiosity powers their learning that my own energies are not needed to push them forward.
I think that this is the point of noticing and wondering. I don’t have to pour my energy into teaching. Instead, I just have to act like a rudder and steer their curiosity with good questions and fun tasks.
I hope you enjoy this free activity. I personally have plans to bring this activity out at least once a year for our family to tackle. As their awareness grows, I am sure they are going to find more and more interesting patterns. I have seen it spark a wonder in them about math and that is a win in my book.