Organizing Math | Gattegno Homeschool
Getting organized for homeschool is always a trial but when you use Gattegno’s methods, organizing math can seem tricky. The reason is that most curriculum are designed using Greek linear thought which is a chronological path of learning. Gattegno uses more Eastern thought of block logic that focuses on relationships to develop a deeper understanding.
In this blog post, I am going to touch on block logic and its benefits. I will also share our latest math struggle. Finally,I am going to show you how I organized Sonya’s manual with the task cards to take advantage of the block logic present in Gattegno’s methodology.
Warning, there aren't many pictures until you get to the end where I show you how I organized my math binder. Feel free to skip my blah, blah, blah, if you are just here to see how I organize my math. My blah, blah, blah might help you see my reasons for organizing math this way, but you are certainly free to skip to the bottom.
Block Logic vs Step (linear) Logic
Linear thinking comes from Greek philosophy and thought. It is a step by step linear movement that we are most familiar with when teaching mathematics. First, addition. Then subtraction. Next, multiplication. Finally, fractions and division. It is a chronological focused presentation.
However, it isn’t the only way to think, and in fact, ancient Middle Eastern thought and formation of civilization began with block logic. It is very much how Gattegno’s methodology of mathematics is presented.
Dr. Caleb Gattegno presents all the general relationships that exist in mathematics at one time. He uses the superior mathematical tool of Cuisenaire rods to present addition, subtraction, fractions, multiplication, and division all at one time.
Much like the base of a pyramid, it a large block of information, and so contrary to linear thinking that it appears to be an impossible feat. But because of the mathematical qualities of the rods, it was easy for Gattegno to present this large chunk of information.
It takes time to build the foundation, but it is a worthy endeavor. Gattegno builds the student’s understanding slowly and surely through hands-on manipulation of the rods and oral mathematical descriptions. Once the foundation is laid, the structure is observed by the student making it easy to build upward.
What are the benefits of block logic?
Western step logic doesn't provide a construct for students to know where all these math ideas lead and how it is all related. They have to trust the instructor entirely to lead them in the right direction and even make the connections for them.
Block logic is great for observing patterns and this is key for organizing math. Many people take issue with the chronological formation of the Bible because their thinking is that good stories are linear. In Eastern thought, chronology takes a back seat to patterns and relationships. It is patterns and relationships that make for a good story, not a linear set of events.
The Bible is set up using block logic to help reveal God’s character through the observation of relationships and patterns present in those relationships. It allows us to see the interconnectedness of events, ideas, and wisdom. True wisdom finds itself patterned everywhere and cannot be revealed through linear thought.
Mathematics is ultimately the study of patterns. Patterns are relational. In traditional mathematics, a student must wait years to see certain relationships hindering the development of a fluid understanding of existing relationships present in numbers. I equate this to building on the third floor without a supporting beam to connect to the first floor. Mind you the second floor hasn't been built at all.
A Deep Web of Relationships
Linear lacks depth because it is always forward moving never making deep connections because relationships are not the objective. Instead, it is the mastering of algorithms that is center stage. It is superficial objective but it will get you a perfect score on the SAT as long as you have a solid memory.
It is like the Greek Aqueducts. Certainly a wonderful marvel but it is limited to the relationship of point A to point B. There may be connections between cities and towns but nature has limited the aqueduct to a very linear direction.
Block logic doesn’t focus on forward moving. While it is forward moving in the overall sense, the progression is focused on relationships and the patterns present in those relationships. Like a pyramid's base, it is broad, providing an understanding that allows a student to build upon ideas easily.
In block logic, algorithms are found conveniently because the mathematical structures present themselves naturally through the observation of patterns and relationships. Memorizing math facts and algorithms are by-products of Gattegno’s relationship focused method.
The World is Relational
Think of the nervous system. It is not linear, but instead a deep web of interconnected information. The more connections the better the memory. Those connections are formed through common ties resulting in patterns of understanding.
Deep mathematical understanding doesn’t develop chronologically as linear thought is understood. It develops through block logic. It is a web of interconnected experiences and awareness.
Linear thought is exactly why I got frustrated with my children using the Saxon textbook. Linear thought assumes that a step by step path will develop a fluid understanding that allows children to transfer understanding to new ideas.
When we deprive them of the greater structure, it is vain to think students can somehow make the necessary connections to manipulate and use math to solve problems.
In a block logic presentation of Gattegno's methodology, patterns are more easily discovered because relationships take center stage. Students easily see that subtraction is fractional, multiplication is a form of addition, and division is a form of subtraction, multiplication and fractions. It is a deep web of interconnected relationships that goes on and on.
Our Latest Struggle in Math
With my own kids, I have watched them grow in their understanding of mathematics in this last year, but most importantly, they have found mathematics to be a joyful subject. Algorithms have more meaning as they have discovered them through patterns. It has revealed to them the beauty of math.
The only fighting that happens in math is telling another child to refrain from sharing their latest mathematical discovery to prevent them from spoiling the joy of another child’s discovery that is on the verge of manifesting itself.
I am happy for this transformation. Fighting to get math done is no longer the issue. When my kids’ eyes light up with each pattern found, I rest easy in our math journey. Now, if I can only keep them from spoil each other's adventures in math.
How I Organize Manual Module 1
We are mostly past Module 1 with the older children, but I am gearing up with a more formal exploration of math with my 18-month-old. Our focus will be on developing his language for mathematics which I strongly believe can begin this early like the acquisition of any second language.
Module 1 and upcoming module 2 is a place I will be in for a long time with my littlest, but I want to keep our journey organized. I placed everything in page protectors. This is a more durable way to manage the material with a toddler.
I used translucent folder tabs to section of each part of Module 1. I keep journaling pages in the folders to easily grab and record information.
One day, we might do rod racing. The next day we may practice search and find. We could even do both in one day splitting it between morning and afternoon exercises. Having journaling pages in each section of the module lets me see where I left off for each section.
Don't DO THIS
Understanding block logic helps you see why I am organizing the manual this way and why you should use the manual in the same way. Whatever you do, don’t use the manual in a linear form. It is designed for you to move back and forth between sections of the manual. In fact, you should return to activities in module 1 even if you are in module 2.
One mistake many may make is trying to use Manual Module 1 in a linear fashion. That is completing module 1 in sequential order. Certainly, you can begin in Module 1.1, then 1.2, then 1.3, but it is necessary once you get to 1.3 to start working in all parts of the remaining module 1.
You may even find it necessary to return to module 1.1 or 1.2. We aren't looking for a superficial understanding. If your child loves the blind find game in Module 1.2, return to it often. It is a great activity for discussing size and using mathematical language. Those task cards are easily extended to include discussion of fractions, subtraction and so on.
I placed task cards into album pages and inserted them next to the corresponding page in the manual. This helps me to do less searching for the right task cards.
I also placed play mats into page protectors. It is more durable than a composition notebook. My littlest isn’t likely to start writing for some time. It doesn’t make sense to put play mats into a composition notebook if our focus is merely oral. I will record his progress in my own journal.
I also inserted play mats from PDL's Interactive Math Notebook Bundle where I thought they fit into Module 1 Task Cards. For example, I inserted PDL's Search and Find for Cuisenaire Rods (also found in the bundle HERE) in Module 1.3 with the search and find task cards.
I also added PDL's Number Building Staircases for Cuisenaire rods to Module 1.4 which is all about staircase. I also added the staircase version of PDL's Search and Find for Cuisenaire Rods. I ran out of album pages for task cards and found zip-blok baggies as a great alternative for storing task cards in the binder.
I added the first few play mats from PDL's Fraction Exploration which are also included in the Interactive Math Notebook bundle here. I added them to Module 1.6 with the Rod Race task cards.
The More You PLAY, The More You DISCOVER
Block logic is better suited to help children see the beauty of math and the joy of pattern finding. Relationships are core to demonstrating and understanding the depth of mathematics, and linear logic is limited in its ability to present math relation-ally.
The more you play with Cuisenaire rods, the more you see how to extend task cards, play mats and activities to expand discovery and mathematical discussion. No one likes printing an activity one time to fill out and be thrown away in one afternoon. I know I don't. These activities are designed for repeated use.
I am going to post soon about my latest free activity found here. You will see how far mat building (which begins in module 1.6) will take you. Even if your kids aren't ready for this activity, I highly encourage you to do this activity yourself. Hopefully, you will discover the beauty of math and the joy of pattern finding.