In order to start talking about the mathematical part of the brain, it is worthwhile to spend a bit of time on basic functional brain organization. How does this amazing 3 lbs of wonder tissue organize the world around us?
The foundation for human experience and learning is sensory input. We have all learned the five senses: Sight, Sound, Touch, Taste and Smell. When you look at the brain, you can see that the sensory information is divided into separate regions. We have all seen the images of the four lobes of the brain. What we see is the outer-layer, or cortex of the brain. The four lobes each received distinct sensory information that is processed there. The lobes next to the ears are called the temporal lobes. These lobes (one on each side) are important for processing sound, or auditory information. The lobe in the back of the brain, the occipital lobe, is important for processing visual information, for sight. The lobe at the top of the brain, the parietal lobe is where we have a full body representation of every part of our body. Here we process touch and other senses from the physical body.
What about the infamous frontal lobe? Most responses I get when I ask educators is that the frontal lobe is about executive function. What I don’t often hear is that the frontal lobe is motor. Well, it is both. The primary motor is right next to the sensory cortex in the parietal. As this information flows forward we have more abstract representations. In fact all the streams of sensory information flow into the frontal lobe. Sensory information comes together in various ways. It is through the synchronous activation of these different regions that we can perceive a coherent whole image, make creative connections between ideas, and abstract conceptual reasoning.
It is the dorsal part of the frontal lobe that serves the executive function, most don’t know as well that the ventral regions of the prefrontal cortex serves complex emotions, and is part of the Default Mode Network (DMN). Yes the prefrontal cortex is also emotions! This is a complex and beautiful part of our brain indeed.
I mention dorsal and ventral with regards to the prefrontal cortex, but this term is used ubiquitously. It refers to direction in the brain. In looking at the mathematical brain, we can make a distinction in how visual information is processed. Dorsal is towards the top, and ventral towards the bottom, or underside of the brain.
Visual information from the back of the brain, the occipital lobe, flows in two primary streams. It flows up through the parietal lobe as the dorsal stream. The dorsal stream is the Where and How pathway since it is critically involved in visual spatial awareness and embodied action. Visual information also flows down through the temporal lobe as the ventral stream. This has been called the What Pathway since it is critical in object naming.
Neurology studies have shown what happens in these separate pathways for naming and doing. Patients with damage to the What pathway are unable to recognize verbally what an object is, but would have no problem demonstrating how to use it. For example, a patient shown a hammer and asked to say what it is, will answer, “I don’t know”. They really believe they have no idea what the object is. However, the same patient will hammer nails into a block of wood with no guidance. This also shows us how the doing pathway is separated from our explicit memory. This separation of the ability to do something and to explain it, or even “know” it as evidenced by being able to recall it verbally, deserves attention as educators, and is critical for understanding why Conceptual Math presents information the way it does.
The input flowing in the dorsal stream, the Where & How pathway, is critical to processing information about mathematics. This mathematical mind is interested first in doing, not naming. Recall this pathway flows from the visual cortex toward the sensory representation of the body. It is where we relate our self to the space around us. It is how we orient ourselves in space. It is how we track and locate objects—our natural physical scientist lives here.
For the most part, our traditional teaching methods emphasize the What pathway. Once a student leaves 1st grade, they must be able to explain what they are doing and why. Name and explain is the name of the game. This is not accessing that Where pathway, the mathematical mind, the stream that is non-verbal, the stream that is about doing.
Conceptual Math is designed to access the non-verbal visual-spatial stream using both visual and embodied ways of representing the base ten system. I have found this embodied approach is particularly effective with the lowest students. I had one student who could not even count by twos, he had a hard time even when it was shown to him on the circle in the array of nine, but when we embodied the numbers, he was able to perfectly complete the twos up to 100 on the first try. It always amazes me when I see the power of embodied learning.
George Lakoff argues that all cognition is based on knowledge that comes from the body and that other domains are mapped onto the embodied knowledge. He also states that most of our thought is unconscious. There is a general acceptance that most of what drives our thinking and behavior goes on unconsciously. This includes our learning experiences. Just think of the possibilities we can tap into when we start teaching in a way that allows for those implicit learning systems to move us into greater understanding. I will talk more about implicit versus explicit learning and memory later in this book.
We need to take embodied practices seriously if we are to maximize the potential of our students, in particular in the areas of math and science which rely so heavily on spatial and non-verbal ways of knowing.
Patterns, Fluency and Number Sense
Dr. Daniel Ansari has been a strong supporter of Mind, Brain and Education, serving as the president of their International society. He and his team of researchers have illuminated much of what we know about moving from quantity awareness to the symbol. To cut to the chase, it is a developmental process. One of my favorite studies looks at the relationship between high school math test scores and simple mental math.
He puts it this way “Arithmetic fluency, the speed and efficiently with which correct solutions to numerical computations are generated is thought to represent a scaffold upon which higher-level mathematical skills are built.” (Price, Mazzocco & Ansari, 2013). Basically lower-level number sense is important for higher-level mathematical success. Now is that really surprising?
But here is the caveat. The way the most successful math performers solved the task, was through utilizing brain regions that were not about quantity processing. What we need to remember in math education is that numbers are not just amounts. Fluency with numbers is in part moving beyond the simple notion of amount, and I believe into understanding relationship of the numbers within the base 10 system.
The focus our teaching of numbers is traditionally on amounts. We use concrete manipulatives and children count the little red chips, or dots on a page. Research shows working with concrete examples and manipulatives helps learning. I agree, we must have something in the physical, visual and embodied domains in early learning. But numbers represent much more than amounts. They represent distances, time, relationships, and it is good to work with numbers in a wider array of experiences than simply amount.
In fact most of math in science and engineering deals with relationships between numbers, how various things interact and impact each other as moveable parts of a whole. It is limiting to a child’s active mathematical mind to focus so heavily on amount.
There is a mathematical mind. Ask any mathematician. It is highly connected to visual imagination and when mathematicians see formulae they see the whole array of what the numbers represent played out in their minds eye. Some even describe equations as beautiful…or ugly.
We have created many varieties of manipulatives to help students understand tens, hundreds, and thousands in a more concrete way. The small plastic blocks that come in ones, rows of ten, squares of 100 (10X10) and blocks of 1,000. These are meant to make the experience more concrete. This has some value. Part of our natural mathematical ability relates to what researchers refer to as our Approximate Number System (ANS). Training in assessing general amounts and discriminating between them has been shown to improve math ability, particularly in arithmetic (see Park, & Brannon, 2013). But there is considerable debate about the relationship between this system and the symbolic system. Overall, there is a great deal of overlap in the activation of the symbolic number system and ANS, but there are also distinctions. The symbolic system appears to be more lateralized into the left hemisphere, and the ANS has a unique area of activation in the right hemisphere (see Sokolowski, Fias, Ononye & Ansari, 2017).
The brain is a network of neural connections. When we activate one region, its activation initiates connected regions. This is necessarily true. What I propose here is that the general approximation system is doing more than calculating amounts. If that were so, it wouldn’t be sensitive to the spatial arrangement of numbers, distance or size (Leibovich, Kadhim & Ansari, 2017).
All of this is to say, that we can move beyond the use of quantity in our teaching of number sense.
One of the findings that has been significant to research in children’s number sense is that they are able to make relative comparisons to a number. Using a scale, and relating the distance from one side to another, helped students see the relationship between numbers.
This is important as it is not just amount, but the relative position of one number versus another to an anchor. In our case, using the base then system, the anchor is 10, or any multiple thereof.
Conceptual math is all about number sense, understanding the base 10 system, and math fluency. It is all about building a foundation. It is about building a network of numeric relationships in the brain. It is not a trick or strategy, it is a pattern generation system designed to activate our brains natural love of patterns.