Here is a bit from the book on Conceptual Math about the way we can learn effortlessly.  Bypassing the need for the frontal executive function, but learning just as deeply.

From Conceptual Math

Psychological science defines two primary types of memory, explicit and implicit.  For the most part, traditional education is focused on what we call explicit memory systems.  Explicit memory systems record memories that can be freely recalled by the learner. Implicit memory, on the other-hand, is mapped in the subconscious.
 
Explicit memory itself has multiple components, and certain implicit memories can become explicit.  In fact this type of learning is the most exciting.  When something brewing in the subconscious suddenly becomes accessible to our conscious awareness we have a burst of dopamine in the brain and a wonderful Ah-Ha moment.  This is why discovery learning is my favorite approach to teaching complex concepts.  More on this is presented in a later section. 
 
The two primary divisions of explicit memory are semantic and episodic.  There is an emphasis on semantic memory in our teaching profession.  Some teachers probably believe it is only way to teach.  Our current obsession with rubrics, “I can” statements, and pre-learning activities that ask student to tell the teacher what they are in the process of learning, or what they will learn are evidence of this belief that learning isn’t happening if it is not explicit.  There is research that shows that having students state explicitly what they are learning increases recall on tasks that are also explicit.  Indeed explicit teaching through semantic knowledge has its merit. But there are so many layers of learning that can aid in building this explicit knowledge, and some of these approaches are the crux of joyful learning.  In all instances, teachers need more tools if they are to meet the needs of all students.
 
Some things are easier to teach in an explicit manner.  Typically when we are layering on new information to something the student is already familiar with, it is easy for them to build on the semantic web. 
 
We can look at this from the perspective of Piaget’s model.  He divides learning into assimilation and accommodation for acquiring new knowledge using semantic memory. Semantic memory is based on creating webs of factual knowledge that are connected in categories in the brain. Assimilation is easier because it places new information into a pre-existing web or neural network. Assimilation requires the inclusion of a few more synapses, and recruitment of a few new neurons, but not a whole new network.  Assimilation takes time, but not nearly as much time as creating a whole new network required for accommodation. 
 
Think about what happens in your own class when you have to introduce an entirely new concept—i.e. accommodation?  Teachers dread this experience, since there is often resistance. We are all familiar with the experience of students shutting down and saying, “Why do we have to learn this?”, or with older students, “Is this going to be on the test?”.  I love that one, because if it is not on the test, the student had decided to just daze out for a moment while I teach.  These responses that students make are the result of the challenge of introducing new material without first priming the brain. 
 
The other memory system that we rely on most as educators is working memory.  Working memory is actually a component of executive function.  As we have already discussed executive function in detail above, we will leave this for now. 
 
Much of the learning and memory demands we place on students use effortful processes.  Our teaching relies heavily on semantic and working memory.  In general, we do not take advantage of the many other non-effortful processes that can make our teaching flow forward from the interest of students thus helping our students advance in leaps and bounds.  Some of these effortless processes are priming, episodic memory, and procedural memory.
 
There is a great deal that can be learned without putting demands on the frontal lobe, and that information can become explicit in meaningful ways, that information can be used creatively, and helps to make sure we reach all students in our charge.

Priming the Brain
 
There are several other tools at our disposal to move beyond the dependence on semantic memory. Implicit memory is automatic and effortless. Priming is one form of implicit memory. It is related to some of the oldest memory systems in the brain, works automatically without any effort on the part of the student, and has powerful influences over what we will learn and how easily we can absorb new information. 
 
Priming happens when exposed to any sensory input.  It is a passive sensory imprint. The sensory information from all around me, simply flows into my brain, automatically and effortless, with powerful effects. Priming sets the brain up to accept certain ideas.  It is like clues that the environment provides.
 
Think of our ancient ancestors, walking through a canyon, they are in search of water, they intuitively know to go one direction rather than another.  Their sensory store is sending them clues about where the water is located, by the patterns of past experience.  The presence of a certain plant, the smell in the air, subtle animal tracks all point them in the direction of water. This can be initially unconscious, but with a powerful influence on choices and actions. It can also eventually become conscious, when we suddenly realize the patterns we have been seeing.  This is a deep, meaningful and powerful learning moment.
 
When we think about the sensory cues our students are responding to, this awareness of priming can be disheartening.  Especially when we think of those students who may have done poorly in the past.  Everything about the classroom may have a negative sensory impact on his/her learning.  Think about what new cues we can put in place. 
 
Priming should be use not just to change someone’s emotional relationship with the classroom, but to prepare for the introduction of new concepts and ideas. Conceptual Math takes advantage of priming the brain, dropping vocabulary words, using color to highlight ideas that will be coming soon and leaving images around in the classroom.  To understand priming, think of subliminal messages.  Plan your lesson with the hard parts in mind, leave your students many subtle clues in the lesson and around the classroom. A trail of breadcrumbs leading to the answer will give them the gift of discovering the path.  Don’t bring the answer too soon.  Timing is everything. 
 
In fact, for some students, bringing the explicit answer too early shuts down the chance for the priming to do its work.  The brain stops looking for the answer. Remember the brain is trying to solve the question of why.  Like the scientist hot on the trail of a new idea, the brain is searching for answers.  If the answer is given, it just says, “OK, no need to look at that anymore.”  You have just killed curiosity.  Priming is a great way to spark curiosity and that desire to know without the student even realizing it.
 
In making use of implicit memory in learning we simply create experiences of hearing and seeing the ideas around the topic in our environment. Naming of new terms should be introduced in a passive/implicit way.  This is can be accomplished by the teacher using the advanced and unfamiliar terminology while the students are engaged in doing something related to the idea.  This natural, light-hearted use of the new terms by the teacher will result in the students learning the meaning of the terms automatically. This is where there will be the linking of the verbal stream to the concept.  This is where the students may be able to initially talk about what they are doing.
 
As you continue to plant the seeds, they will begin to flourish in the students language, actions and especially the questions they present.  As they come up with questions, you can begin to increase your demands on their explicit memory.  Don’t push this, it will cause the students to shut down, especially if you have students with math anxiety.  By pushing, you may cause fear around something they have already been doing.  Watch how effortlessly they already are performing their multiplication tables, don’t judge that they can’t explain yet.  Wait, give them time. 
 
The time necessary for the idea to take hold is completely individual.  Some students may be able to give explanations right away, making the connections immediately.  Other students may take much longer, but when prodded or probed about what they know they may freeze up and go blank and be unable to solve even the simplest equations. This simply means the student is not ready yet. Give them time! Keep using the implicit approach until they are ready.
 
Episodic, and Procedural Memory
 
There is yet another powerful non-effortful learning approach that in fact relies on explicit memory.  This is the use of our episodic memory system. Basically, the approach to engage episodic memory is nothing more than storytelling.  Episodic memory is actually an explicit memory form, meaning it is not subconscious, but easily retrievable by our conscious mind.  Nonetheless, it is a non-effortful form of long-term memory.  “What’s that?” you say.  “Can it be true?” “Non-effortful, long-term?”
 
Yes indeed, episodic memory is non-effortful and long-term and in my opinion should be used as a standard way of introducing new content. It is how the brain was designed to remember. Before the written word we had the story.  Before the story we had the experience that needed to be recounted.  Our brain has designed a special system for recording events that are remembered as a whole storyline.  This is episodic memory.
 
Just as easily as the brain records the events of our lives, it engages that episodic memory system to record stories that are told to it.  Storytelling then becomes this powerful way to get students able to enter into a new idea.  Just test for yourself what happens when you hear the words…Once Upon a Time…
 
We exhale, relax and get ourselves ready to receive something magical.  I know what you are thinking, “How can we use stories for math?”  I have written a number of stories I am happy to share, but I am sure you can also make up your own.  Integrating math lessons in the history of mathematicians, like Euclid and Pythagorus can really enhance you lesson and create meaningful connections with other historical events. This will help link some of the non-verbal ideas to more of the social-conscious brain.  This helps students consciously access the ideas. 
 
Why not have your students make up some stories together.  You can then elaborate on these ideas in the class.  Making up stories, doing artwork, creating theater plays.  All of these fabulous ways of learning can be employed in math just a well as in any other subject.  I usually link up the stories and songs with the emotional aspects of the HNE program.  I will present this in a separate format.  Story is no mystery.  Simply create a beginning, middle, end, some suspense, characters and a resolution.  I particularly like creating characters around fractions.  It helps so much with the terminology.
 
The final form of implicit memory discussed here is that of procedural memory.  Procedural memory is motor memory, and probably the most central to the initial phase of the conceptual math approach.
 
Procedural memory, like the other forms of memory discussed here, is effortless. It is deep and long lasting. What’s more, it is fun. When you think of procedural memory, think of riding a bike. You never forget.  Conceptual math takes advantage of procedural memory systems and therefore helps students retain their learning for a lifetime.
 
One of the earliest studies showing how procedural memory is distinct from explicit memory came from the famous patient H.M.  H.M. had suffered from extreme epileptic seizures and the famous neurosurgeon Wilder Penfield made the decision to remove the foci (central source of the seizures).  It so happened that this foci was in H.M.’s temporal lobe, and specifically including his hippocampus.
 
When the patient came too, he seemed fine. His intelligence was intact. He could walk and talk, but he had one serious problem. He had complete amnesia for anything new he experienced.  If he were to meet you in his room, and then you were to leave the room, when you returned five minutes later he would have no recollection of ever having met you. 
 
H.M. became a fascinating subject for the neuropsychological community. He participated in thousands of research experiments.  One of the most famous was examining his performance on procedural memory tasks.  During these experiments the research scientist would give H.M. a procedural memory (or motor memory) task.  My favorite task asked H.M. to draw a star image that was being reflected in a mirror.  This task is trickier than it looks.  When you try to draw, you tend will move your hand the direction of the line as you see it.  You need to practice to learn that the line is a mirror image, and you must move your hand the opposite direction of what you see.
 
Each day, when the researcher would show him the task, H.M. would say he had never seen the star, nor the researcher for that matter. However, despite his claiming he had never done the task, each day he would show improvement on his performance.  This showed the scientists that procedural memory was processed in a different part of the brain as explicit memory.
 
Procedural memory does not require the involvement of the hippocampus but rather involves regions motor regions: the cerebellum, basal ganglia (striatum) and the motor cortex. Each of these regions are involved in motor control. We will come back to the importance of the basal ganglia in attention in a following section of this book.
 
Procedural memory is based on doing.  So much of what we present in conceptual math is learning through doing.  The moving of the patterns, embodying the number position and drawing the array of nine all utilize procedural memory.  Students develop this skill, sometimes without realizing they are learning the times tables.
 
At times working with students, I have noticed that when I ask if they know certain times tables, they will say, “No”, but when asked to perform them, they do so with ease.  This is usually an Ah-ha moment for these learners. For most students who have struggled at one point or another, it is after they see themselves being successful that they realize they can do it.  It is ironic that when using procedural memory, students learn it even before they realize they have learned it.
 
The system of conceptual math helps to build a foundation.  It is a bottom up approach. It works through repetition, imitation and exposure.  It does not expect the student to explain what they are doing or why.  It is important to recognize that as a teacher or a parent, it sounds strange that you can teach a lesson and at the end of the lesson if you were to ask a student “what does that mean?” they wouldn’t be able to answer you.  But mathematics is unique in that it is processed in non-verbal streams that are connected to motor, and embodied aspect of being.  These streams will develop through rhythmic repetition.  They will provide a powerful conceptual foundation for the explicit verbal brain to draw on.  Never doubt the power of the embodied mind.

 

In 2019 I published a book on what I refer to as Conceptual Math.  This approach to learning math is to me the equivalent of phonics in reading, or learning the alphabet for writing.  I still think it is the greatest thing I have yet created.  

Here is a bit from the book on Numeracy:

​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.  ​

 

I recently ran across one of my early emerging curricular models.  These five concepts seem to reflect what I believe would be a revolutionary approach.  Although it hasn't seen the light of day in quite a few years, in looking over this model, I see it holds promise.  Perhaps coming back to it now, I can find ways to integrate it into some of my other curriculum.  At the very least, this reflection will keep me connected to these once inspired ideas.  It is also a great way to track my own progress and thinking in these areas.

I actually created the model through conversations with my friend Brandy McCray.  What an incredible thinker she is.  But for the most part, it was an inspired look at what seems to be an evolved way of imagining education.

Rather than academics, this looks as five domains for personal development.  It seems to me, looking at school not as the way of developing academic skills, but developing cognitive capacities would lead us back on track.  It doesn't mean academics don't come into play, but these are at the service of greater goals.  They are simply the basics, the goals are much deeper.  A child who has an idea for a creative project aligned with a deeper purpose will be motivated to read materials on the subject, write up their plans, calculate what finances they need, etc.  

But this is getting ahead of myself.  Let me just go over these areas briefly, then I will take some time to break them down and see where I stand in my own thinking at the moment.

The Five Foundational Concepts:

Self Directed
Creativity
Critical Thinking
Beauty
Community

Maybe changing the form of the words so they are all the same...so the noun Self-Direction, instead of the adjective Self-Directed.  

For now, let me just review what the meaning provides.

Self Directed is number one.  It reflects more than just students ability to select what they want to learn, and not feeling the oppressive control of the teacher.  It comes from a deep awarenss that we each have a deeper purpose.  

Creativity is number two.  It is through creativity that we can express our inner being.  Creativity is a means of self-discovery, which is a critical piece to know what it is one is moving towards.

Three is Critical Thinking.  This type of thinking is more analytical, it is grounded in logic and rules.  It is important for linking creativity to the world.  

Four is Aesthetics, or Beauty.  Why would this be here?  Maybe I could change this to Emotions?  Aesthetics and Beauty are most definitely connected to emotions, but there is more there.  There is the sensory connection.  I will explore this more before making changes.

Five and finally, there is Community.  This is the social element.  This is the moment, I live not just within my unique being, but in full connection with the other.  Relationships can be a challenge.  But when I know who I am and am acting in integrity with my deeper purpose, relationships become the source of great endeavors.  

A little short review of an old model, but I like it so far.

I will post on the individual ideas later.