Pedagogical ConsiderationsMy background in the neuroscience of attention and my experience teaching non-traditional learners led me to my current path in curriculum development. The more I studied the efficacy with which some previously struggling learners could engage and learn using game-based, play-based and embodied approaches, the more I realized the power of this approach for all learners. I find this is particularly true for teaching mathematics. Conceptual math was designed as a way to better align the way we teach with the way numbers are represented in the brain. Conceptual Math understands that the brain has an intuitive sense when relating to numbers. The goal of Conceptual Math is to tap into these intuitive centers and build on them. These neural pathways are the foundation for embodied cognition, and visual spatial awareness. Conceptual math works through these pathways to access the mathematical mind. There are several other basic pedagogical concerns of Conceptual Math There are some basic pedagogical frameworks that underlie the Conceptual Math approach. Learning by Doing I wrote quite a bit about the reasoning behind learning by doing in the section on implicit and procedural memory. This as it is related both to the way in which math is processed in non-verbal streams, as well as the emphasis on of engaging procedural memory in the skill building activities. It is hard for teachers to fully grasp that the need to explain before-hand is not only un-necessary, but distracting. So many students will start to tune-out when the teacher begins to talk. Of course, I am most familiar with working with reluctant and definant students, but some of this reluctance and defiance is simply a response to being talked to so often. Just moving into the activity, modeling what response you want, has such great power. Joy and Fun Jung-Beeman and Subramanian have researched the Ah-ha moment and found that it happens more often when we are in a positive mood. Why do we equate learning with being serious, when in reality learning happens most easily and effectively when we are having fun? I recall a parent comment on the national school registry talking about the local Waldorf charter school in his neighborhood. If your main interest is knowing that school will be fun place for your kids. This is a good school… if one's priority is in learning the basic educational curriculum, then this will not be the school for you. However, if one's priority is that your child thinks that school is fun and that people are nice and, if that is good enough for you, you will more than likely like this school very much I just think this quote so epitomizes our current view of education. If it is fun, it can’t be learning. My own research on the school above showed that these students, besides showing greater levels of school enjoyment and better emotional well-being, also were the top of their district in reading and math by the 8th grade. We must remember that learning is developmental, and if we get students on the right track early, they will propel themselves to heights we can’t begin to imagine. Even more disheartening, regarding another Waldorf charter, one parent wrote a revealing view on play. I like the playful, cheerful environment, but it is a lot of play and not a lot of work. If the purpose of school is to prepare our children for the future and to build the foundation for a strong work ethic and a successful career, then [Waldorf School] falls dreadfully short. The influence I see being indirectly taught to the children is that of mediocrity and laziness, which is such a shame. What truly is a shame is that we as a society have become gluttons for punishment of our children in schools. It is no wonder that the incidence of depression in our teens is so high? Research on play-based learning shows that play is a powerful tool for learning and building capacity. It also shows that play leads to greater empathy, stronger leadership and social skills, and overall better life outcomes. Research on toxic stress, on the other hand, shows us just how damaging this misguided societal view on pressure to succeed can create. Robert Sapolsky has spent his career showing how stress causes degeneration of brain cells, in the critical brain region, the hippocampus. Stress also causes changes in our neural processing. We move into a defensive mode and shut down the higher cognitive centers of the brain. Some students are more sensitive to stress, particularly those who have already experienced trauma in their lives. For these students, even small stressors shut down the capacity to learn. But watch what happens with those same students when you start to include music, and art and games in the learning. It is as if they wake up out of their fog and can now come to the task of learning with ease. Happiness and play are not what our society thinks when it comes to learning. They are the best way to access our highest thinking in all students. Counting Counts One thing I cannot emphasize more is the value of counting at all ages. Counting forwards, counting backwards, counting by 1s, 2s, 5s, 10s, but also by 3s, 4s, 7s, 9s, by 60s, etc. Counting is truly the foundation of it all. It is also fun! Children love to count. Try using counting during a timeout on the playground, and watch how, instead of the one child having to sit by himself on the bench, a whole group of children will come and circle around and count with him. There are several activities associated with counting that I encourage. I encourage choral counting for the youngest children. Choral counting with movement in a circle is a fabulous first activity. Counting and doing jump rope is a fantastic activity. I was at a Waldorf school where they had the jump rope challenge. The youngest grades were out there everyday counting and jumping, both forwards and backwards. I have heard people complain that when children who are very young count don’t know the meaning of what they are saying, but the truth is that is doesn’t matter. The familiar sequence being engrained will be used again and again in more and more complex scenarios. It would be as if we were to say that learning the alphabet doesn’t mean a child can read, so we shouldn’t teach them the alphabet. Learning the sequence of numbers is a rote activity, and yet, there are repeating patterns that the brain is picking up on. The fact that we are in a base 10 system, the brain will automatically recognize these patterns, it does not need an explanation first. In fact, it is better if it is explained indirectly, since as I have mentioned the mathematical centers are non-verbal. The joy of discovery is the basis for powerful learning that inspires curiosity. Neuroplasticity and Discovery-Based Learning. Why discovery based? In fact this is the way we learn deeply. When learning is intrinsically self-motivated it has personal meaning. The brain is designed to support the human being, by this, whatever is serving that person becomes relevant and a point of interest which the brain will subsequently work to learn about. Learning comes from having a question. In a discovery based approach, we seek to incite curiosity, rather than memorization. We encourage exploration of an idea, rather than focus on getting a right answer. Learning is also a developmental process. It happens because we are creating new connections in the brain. Working from procedural and implicit learning, we are laying down new neural pathways. The time spent laying down these pathways is only the first step to creating joyful learning experiences for our students. The second step is to allow students to experience the Ah-ha moment. This is when the connections from the implicit learning that has been laid down are brought into conscious awareness. This moment of moving from implicit to explicit learning is associated with a burst of our reward neurotransmitter dopamine. Learning really is that naturally rewarding and motivating when we create the right conditions. In education we have taken away the most powerful part of the learning by giving the explanation up front. Learning should be fun. Learning releases dopamine! When you have the Ah-ha moment it is like a burst of enthusiasm and a natural high. So why are our students so apathetic? Because they don’t often get to experience that dopamine burst. They are told the answer. We don’t wait or have the patience for them to learn in the natural way. Learning becomes a chore without the natural reward. Then, we build in artificial rewards to take the place of that loss, but these rewards are not connected to the learning itself. The dopamine burst is there not just to create pleasure for no reason, but it helps to strengthen the new learning. This produces “deep learning”. My move into the field of education was a direct result of the overzealous prescribing of powerful medications that affect the primary motivations systems in the brain. I shared about dopamine’s role in learning. These drugs artificially increase dopamine, but there is no context, or personal motivation, and the system becomes dependent on this outside supply of dopamine rather than being allowed to develop and respond naturally to what is truly personally motivating. Our motivation to learn comes from a desire to answer a question. As Allison Gopnik says, we are all scientists from the day we are born, seeking to understand the world around us. At the very core, simply by doing, we create the desire to be able to explain and understand what we are doing. This is the moment at which explicit learning becomes a critical component of Conceptual Math. I have said earlier, that knowing when to move from implicit to explicit learning is a sensitive issue. If we force students to give us the explanation too early, they can emotionally shut down. Frequently, we as teachers become impatient and simply give them the answer. This steals the wonderful and amazing Ah-ha moment of discovery that anchors in the new learning for life. When do we start pushing for explicit answers? The best indicator is the moment is when the student has the question. Once the student has the question, once the concept has been built through experience, the teacher can help create the support to draw out the answer. At that moment, the student will have discovered a truth. They will have the experienced the Ah-Ha moment. They will have connected the dots from all of the pieces of information, including the words and names they may have already heard. This is the moment when deep learning happens. Developmental and Neurosequential Finally, this program is a part of a larger developmental framework I have developed based on a neurosequential model of learning. This I call Holistic Neuroeducation. What I present here is mostly working on the somatosensorimotor system. I move students into emotional connections when working with fractions. This I will share with you at some near date. Repeated Phrases and Implicit Learning My experience working with students with learning difficulties is that when I could give them a phrase, especially one that rhymed, they could remember concepts that otherwise baffled them. The use of repeated phrases that can be said together as a group or presented in a regular manner is a powerful learning tool. Adding a special intonation and gesture can also aid in enhancing the memory for the fact. This should be worked into the lesson. The students should get used to that question or phrase appearing at the critical moment, so that when they are on their own, this same idea will occur to them automatically. The Three-Day Lesson Plan Neuroplastic changes occur due to a physical change in the brain. Growing new synaptic connections takes times. There are some basic principles of how much time is needed in processes of neuroplasticity. Three days is when things we are learning start appearing in our dreams. Three days is when we have made those new neural connections and are ready to make the conscious connections. Creating a three-day plan where you intentionally delay activities that require students to recall or integrate the new ideas until the third day will ensure everyone has the chance to learn deeply. Creating Patterns The brain loves patterns. These are especially so in problem solving. It seeks out patterns to predict and solve problems with ease and grace. The multiples represented on the base ten circle create beautiful patterns. In considering your presentation of the exercises, always include time for students to recognize and play with their awareness of the patterns inherent in these exercises. Movement, Action and Mirror Neurons As teachers we want to make sure we are not becoming overly reliant on the technology to do our work. If we work from the desk and project things on the board, the students don’t see us, the human being. Our brains are wired to map human movements. We have special regions of our brain that map “biological movement”. This is how important it is to us. More than just special human movement regions for others, our own motor and movement neurons are activated when we see others performing movements. This is the true source of the word mirror neuron. It came out of the lab of Rizzolati. His finding sparked a revolution and a stream of studies showing how our brain is activated by other people’s actions, and subsequently emotions. The mirror neuron system has gotten the most attention in the research on empathy, but it came out of research based on attention and action. If you are sitting behind the desk, you are not activating the movement related neurons. The students are missing the opportunity to activate those neural pathways involved in procedural learning. Bottom line, it will be much harder for the students to later perform the actions themselves. So whenever possible, let them see you do the movement and actions with your body. Let them see you draw in real physical space. Learning will be easier, and your job of teaching will too. Building Enthusiasm and Curiosity Mathematicians LOVE numbers. Besides making learning fun for the reasons stated before, we want to help instill a love and wonder of numbers. One thing I always do is the 100 dance. Whenever we get up to 100, we do a little dance. It gives the sense of accomplishment and marks a good ending point. I also have special numbers I like to emphasize. I make a big deal of the number 144. It is a powerful number as it factors in so many different ways. Whereas 100 on lands perfectly for 2, 4, 5, 10; 144 lands perfectly for 2, 3, 4, 6, 8, 9, 12. A lot of our numbers land here. It is also in the Fibonacci sequence, which I like to introduce to kids for fun. You can have kids choose their favorite number and have them make up a special move for when that number comes up. So much fun for them to create relationships with the numbers. It helps make math more personal connecting math to emotional networks and drawing their attention to points in the sequence, thereby increasing engagement. The Importance of Working with Larger Numbers Conceptual Math emphasizes moving up through the patterns of the multiples to higher and higher values. This is a natural way to build place value understanding and takes away the fear of higher numbers. Having student take on a 5,000 challenge or trying to break each other’s record for any given multiple is a great way to promote this. I have noticed students who have been successful in memorizing their times tables up to 12 have difficulty when moving beyond those, especially when they reach triple digits. It is interesting to watch these students as they realize their peers are out-pacing them at what they thought was their forte. It does not teach students number sense to memorize their times tables. Working with larger numbers benefits building number sense and also basic cognitive function. When using Conceptual Math, students challenge themselves to move into triple digits, quadruple digits, and even the 10,000 challenge. This requires grit. As these numbers get higher and higher, students must remember three place values or four place values as they move on. This is challenging, and a powerful exercise for our working memory systems. Final Thoughts on Making Conceptual Math Work for YouThis new way of representing the number line as a circle is one that has many avenues for further development. My hope is that you will see this as a flexible tool and make it your own. None of the exercises here are meant to be viewed as some sort of immutable dogma. Last year I attended the Mind, Brain and Education conference in Southern California. The opening key-note speaker talked about fidelity. Fidelity has been a thorn in my side ever since they began using it with regards to curriculum implementation. Fidelity is usually equated with rigidity. You must follow the script to the letter or they will not guarantee that the program will work. This is ridiculous. What this keynote speaker proposed was a new view of fidelity. He suggested that our measure of alignment shouldn’t be in the script, but in the ultimate goal. My ultimate goal is student engagement, enjoyment, self-direction. My ultimate goal is healthy attention and emotional systems. My ultimate goal is students learning the relationship between numbers, being comfortable and fluent with their use of numbers, and exploring numbers through math. What I need to do each day to determine fidelity is to assess how well I am meeting those goals. Are my students using numbers fluently? Do they love learning and seek out learning on their own? Do the seek challenge instead of shying away from it? These are my measure of fidelity and success. If there is something that works in this program, keep it. If there is something that is not, leave it. |