Which problem would you give a child:
a) 4 + 8 = __
b) __+__= 12?
Why?
What’s the
secret to
teaching kids
as if
they have
a brain?
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{ 23 comments… read them below or add one }
Geez, Rick…You always ask such hard questions….
I think as a parent, I try and teach on a theme of factors…there are more ways than one to arrive at an answer. I try to show that there are at least two sides to every story.
But I guess I see those two problems as very different. The first one is a math problem, the second a philosophy question. I think both are important, but I think the second has more utility and takes you further in life.
Nice, Shirley. You always have such good answers.
Hi Rick,
In the math problems you have stated, the second equation would enable the child to explore several options of the combinations of 12. In order to assist the child in these discoveries and help them use thinking skills, the use of concrete materials will be very helpful.
As adults, we tend to treat math as a set of abstract techniques of calculation. However, the gifted mathematician and educator, Dr. Maria Montessori, emphasized the importance of concrete experiences that lead children to search for measurable mathematical relationships.
In the Montessori method of education, children learn mathematical concepts and functions through manipulation of concrete materials and personal discovery. Cardinal and ordinal numbers, the hierarchical order of the Decimal System, arithmetical operations, congruence, equivalence, similarity, algebraic functions and geometry are all experienced through Montessori sensorial materials.
Right you are, Chandra. I am so glad you brought Maria into this. Her math materials are terrific–though I do wish her curriculum had more challenges, the answers to which are infinite.
Thanks, Rick. Most people do not realize that the Montessori “curriculum” is as vast as the cognitive spectrum, potential, and interests of individual children and the way their teachers foster, encourage and facilitate their learning. The materials are tools in this process. I hope you will visit some of our Montessori elementary and Middle schools to witness the active learning taking place in those environments. Thanks again, for your comments.
Hi Rick,
I am intrigued by your challenge to Chandra that Montessori math should be more challenging. Personally, I find many challenges in this curriculum. Some examples:
- pre-school children are asked to learn to count to 1,000 and are introduced to addition, subtraction, multiplication and division with four-digit problems; they learn to recognize a wide range of 2- and 3-dimensiional figures, including trapezoids and quatrefoils, ovoids and ellipsoids.
- Children in grades 1-3 learn to add, subtract, multiply and divide with numbers in the millions (bead frames, checkerboard, test tube division); they solve simple equations for a single unknown, create algebraic formulas for binomial, trinomial and decanomial squares; they construct – and learn the qualities – of the whole range of 2-dimensional figures and explore the concepts of congruency, similarity and equivalence.
- Students in grades 4-6 learn all four operations with fractions and decimals, with decimal problems running into the millionths; they learn to do operations with integers, they can multiply four digit problems without partial products, they learn how to extract square roots and cube roots from numbers in the millions, they create algebraic formulas for binomial and trinomial squares and cubes; they derive the formulas for area and volume of the whole range of 2- and 3-dimensional figures.
All this is done through a process of auto-education, or self-directed learning by manipulation of the materials, which I believe meets your distinction between learning and education. Maria Montessori believed the purpose of math was not just to get the right answers but to build the mathematical mind of the child.
Michael Duffy (author of Math Works: Montessori math and the developing brain)
Michael and Chandra,
Thank you for starting a very interesting and important conversation. I absolutely know that the Montessori Curriculum is very challenging and yields students who are often way ahead of their peers in traditional schools in the academic curriculum.
I was not saying it was not challenging, I was saying I wished there were more challenges that challenged them to think creatively (a large set of playschool blocks, for instance).
I know Montessori schools–was the principal of one, was on the board of another.
Everything can be improved, and I am suggesting that there is room to grow in the creativity department. I am not even saying that I think Montessori is deficient in this regard. I know many montessori kids and they have grown up to be very creative, indeed.
…and yes, Michael, your examples do meet my criteria for the difference between education and mere schooling.
So, thank you, and I am glad you both gave such great examples. It’s wonderful to see how the revolutionary work of Maria Montessori lives on. I only wish it would catch fire everywhere. It bother’s me that it is still “alternative.” The principles that she uncovered through her scientific approach to education need to be instituted everywhere.
Rick,
I believe there is a lot of creativity built into the Montessori materials, particularly at the elementary level. One example that comes to mind immediately is the work with the second level constructive triangles. Students construct new figures from sets of triangles, and there are a virtually infinite number of combinations they can make to explore equivalency, well beyond anything in our training. My students always surprised me with the creative figures that they constructed.
Thanks for recognizing the contribution of Montessori to the conversation.
Michael
Hello Rick,
There are many ways to define creativity. Creativity focuses on the process of forming original ideas through exploration and discovery. In children, creativity develops from their experiences with the process rather than concern for the finished product.
When students are being creative in the classroom they are likely to: Question and challenge, envision, make connections and see relationships, explore ideas and options and reflect critically on ideas, actions and outcomes.
In a Montessori classroom, the mathematics curriculum is reality based, concrete materials are used to present abstract mathematical concepts. The teacher creatively uses appropriate and beautiful didactic materials to lure the children to understand mathematics, the logical thinking, not just in computing.
Number and mathematical concepts and relationships are constructions of the mind and they cannot be transmitted verbally or taught directly to the child, they require the coordination of mental and physical activity and action with Montessori materials. These activities develop coordination between the brain and the senses.
The development of logical mathematical reasoning unfolds from a concrete reality to abstraction, which the child derives from the action on things. The child’s progress of counting, numbers and mathematical concepts vary according to their individual ability and developmental level. The child is introduced to new lessons when the previous activity has been mastered.
I am interested to know your thoughts on creativity and how the Montessori math curriculum can be improved.
What do you suggest?
Munir Shivji
Right you are, Michael. a good example of what I have in mind.
Munir, Your descriptions are excellent. I wish all teachers (in or out of a montessori classroom) had the grasp you have of how learning academics is actually just an excuse for full brain development.
You do a nice job of defining creativity–I like Sir Ken Robinson’s also: “A new idea that has value,” but I like the multidimensional way you describe the creative process.
The things I activities I have in mind are where kids practice creatiing something out of nothing–like a blank piece of paper and paints or crayons–where the starting point is the child’s imagination–or a set of blocks or legos (not legos that come in a kit that shows you already what you are supposed to make, but rather a random set. Or just plain the creativity that comes from a few kids in the yard making up a game–from nothing–or a simple prompt.
(Only problem is that in most Montessori classrooms I have seen, there isn’t much room for more stuff.)
activities where the starting point is the child’s imagination are also good.
Hello Rick,
Just to comment on the open-ended creativity part in your last reply. I actually have plenty of room in my class for that kind of creativity. As a montessorian, I am not shy to say that I have blocks in the classroom right next to the authentic Montessori works. The children are free to create whatever they want from those blocks. We actually picked random shaped scraps of wood that the kids sanded themselves. We all then decided on the ones we liked and put them in a basket on a shelf.
Another way we have that kind of creativity in our evironemnt is in our school yard. Our outside time is an open-ended where the children decide and direct play. We don’t have a playscape or certain yard toys that dictate what to do with them. We have tree cookies, stumps, logs, and tires. We have dirt, sand, mulch and grass. And we have a rain barrel. The children create from scratch what they want, they build brideges, forts, water channels, and pirate ships. They dig tunnels, and make nests. So yes, that type of creativity where the children, as we say in our school “in charge of their own imagination”, exists in th Montessori environment.
Lulu, so glad to hear it. Not all Montessorians agree with you, Right?
One of the things that parents loved about my last school is that we had an acre. Someone came up with the slogan “It’s in the dirt.”
Hello Rick,
I appreciate your feedback and acknowledgement. Thank you. I have some additional thoughts. Montessori environments may appear in disarray to a person that is not familiar with the methodology. The environment, however, is prepared in a very precise manner. The outsider cannot see within the child as to what and how the child is constructing. Observation is the key.
Montessori teachers recognize that their role is not so much to teach as to inspire, mentor, and facilitate the learning process. The real work of learning belongs to the individual child. Because of this, Montessori teachers remain conscious of their role in helping each child to fulfill his potential as a human being.
Montessori presentations are designed to inspire independent exploration and curiosity. Teachers are conscious never to present all possibilities, leaving the child free to make his own discoveries.
Creativity takes place in all areas of the Montessori classroom, across all the ages. Montessori teachers encourage creativity by emphasizing process rather than product, provide a classroom environment that allows children to explore and work without undue restraints, adapt to children’s ideas rather than trying to structure the children’s ideas to fit the adults, use creative problem solving in all parts of the curriculum and continuously ask open ended question.
Montessori teachers facilitate creativity during the work period by providing ample uninterrupted blocks of time for creative expression. Children need large blocks of time to create and re-create; being hurried or pressured to begin or finish anything works against the creative impulse.
Specifically, we provide ample space indoors and outdoors for children to express their creativity. We also provide additional resources (outside the required Montessori manipulative materials) including open-ended toys, props, materials, and creative scraps for children to explore and discover. I agree with you.. children need paper, paint, blocks, legos, etc. as creativity involves the whole curriculum. I truly believe children cannot create from coloring in sheets or sitting down and being told what to do. A powerful way to develop creativity is to be a role model. Children develop creativity not when you tell them to, but when you provide them with a meaningful prepared learning environment that has ample materials for exploration and discovery. Children will more easily make connections between things they’ve learned if they environment encourages them to do so.
Munir Shivji
Hi Rick, Great information and conversation about our Montessori curriculum coming from many places. You stated “It bother’s me that it is still an alternative” and “it needs to be instituted everywhere.” I certainly agree and here is some great news. Cincinnati Public Schools have more that 2,500 elementary children in five new LEED Silver or Gold Green Buildings plus more than 1000 secondary students in two Montessori public secondary schools. Many of these schools are ranking in the highest categories for state academic testing. One is a completely neighborhood school. CPS 2011-2012 serve 69.7% economically disadvantaged children, 67.9% are African American children, and 21% have special needs (not counting all the children before 3rd grade who have not been identified yet). As a Montessori professor at Xavier University for the past 21 years, I have the privilege of being in many of these schools daily to observe our interns. What a fantastic experience because of the dedicated, highly qualified, and talented teachers and supportive parents. It is “every where in Cincinnati”. Ginger
Ginger, I am so happy to hear that Montessori is alive and strong in Cincinnati–I bet you had a hand in it. There is one Montessori public school in Decatur and one private pre-school.
Munir, everything you wrote is true–I know, I have seen it myself. Only one statement, I am not so sure of. I never heard anyone use the word “disarray” in relation to a Montessori classroom. Just because education is coming from the inside out, doesn’t mean it automatically looks like disarray.
As a Montessori teacher for over 20 years I find myself observing children with the math materials just going through the motions. It seems to be happening much more now than decades ago. There is not any learning taking place then, just as I learned an algebraic formula in high school but did not really understand it. So, it is the teachers priority through observation to ensure that learning is happening and that the students have not turned the materials into a video game! I’ve come across some other lessons that I incorporate into my curriculum to enable thinking and more discovery and learning. This is a very important part of teaching.
Rick,
You are correct. Not many Montessorians agree with the open-ended blocks and Legos be out on the shelf side by side with Montessori materials. However, I have known and was mentored by the kind of Montessorians who would let the child explore the materials without taking a lesson/presentation first. They are the teachers who would be ok if a child uses the red rods as measuring tools around the classroom, or uses the brown stairs as catapults or mazes for marbles. Those teachers exist and they are still capable of presenting classical Montessori lessons just the same if needs be. However, You’ve got to let the child lead.
Thanks for creating this whole discussion. It’s great to read many great comments from various points of views.
“Let the child lead.” Is that at the heart of Maria Montessori’s work? Is that something that binds all educators together?
Hi Rick,
You initiated a valuable conversation! Yes, “Follow the child” is a Montessori motto. It is also tied to ” Follow the child’s level of development”. Being trained in both Montessori and Piagetian studies, I can see parallel and divergent thoughts on these topics.
Thank you for a great conversation on creativity as well. Yes, we can certainly improve on the practice of Montessori philosophy. Dr. Montessori compared the Montessori adult to a scientist and her classroom to a lab where experimentation takes place each and everyday! This is the reason Montessori education has lasted for more than a century. This experimentation, however, needs to be conducted in an environment in which the conditions are carefully prepared for the child’s success.
I came away from studying Piaget and MOntessori wedded to the idea that being an educator means putting yourself in a scientific frame of mind so that everything you try you evaluate based on the reaction you get from the student. You might even learn that (piaget) thought you originally thought there were at “formal operations” in this particular challenge they are at concrete operations, or (montessori) those materials that worked so great in the past, aren’t working so well with the student(s) I have in front of me now.
The room was carefully prepared, and ooops, guess I have to do something different next time.
If a teacher gives a child 4 + 8 and the child comes up with nothing over and over again, standard proceedure is to get the child tested. An educator tries _____ + _____ = 12 to see what happens, and if that “doesn’t work” she comes up with something else until something “works.”
Right?
As you have pointed out yourself, Rick, you need to decide which of the two BEST SUITS the child. Maybe they are better at formal operations or maybe they are better at wanting to learn how to come up with many options for their answer. As you also said, an educator comes up with something else until the educator finds something that works for each individual child. Chandra also makes very good points in that the Montessori directress or “guide” has to be like a scientist and to learn how to observe the students of the classroom.
Nice, Sally. Thanks for pitching in.
Thanks, Rick and Sally. I just landed in the US and read the last couple of entries. I followed these conversations from 10,000 miles away!
Thank you, Rick.You should think about attending the Montessori conferences in Orlando in March or the Montessori Congress in Portland in July!