Geometry in 6th Grade
I am presently (late February 2019) working on the geometry section of our sixth grade syllabus. Thought folks might be interested in a peek…Here is an excerpt from its opening:
In a Waldorf school, geometry holds a special place in the curriculum. Although ordinary Euclidean geometry is certainly taught, the mysterious beauty of numeric relationships as expressed in form is the main emphasis and is where we start in sixth grade. Though we will touch on the Pythagorean Theorem this year we will not work with proofs in any way. Rather, the goal of sixth grade geometry is for your child to experience the harmony, order and balance of the Universe. Solving proofs and the practical application of geometry is for later year, giving a solid foundation for trigonometry, among other disciplines.
Geometry calls for accuracy, patience and perseverance, all soul qualities which every parent wants to stimulate in their child and which lay firm foundations for learning. Geometric forms quickly go very wrong if one is not absolutely rigorous in one’s accuracy and wellsprings of patience are required when one has to do a form again and again. Do not shortchange your child of these character-forming lessons. Rescuing him in geometry is not the teacher’s job—but good-humored guidance and knowing when to step back and watch silently as one’s child explores geometric constructions, is.
Indeed, this 4-week main lesson provides a number of new ways for you to teach your child. As in second grade when I insist that a parent ensure that her child has ample opportunity to work with symbols (numbers on the board), manipulatives (the pebbles or acorns one uses) and verbal instruction and, further, that she learn to move easily between each mode of mathematical working, so in sixth grade geometry, we at Christopherus provide your student with another experience of multifaceted learning. Your child will work on constructions with you; she will create forms as you dictate the instructions; she will look at a form and figure it out herself; she will read instructions and work alone; and she will have the opportunity to talk you through a form, figuring out how best to order the steps.
We are well aware that many of you have never held a compass in their life, have never drawn a geometrical construction. Or that some of you younger folks have experienced geometry software, a poor substitute for work with an actual compass and straightedge. And so we have provided a companion video to this main lesson which talks you through using a compass and straightedge and a few basic constructions. We hope people find this useful!
And of course, this video is solely for you to watch. An important aspect of Waldorf pedagogy is that a teacher—you!—teaches, imbuing her lessons with love, warmth, excitement and interest. There will be many times in the future when an older student must learn via a video. Please make sure this is not one of them!
Geometry needs full concentration—this is a subject that does not really work with little ones around. You know from this syllabus (and Christopherus in general) that we try as hard as possible to make homeschooling with Waldorf doable, making suggestions where you can combine different grades or cope with little ones whilst teaching an older one. But there are a couple of subjects that, for serious pedagogical reasons, require the kind of focused concentration that is not possible when a parent—and sixth grader—are distracted by younger siblings. Form drawing is one of these subjects and geometry is another.
It is absolutely vital that your sixth grader be able to give all of his attention to his geometry work. Geometry is a schooling in precision, accuracy, mindfulness and perseverance. He must be able to think through what he is doing, to think into the forms and to quickly see when he goes wrong.
And so although we give detailed instructions for a four-week geometry main lesson, for some families, geometry lessons must take place on the weekend or in the evening or when younger siblings are not present. Such an arrangement is far more important than the integrity of a geometry main lesson. However, since the sequence of the forms is very important, do ensure you stick to how they are laid out. Again, we don’t say this very often in Christopherus—we like to explain things and then say ‘knock yourself out’ to parents. But the inherent lawfulness of geometry requires a step-by-step approach. Such an experience will give a solid counterbalance to more free-flowing lessons in other areas of your child’s education. This is an example of the breathing in and out of lessons and is an excellent therapeutic practice.
Further, the way we have set out our geometry lessons demands focused one-on-one work between parent and child. Again, there are many times in many lessons this year when this is not so rigorously required. But the way we have crafted these deeply pedagogical lessons demands this. Sometimes you will teach, sometimes your child discovers, sometimes your child teaches/leads you and sometimes you explore together.
We have varied the way the different constructions are taught. Geometry will carry many lessons for your child—and for you! And one lesson which needs to be clear from the get-go is that there will be times when you will be watching what your child does, helping him see where he is going wrong (other times you will leave him to figure this out himself). Your child is simply not allowed to say ‘go away—I got it’. Here is another area of discipline for her to tackle: she needs to learn the art of sometimes swallowing pride and graciously accepting instruction and guidance. This is an area that many homeschooled children, always allowed to call the shots, can find extremely challenging. Instead of making this a parenting issue, here one has the opportunity for the work itself to demand this situation.
Alongside the discipline of the work itself, our geometry lessons also provide an opportunity for your child to learn to schedule his work and gain insights into how long it takes him to complete certain tasks. I strongly suggest that you allow no more than 90 minutes per day for geometry (either as part of a main lesson or in the evening or on the weekend). Have a goal in mind—we have provided a detailed schedule for the progressions of lessons, but not how long each will take. It is safe to say most children will not satisfactorily complete all the work outlined for each lesson, each day. So then you need to define what a reasonable amount of work is for your child and be clear in expressing this to him.
Then you can do two things: 1) extend geometry lessons into an afternoon slot and 2) insist that some of the work (such as the copying into the MLB and coloring) take place in your child’s own time (ie outside of lesson time).
Your child is now in middle school and learning about pacing work, juggling work and other commitments, scheduling and meeting deadlines, all realities which need to be taken on, whether your child attends school for high school or remains at home. We have a teaching video on preparing for high school which deals with this very subject and which might be useful to you perhaps at the end of sixth grade or the beginning of seventh.
Coloring in lovely geometric forms can take a very, very long time for some children. It is imperative that this not take up precious teaching time during main lesson. And…if your child sees that her perhaps overly leisurely coloring is infringing on her playtime, she might speed things up a bit. Or she might just prefer coloring to play—which could be just fine!
As for those children who rush through their coloring, they might need to do their coloring during main lesson time simply so you can say things like ‘hold on there Speedy! That triangle needs more work—look at all the rough patches and where you left bits white’ or similar.
Please note: Do not let your child see the forms. Some lessons require that you read the directions to your child and that he follows, seeing in his mind the form as it unfolds and not having the form there already. For other forms, he will be asked to figure out how to construct them—ensure he does not ‘cheat’ by holding his compass against the form and copying the construction in that way! This will work (and it’s clever he figured that out!) but it robs him of the more intensive thinking required to work out and understand the steps in constructing a form.
Lastly, the expectation throughout these lessons is that you are working on paper, not a blackboard. To draw geometric forms on a blackboard requires the use of a special blackboard compass, something not readily available and something which requires quite a bit of practice to use properly.
Posted on February 25, 2019 in Uncategorized