We spend probably the largest portion of our school time working with numbers, taking a three pronged approach that addresses conceptual understanding, application, and numerical sense/speed. While this structure may come across as serious or intense, it actually works to free us up quite a bit to be playful and joyful in our numbers work together. I'll address each prong below for each of our children.
This is where I introduce new concepts to the children in a concrete fashion and let them explore and work towards understanding. I do not teach concepts, but guide the children through an independent discovery process. I find that Montessori lessons and materials fit our needs well here.
I obtained our Montessori lesson albums from the North American Montessori Center (for grades 1-6) and NAMTA (for grades 7-12). While costly, I cannot say enough positive things about these albums. They provide clear and thorough explanations of how best to guide the child to discover and understand for themselves the concept at hand. If there is interest in hearing more, I can provide a detailed post in the future summarizing my full opinion of their value.
I obtained our Montessori materials from Alison's Montessori, Adena Montessori, and Montessori Outlet. With very few exceptions, I wasted quite a bit of money here. Most of the materials I bought could have been made at home successfully for a fraction of the cost. I can provide a post in the future summarizing my opinions concerning which materials are a "must buy" and which can be made successfully at home (and provide directions or links explaining how this can be done), if anyone would like guidance here.
While I am prepared at the start of the week to give each child 4-5 Montessori math lessons, I follow the pace of the child (considering both their wants and needs). This means that I will support a child's lingering on a concept for several days if there is interest (want) or I may encourage a child to linger on a concept a bit longer if I see that they are on the cusp of a breakthrough (need). We will also skip over material or cut short lessons if there is little interest or learning occurring from the work. When this occurs (and it does, despite my best efforts), I simply make a few notes of my observations in order to learn more about my children and where they are in their conceptual development and learning styles. I then use these notes to determine if, when, and how we should circle back to the lesson(s) we cut short.
The atmosphere when the children are completing this work is one of focused concentration. The kids seem quite absorbed in their work and pursue it with quiet intensity. I step back and simply observe their process, sometimes jotting down observations as I discover more about the children and how they learn. I do sometimes feel bored while they work (there is only so much to observe) and so also spend this time quietly preparing materials for future work, returning emails, etc.
Here are a few pictures taken this week of the children working independently following their Montessori lesson. Avery is finishing up "grade 3" and Alexander is beginning "grade 7". Do note that, for Avery, I must put the traditional Montessori "command cards" into a context for her to find math interesting.
Here is Avery's work with the checkerboard material to deepen understanding of how to use multiplication when a two digit multiplier is encountered:
Alexander using the cubing material to explore what exactly negative and fractional exponents mean.
This is where we get to see how useful and important math is to our engagement with the world. This is what many call "living math". The children pursue this application work after their Montessori lesson has been completed. I find that this allows the child to "pull up" from their conceptual work and encourages them to continually seek linkages to what they are learning. While a child may spend just 15-45 minutes on their Montessori work, they may linger anywhere from 30 minutes to 4 hours on applications. The atmosphere as the children complete this work is lively, engaged, and noisy as we discuss, debate, try things out, fail, succeed, build, model, and USE math.
The books and resources I have found for Avery (grade 3) are wonderfully abundant and usually focused on a single concept. This makes it easy to link an "application" book(s) to what she is currently conceptually learning. Here are some of our favorites: MathStart books with the wonderfully done MathStart Teacher's Activity Guide, the Pythagoras series, and the Warlord series. While we are not fans of Life of Fred for learning a new concept, we enjoy Life of Fred tremendously for applying newly learned or already mastered concepts.
While Alexander often enjoys working with Avery on her application work, he does have his own to pursue independently or to work in collaboration with me. Resources for Alexander tend to combine several concepts at one time and use puzzles and brain teasers to challenge and dig deeper. Here are a few we currently enjoy: Murderous Math, Family Math the Middle School Years, and Life of Fred (with the above mentioned caveats). Below are some pictures of the kids applying what they have learned.
Avery applying what she learned regarding geometric solids (from a few weeks past)
Alexander applying what he learned regarding fractions (from a few weeks ago)
This includes the area of learning that many of us home educators (myself included) consider "drill and kill" and avoid at all costs. In the last two years, I have circled back to this area to try and salvage what is beneficial here to my children's learning. I do believe that lots of practice with basic math skills will lead to an instinctive numerical sense and a computational speed that increases enjoyment of mathematics. But how to do this without killing a child's love of math? Our answer is games. Lots and lots of games. If there is interest, I will write a detailed post in the future that shows the many ways we do this, but here are the resources that I find myself frequently drawing from: Family Math series and Peggy Kaye's Math Games. For most of the games, I have the flexibility to adjust the material practiced to best fit each child. And, lest I not forget, Scholastic has also produced some thoughtfully done workbooks that provide practice in a silly, fun way. The atmosphere while the kids pursue this work is joyful, silly, competitive, and often loud! Below are some pictures of the kids playing games to bolster mastery and speed.
Avery's animals ready for skip counting games.
A home made board game with practice cards made for Alexander