Hidden in plain sight

I was doing a math moment with the kiddos at Ethel Streit this week. We have a couple who have had negative experiences with math, some who have only experienced math in traditional public school settings, and some who are quite gifted in mathematical thinking but have little facility with math facts or algorithms. To try to set the stage for the idea that we all explore big math ideas together, I made a “simple” drawing on the whiteboard:

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I asked the kids how many squares there were.

The answers were immediate: 9. No, wait! 10! As the kids were thinking, I heard L say”9 times 5…?” I have to admit: I had no idea what she was talking about. She tried to explain to me that there were 9 on top and 9 on the right and 9 on the bottom and 9 on the left. She said that wasn’t quite it, and I chose to move on.

Eventually, the kids figured out there were 14 squares (for those of you following along at home, that’s 9 1×1 squares, 4 2×2 squares, and 1 3×3 square).

L piped up again with “times 5?”

I still had no idea what she was talking about. Luckily, my incredible co-teacher, Sara, looked at L and asked if she meant like a cube. Yes! L’s face lit up. She explained that we had only counted the squares we could see, not all the ones we couldn’t see. She was seeing the visible squares as the front-facing side of a 3-D cube…

I didn’t want to lose the other kids, so I decided to stick a pin in that idea. I told L we would talk more about that later, erased the drawing, put up a set of triangles, and moved on with the lesson. L clarified, “We should only count the triangles we see, right?” Right!

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A trickier picture with triangles – we found 48. How many can you find?

On the way home, L piped up again. “It’s not times 5! It’s times 6!” She had realized that you had to count the squares on the face you COULD see in addition to those on the faces you couldn’t. I realized that clearly, L wasn’t done with this yet.

When we got home, I printed and cut out a cube net.While I was doing that in the office, I gave her a piece of scratch paper and asked her to represent what she meant mathematically. This is what she created:

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I then gave L the cube to represent her work in three dimensions. She asked me to draw the 3×3 grid on each face and then proceeded to complete her calculations.

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She began by counting each small square, realizing that her total was 54. I had her check using multiplication to see that 9×6 is 54.

She then said, “But there are 14×6 squares,” remembering that we had found 14 on each side. I gave her a high five and we decided to multiply it out to find the total. We broke 14 into 10 and 4, giving us two problems: 10×6 and 4×6. She immediately knew that 10×6=60. 4×6 was harder for her. She did know that 4×2=8, so we used that fact three times to compose 4×6=24. We then joined the 60 with the 24 to learn that there were 84 squares on the six faces. There are our written notes from the process:

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Because L is L, though, we weren’t done yet. She wanted to figure out how to build the cube out of smaller cubes. As she was considering the build, she gasped, “The cubes are all hidden in plain sight!” When I was done laughing, I told her she was right.

We pulled out Cuisenaire Rods to build. I wouldn’t usually use cuisenaire rods for this build, but we had recently taken our base ten blocks into the school and I could tell we needed to do this build right now. I was particularly aware of the fact that we don’t have 27 1-unit rods. I decided to let it ride, though, and see what she did with it.

L began to construct a 3×3 base out of the 1-unit cubes. I went outside to collect cherry tomatoes from the garden (yes, the garden is still crazily producing). L ran up to me and said, “I don’t have enough cubes!” I said, “Oh.” As I was thinking of what to say, she spurted out, “But it’s ok! I can get one that’s 3 long and use it in place of three ones!” She turned around and ran into the house. Ok. I thought. She’s got a plan.

I came back in to find her build complete.

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As she began to figure out how many 1×1 cubes were in the whole cube, I kept quiet and watched. She sorted the white cubes from the purple rods. There were 12 white cubes and 5 rods. She counted the white cubes one by one, moving them over as she counted them. When she got to 12, she began touching the first purple rod. She touched it on the end, the middle, and the other end as she counted, “13, 14, 15.” She moved that rod over with the white cubes and repeated the same process until she got to 27 as her sum.

The look of pride on her face was palpable. I loved that this was all self-driven and to solve a question that she had been puzzling over. She’s also been talking about the math on the board and the cube daily since then. Hidden in plain sight, indeed!

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I can’t keep up with my kid!

Why am I writing at 1:00 in the morning, you may ask? Midway through the first week of teaching classes at the university? Just a little freaking out, of course.

I’ve been really worried lately about how L will respond to what’s coming her way – her involvement with a micro-school designed for a handful of kiddos to work at their own pace. That part sounds great – the part I’m worried about is that I’ll be leading instruction there two days a week and she and I still haven’t figured out how to work together on a daily basis.

L doesn’t respond the way I anticipate a child will respond. Doesn’t matter what I incentivize her work with. Doesn’t matter what I attempt to use as consequences.

She will do what she will do when she wants to do it.

Making steady progress

For the most part, I try to breathe deeply and trust that. It’s easy for me to do with reading, since she reads voraciously. Recently, I attempted to remind myself how to do running records by practicing on her. I downloaded some grade-leveled passages and sat down with her, asking her to read them out loud so I could practice taking running records. I anticipated that the second grade passage would be too easy and the third or fourth grade would be just about right.

Grade level Words per minute Accuracy Comprehension questions
2 102 98.6% 2/2 questions
3 69 (proper nouns were a struggle) 96% 2/2 questions
4 92 97% 2/2 questions
5 73 96% 2/2 questions
6 74 96% 1/2 questions

She turned 6 years old last month. So, yeah. I don’t worry about her reading. She is constantly lugging a book around (DK Eyewitness Books: Ocean and others in the series are a huge hit these days).

Her writing is a concern, for sure. She struggles with letter formation and isn’t drawn to production of text. I am playing around with cursive for her (which has been better than printing) as well as typing, but we don’t have a magic bullet yet.

I don’t worry about science. At all. She is constantly reading and watching videos and documentaries and conducting experiments. She’s probably most advanced in science, which is ironic given that I am not drawn to natural sciences. She came by that one all on her own!

Social studies? Well, I would have argued that she’s not interested in social studies, but I think I’ve been too narrow in what I defined as social studies. She knows where many countries are and what their flags are from looking at a map on her wall. Same for the states and the state flags. She was deeply interested in the American enslavement movement and Civil Rights era for awhile and read deeply on those topics. She has onboarded much of the events covered by the musical Hamilton and has enjoyed reading the lyrics and asking questions about them. She has also followed along with NPR coverage of the presidential election season and has explored the ideologies behind the Democratic and Republican parties, even going so far as to define herself as an “environmental one-issue voter.” So maybe that’s pretty good coverage of social studies at age 6?

But math!

Anyway, this brings us all to math. Math is the source of today’s hijinks and is why I am still up (it’s almost 2 now, for those of you following along).

L has been “fighting” about math, pretty much always. Her fight is saying she doesn’t know how to do something or she’s bored. There was a great period early on when she used Todo Math (which she blew through but enjoyed), Slice Fractions (same) and Dreambox (which was new for awhile but she became bored with it). Even Beast Academy, the “go to” for elementary g&t kiddos, was interesting to her to read (she lugged books 3A-D as her bedtime reading for awhile), but she was never interested in the practice books.

And yet, we’re going to be working together in this micro-school, so I need her to be able to sustain work in math. I grabbed a couple of story problems from a third grade problem set and gave them to her as today’s work.

Let me be clear: I was at work and she was in my office with me. My attention was clearly divided and I couldn’t reinforce to her that she should continue working. However, she essentially didn’t do anything for long enough that I gave up. I decided that the battle wasn’t worth it today, that I needed to get a bit of my work done, and that we would try again tomorrow.

However, after school, she was fighting my husband as well, and he decided that she was going to work through these 6 problems because, well, sometimes you need to listen to your parents.

So she sat down and figured out the answers to all six problems. Most of them in her head.

Sample 1: There are 2,532 students at a school. 1,312 of them are girls. How many of the students are boys?

She picked up her pencil and wrote down 1,220. Didn’t write the problem out. Didn’t use manipulatives. Nothing.

Sample 2: Mom has 11 apples. She needs 5 apples to make 1 pie. She wants to make 5 pies. How many more apples does she need?

She circled the words “5 pies” and wrote out “25-11=14”.

I posted about this on my favorite facebook group and another mom suggested that maybe she is bored. Her kiddo presented with “it’s too hard” when it was really boredom.

Bored? No! She couldn’t possibly be bored! She is just now 6 and has had almost no formal math instruction. It’s all been picked up through apps and occasional problem-based lessons. And I pulled those from a third-grade book.  When she got bored with Dreambox last year, she was 84% finished with second grade.

Seriously?!?!

The freakout

How am I supposed to stay ahead of this kid? I feel like every time I have an idea of where she is, I turn around and she’s past it. I have felt that way for three years now.

I think I’ve made significant progress in changing how I think about education as it relates to L. Instead of thinking of myself as a teacher who sets out the path, I think of myself as an intense kid watcher. I watch her for emerging interests and skills and then scour the world for the most appropriate resources, which I place in her path in the most time-efficient manner I can manage. She is thus constantly picking up high-quality, high-interest materials and, since I know her pretty well, they’re typically in alignment with her interests. I don’t lead her down a path. I don’t even walk next to her on the path. I walk behind her on the path and slip goodies onto the path, hoping not to be seen.

But I’m at a loss here.

I don’t even know how to assess her appropriately to figure out where she is mathematically. I don’t know what curriculum to turn to. I don’t know how far off my estimate of her progress is.

I was re-reading an article about the opportunities the internet allows for gifted kiddos. The article refers to the Art of Problem Solving, a name I’d certainly heard bandied about (and the middle- and high-school wing of the Beast Academy company).

Tonight, I checked the diagnostic test to see if she’s ready for their first class – prealgebra (for students who have completed elementary math – grades 5/6). To be clear, I know she wouldn’t pass all of it right now, because we haven’t talked about division. She could nail about half of it (basic arithmetic with negative numbers, addition and subtraction of fractions with same denominators, basic fraction comparison, and some of the word problems).

Dear me. She can nail about half of the assessment for the prealgebra class “specifically designed for high-performing math students”. And she’s 6. And we don’t really do math in a sustained way.

But I’m pretty sure that if I introduced division and decimals, she could be ready in a month or two.

And then I look back at the online prealgebra class schedule and it occurs to me: she can’t take this class anyway, because it goes past her bedtime! How ridiculous is that? A class that she could be ready for in a month or two that she can’t do in a month or two because she still goes to bed at 8 BECAUSE SHE’S 6 YEARS OLD!

And it hits me again. She is atypical. She is asychronous. I am so lucky to get to be her mom. It is so terrifying to be her mom. And I’ve simply got to get some sleep.

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Getting out of the way

Why is it so hard for me to learn some basic ideas?

Seriously. I’m pretty bright, generally speaking. Ask around. People will tell you I have many faults (and I’ll agree with most of them!). However, a lack of intellectual firepower and/or curiosity is not generally one of them.

And yet, I still can’t seem to remember one simple thing: L likes to do it herself!

Very early on, my handsome husband was attempting to transport L from one place to another. She replied in indignation, “No mommy carry you! No daddy carry you! L do. A-self!”

She likes to be independent.

I have previously written about how difficult it is to divorce myself from the processes and norms of school. As a former teacher and teacher educator, it is particularly hard for me to let go of the idea of gradual release of responsibility.

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The basic idea behind gradual release of responsibility is that the teacher is the mentor in the classroom. As the resident expert in a given content area, it is the responsibility of the teacher to support the learner in their apprenticeship in that area. As a part of the apprenticeship, the mentor begins with heavy support and eventually, the balance shifts to the apprentice having the majority of the responsibility.

Another way to think about this model is that first, the teacher models the content. Then, the apprentice tries the content while the mentor giving hands-on with assistance. Finally, the apprentice is able to demonstrate mastery over the content.

This is such a great model. I believe in it strongly, particularly in my experience teaching middle grades in a workshop format. The only big problem? L hates it.

We get a monthly delivery of Tinkercrate. It’s a great hands-on, exploratory, all-in-one service. After looking closely at the different levels and formats the service offers, L decided she wanted the Tinker version, which is building and experimenting with projects. It’s advertised for children ages 9-16+. L’s hands aren’t particularly strong, so I suspected she would need support with some of the physical work.

Our delivery last month was the automaton.

L grabbed the crate out of the mail as soon as it arrived. We sat down and looked at the project, reading the first few pages of the guide. I told her there was a video on youtube, but she wasn’t interested in looking at the video. Instead, we pulled out the materials and got started.

By “and got started,” I mean that I asked her thoughtful questions and offered help as I thought she might need it. She replied with feigned helplessness.

She didn’t know how to pick something up. Or how to peel the backing off a sticker. She looked at me – no, through me – when I asked her questions. I felt my frustration growing, but knew that she needed me to help her, as I am the mentor and she is the apprentice.

I persisted with my helping. I am very persistent.

She persisted with being floppy and without ideas. She is very persistent.

An hour passed.

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As far as she got with an hour of my help

I finally got so frustrated that I walked away. I told her, “L. I know that you know how to ask for help if you want it. I will be in the other room, reading my book and drinking coffee.” I then went in the other room, feeling quite smug. Ha! I showed her! She needs my help and soon she’ll come and ask me for it. Then, I will help her and things can proceed.

I was about 15 pages into my book when I happened to glance back into the kitchen.

She had completed about three-quarters of the project now that I had simply left her alone. I decided to walk over and refill my coffee, snapping the picture below.

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As far as she got after 15 minutes of me leaving her alone

My own ego gets in the way of homeschooling (heck – of parenting!) on a pretty regular basis. I’m going to continue applying my intellect to the idea that I need to trust my child. If something is too hard, she’ll let me know. If she wants help (not needs it, but wants it), she’ll let me know. My expectations and routines are just that – my expectations and routines. It is my work to provide her with an environment that supports her inquiry and curiosity. Beyond that, it is my work to get out of the way.

If you need me, I’ll be in the other room, drinking coffee and reading my book.

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Faulty socialization and homeschooling

“But what about socialization?”

That’s usually the first question I hear in response to the information that we’re homeschooling, as I know it is for lots of other homeschooling parents.

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There are lots of great posts about there on the subject of socialization, including passionate posts about what’s wrong with the school-based socialization in the first place, those sharing results of research on socialization of homeschoolers (both pro- and con-), reasons its hard to avoid socializing homeschooled kids, tips on how to find opportunities for socialization, and those embracing a lack of socialization as an advantage.

I’m not going to weigh in on any of these aspects of homeschooling. Frankly, the territory is well-covered.

Those who know us know that L is often out and among others – those younger than her, her age, and older than her. Those in multi-age groups. Those in formal lesson-based settings. Those in open-ended, creative settings. Weekly brunches. Daisy Girl Scouts. You get the idea. And anyone who is still concerned about our form homeschooling and socialization is likely not going to be convinced by anything I say here, anyway.

I want to explore a different perspective the problem of homeschooling and socialization.

The problem isn’t with L’s socialization. It’s with mine. It’s the extent to which my socialization gets in the way of homeschooling.

I attended traditional schools from primary school through earning a doctoral degree. I taught in middle schools. I teach teachers how to teach in schools. I am firmly socialized into the context of schools.

I am socialized into the idea that learning happens on a schedule, whether daily, weekly, thematically, or otherwise. The way I experienced school and enforced school on others, there were defined times for defined subjects. The time and schedule were the constant, with the learning as the variable. As a student in a class of 30, my individual needs and preferences weren’t the primary drivers of instructional planning (nor is it realistic for my child’s to be – particularly given how specific and different from the norms of school they are!).

I am socialized into the idea that a child’s grade matches their chronological age. I struggle with not knowing how to answer (or help L answer) when people ask what grade she’s in. Typically, we fumble around for a minute and then mumble something about her being 5 and homeschooled. People feel awkward enough at that point to let us off the hook, I think.

I am socialized into the idea that we must all achieve a set amount of learning in each subject in each year. I have no doubt that L will eventually even out in terms of her interests, and she learns quickly and easily enough that I don’t fear that she’ll reach college and be functionally illiterate in any subject. However, she’s simply not interested in social studies right now. My urge is to enforce pursuit of everything instead of trusting that my child has wide-ranging curiosity and that curiosity will lead her to in-depth understanding.

I am socialized into the idea that learning must produce something. Something that I can judge or grade or assess or whatever you want to call it. Today in co-op, there were a ton of great activities out for children to explore. Making an articulated hand from drinking straws. The book Stick Man complete with a huge pile of sticks and material and hot glue guns to make your own. 3-d construction with paper tubes. Painting sticks. Valentine’s mad-libs. Instead of trying any of them, L engaged in three hour-long open-ended play sessions. She spent significant time playing with Playmobil animals and creating an imagined reality. This time was partially alone and partially with other children. She created an open-ended set of creations with Duplos, an activity we have at home but she rarely gravitates toward. This was time mostly spent with other children. She pulled other children in a wagon and was pulled in turn by them. I heard lots of narration about the ocean and she sang songs constantly.

I laughed about it with some other moms, but I genuinely had to check my feelings that she should have been creating some products that we could then have as evidence of her learning. I know she learned a ton! I know that free play is essential for learning and there’s a strong link between free play, the development of social and emotional skills, and achievement. But I’ve been socialized to expect a product.

I am so lucky. As Karen Maezen Miller reminds us in my favorite parenting book (Mother Zen: Walking the Crooked Path of Motherhood),

“Your child is a tireless teacher, constantly probing your self-imposed limits boundaries, your self-centeredness, your sheer stubbornness. It is a thankless job, and who would want it? But each day your child comes to work again, taking up the monumental task.”

Thank you, L, for reminding me every day that my own socialization is simply that: the model I’ve internalized for what education ought to look like. There’s nothing inherently right or true or good about it. It’s not more natural or more effective or better than all the alternatives. It’s simply what I know. What I’ve learned and now believe to be true.

I am socialized to believe that as the adult, I teach, and as the child, L learns. I am so lucky to get to re-learn this relationship.

MC Escher and tessellations: Where math meets art

In our ongoing quest to keep L engaged with math without necessarily pushing her through more and more abstract concepts. I still harbor fantasies of her going back to school at some point, and I worry that the growing disconnect between her age and her abilities is only going to make finding a fit harder. However, I want her to continue to push past the zone where things are easy and have to persist on some difficult tasks, too. She already struggles with shutting down if things don’t come instantly to her (or if she doesn’t do them “correctly”) so one of my goals for her educationally is to grapple with that which is just out of reach.

We recently completed a lesson in Beast Academy related to using polyominos to fill defined spaces. We’ve also been using pattern blocks in relation to our study of fractions, so it occurred to me that we could use pattern blocks to begin to explore tessellations.

A tessellation is a repeating pattern that has no overlap or gaps between the pieces. You can tessellate lots of shapes, but if you want to see how cool tessellations can be, you’ve got to check out the artwork of M.C. Escher.

I found a really cool link that shows how to make your own tessellating shape, but I knew that opening with that level of open-endedness was likely to freak L out. Instead, we started with our pattern blocks.

I took a cookie sheet and used washi tape to define a small (about 4″) square on the cookie sheet. We defined this as our field. We then sorted the pattern blocks by shape. L chose a shape to begin with and we began seeing how we could cover the entire field with that single shape with no overlap and no gap.

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Tessellating squares is easy!

We then moved onto hexagons, which were also simple to tessellate.

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We had a nice connection to the honeycomb in nature when we did this one

We then moved onto a shape which I’m not sure they had “when I was a kid” – or if they did, I certainly didn’t know anything about it… rhombuses! L loves the shape and the word – and I love the way she says the word (a mildly trilled “r” and like rum-busses). She first arranged the small rhombuses in a non-standard pattern, which we decided also looked like nature.

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Like the wing of a bald eagle!

When we moved onto the larger rhombuses, I asked her to arrange them differently than the previous set of rhombuses. One of my strategies with her is always to ask her to reflect on what she’s just done and find a slightly different take on the task. Here’s what she came up with for the larger rhombuses.

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A different arrangement of rhombuses

I decided at this point that she clearly understood the basics of the task. I asked her to remove most of the blue rhombuses from the field and instead, use a few rhombuses to make a different shape. Instead of tessellating rhombuses, we would tessellate this new shape she created.

L put together three blue rhombuses to create a hexagon. She was concerned that they didn’t fit together perfectly, but I told her that we could pretend there were tiny white rhombuses filling in the gaps because the gaps themselves were regular. She then began tessellating the sets of three rhombuses and came up with quite a cool pattern.

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Hexagons made of rhombuses

As we were admiring the work, L decided that we could now add some of those whole yellow hexagons to the field. I asked her to think about how to add them in a pattern, like she might find on a floor or a wall. She came up with stripes.

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Yellow and blue striped hexagon tessellation

And then, of course, she decided to input the red half-hexagons in sets of two to complete the stripes.

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Full on hexagon stripes

Very cool!!

Building off the idea of altering patterns, we then picked up the final shape we hadn’t yet used: the humble equilateral triangle. She designed a tessellation in which the vertex of one triangle rested at the midway point of a side in each line.

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Each line is the same with the triangles in the same places

She then pushed over lines two and four to line up the lengths of opposing triangles with one another to form a slightly different pattern – and in it, she found hexagons! We had a conversation about how we could re-create the three-lined hexagon tessellation above with additional green lines or how we could use three triangles in the place of any one of the red half-hexagons to complicate it further.

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Look, mom! Hexagons!

I was feeling pretty good about the open-ended result we’d experienced so far on this day, and I stepped away to take part in a quick phone conversation. When I returned, she’d created this tessellation. The green triangles are the wingspan and the single triangle above them serves as the head of one bird and the tail of the next bird.

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The birds in mid-flight

She also used this time to find tessellations on the floors and walls of our bathrooms. Since she was still really into it, I pulled out a recent supply I’d ordered from Nasco, anticipating both her enjoyment of this concept and her love of animals.

Animal. Tessellation. Templates.

I kid you not.

I mean, what in the what? Right?!

Anyway. They were a hit!

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Look at how fun these are!!

Let me be clear: I am jealous that we didn’t have these.

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She tessellated fish

The fish was the end of it for the day for her – I mean, she had been at it for a solid few hours. However, a few days later, we revisited the templates again. This time, I urged L to think about coloring in a pattern to enhance her tessellation. She picked up the dog and came up with this.

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Red and black tessellated dogs.

We’ll get to the self-made templates in the coming weeks. Overall, I feel relatively certain that she engaged her pattern-making brain, build some fine-motor skills, and also had a pretty darn good time, too.

#HomeschoolWin

 

A note on our curriculum choices

As a teacher by training, I like curricula. I like pacing charts and a spiraling curriculum. I like scaffolding big ideas. I like anchor charts. However, I already know that when I’ve tried to use curricular resources, we do a big lesson and then skip the next 10% because L’s already figured out and applied the pattern. Have I mentioned that she excels at patterns? It doesn’t help that she’s got massive gaps between what she can write and what she can express, or that she focuses on subjects of interest to the exclusion of everything else. These traits make it hard to find an “all in one” curriculum. Instead, we would be best described as eclectic (a word my husband hates), relaxed, thematic, and passion-based homeschoolers. Here’s what we’re using now (age 5).

Language Arts 

I could care less if she can spell words right now. She’s 5. Spelling will come. It may be an area of strength for her (as it was for me) or an area of frustration. Just as she learned to read when she was ready and using her own method, so too will spelling come. I could care less if her handwriting is legible right now. A huge part of that is that she’s 5. I want her to focus on the idea that she has important things to say and recording them allows them to be shared with others who aren’t with you right now. So we jot things down all the time and I scribe for her when she requests it. Our house is littered with notes.

Th closest we come to spelling practice is pointing out word parts like prefixes, suffixes, and word roots as we decode complex texts. We also read Grammar Island  on the couch as if it’s a read-aloud. We have a moveable alphabet and stamps. We don’t use them often. We play Wordsearch and Pathwords Jr.

L reads. Beautifully. Narrowly. And far above grade level. She hates most fiction. She doesn’t like reading for strangers. And she absorbs everything she reads, so working on non-fiction graphic organizers is a waste of time right now. We just read. A lot. Vociferously, in fact. And I often curl up and read my own books while she reads hers. (The link at the beginning of this paragraph is her reading a page from Almost Gone: The World’s Rarest Animals – Lexile AD1020L, DRA 34).

We also do a ton of stuff that strengthens her hands. Sewing. Play doh. Drawing. Snap circuits. Lego. Her handwriting will come along as her hands grow. So I call all of that hand-strengthening stuff “language arts” too!

Science 

We don’t go more than a few hours ever without doing science. She reads science books. Plays science apps. Watches documentaries and other science shows. Is known by name at both our zoo and natural history museum. Her pretend play with animal figures is based in actual appropriate behaviors for the species represented.

Oh, and we make stuff. Snap circuits. Building with “garbage”. Tinkercrate. Chemistry kit. Marble run. I think we’re good on science.

Math

I have really struggled with math for L. One issue we consistently have is that she gets bored (habituates) quickly. This manifests itself in a number of ways.

First, she is not interested in repetition, especially if the repetition is intended to do things like teach math facts. Exploring addition? A-ok. Practicing number pairs which compose 10? Only in context, my friend.

And how does she manifest said habituation? Well, through stubborn shutting down, of course! I will not be moved, her behavior says. One thing we work on is helping her learn to tolerate (and it is literally that – tolerate) that which she doesn’t find engaging. However, I feel pretty strongly that my 5 year olds world shouldn’t be primarily or even a lot about learning to tolerate disengagement!

Where that lands us, then, is in a land where what we find that appeals to her get used manically for some period of time and then discarded, never to be touched again.

Good times. Expensive times.

Some of you may recall how L was obsessed with dinosaurs for almost 2 years. She now won’t look at them. Literally. She keeps her eyes down at that part of the museum. Because she’s done with them.

In the past, she worked through all of Todo Math which contains PreK-2nd grade concepts aligned with several domains of the Common Core (which, to be clear, I’m not opposed to as a set of standards). She has outgrown that app. She tried the Redbird mathematics curriculum and didn’t respond well to the format. That was very quickly a struggle. She responds really well to the Dreambox Learning app (aligned pretty broadly to Common Core), but she powers through it in pretty big chunks. A few months ago, she walked through a review of kindergarten, all of first grade, and most of second grade in about 6 weeks. She then got stuck on a particular concept the app had her working through (perhaps it was at the edge of what she could do?) and started fighting about it.

We use a variety of approaches right now, mostly low-cost ones! As I mentioned in an earlier post, we have some mad love for manipulatives right now. We also continue to use Beast Academy on a very casual basis. I also surf pinterest and grab ideas that look interesting. Anytime we come across a math app that looks interesting and seems reasonably priced, we grab it. Right now, we’re playing with Slice Fractions, Quick Math, Jr., and Attributes.

We also play lots of logic games. We love Rush Hour, Blokus, Kanoodle, Pattern Play, Set, and chess.

Social Studies

L was interested in enslavement earlier this year, so we spent an intense 6 weeks learning a bit about US history and specifically those who didn’t escape on the Underground Railroad (she told me that since so few people ever escaped, it was better to really focus on those who didn’t escape). Luckily we live very close to the National Underground Railroad Freedom Center, so we were able to visit several times and obtain lots of information.

Very quickly, that interest passed.

This will shock you, but we listen to NPR. I’ll give you a second to recover from your shock. In any case, we were listening in the car and I attempted to talk with L about one of the stories. She cut me off and said, “Mom. I only care about the people I already love and nature. Not anyone else.” Ok. We’ll come back to social studies another time.

So overall…

Follow her passions. Watch closely. Be prepared with lots of high-quality, open-ended, inquiry tools. Drink a lot of coffee. These are the things that make our homeschool work.

At least, it works sometimes.

Exploring equivalent fractions: I’m 5 1/2 now!

This is the first in a series of posts about the mad love we have in our home for math manipulatives.

I’ve spent some quality time thinking obsessing about how to deal with L’s math brain. She doesn’t deal well with repetition and likes to think “big picture” things. I have had to deal with the idea that her automaticity will come through repeated use of facts rather than memorization simply because she shuts down if we attempt to do anything resembling math facts. There’s another blog post brewing in my head right now about the approaches we’ve tried…

In any case, our idea right now is that we are exploring in-depth topics that she seems ready to absorb.We’ve been playing with fractions already, especially as we cook and bake pretty frequently so she’s used to seeing them in recipes and on cooking tools.

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A previous day’s work: Find 10 different ways to make 1 using halves, thirds, quarters, and sixths

Recently, L turned 5 1/2. When I was a kid, we celebrated my summer birthday at the 6-month mark at school. L is also a summer birthday, and at the awesome play-based nature school she attends two days a week, they celebrate half birthdays.

In the car, we began discussing how L was 5 1/2 and how there were other ways to express that same idea. I didn’t delve into the conversation at that moment, but knew what the next day’s lesson would be.

I started by layout out for her the conversation we’d had the day before, telling her that there were lots of fractions that would express the same amount. I wrote out for her a variety of fractions with 5 as a whole number and 3, 4, 5, 6, 8, 10, and 12 as denominators (with no numerators). I then asked her to use her fraction circles to figure out which of those denominators could be made into a fraction that showed the same amount as 1/2. I very specifically wanted to include denominators that wouldn’t work to help her think about what fractions mean, not just how to work with them.

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Playing with the fraction circles

She used a few really interesting approaches, including laying pieces on top of the 1/2 piece (so putting the 2 1/4 pieces on top of the 1/2 piece to show they were the same) as well as using the additional pieces to complete a circle along with the 1/2 piece.

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Completing the circle with tenths

She was pretty quickly able to record 2/4, 3/6, 4/8, 5/10, and 6/12 as representing the same amount as 1/2. I told her she’d discovered a really cool idea called “equivalency” and asked her what word she could find in the beginning of “equivalent”. She found “equal” and we discussed how equivalent fractions represent the same amount as one another.

L showed me how the 1/3 pieces couldn’t be made to be the same amount as 1/2. She showed me how one 1/3 piece was too small to fill up the same amount of space as a 1/2 piece while two 1/3 pieces filled up too much space. We set those aside. She repeated the same reasoning with two 1/5 pieces and three 1/5 pieces. We put those in a pile we called the “crying, sad fractions” and crossed them out on the paper =)

We then looked for a pattern that emerged between which fractions could be made equivalent to 1/2 and which couldn’t. I introduced the words “numerator” and “denominator” so we could talk. We looked at the denominators and noticed that each of the denominators in fractions that could be made equivalent to 1/2 were even. We decided to test this theory by taking all of the pieces of each type of fraction and putting even amounts of them on each side of a dotted line.

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Checking our theory

After getting the even fractions divided into twos, we then considered the third and fifth fraction pieces again. L put one 1/3 on one side of the line and another 1/3 on the other side of the line. She sat there holding the third 1/3.

I wrote out for her 1/2 = 6/12 = 5/10 = 4/8 = 3/6 = 2/4 and we discussed again the pattern. This time she noticed that if you added the numerator to itself, you would get the denominator. I told her again what a cool discovery this was, and then directed her to this question:

If you notice that all of the denominators in the fractions equivalent to 1/2 are even, why can’t thirds or fifths be equivalent to 1/2?

It was a big question. She sat there. She put the final 1/3 underneath the dotted line. I worried that I’d pushed too far.

“Their denominators are odd!! They aren’t equivalent to 1/2 because they are odd!”

Right, kiddo.

“And you can’t divide 3 or 5 by half and get even groups!”

Precisely.

“And you can’t add any numerator to itself and get the denominator!!”

Amen, sister.

As we were high-fiving, I decided to double check her understanding of the underlying concept, so I wrote up a quick challenge sheet: Can you make an equivalent fraction to 1/2 out of sevenths? fourteenths? ninths? twentieths? What’s another denominator that you can make an equivalent fraction to 1/2 of?

She spent a minute puzzling over the first four, figuring them out correctly. Then, I heard it in relation to that open-ended question: “Well, I’ll just start in the thirties. 30, 32, 34, 36, 38, 40… any even number would do it.” Eureka!

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A hard day’s work

In true form, though, we got to discuss one more great math term. L said, “So I could say that I am 5 6/12, but I never would.” I asked her why, and she told me, “It’s too hard to understand that one. 5 1/2 is easy.” I reminded her about the word simple, which she told me meant the same thing as easy. It was a quick hop over to writing down “simplification” and defining that as a fraction in the form of the smallest numerator and denominator that represents the same amount.

All in all, a solid day’s work.