Review of Elements of Mathematics: Foundations

About two years ago, Peatie hit a wall in math. He had been using and enjoying Art of Problem Solving materials for years, completing Beast Academy 3-5 and (to my surprise) enjoying the hefty tome that is AoPS PreAlgebra. But partway through the Intro to Algebra book, something just wasn’t clicking: some chapters he could breeze through with ease, while others just didn’t seem to make sense for him.

Of course, I did what any sane homeschooling parent would do: I panicked. Peatie has loved math since the time he first discovered that he could count his fingers, and back in kindergarten he was clear enough on his preferences to inform me, “I want HARD math, Mommy!” In fact, math is one of the few areas in his life where he seems to thrive on challenge–but he was no longer thriving with the challenge of Art of Problem Solving, and my math skills were too rusty to be of much help. But the general consensus of everyone on the internet is that if you want your kid to have the best, most challenging math, AoPS is the way to go. I was clearly failing my child!

In the midst of this weeks-long panic, I somehow ran across a mention of the Institute for Mathematics and Computer Science (IMACS) and their challenging middle school curricula, Elements of Mathematics: Foundations. I could find very little information about the course, but what little I found seemed promising. Noting that the first course (of 18, intended to cover a three-year period) was offered for free, I thought I’d let Peatie try it out. He loved it!

Here’s a sample screenshot from somewhere in the middle of the second year of material, just so you can get an idea of the format of the program.

For the past two years now, Peatie has been working through EMF courses. Since I still rarely see these mentioned online, I thought maybe someone would benefit from a review of them.

EMF consists of 18 courses of varying lengths, intended to take a middle school student through what’s traditionally considered Prealgebra, Algebra 1, Algebra 2, Geometry, and Precalculus over the course of three years.

There are a few downsides to this curricula:

1. While there are in-person classes for those local to IMACS, most of the world would be accessing these classes via the internet. You need a reliable internet connection with speed adequate for downloading the videos in order to do this course. If your internet goes down, you’re outta luck.
2. The symbols and terminology used in this program are so vastly different than those in traditional math classes that parents will likely be able to offer very little help if their student is stuck. Sometimes my son simply needs me to stand in the room as a sounding board, and sometimes I can offer suggestions like, “Why don’t you start over? There’s a chance you made a mistake that you aren’t noticing.” Other than that, I’m not much use. There is a help forum available, and many questions have a little “?” icon with past students’ questions and the suggestions offered to them, but some kids might find this to be too little direct assistance if they’re used to a solution manual or the possibility of parent assistance.
3. Because this program is so different from others, there’s no way to start midway through the program. Every student must start from the very first course. This caused me some hesitation: Peatie had already completed a very solid Prealgebra course and was most of the way through what would be considered Algebra 1. It felt like I would be consigning him to repeat old material and generally slowing his progress by having him start over. I eventually came to my senses, reminding myself that education isn’t a race and it would be far better for him to repeat material and know it very well than to rush ahead just so he could complete Calculus as a high school freshman (and why?). For all my agonizing, he had no complaints about what he was doing. Despite the fact that the first courses he was taking were part of a “Pre-Algebra Plus Course Pack,” the approach and scope were so vastly different from what he’d been doing (with titles like “Operational Systems,” “Ordered n-Tuples,” and “Number Theory”) that it didn’t feel at all like repetition.
4. The pacing for this program is a challenge. Though you know how many courses they expect you to complete in a traditional school year, some courses are much longer or more complex than others, so being halfway through the courses might not mean that you’re halfway through the curriculum. For the first year, IMACS offers a week-by-week progress guide to help the students pace themselves. After that, however, there’s no more guidance offered. While I realize that it’s wonderful to work at your own pace, it’s nice to have some idea how long the course creators expect the material to take. Particularly since my son is a slow worker who is easily side-tracked by his ADHD, it’s nice to have a goal to help him stay on track (and give me an idea whether those three pages took him all week because the problems were especially hard or because he got distracted by the graphing process).
5. It’s not inexpensive. Currently the courses are listed at $60 apiece, which adds up in a hurry; HOWEVER, you can save a lot by buying a “Course Pack,” which bundles all the courses for one year into a single purchase. (The first year, made of Courses 1-9, rings up at $360, while the next two years are around $250 each.) That said, if you’re considering getting instructional help for AoPS, even their self-paced courses are $460 per semester–not including textbooks–at which point $350 per year doesn’t sound so bad.

Those were the drawbacks: now for the positives of EMF.

1. It’s less expensive than many of the alternatives. I researched online AoPS instruction, both from AoPS itself and from other providers, and I couldn’t find classes from institutions I’d heard of for less than $800/year. (I did find an obscure provider offering $25/month classes, but I was concerned about paying $200–plus the cost of the book–for an instructor I knew nothing about.) Local math tutors offered services starting at $25/hour. Again, even $350 sounded like a bargain compared to the alternatives.
2. It’s challenging. My son is thriving because EMF is stretching him to think of math in new ways, and he’s loving it. Though AoPS is an excellent program, it’s not the only amazing program out there.
3. It’s fun. While EMF has a lot of text, similar to AoPS, there are also silly little videos (okay, mostly audio) featuring students and a professor discussing the concepts that are being taught. My son enjoys the humor these characters add to the learning process, and he gets a kick out of the fact that he can “level up” as he learns. In addition, EMF sprinkles in silly problems (similar to AoPS) and offers interactive elements so you can better visualize the math you’re doing. (The one downside: sometimes my ADHD son gets so into testing different things with these interactive models that he completely loses track of time.)
4. Remember how I mentioned that I couldn’t really help my son with math because the symbols and terminology are so different? While that’s a drawback, it’s also a positive. Because he can’t expect me to rush over and clarify, Peatie has had to become a little more independent. For a child who loves my apron strings, this has been great. He’s learned to re-read confusing passages, look up terms in the index, re-work incorrect problems, and check out the help forum–all skills that will serve him well as he faces challenging courses in the future. And the fact that he’s experienced success in problem-solving has bolstered his confidence that he can do this on his own.
5. It seems to be effectively developing his understanding of math. I have my kids take a standardized test each year to make sure they’re comfortable with the format of major tests and give me information about their strengths and weaknesses. Last year I had Peatie take the MAP Algebra 1 test, figuring that even though he was still finishing Pre-Algebra in EMF, he’d already taken most of Algebra 1 through AoPS. This year I was scratching my head at what to do: I didn’t feel like he should re-take the Algebra 1 test that he’d done well on, but he technically hadn’t completed the Algebra 1 material in EMF and hadn’t even touched on Geometry yet. In the end, I signed him up for Algebra 2 and told him to do his best and not worry about it. He scored in the 98th percentile on the test, despite technically still being in the middle of Algebra 1.

Well, that’s all that I can think of for now. I hope that this report is helpful to someone else who is looking for a challenging math alternative to AoPS.

2021-2022 Curriculum

I can’t believe that it’s this time of year already! I feel like this past year, insane and limited as it was, flew by. I’ve had a hard time choosing curriculum for this coming year: I want it to be rigorous since my older two are in middle school, but I still want them to enjoy learning. Add to that the fact that I’m not sure what extracurriculars are going to be available to us because of the ever-changing nature of this pandemic, and I’m not sure how much at-home vs. on-the-go time to plan on. But I’ve done my best, and here’s what I’ve come up with:

2020-2021 Curriculum

I love the start of a new semester! The excitement of curriculum planning has always appealed to me. This year I’ve decided to steer my kids towards a bit more rigorous and independent work, both because they’re reaching middle school and are ready for more independence and challenge, and because we have no social life vying for our time. So without further ado, my curriculum choices for this year:

2019-2020 Curriculum

This year was a hairy one, since I was working valiantly to balance academics with extracurricular activities and social opportunities. Here’s our curriculum for 2019-2020:

2018-2019 Curriculum

It’s that time again: time to list another year worth of curriculum! Read on for our choices for another year of mid-level elementary schooling.

2017-2018 Curriculum

This is the first year I’ll have three kids fully in the work rotation, though Pookie did some light daily work for the last half of the previous year. Here are my curriculum choices for everyone:

changing direction: when tried-and-true stops working

I used RightStart Math for my kids’ introduction to math mostly because I could do it completely through play. Peatie has always had a passion for numbers, and by the time he was 4.5 he had me scrambling to figure out how to help him feed his passion. Since I wanted to delay formal instruction and keep his learning play-based, I settled on RightStart as an ideal program.

Once upon a time, everyone was contentedly learning with RightStart Math.

We went through RightStart A completely informally; I would read through a few lessons on my own and then implement the activities as they fit into our play. The kids loved making patterns with the colored tiles, testing out the math balance, and playing the various card games. When we finished with level A, I happily moved us on to level B. There ensued great glory. Level B is brilliantly structured, perfect for laying a strong conceptual foundation for future math skills. It also strikes a perfect balance between hands-on discovery and games sprinkled with just enough practice pages to prove mastery and make your kid feel smugly grown-up about having occasional written work.

About a year after we began using RightStart B a few days each week, we finished the material and moved on to level C. I quickly began seeing red flags. It began with too much review, but that’s to be expected; when you school year-round, you don’t need the beginning-of-year review necessary for those who have taken an extended summer break. But when the algorithm for addition was introduced and taught exactly as it had been in level B—as if it were an entirely novel concept—that’s when I began to have misgivings. Peatie’s mutiny was constant sighing about wanting hard math, not stuff he already knew. Goobie’s mutiny was throwing fits at the very idea of math and grumping like nobody’s business through each and every lesson (as much as she believed she could get away with, since in our house too much attitude will result in your work being put away and a subsequent forfeit of your allotted Technology Time for the day). She was convinced that she despised math, but she’d also somehow decided that she was also bad at it. Math was no longer the bowl of cherries I’d been enjoying.

I sped Peatie through level C, skipping all review and only staying on each concept long enough for him to prove understanding. He finished in three months of three- or four-day weeks. I tried giving Goober options—game or worksheet? Math first or last? In the end, we bailed before she finished.

That’s right, I’ve given up on RightStart. Oh, I still like level A as an introduction, and I’ll never lose my passion for level B, but I finally came to accept that a program that worked spectacularly for a season may not be the best fit forever.

Peatie is now working through the Beast Academy math books by Art of Problem Solving. For a kid who seems to thrive on challenge, they are perfect for him. They offer problems that make you apply your knowledge in a variety of contexts, truly owning the information and showing how different aspects of math must be used in tandem—like knowledge of geometry and multiplication to determine the area of a shape.

I backed Goobie up to Singapore 2A, which was vastly too easy for her. This, however, has been the perfect solution for rebuilding her mysteriously-waning confidence, and the puzzle-y format has rekindled her love of math. We only did about half of the pages in Singapore 2A before moving on to B. Yesterday morning this math-averse child of mine came dancing from her room. “I love math so much that I was doing some leftover pages of my math book in my room this morning before it was time to get up!” And sure enough, there were her workbook and pencil, positioned on the floor in the glow of her nightlight. Today she begged me to make her some extra math pages like one puzzle she had particularly enjoyed, and when I did so later in the day, she gleefully spent some of her playtime on doing them.

Sometimes it’s hard to change course once your sails are set, but making that change might be the best possible thing to do.

a dozen activities to develop early number concepts

Exposure to numbers doesn’t have to be formal and serious. Even silly experiences can be full of learning!

Learning about numbers is one of those foundational concepts, necessary for understanding so much about the world.  For this reason, we diligent parents are eager to ensure that our tykes are on-track for numeracy skills from the very start.  These ideas, for the crowd that isn’t quite ready to write numbers or study formal math, will build a foundation for later success–without the need for any formality or structure.

1. Make counting a regular part of your life, from how many baby carrots your child wants for lunch to how many strokes it takes to mix your brownie batter. The more opportunities you take for counting (within reason, of course!) the more familiar your child becomes with numbers and their order.
2. Master the “concept” of number. Talk about how many cups you’ve stacked, how many blocks are in a line. Help your child develop a mental image for each number—especially for 1-5.
3. Compare groups of items. Line up all your red Hot Wheels and park the black ones in a row beneath them. Decide which group has more, and how many more there are.
4. Have fun with simple word problems. “Look, you have five grapes left! How many will you have after you eat another one?” In this way, your children will learn that numbers and math help them to describe and understand their world.
5. As your child ages, “their” number is infinitely special to them. On each child’s birthday, I make a sign saying, “[Kid] is [#] today! Happy birthday!” This sign starts prominently displayed in the kitchen, but after a week or so it moves to their bedroom, where it hangs all year. If nothing else, my kids master recognition of one number each year!
6. Point out numbers in your child’s world. Count down that last minute on the timer until the cookies are done, talk about the price of items you’re buying at the store (as you point to the numbers on the price tag), inspect the page numbers of the books you read. Start to nurture familiarity with the written symbols we use for numbers.
7. Post a number line or 100 chart for your child to ponder. Show them how it’s organized, and refer to it when numbers or comparisons arise. They may love to count while pointing to each number—but if they don’t, that’s fine, too!
8. Combine an understanding of number order with some fine motor work: get simple dot-to-dot pages for your child to complete!
9. When they’re interested, play a matching game. Place groups of objects on the floor, a different number of each. Give your child a set of number cards (marker on a blank index card or bit of scrap paper works great), and help them match each number to its corresponding group of objects.
10. Here’s another matching game: Place the number cards on the floor, and see if your child can create an appropriate group of objects for each card.
11. Using Uno cards or homemade number cards, have your child try to put the numbers in order. Be sure to reinforce the left-to-right order they’ll use for reading!
12. Read fun number books. A search of your library’s catalog is sure to turn up at least a handful of colorful or silly counting books.