Category Archives: Educational philosophy

I remember the moment well – it was almost bedtime, and my son, already in pajamas, asked me about the derivation of the quadratic formula. Yeah, yeah, that’s kind of thing just happens in our family. Since it had been a while (like 25 years) since I had studied this myself I looked on the internet for the best way to present the idea of completing the square.

I loathe Khan Academy, but decided it might be useful for this kind of refresher. Boy was I wrong! The opening of the video started off ok as he discussed the fact that quadratic equations that are perfect squares are super easy to solve.  Check! If we can make any quadratic look like its a perfect square we can solve that easily too. Great! I won’t bore you with the details, but needless to say it went downhill from there as the video immediately dived into a bunch of manipulations and never once explained things conceptually. You see, completing the square is actually really easy to understand if you actually draw a square. Hey look, we’re taking a rectangle and cutting off some bits to make it a square and adding another little bit to finish it off! You can draw it out and show the student visually exactly what’s happening. Here’s an example of a site that gets it right.

This is the crux of my problem with Khan Academy – a lot of procedural mathematics with little focus on understanding. It might be ok for test prep review when concepts have already been mastered, but I absolutely cringe when I hear of homeschoolers using it as their primary math program or parents of gifted kids using it to accelerate. These kids need to be challenged, inspired, and taught problem solving skills, not memorization of mindless algorithms. For a child that is behind or already turned off from math, seeing numbers fly around the screen with no idea of the underlying concepts could lead to even more confusion. Real math is not about manipulation, its about knowing how to think and set up a problem.

I do think there are some good videos out there for learning math, for instance AOPS has some that follow along with their booksEducation Unboxed has outstanding teacher training videos for using Cuisenaire rods, and The Great Courses has some cool lectures that we’ve really enjoyed. Additionally, online courses can be a great way for motivated older students to learn. However, when it comes to elementary and middle school math in particular, I really believe that human interaction, discussion, hands-on manipulatives, and instant feedback are essential components of learning how to think about numbers. For a more detailed and articulate critique of Khan Academy I highly recommend this article from the Washington Post: Khan Academy: the revolution that isn’t. The title says it all.

Filed under Educational philosophy, Gifted math, Uncategorized

Variety is the spice of life… and the key to math acceleration

Sometimes I get asked how my son came to be so accelerated in math. I’m sure some assume we must be doing a shallow just-the-basics curriculum or that I stand over him Tiger-mom style as he whips through pages of worksheets. Neither is the case, and the math study we do is both very deep and the most joyful part of our school day. There are a few secrets though:

• Math is his thing. It happens to be my thing too, so this creates a synergy between teacher and student that is hard to force. Simply put, it makes magic. This is the most important ingredient to math acceleration. I would not be able to coerce the amount of learning we do: it comes from the intrinsic interest of the student.
• We homeschool, and do so year-round. Nothing beats a one-on-one learning environment. With homeschooling we can take time to work through concepts that require more attention and skip the busywork, all with my student getting immediate feedback on mistakes. The lack of a long summer break means we can keep moving forward without a loss of learning or time wasted on extensive review, and although we do take plenty of vacation time, we also do math on most weekends and while traveling. I estimate the combination of summer and weekend work probably doubles the amount of math we get done in a year compared to a student in public school.
• I make it fun. Hands on activities, games, and a unique approach to skill-building along with humor and a light hearted attitude by me, keeps us moving forward even when work gets tough.
• Variety. The first three points might be obvious but the issue of “what to study” in the first place is not quite as clear and what I’m here to talk about because it applies to all learners, accelerated or not, math-loving or less so.

The biggest lesson I’ve learned in working with gifted children is that no matter how brilliant the student is, stamina and attention span are still strongly tied to chronological age. Without a sustained attention span the amount of material one can cover is going to be limited.  What does this mean in practical terms? It means even if you have a rare Kindergartner who can do long division, they probably can’t do a whole page of practice problems the same way a fourth grader could. In fact, they can probably only do one problem before they are bored or mentally exhausted. Even students who are not extremely advanced will get bored with too much repetition, and this can hinder math progress as well as leave students thinking math is uninteresting and uncreative when nothing could be further from the truth.

My radical solution is to give up on the idea of mastery, albeit temporarily. Some educators (though fortunately not all) are obsessed with the idea that students must memorize their multiplication tables, perform spectacularly on timed tests, and get 100% on assigned work before they can move onto more interesting stuff, but this is entirely backwards. The more advanced the student is, the more backwards it is. To let you in on a secret: the narrow track we’ve created for math education is almost entirely artificial. Math is such a broad and rich field that there’s simply no reason you have to stay on path. You can mix arithmetic practice with probability with some beginning algebra with number theory and cover more than one topic at once. I have especially high regard for discrete math which is all but ignored by traditional K-12 schooling. Of course all students should master multiplication tables and become automatic at traditional algorithms, but not to the exclusion of other learning – by exploring multiple threads of math at the same time its much easier to keep interest high, tedium low, work longer, and gradually build stamina for more difficult work.

Just the other day I came across this profound statement in Art of Problem Solving: The Basics, Volume I:

We strongly feel that a student should learn all subjects simultaneously. There are two reasons for this. First, it helps to convey the interconnectedness of it all; how geometry naturally leads to coordinates and how those coordinates make it easy to define conic sections and the complex plane; how counting leads to probability, the binomial theorem, and number theoretical ideas. Second, it all sinks in better. Overloading on a single subject can cause students to acquire a surface understanding which doesn’t connect to any deeper comprehension, and is thus rapidly lost.

In my next post I’ll give some specific examples of how to include more variety in your students math education.

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Filed under Educational philosophy, Gifted math, Uncategorized

Starting after school enrichment with math-talented students

If you have an elementary student who is talented in math (or just simply loves it), figuring out how to supplement their math education can be tricky. You may sense that they need more than what school is providing, but not know where to begin. With remedial students it’s usually clear which skills need to be shored up or which gaps in conceptual thinking need to be bridged. However, for a student who is great at math and is under-challenged in school, the answer is less obvious. Doing more of the same is rarely the answer. Math-talented students don’t need more work, they need different work; work that really stretches their brain, but not so tough that it’s discouraging.

Acceleration is usually the first option considered, and it can be part of the solution, but its not always the perfect choice. Frankly, most math programs don’t actually get much harder as they go along. They introduce new topics and procedures but rarely delve into the kind of multi-step problem solving or open ended questions that will really develop math ability.

Also getting too far ahead in the school curriculum can be a real concern for some parents. This isn’t a problem if you’re a homeschooler, have a very flexible school that will work with your child’s acceleration, or are willing to continue supplementing at home until high school, but otherwise there can be consequences to working too far ahead and having a child bored in the classroom.

Fortunately it doesn’t have to be an either/or decision. There are many topics in mathematics that are both broader and deeper than basic arithmetic and that go way beyond the standard pre-algebra-to-calculus pipeline. I’ll be talking more about this in future posts. However, my first recommendation for any student, of any age, is to improve their problem solving skills.

An easy way to get started is with the Singapore Challenging Word Problems (CWP) series. A great first step is to simply buy the CWP book that is at a level appropriate for your child, tear out a page, grab a magnet, and put it on the fridge. Make this “The Problem of the Day.” Do it before school or after school or whatever time of day is best for your family. Perfectionism among gifted students is common, so I believe it’s important to keep it fun and not get caught up in whether your child gets it right or wrong. Doing math outside of school should not be a chore, but about learning to love problem solving.

There are other great problem solving books out there, but this is the one that I’m most familiar with. The problems in CWP have incredible depth and are extremely well written. The best thing about this is that you’re hitting a lot of different skills in one time-efficient package. You get multi-step problem solving, a gentle introduction to algebraic thinking, and basic arithmetic practice on a variety of topics. My only caveat is that CWP, as the name says, is challenging, especially if you haven’t had experience with Singapore math, so I often recommend buying a grade level down to build confidence and patience.

The reason I like the “problem a day” approach is that it fits in with my two basic educational principles.  The first basic principle is that starting small is the best way to begin. There are a few people who can radically overhaul their lives in a small amount of time, but its rare. For most parents, finding the time and energy to squeeze in some math enrichment between sports and music and homework and dinner and bedtime is tough. Starting small means you will actually do it.

This leads into my second basic principle, which is that a little bit every day is better than a lot all at once. With daily work you have higher retention and the extra minutes quickly add up. I know the idea of starting with 10 minutes a day of extra math sounds like the too-good-to-be-true ab workout on the cover of a fitness magazine in a supermarket checkout. However, that extra problem a day can painlessly lead to an extra hour of math per week, and if you keep it up all year round you’re potentially talking about the equivalent of two full grade levels in the course of an elementary education.

So if you’re not sure where to start with your math-talented student start simple with a daily word problem.