Category: Math

You’re Smarter Than You Used to Be

Posted on by Chris Loper

A middle school student working hard

There’s an old math topic that was really hard for you. Maybe it was fractions. Maybe it was long division. Maybe it was memorizing your multiplication facts.

Now here you are in middle school or high school, and this topic still haunts you because new math is always built on old math. You know it would be helpful to patch that hole in your knowledge, but you don’t think you can. You think it will still be too hard for you.

But you’re wrong.

You’re way smarter than you used to be.

How could you not be?

That “super hard” math topic was from years ago, and your brain has been getting stronger the whole time. You’ve learned far more difficult things since then. So you’re more than capable of going back and learning that old topic.

Put in a few minutes a day until you’ve got it down, and you’ll have easy access to that knowledge for the rest of your life. I promise you won’t regret it.

Why Calculus is Hard

Posted on by Chris Loper

A blackboard with calculus work in progress

Calculus is widely regarded as a very hard math class, and with good reason. The concepts take you far beyond the comfortable realms of algebra and geometry that you’ve explored in previous courses. Calculus asks you to think in ways that are more abstract, requiring more imagination. You have to wrestle with new vocabulary, new symbols, and new processes. The problems are often longer and more involved, sometimes taking a full page or more of written work to complete.

Those are the obvious reasons why calculus is hard. But there’s more to it than that.

The Main Reason Calculus is Hard

In my experience as a tutor, the primary reason calculus is so difficult is that it is sitting atop an upside-down pyramid of previous math concepts. Calculus ties together everything you’ve learned (or were supposed to have learned) in algebra, geometry, and precalculus, as well as the more basic math from elementary and middle school.

Everyone accumulates knowledge gaps as they progress through math, and for many students, calculus is the course where these gaps finally come back to haunt them. At the same time that they’re trying to learn the new, weird concepts that make calculus what it is, they’re also struggling with all the old things they don’t know. This makes new concepts more confusing. It makes it hard to keep up with lectures. And it makes doing the problems very challenging.

Why do students have so many knowledge gaps?

First of all, the way our school system does math makes knowledge gaps inevitable. Let’s say the average student finishes a typical math class with an 80%. We say to them, “You got at B-. Not bad. Now move on to the next course.” So next year, they start out missing 20% of the information they’re supposed to know! We do this year after year, until their math knowledge pyramid looks like Swiss cheese.

But it’s worse than that, of course, because students routinely forget things that they used to know. The mastery path is a muddy slope, so even if you understand a concept at test time, you’ll quickly forget it without additional practice. This forgetting happens during the school year as well as over the summer.

And right now, in 2022, knowledge gaps are at an all-time high because students didn’t learn as much as they were supposed to during COVID. So if you’re going into calculus this fall, you can be sure that you have knowledge gaps that are going to make your life difficult.

Don’t be ashamed of your knowledge gaps.

You might be ashamed to admit that you don’t remember quadratics or trigonometry, but it’s okay. I’ve worked with calculus students who didn’t know how to do basic algebra. I’ve worked with calculus students who didn’t know how to add or multiply fractions. I’ve worked with calculus students who didn’t know their multiplication facts.

Having these gaps doesn’t make calculus impossible, but it does make it a great deal more difficult. So if you’re about to take calculus, or even if you’re in the middle of the course, consider hopping on Khan Academy and patching up some of the holes in your math knowledge. Yes, it’s extra work now, but it will make things easier for you in the future.

Don’t be ashamed of your knowledge gaps; deal with them. If you do, calculus won’t be quite so hard. You’ll avoid slipping into a downward spiral of math avoidance by transforming this daunting course into a manageable, even enjoyable, challenge.

How to Patch Holes in Your Math Knowledge: 3 Steps

Posted on by Chris Loper

Every math student collects knowledge gaps as they progress through school. Sometimes you don’t understand a concept when it gets taught, and the class moves on before you figure it out. Sometimes you get it, but since the mastery path is a muddy slope, you quickly forget. And some key pieces are, for whatever reason, never taught at all. This year, with limited schooling due to COVID, that final case is more common than ever.

And if you understand that math is an upside-down pyramid, then you know that knowledge gaps make a very unstable foundation for future learning. They increase the likelihood of getting confused, falling behind, and slipping into a downward spiral of math avoidance. So patching holes in your math knowledge is critical. But how do you do it?

What follows is a step-by-step process for identifying and filling knowledge gaps in mathematics.

1. Identify

Step one is to identify what you don’t know, back up to the earliest, easiest things that you don’t have mastered, and start from there. To do this, I recommend using Khan Academy’s course challenges. These are like miniature final exams for Khan’s math courses. There’s one for every grade. You can also take the unit tests that are at the end of each major subject within these math courses.

When you take a course challenge or a unit test, Khan keeps track of what you get right and what you get wrong. (You have to be logged in to your free account for this.) Then, when you scroll through the exercises of the course, each activity will have a little icon next to it that will have zero, one, two, or three purple bars filled in.

  • Three purple bars represents mastery: You’ve gotten it right several times and haven’t recently missed a question.
  • Two purple bars represents proficiency: You’re approaching mastery.
  • One purple bar represents familiarity: You’ve gotten a couple questions on this topic right, but you’ve also missed some questions, so this needs work.
  • Zero purple bars means you haven’t demonstrated any understanding of this topic.

Topics with zero or one purple bar are your knowledge gaps.

Let’s say you’re about to take high school geometry because you’ve just finished Algebra I. You might take the Algebra I course challenge. And if you have a pretty good memory of what you learned this year, you might miss only five or ten questions out of 30. If that’s the case, those five questions represent content areas for you to target. Go back into the course and work on them.

But if it doesn’t go so well, and you miss say, 15 or 20 out of the 30 questions, you probably need to back up further than Algebra I. Drop back to the Pre-algebra course challenge, or go back to the 5th grade course challenge. Feel it out. Maybe you need to go way back and relearn your math facts. Maybe your math comfort zone is so small that the right move is to start at the beginning and work your way up.

I’ve worked with Algebra II students who didn’t know their multiplication facts. I’ve worked with calculus students who needed to relearn fractions. And I myself had to relearn subtraction with borrowing at the age of 27. It’s all okay. There’s no shame in not knowing something. You just need to identify what’s missing from your math knowledge so you can go get it.

2. Understand

Once you’ve identified your knowledge gaps, it’s time to get to work. The first milestone on the mastery path is achieving understanding. For this, Khan has videos for every topic. Watch them, and pause often to take notes and try things out yourself. If Khan’s videos aren’t sufficient to “get it,” there are numerous articles and videos online that teach math, along with workbooks and textbooks. And, of course, you can ask other people for help.

Being confused is normal. Work through it by using resources and making written product.

Once you think you understand, try the exercises. If you get stuck, consult your notes or use the hints provided with each problem. The hints walk through the questions step-by-step. And don’t just look at the solutions – write them out.

3. Master

After you achieve understanding and do an initial round of practice, you need to follow up on that with spaced repetition in order to solidify both your memory and your comprehension of the topic. Revisit this topic several times in the days after you learn it and then for several weeks after that.

Start with massed practice. For this, Khan isn’t the best choice. Worksheets are a better option. Do a quick google search for the topic plus the word “worksheet,” and you’ll find plenty of choices. Include the words “answer key” if you’re not immediately finding worksheets that come with answer keys. For high school math, kutasoftware is great. Hit print, and get to work. But don’t necessarily do the entire worksheet in one sitting. It’s better to do a little bit each day than to do it all at once and then ignore the topic for a week.

Repeat this process for several topics within a unit on Khan, and then take the associated quizzes and the unit test. These present you with a variety of problems, forcing you to remember which methods apply to which situations. This is called “interleaving,” and it is critical for advancing as a math student. Just remember that you don’t want to go straight to interleaving; it is the combination of massed practice and interleaving that cultivates deep mastery.

Make Progress

Patching holes in your math knowledge is hard work. So if you engage this process, and you discover that you have many knowledge gaps, you might be disheartened. But please don’t see this as an all-or-nothing endeavor. Any holes you patch will make your life as a math student easier. Any progress you make here is worthwhile. Everything counts.