Just teach the math, thanks!

Every time I see “multiplication math tricks” like the lattice method or this line trick, it really boils my blood.

The problem with tricks like these is that they aren’t fool-proof, nor are they scalable. Don’t believe me? Ask a student who has only been taught to multiply using the lattice method to multiply two decimals. He’ll either ask for a calculator or throw the paper back in your face. Neither is good for the student’s self-esteem.

The reason why we teach algorithms is so the student has something scalable that can be adapted to the type of number he’s working with. Even if it’s a bit daunting to the student at first, with practice they get the hang of it. When they run into problems multiplying fractions or decimals down the road, you can then tell them how the rules change (and those are pretty slight changes) and they’ll be far more successful.

Tricks are an easy out, but they don’t do the student any favors. It’s far better, both for you and the student, to just teach the algorithm from the start and then give the student ample opportunities to practice the algorithm.

Sounds like my students

Let’s hope this year proves my students, and math students everywhere, very wrong!

What’s the point of homework?

The school year is slowly beginning, and my students are starting to bring in their homework. It seems like the ideal time to remind everyone what homework’s purpose is in this life.

For those not familiar with me, I work in a K-12 learning center. I spend a lot of my time with the advanced math students, many of whom bring in their homework either for me to check over or to explain.

I know what you’re thinking. They’re teenagers. They probably didn’t pay attention in class and that’s why they don’t understand. In some cases, that’s true and the student admits it, but we’ve had a startling trend over the past couple of years (as the integrated math program has invaded the younger grades, coincidentally) where math teachers are assigning middle school and high school students homework for the skill they’re studying the next day. In a frightening number of cases, the teacher then fails to either check the homework or teach the skill.

Naturally, I have some very frustrated kids who just want to understand how to do the math being shoved on them.

The point of homework is not to replace instruction. That’s not what the school district is paying for. That’s not what parents want. Amazingly, that’s not what students want, either. Sure, you have the slackers, but eventually even the slackers want to be able to pass the class and not fail the next one.

The point of homework is to give students an opportunity to practice the skill you taught them. It’s a chance to make sure they understand it before you give them a harder skill built upon this skill. It’s a chance to make sure you taught the skill to them in a way the majority of the class understands.

If you’re expecting your students to teach themselves, perhaps it’s time you reconsidered your career choice, if only for the sake of the students.

The struggles of scaffolding

Last year when I started working on Dead Bunny, I realized very quickly that part of why my students struggle with new math concepts is because they often have to learn not just the new concept but a skill they probably should have been taught earlier.

In some cases, a teacher just failed to introduce the earlier skill at all. In others, it’s been a curriculum issue. At any rate, the poor student needs to understand something new to learn the new skill you’re teaching them, which introduces a small overload.

When I noticed that there were skills I was perpetually teaching my students so they could work through a new skill, I grabbed a stack of index cards, wrote the skill at the top, and all the skills the student probably needed to know before successfully being able to learn the current one. Then, I attempted to sequence the cards based on this list of prerequisite skills.

It was an eye-opening experience. I’m pretty sure I’ve blogged about the need to re-order fraction skills (an experiment that met with mixed results in my learning center due to less than ideal conditions), but there are other skills that could benefit from being presented in a different sequence.

For example, I’ve found that if I approach the number line as an example of the x-axis with students just starting with integers or early algebraic reasoning skills, they’re less likely to confuse the x-axis and y-axis when they start learning about coordinate geometry.

Another example (and one I’d love to hear more about how it aids later learning) showed up when I was catching up on my “to read” links the other day. Denise over at Let’s Play Math uses the distributive property to help students better understand their nines facts in multiplication. My high schoolers struggle with the distributive property, so this is pretty interesting to me.

What’s funny is that reading about Denise’s use of the distributive property came right on the heels of a discussion at work about whether it was reasonable or necessary for a second grader to understand the concepts behind the associative and commutative properties as well as their proper names. But if we encourage that understanding in our younger students, would I have fewer high schoolers struggling with the same skills at a more advanced level?

Again, it’s a question of how to best scaffold these skills to really help students not only learn, but retain these skills through their math education and beyond.

Dead Bunny on YouTube

Sorry I’ve been away, everyone. Work has been just a little crazy. Actually, it’s been a lot crazy. On top of that, my computer had problems and had to go away for a long time…with all of my current proijects on it, including Dead Bunny’s book. The computer’s back now, and I have a number of things to learn and work on to catch up those lost months.

While I was on a borrowed computer, though, I did manage to step outside my comfort zone and make some things for you. One of the things I’ve always wanted to have for this site is a series of videos teaching various bits of math. I’m pleased to announce the first five are completed, and available on YouTube.

Please forgive the quality. I’ve never done anything like this before, so it’s been a real learning experience. And as you can see by looking at them, I still have more to learn. So, please forgive the audio and video quality as you enjoy the videos.

Dead Bunny’s Guide to Algebra on YouTube

Book review: Math Doesn’t Suck

Last week, I finally managed to get my hands on Danica McKellar’s book Math Doesn’t Suck, and now I’m recommending all of my girl students read it.

In the book, McKellar offers explanations, tips and practice problems for some of the topics that give most kids fits in math class- factors, fractions, decimals, percents, proportions, and encountering variables for the first time, and she frames it all in stories just about any girl can relate to.

Along the way, she offers insight into her own experiences with math and being a smart girl, and encourages the reader to find their own confidence and to not give that up for anyone.

The book also has an accompanying website that offers full explanations of the practice problems in the book, and provides a forum where students can gather to talk about math education, math and science careers for girls, and being both beautiful and smart in this day and age.

Maybe the stick figure in the second panel above should read it, if only to get a clue!

Combining like terms

To make your math life simpler, we’re going to attack combining like terms. Let’s start by defining what we mean when we talk about terms. A term is a mathematical unit that can be acted upon by other units through the four basic operations.

Okay, that sounds pretty scary. It’s not really that bad. A term is simply each bit of an expression or equation. Terms can look like 7, 5y, 2b², or x³y⁴.

So, now that we know what a term is, what makes terms “like”? Like terms have the exact same variables, raised to the exact same powers. For example, 2b² and 6b² are like terms because they both have b² in them. 7 and 5y, however, are not like terms because 7 has no variable.

Why would we want to combine like terms, anyway? Imagine you had the following expression to deal with:

2b² + 7 + 6b²That’s pretty messy. If we combined those b² terms together, the expression would look a lot neater, and would be easier to work with. So let’s combine them.

We can’t work with the 7 because it doesn’t have a b² term, so we’ll just ignore it for now. Instead, we’ll focus on combining 2b² and 6b². What we really have here is:

2b² + 6b²which can also be looked at as:

(b² + b²) + (b² + b² + b² + b² + b² + b²)We’re essentially just adding b² terms together. How many b² terms do we have all together?

There are 8 b² terms, so we can combine 2b² and 6b² into 8b². Our simplified expression now reads:

8b² + 7 which looks far less scary to work with.

The order of operations

The order of operations tells us how to handle complex expressions to make sure we get the correct answer. This order is: parentheses, exponents, multiplication and division, addition and subtraction. We remember this order through the acronym PEMDAS.

For example, the expression (5 + 3) * 4 would become 8 * 4, which equals 32. If those parentheses weren’t there, the problem would be 5 + 3 * 4, which would become 5 + 12, which equals 60. It’s an entirely different problem.

You might notice that I have multiplication and division in a group together in the list, and I’ve also grouped addition and subtraction in the same group. This is because multiplication and division are done together as they occur starting from the left.

Let’s look at the problem 28 / 7 * 4. Your first instinct might be to multiply 7 and 4 together, which would give you 28, thereby making the answer 1. However, because the division comes first, you have to divide 28 by 7 to get 4, and then multiply that 4 by 4 to get 16. Not exactly the same as 1, is it?

The same is true for addition and subtraction. Look at 9 – 3 + 2. Again, you might try to start by adding the 3 and 2 together to end up with 9 – 5, which leads to the answer 4. But, you should start by subtracting 3 from 9. This gets you an answer of 6, which you then add to 2 to get the correct answer of 8.

It would be very confusing to try to show this in our acronym, though, so you’ll have to remember this on your own.

Tips for working with integers

As part of working on Dead Bunny’s Guide to Algebra, I’m trying to develop tips and tricks for handling a variety of situations. While it’s helpful to understand why these work, sometimes it’s much more useful to know the tips themselves.

This week,I was working on tips for working with integers, since those often stump my students. I’m asking them for feedback, and I’m asking you to tell me what you think, too.

Adding Integers

• When adding numbers with the same sign, add the numbers together and put the sign in front of the answer.
• When adding number with different signs, subtract the numbers and put the sign of the larger number in front of the answer.

Subtracting Integers

• When subtracting integers with the same sign, subtract the integers and put the sign of the larger integer in front of the answer.
• When subtracting integers with different signs, add the integers together and put the sign of the larger number in front of the answer.

Multiplying and Dividing Integers

• When the numbers being multiplied or divided have the same sign, the answer is positive.
• When the numbers being multiplied or divided have different signs, the answer is negative.

An announcement and some math puzzles

I’ve done something terrifying. I’ve finally found a bit of time to start working on the book. I’m learning some interesting lessons along the way, rethinking how I’m presenting certain topics. I may be posting bits and pieces of the book as I work through things, so expect a reposting of certain topics, hopfully with better explanations. Don’t be afraid to leave me comments and tell me what you think.

For today, however, I leave with a link to a treasure trove of math puzzles. Who says learning math can’t be fun?