For a while, a long while, I let this math blog just sit. I didn’t want to delete it. I spent too much time on it and I knew it was full of great stuff that was still being used. But I wasn’t in love with writing on it.
In my lethargy, the hackers and jerks managed to infiltrate and completely take it down. That was on October 23, 2025.
Today is December 30, 2025. I rescued all the posts and will gradually get all the good ones reposted. But I have to start over, creating the whole website from scratch.
Luckily the wayback machine (internet archive) has a nice snapshot of the images and layout. It’ll take me a while, but I’ll get it all back up. Maybe even with some improvements!
If you are looking for a specific post – something you remember from long ago or something you want to know about – leave it in the comments. I’ll hunt it down and get it posted.
I’m looking forward to having all those great math stories, thoughts and tips live again – I hope you are too.
Reading through it, the thought that came to mind over and over was, “Finally.”
Here was my favorite part:
…the subject matter itself isn’t all that hard. What’s hard, virtually impossible, is beating it into the heads of youngsters who hate every step.
Indeed there is a gracious sufficiency of beating – and resistance – when teaching math.
But everyone already knows math!
In the book, Albert wrote:
If you never teach a stitch of math, in a mathematical culture your kids will learn heaps of it anyway. …learning math along the journey is a difficult thing to avoid.
You can’t not do math and exist. It’s like not breathing. Impossible.
We survived for a very long time without the written word. But we’ve never existed without math.
Early humans knew that to divide a chunk of meat between two people would yield too little:
He might not have have had the fancy way to write it – but he could learn how to write it in 8 weeks.
Daughter playing with washers: counting them and learning math in her world.
Can you teach all of K-12 math in 8 weeks?
The short answer – it depends on to whom. In Albert’s article, he’s teaching children. You can totally do this with kids. And cavemen, I suspect.
Grownups are a different story. Innumerate adults already have anxiety, anger or fear associated with math. There’s a whole lifetime of un-doing that would have to take place.
But once you undo this, then the 8 weeks would work.
The way I teach college classes is this: we’re going to spend 48 contact hours together – I’m going to wow you in ways you’ve never expected. When we’re done, you’ll be much calmer. You’ll be able to learn math much better, both on your own or in a classroom.
I don’t force learning. I don’t even require learning. I let it happen. It’s the closest I can get, in the classroom, to what Albert is saying.
Do you struggle explaining to your students what “simplify” means? Do you wonder why they won’t reduce fractions to their lowest terms?
Here’s how to explain and teach some of the top confusing terms in mathematics – without throwing you or your students into a tizzy!
Where
Really! “Where.”
As in “Where do the lines intersect?” or “Where is the highest point of the graph?”
“Where” in math almost always means “at what x-value?” Every now and then “where” can mean “at what point?” In the first case, “Where do the lines intersect?” can be just as easily satisfied by a point (x, y) as the x-value only.
When you get into calculus, the word “where” is used everywhere (pardon the pun). “Where is the slope of the tangent line equal to zero?” and “Where are there discontinuities?” among others.
Keep your eyes peeled in the textbooks for this tricky term.
Reduce
Typically this means to reduce to lowest terms. Which isn’t really helpful, since you still need the definition of “terms.”
Numerators and denominators (tops and bottoms of fractions) that have one or more common factors are not officially “reduced.” Once they have no common factors at all (except 1), they are said to be reduced or reduced to lowest terms.
A reduced fraction helps in many ways. You can “feel” the value more easily: 2/5 is a lot easier to picture than 14/35. Also, when you add fractions, starting out with all the numbers in their reduced form will often help to find common denominators.
Zeros, Real Zeros and Roots
Zeros, real zeros and roots all mean the same thing. It is where a graph passes through the x-axis. We can write “where are the zeros” meaning at what x-value (“where” from above) does the graph pass through the x-axis? We also will write “what are the zeros” and mean the same thing.
To find the zeros or roots, we set a function (or any equation solved for y that involves x) equal to zero. Then we solve for x. This will give the result that we want – the x-values where the graph will go through the x-axis.
We call them zeros because they are where y = 0. We call them real zeros because there can be complex or imaginary zeros that don’t exist on the graph.
Finding the zeros or roots of a function help when graphing it. If you know all the places it goes through the x-axis, you can then determine where it’s above or below the x-axis. Then you can roughly sketch the graph.
Simplify
Of all the terms in mathematics, this one is the most slippery. This meaning of this term is always in context to the book or situation. What it means is “manipulate the expression.” The end result may or may not be “simpler” than the starting expression.
Better textbooks use phrases that are clearer, like “multiply the polynomials” and “combine like terms.”
If “simplify” is the instruction to a problem set in a textbook, consult the examples to see what type of manipulation to do.
In later courses, students have to recognize when a situation requires a different form of the expression. Practicing various forms of manipulation (or “simplifying”) gives them tools to manipulate the expression to satisfy their needs.
Do you run into other terms in mathematics that are hard to explain to students? Hard to understand yourself? Share them in the comments!
Have you seen the kid that isn’t getting the attention he needs at home? Wonder what’s going to become of him?
Well, the “it takes a village” concept isn’t far off. You might not be able to change his world, but sharing a little about math might help with one corner of it.
Share math in small bites.
Every time you see the child, pose a fun puzzle. It doesn’t have to be blatantly mathematical. Anything to get his logic working helps. I love the Childcraft Mathemagic book for puzzle ideas.
Ask her random math facts. Tell her you’ll give her something for each one she gets right – or each 10 she gets right in a row. Find out what’s important to her that you can’t get in trouble sharing. Like money, chocolate, baubles, etc.
Tell him that when you were growing up you wanted to be a mathematician. Ask him to find his favorite mathematician online. Find one yourself that you can discuss – I’m partial to Abel and Galois.
Be positive about math.
There might be many things wrong in a child’s life. Let math be the one that isn’t. Be positive. Help make math the thing a kid can hold on to.
Be the village because it takes a village. Share your stories in the comments!
I’ve learned of a thing called “What can you do with this?” from dy/dan. This teacher sets up a situation so that students can ask questions.
I’ve been pondering the effectiveness of this for a while.
The thought is that if you allow students to observe something interesting and ask them “What can you do with this?” then they’ll create their own word problems.
This is in response to the fake or “made up” word problems from a textbook which mostly don’t work for teaching thinking skills.
But the issue remains the same. If someone presents a student with a video of Coke vs. Sprite and the student lacks curiosity about that subject, then it’s still a contrived problem. Or a contrived situation.
The only reason to do a word problem is if you’re emotionally attached to it.
Husband and I were talking about word problems the other night. After my demonstration about using to teach math, he said he wished he learned math that way. He needed something to hold on to. A reason for doing it.
He’s a set dresser in Hollywood for part of each year. Which means that he has to hang pictures on movie sets. And they have to be 55″ above the ground – at the center of the picture.
Not hard to measure, but there’s also the wire on the back to consider. Is the wire dead center? No. It’s probably above the center of the picture.
It becomes one giant word problem. But it isn’t written in a book. And it isn’t videoed by a teacher. It isn’t fake. There’s a real reason for him to do it.
Which made me realize that there’s only one reason to do word problems: if you’re emotionally attached to it.
If you need an answer to a question, you attach to it emotionally.
Parent: You’ve got 45 minutes to clean the kitchen before we leave for softball practice.
Kid: If I finish the kitchen before we leave, can I watch TV?
Parent: Sure, but the kitchen better be spotless.
Most likely the kid has a plan for TV – like watching his favorite cartoon on DVR that takes about 30 minutes. So he works out how fast he needs to clean the kitchen so he can get in his cartoon before leaving.
This is a real problem. His problem.
Watch your kids intently. See where they are doing word problems in their heads. Ask them to explain them. Give credit for work done – especially when self-created.
If someone else needs an answer, you attach to it.
Being helpful is a powerful motivator. Try this: with a pencil and paper sit in a public place. Act like you’re writing something important. Then ask out loud, “What’s 87 minus 13?” $5 says that at least four people will chime in to be helpful.
Let your kid help with balancing the checkbook or creating the budget. If you’re a classroom teacher, let the kids help figure out what teacher supplies to buy. Give them a limit on what to spend and the catalog and some guidelines.
If someone you like wants an answer, you attach to it.
I couldn’t have given a feathery duck’s tail about biology, but the teacher was crazy cute. So I wanted to please him. So I worked. Hard. And had a 100 average.
I suspect this is why the teacher at dy/dan is so successful. He’s cute, compelling and cool. Who wouldn’t want to engage with him?
If you have carisma and charm, use it. This might not work as a parent but will definitely work as a classroom teacher – at least for some students.
Give it a try. Tap into the emotion. And share your success below!
I discovered this video about the philosophies of teaching math from the blog Learning to Teach, The Empty Set of Education. Having degrees in math and not teaching I found it very interesting from a theoretical point of view.
And I totally love his conclusion – the 5th philosophy!
What is your philosophy for teaching math? Share your thoughts in the comments.
If you’ve ever taught or tutored math you’ve encountered the question, “When am I ever going to use this?” Maybe even hundreds of times.
And no doubt you’ve tried the answers that you’ve heard your math teachers give:
You’ll need it in a future job.
You’ll want to balance your check book someday.
Blah, blah, blah.
I was on the Teachers.net chatboard last night and there’s a discussion in the math teachers section about how to answer this question.
I was horrified to read that some teachers actually respond with, “How about as homework, you find the answer to that question.”
Egad!
We all know it’s a discrationary tactic. We know that there are lots of good uses of math. And we’ve experienced our answers shot down with, “I’m not planning on doing a math job for a living, so I won’t need it,” or “I’ll hire a CPA to do my checkbook.”
There’s only one right answer to this question.
“You’ll never use the math I teach you. Ever.”
I offer $10 to anyone who can come back to me in 10 years and tell me that graphing functions (or whatever we are learning that day) has actually had an applicable use in their life.
Of course they’re horrified at this answer. They give me looks like, “What? Are you an alien here to invade our classroom. Did you eat the real Bon?” No teacher has ever been that honest.
Graphing functions is virtually useless as a real tool. As is most of what we teach.
I used to get phone numbers from men at bars with my amazing use of the quadratic formula, but that’s only something you can tell college students. And they don’t buy it anyway.
Teaching math is teaching brain exercises.
The reason we teach and learn graphing functions (or other math) is to exercise a part of the brain that we rarely get to use. A part that will get used sometime later in a weird way.
We’re building new paths in the brain. We’re carving a way to alternative problem solving that might one day be useful in solving interpersonal, business, automotive, or other type of problems we have.
I tell them that math class is a game. A fun time to escape once a day. This is a play time to stretch their brains and do something completely different.
And I certainly don’t pile pissiness upon pissiness with the attitude of “If you’re going to challenge me, small menial student, then I’m going to give you extra homework.” That really motivates students… to hate math.
How about you? How do you answer the question? Are you supporting future math happiness? Share your thoughts in the comments.
I found an EdReach article through The Republic of Math‘s site and was so relieved. I’ve been hearing so much about the Khan Academy. Every time I look at it, I can’t help but think, “Same stuff, different medium.”
Math teaching is about connecting math with people in a way that works for them. If the Khan Academy helps a kid get it, then great. But if that’s the case, then it’s likely that they’ll get it regardless of the medium.
It’s time to step it up.
I’ve been hearing about experience selling for a while. You don’t sell a product, you sell an experience. I’m not typing on a Mac right now because it’s a better computer than a PC. I’m typing on a Mac right now because if I have a problem, the cute purple-haired kid at the Apple store 3.2 miles from here will help me fix it.
The same holds with math. We need the experience. Kids need the experience. It no longer works to sell “you’ll need this in life.” You have to sell the emotion behind it. The “what’s in it for me?”
And that’s easy – if we just step it up a bit.
This ain’t your daddy’s math class.
Nope – not anymore. Once kids are old enough to get it, they need to be allowed to get it. Until then, we can sell them the algorithms and the memorization. But once they hit that threshold (which is different for every kid), they have to be allowed to do it their way.
Which means no more rules. Math teaching is now about facilitation of learning. No more, “You must do it this way.” We’re losing them with forcing them to rationalize the denominator, simplify the fractions and write polynomials in descending order “because they’re supposed to.”
They should rationalize the denominator because it compares better when put next to another number. They should simplify fractions because it’s easier to wrap your brain around 1/4 instead of 13/52 (unless you play ). And polynomials add easier when you write them in some designated order.
Siggi over at Turkeydoodles wrote a post about her preference of calculators over flashcards. It’s her preference, but her arguments seem to be founded on the misuse of of flashcards. Furthermore, it seems she’s not seen the detrimental effects of early calculator use, yet.
When you should give a kid a calculator is a topic for another article. In the meantime, here are some tips on good flash card usage.
Flashcards are educational toys.
Flashcards shouldn’t be used as testing devices. They’re educational toys. They’re exploratory devices. Let them “peek” as much as they want.
As a first introduction, use them to build houses of cards. They should be fun and comfortable.
They are limited in scope.
The 6 x 8 = 48 card will never be able to give the cosine of 60 degrees. This makes the flashcards so beautiful. Once you understand what happens when you create a calculator addict, and see how that works as the kid enters college, you’ll know how important this limitation is.
Encourage variation to limit boredom.
I distinctly remember using flashcards in my dining room, sitting next to the sliding glass door. I was reading them. But because they would get boring, I would chant them. It became sing-songy and fun. I could go through them quickly this way.
And I looked forward to the ones that rhymed.
Let the flashcards be rejected.
If a kid really hates them, let it go. There are other ways to get that information across. Schoolhouse Rock’s Multiplication Rock is a fabulous tool for this.
And you can sing or chant multiplication facts yourself. My mother learned the most common prepositions by saying them as she jumped rope. You can vary some skip counting with jumping rope to learn multiplication facts:
3 x 1 = 3
3 x 2 = 6
3 x 3 = 9
and so on…
Allow the flashcards to be the context in and of itself.
It is not necessary that math be learned in context. So many people keep pushing this. Sometimes it’s just fun to know random stuff – including some quick and nifty facts.
Don’t push math for math’s sake, but offer it. There are kids, lots of them, who just like to do puzzles. Plain math – arithmetic and facts – is a great puzzler.
What do you think? Is this a better use of flashcards than the ones you’ve seen? Share your thoughts in the comments.
Actually, you get only one chance when teaching anything.
I was in an internal adult training class for corporate websites. In the middle of a discussion one student said, “How about if we each take something about Sharepoint that interests us, research it and report back to the group?”
Great idea, right?
The instructor put her face in her hands, breathed deeply, ran her fingers back through her hair and said, “Well, I want to make sure you get the things you need, so hold off on that.”
CRUSH.
End of learning.
Game over.
What are your thoughts on this? Share them in the comments below.
