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• Hi Dan (great name BTW 😀 ), how do you deal with math misconceptions with young students? Especially when maths is usually confined to one hour blocks with different types of misconceptions.

• The answer depends on the age. For preschool age kids, it's not really a big deal with math misconceptions happen, since they're going to be learning and relearning so much anyway, and there are going to be numerous wrong answers along the way. For older kids, you sometimes want to catch this misconceptions earlier before it becomes a habit.

The best way to handle math misconceptions is to try it out and see if it works. This is one reason I like games: if you think you have a winning strategy, play me and beat me! If I win, I've shown you your strategy is insufficient without having to tell you.

But it works for arithmetic too. If a kid subtracts 55 - 4 and gets 15, I have a guess what the misconception might be. I could try to explain, but it likely won't work, since the error is based on a more abstract misconception. So I might reach for another tool (building or drawing 55 and crossing out 4, going to the number line) that exposes the error without it just being me saying "No that's wrong. See what you did wrong here?" Getting kids to understand that something is off with their own understanding and that there is a possibility to fix it is really the key move.

• Hi Dan & Stephen!

I think that many teachers and families have been inundated with lots of resources and “how to” guides. It can actually be overwhelming. I am wondering about how to help folks wade through this... what are some key things to focus on?

• This is one of the things I've been most concerned about: the "here's forty great websites to help do math at home" problem.

I think the answer is to focus on just one thing at a time. One activity, one questioning style, one innovation. Don't try to do it all at once.

I tried to do it this way in my emails: one idea or game or activity at a time. Now there's a stack of nine of them, though! So if that's what you want to look through, I'd recommend just starting with the first, and not trying to plow through all nine at once.

• How much of the difficulty with children and math is really
with the parents? As children are not being schooled at home, I have seen that Math has become a bigger issue. The more I think about this the more I wonder if I am the problem. When my attitude is of course you can, the less drama there is.

• These things become really specific to the situation. Some parents have real trauma around math, and that can get passed along to kids, depending on how they hear adults talk about math and approach math. But sometimes stuff happens at school that's totally unreleated to parents.

The present moment is unusual because school is just over for so many people right now. But I think that presents an opportunity for parents to approach math in a positive way at home, to stay light, to stay playful, to stay curious.

It's not a panacea. But we're not looking for perfection. We're looking for small positive steps we can take. And if we keep taking them, things will get better.

• Second question – I have always thought that math basics
(times tables, math facts) are vital to success, but I wonder if I am wrong?

• I'm inclined to agree, actually. There is a collection of facts that are very important to further work and understanding. It's just a smaller collection than you might think.

We delude ourselves if we think all math is just memorizing a larger and larger body of isolated facts. In reality, we explore and make connections, and our understanding grows. We need to know certain facts and algorithms by heart (i.e., sums & products of one-digit numbers), but it's in connecting those facts and algorithms through observations, ideas, arguments etc. to the wider body of mathematics that they stay meaningful and at our fingertips.

How can teachers help caregivers facilitate beautiful tasks, such as the ones on your site, without any face to face conversation? And what if there are language barriers?

• This is the tough question right now, and frankly, there's no easy answer. Remote learning is hard. We're not prepared for it. It's just deeply challenging.

Ideally, if you can get a child curious about something, they might be motivated to go off and try to mess with it on their own. So there's an art to helping kids see that there's something they don't get yet, but that they can play around with it and learn more.

I think choosing tasks that require less explanation, and have a flavor of the internet meme of "here's a weird math trick. Does it really always work?" might be helpful right now. And the fewer words involved, the more accessible. But frankly, I'm trying to figure out the answer to this question too.

• Hi Mr. Dan, I like that you advise parents to start where the math feels comfortable. This made me think about the terms ‘struggle and productive struggle’. Would you define these as they pertain to math? How would a parent or student know the difference?

• Thanks for this question. I think of struggle as pertaining to any situation that challenges you. And challenge is good! But our goal is to help the struggle be productive, because that's fundamentally where learning happens. It's the balance between challenge and success.

So what makes struggle productive? Most important is a sense of movement, of not being totally stymied. The key is to have experiences where kids can have initial success, have a sense of possibility about what to try, but not necessarily know all the answers.

Think about playing a game. If the rules aren't clear or if you lose every time, it's dispiriting. If you win every time, it's boring. But if the rules are clear and you win sometimes, but also lose sometimes, that's when things get interesting. That's when the struggle gets productive.

Some kids thrive on challenge. But for a lot of kids, I'd help with that sense of early success, and making sure the "rules" of the game, or whatever challenge you're working on, is clear. If you check out my subtraction challenge problems, in the links above, I've got one on Diffy Squares. https://mathforlove.com/2020/03/diffy-squares/

Kids should be able to do a Diffy Square on their own. That's understanding the rules of the game, and that's the entrance for playing. But how in the world can you make a Diffy Square go on for more levels? Super challenging question. It's between those two places that productive struggle can happen.