Best conceptual math games

[Eilonwy]

8 hours ago, Xahm said:

We got our version of Go Fish for Ten from Addition Facts that Stick.

We did this too and then used the Tiny Polka Dot cards to give different representations of the numbers. Has anyone used the games from Subtraction facts that stick?

[Not_a_Number]

Just now, Eilonwy said:

We did this too and then used the Tiny Polka Dot cards to give different representations of the numbers. Has anyone used the games from Subtraction facts that stick?

We've played a few games with the Tiny Polka Dot cards, although they seemed to like Concentration-style games best. How did your "Go Fish for Ten" work? Any links to instructions? I'm going to try to compile all the ideas in the top post with instructions included. 

[Eilonwy]

18 hours ago, Not_a_Number said:

I actually kind of disagree with that, believe it or not. I don't think it's a great idea to think of 63 as anything other than 6 tens and 3 ones. it muddles your thinking. 

By definition, it's 6 tens and 3 ones. Finished, complete, end of story. 

I think there are other ways to think of these numbers, though. At work, a number is nearly always a dimension, a speed, a flow rate, and it has a very specific connection to the physical world. Our different ideas about the importance of this probably illustrates the gap between pure and applied math, but kids who like the applied side have to translate all the time between the two. 

[Not_a_Number]

4 hours ago, Eilonwy said:

I think there are other ways to think of these numbers, though. At work, a number is nearly always a dimension, a speed, a flow rate, and it has a very specific connection to the physical world. Our different ideas about the importance of this probably illustrates the gap between pure and applied math, but kids who like the applied side have to translate all the time between the two. 

Of course there are other ways to think of numbers. And no, this is not a pure versus applied math thing, since there's plenty of "continuous" pure math. However, I do think that a small child's natural model of number is the countable numbers. That's the natural base on which people build. I don't think the distance model of number is nearly as natural for people. 

Also, 63 is 6 tens and 3 ones and that's the right model, whether you're adding centimeters or bowling balls. Because it's the definition. 

[Eilonwy]

22 hours ago, Not_a_Number said:

How did your "Go Fish for Ten" work? Any links to instructions? I'm going to try to compile all the ideas in the top post with instructions included. 

I couldn’t find a link, but this is what we did: 

Using a full deck of shuffled Tiny Polka Dots cards, we dealt five cards to each player. The rest go in a fishpond spread out pile in the centre. In turn, each player asks for the number that will make 10 with what they want to match, from a specific player. If they get it, they can put it down as a pair and ask again. If the other player doesn’t have the card (any version of it) then the first player gets to Go Fish and draw a card from the pond, and their turn is over. If someone runs out, they can draw 2 new cards from the pond and keep going. Most pairs (highest stack) wins.

[Eilonwy]

17 hours ago, Not_a_Number said:

I don't think the distance model of number is nearly as natural for people. 

Did you come across kids in your classes that did find the distance/continuous model more intuitive? Some of my kids seemed to find it quite useful, though I couldn’t say it was their primary model. 

I agree that 6 tens and 3 ones is the definition of 63 in base 10.  Nonetheless there’s still another level to the connection to the physical world for me that I don’t think I can explain on a board. 😕 

[Eilonwy]

22 hours ago, Not_a_Number said:

Yeah, that's a great idea. That's actually kind of my older girl wound up learning negatives! We would put a question mark for questions that were "less than 0," just to remind her about the order. Soon, she couldn't wait to find out what all those mysterious question marks were. 

Oooh. Or always take the red from the black?? Pull from two decks of different colors? 

We tried subtracting in order, and there are a lot that turn out less than 0.  We played using negative numbers, but if you didn’t, you would have a lot of less than 0 ties.  You could do it by colour, too, with 2 decks. That could be easier for some. 

[Not_a_Number]

14 minutes ago, Eilonwy said:

Did you come across kids in your classes that did find the distance/continuous model more intuitive? Some of my kids seemed to find it quite useful, though I couldn’t say it was their primary model. 

I only ran classes for little kids for a bit 🙂 . So most of my teaching experience is older kids at colleges and AoPS. However, even those kids find counting numbers easier than reals! Like in linear algebra, I have to remind kids that we’re allowed non-integer coefficients. 

I do think it’s a very helpful model once you get to fractions and reals 🙂 . I just don’t think it’s the primary model.

 

14 minutes ago, Eilonwy said:

I agree that 6 tens and 3 ones is the definition of 63 in base 10.  Nonetheless there’s still another level to the connection to the physical world for me that I don’t think I can explain on a board. 😕 

I do think that there’s a connection to the physical world! I am just not sure it helps kids internalize place value. 

We’ve needed a physical model for division by fractions, actually. And I like it for multiplication by reals, too. And I like very visual, very distance-based stuff for trig and linear algebra, too. So... whatever works. But I don’t love distances for early math.

[Farrar]

I didn't read through this whole conversation, but it seems like no one mentioned the two things that immediately come to my mind, which are Corners and Muggins. Corners is the Right Start game that is most worth buying. You cannot play it without the special Corners cards. We played this game nonstop for awhile. It got way more use than any of the other Right Start games. It was so good for number sense.

And, of course, Muggins is similarly about combining possible numbers in various ways.

[kristin0713]

I didn’t read every post, either. My DS used to really like 24 if this is the sort of thing you’re looking for. 

[Not_a_Number]

3 hours ago, Farrar said:

I didn't read through this whole conversation, but it seems like no one mentioned the two things that immediately come to my mind, which are Corners and Muggins. Corners is the Right Start game that is most worth buying. You cannot play it without the special Corners cards. We played this game nonstop for awhile. It got way more use than any of the other Right Start games. It was so good for number sense.

And, of course, Muggins is similarly about combining possible numbers in various ways.

I think someone did mention Corners somewhere, but I don't remember if it was on this thread or the one that started this thread!

 

1 hour ago, kristin0713 said:

I didn’t read every post, either. My DS used to really like 24 if this is the sort of thing you’re looking for. 

Oh, lol, I used to play this game at math camp back in my math contest days. We just played with a deck of playing cards, nothing fancy. I tried to introduce it to my homeschool classes, though, and the kids were too young -- even the strong kids were too slow and didn't have fun. 

I'm curious -- do either of these games demonstrate the concepts, or do they just require kids to practice them? I know 24 basically requires you to know facts already, although it's tremendously fun as drill if you're fast enough. But it doesn't illustrate the operations, at least the way I've played. How about Corners? 

[Farrar]

32 minutes ago, Not_a_Number said:

I think someone did mention Corners somewhere, but I don't remember if it was on this thread or the one that started this thread!

 

Oh, lol, I used to play this game at math camp back in my math contest days. We just played with a deck of playing cards, nothing fancy. I tried to introduce it to my homeschool classes, though, and the kids were too young -- even the strong kids were too slow and didn't have fun. 

I'm curious -- do either of these games demonstrate the concepts, or do they just require kids to practice them? I know 24 basically requires you to know facts already, although it's tremendously fun as drill if you're fast enough. But it doesn't illustrate the operations, at least the way I've played. How about Corners? 

Corners, 24, and Muggins are all similar in terms of what they have you do. You have some numbers and you have to figure out the best way to manipulate them following the rules of the individual game to get what you need - choosing operations to get the highest points, the specific number 24, etc. They don't directly teach skills the way going to the dump does, so maybe it's not what you're looking for actually.

 

[Not_a_Number]

2 minutes ago, Farrar said:

Corners, 24, and Muggins are all similar in terms of what they have you do. You have some numbers and you have to figure out the best way to manipulate them following the rules of the individual game to get what you need - choosing operations to get the highest points, the specific number 24, etc. They don't directly teach skills the way going to the dump does, so maybe it's not what you're looking for actually.

Yeah, I'm thinking more about games that actually help build mental models, as opposed to practicing the mental models. My observations with mental models was that once you HAVE a mental model, you tend to use it for lots of things, but building one is tricky. 

So I very much like games like this for working with numbers, but I've also found that once kids have, say, a good model for place value, they'll use it for lots of things. So then lots of things become practice for the mental model, and of course games like this are really excellent practice, because they are backwards and not forwards reasoning (which I do find to be MUCH more demanding.) 

Am I making any sense here? I feel like this post has kind of gone in a circle... I really appreciate the suggestion! 

Edited February 9, 2021 by Not_a_Number

[Farrar]

2 minutes ago, Not_a_Number said:

Yeah, I'm thinking more about games that actually help build mental models, as opposed to practicing the mental models. My observations with mental models was that once you HAVE a mental model, you tend to use it for lots of things, but building one is tricky. 

So I very much like games like this for working with numbers, but I've also found that once kids have, say, a good model for place value, they'll use it for lots of things. So then lots of things become practice for the mental model, and of course games like this are really excellent practice, because they are backwards and not forwards reasoning (which I do find to be MUCH more demanding.) 

Am I making any sense here? I feel like this post has kind of gone in a circle... I really appreciate the suggestion! 

My experience has been that kids often don't build those models very well until it's a game. I spent a year making my 6th grade remedial math class play 24. It took them that long to get good. And then they proceeded to whomp the other kids in a school tournament. The other kids were more advanced in math as this was the "lowest" math group, but they also had spent all year practicing 24, unlike the other kids.

We played some of the Cuisenaire rod games when my kids were little, learning some place value things. Those are definitely teaching games. Education Unboxed has videos of nearly all of them. I wish that resource had been around when my kids were little.

[Not_a_Number]

1 minute ago, Farrar said:

My experience has been that kids often don't build those models very well until it's a game.

I haven't found that. I've found that to build the model, you actually need to make the kids use the model. If you want kids to use the "x 10s and y 1s" model for place value, they have to use that for months to add, subtract, divide and multiply. At the end of year or two, they will be very good at place value 😉 . But not if you tell them about it and then have them use shortcuts instead of the model itself. 

It goes for pretty much everything, not just basic arithmetic. If I wanted my AoPS precalc kids to USE the unit circle for trig, I had to give them questions that... required them to use the unit circle for trig, not whatever weird tricks they were using to memorize the numbers without the unit circle. No amount of practice solidified the "unit circle" mental model unless it actually explicitly used the unit circle. 

[Not_a_Number]

Basically, humans are innately lazy and will not use a new mental model unless they have absolutely no choice 😛 . 

[Farrar]

3 minutes ago, Not_a_Number said:

Basically, humans are innately lazy and will not use a new mental model unless they have absolutely no choice 😛 . 

I mean, "I must beat the other children and get that Hershey kiss," seems to do it. 🤷‍♀️ 

[Not_a_Number]

Just now, Farrar said:

I mean, "I must beat the other children and get that Hershey kiss," seems to do it. 🤷‍♀️ 

What model specifically were you trying to communicate via 24? 

[Not_a_Number]

Having played 24 a lot... to beat most 6th graders, I would guess you mostly need to remember that most of the time you need to make 6 and 4 or 8 and 3, then to be able to see ways to do that. 

[Farrar]

1 minute ago, Not_a_Number said:

What model specifically were you trying to communicate via 24? 

I was mostly being flip. For these students, just mental calculation that didn't rely on constantly counting slowly on their fingers, so not really a model. And yes, the key for the vast majority of cards is to remember those basic ways to get to 24 by narrowing your two numbers down to 6 and 4, 8 and 3, or 12 and 2. 

[Not_a_Number]

Just now, Farrar said:

I was mostly being flip. For these students, just mental calculation that didn't rely on constantly counting slowly on their fingers, so not really a model. And yes, the key for the vast majority of cards is to remember those basic ways to get to 24 by narrowing your two numbers down to 6 and 4, 8 and 3, or 12 and 2. 

Yeah, I'm a big fan of drill -- the more facts you can recall quickly, the more your brain is free to work on other stuff. 

However, thinking of mathematical teaching as "communicating mental models" has really revolutionized how I teach 🙂 . So I talk and think a lot about that!  

[Not_a_Number]

6 hours ago, Eilonwy said:

I couldn’t find a link, but this is what we did: 

Using a full deck of shuffled Tiny Polka Dots cards, we dealt five cards to each player. The rest go in a fishpond spread out pile in the centre. In turn, each player asks for the number that will make 10 with what they want to match, from a specific player. If they get it, they can put it down as a pair and ask again. If the other player doesn’t have the card (any version of it) then the first player gets to Go Fish and draw a card from the pond, and their turn is over. If someone runs out, they can draw 2 new cards from the pond and keep going. Most pairs (highest stack) wins.

Cool, thanks!

That sounds similar to what we did, but I remember the kids didn't like it. I am not sure why... I should try to remember. I'm not sure if we did precisely these rules or something slightly different! I just remember the games going VERY quickly, because instead of getting four like in real Go Fish, you only needed pairs. And the kids already knew how to make 10. 

6 hours ago, Eilonwy said:

We tried subtracting in order, and there are a lot that turn out less than 0.  We played using negative numbers, but if you didn’t, you would have a lot of less than 0 ties.  You could do it by colour, too, with 2 decks. That could be easier for some. 

Yeah, I would imagine that you'd have about half of them be negative 😉 . Slightly less than half due to the zeroes. 

I think you could stack the decks a bit so that if it's left minus right, the left deck has slightly bigger cards? 

Thanks for brainstorming with me! I haven't had great subtraction games, and this one seems like one I didn't think of for no good reason. 

[Eilonwy]

12 hours ago, Not_a_Number said:

Yeah, I would imagine that you'd have about half of them be negative 😉 . Slightly less than half due to the zeroes. 

I think you could stack the decks a bit so that if it's left minus right, the left deck has slightly bigger cards? 

That is exactly what happened, as you would expect. But, I actually found that my youngest can also calculate negative answers to subtraction questions, though we haven’t covered that with her  at all. And that it gives practice at comparing negative numbers.

You could absolutely stack the deck if you just wanted to do positive subtraction practice.

[Not_a_Number]

17 minutes ago, Eilonwy said:

That is exactly what happened, as you would expect. But, I actually found that my youngest can also calculate negative answers to subtraction questions, though we haven’t covered that with her  at all. And that it gives practice at comparing negative numbers.

You could absolutely stack the deck if you just wanted to do positive subtraction practice.

Hey, cool! How does she know? Number line exercises from somewhere?

I modeled negatives as “debt” with DD8. It worked well... I recommend 🙂 

[Eilonwy]

5 hours ago, Not_a_Number said:

Hey, cool! How does she know? Number line exercises from somewhere?

I modeled negatives as “debt” with DD8. It worked well... I recommend 🙂 

I’m not sure, and she said she heard about it somewhere but doesn’t remember where.  She has done a little bit with number lines in BA 2A, and her older sister had a unit on integers a few months ago.  Pleasant surprise, anyway! 😀

I think the debt model would work, I will try that, thanks! 
 

You wouldn’t even have to stack the deck, you could add a step (depending on age/skill) of choosing the largest number and moving it to the left of the other number. 

[Not_a_Number]

1 minute ago, Eilonwy said:

You wouldn’t even have to stack the deck, you could add a step (depending on age/skill) of choosing the largest number and moving it to the left of the other number. 

Hm, good point, yes. Or you could add a third deck with operations and have them arrange it into whatever they wanted to answer! 

[Not_a_Number]

3 minutes ago, Eilonwy said:

I think the debt model would work, I will try that, thanks! 

We did a few months worth of work on it with DD8 at age 5. DD8 is generally very mathematically gifted, but I've been surprised at how well it stuck. It all made a lot of sense to her. "OK, if we start out with 8 dollars of debt, and we take away 4 dollars of debt, clearly we're still in debt but only by 4 dollars." Therefore, 

-8 - (-4) = -4. 

Although, actually, we did "debt" in apples 😉 . So it was more like the above sentence, but with "apples" instead of dollars! 

Edited February 9, 2021 by Not_a_Number