When: at dinner
What materials did we use: our mouths (just talkin') and then a bunch of miscellaneous items later
J1: [Friend X] isn't good at multiplication.
M: That's not surprising. Has he started learning multiplication in school?
J1: Actually, he's not so good at addition either. [Friends Y and Z] also don't really understand addition.
D: Great, so maybe you can help them. Is there a picture you can draw that shows how you think about addition? For example, 4+3?
J2: (looks at his fingers for a moment) that's 7! (exclamation, not factorial)
D: So what does that mean, when you add?
J1: (making a big gesture with his arms) Gathering things together, collecting them.
D: Are there other models of addition that you know? Other things you do where you need to add?
J2: counting, then counting some more
. . . tbc
So, how many models of addition have your kids seen? How many are there? Let me know what you think in the comments.
For a hobbyist counter, like J3, this activity is an excuse to count up to 20. Twenty? Well
5 fingers + 5 almonds + 5 fingers + 5 almonds = 10 fingers + 10 almonds = 20 etwas.
My argument is that the fingers and almonds aren't really being combined, they are all serving as objects for counting, then counting some more.
On the 4 different stems, we see 1 longan, 2 longan, 3 longan, and 4 longan. How many ลําไย are there all together?
I'm calling this distinct from repeated counting because the 10 ลําไย do make a coherent collection all together in a fruit bowl, while their separation into 4 subsets probably won't be relevant for their future destiny and, in fact wasn't relevant to why they came to our table (they were all bundled together in a cluster).
You've seen this game before: roll the dice and move your fierce dinosaur closer to the finish. A number line gives you a similar model, with the benefit of fractional steps, but I liked the fact that the motion is only sequential by convention here.
A lucky one as the measurement error doesn't destroy our perfect whole number addition here.
Yes, it was necessary to pour out 1/4 cup + 1/2 cup = 3/4 cup of chocolate milk in the course of this investigation!
As with the mass model, note that this is imprecise and the experimenter must decide from the context what gap can be tolerated between the theoretical and empirical results. This is especially critical for valued substances like chocolate milk!
One quarter note plus four eighth notes get played for a time lasting one measure, in 3/4 time. Or, if you want to be measure independent, four eighth notes together last the same time one half note.
This is a bluebird of happiness. It is not a model of addition . . . or is it?