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.
Repeated Counting
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.
Grouping Together
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).
Forward Movement
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.
Combining Mass
A lucky one as the measurement error doesn't destroy our perfect whole number addition here.
Combining Volume
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!
(Musical) Time
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.
Silly
This is a bluebird of happiness. It is not a model of addition . . . or is it?
Another observation about these models:
ReplyDelete(1) repeated counting forms a bridge between counting and addition. It is also the easiest way for a beginner to calculate an addition fact.
(2) grouping together is mathsemantically and philosophically deep. It is really about finding a common unit. For example, if I have two tubs of playdough and you have three, how much do we have together? We could use tubs as the unit, so we have 5. Alternatively, we could use mass as the unit and would get a different sum.
(3) the number line model (location and distance) leads to negatives (location, distance, direction) and naturally to vector addition. That also links with polynomial addition, addition of complex numbers, matrix addition.