I absolutely love the area model for making our thinking visible. The standard algorithm for multiplication relies on students using their basic facts which many don't know...so they skip count which is not the multiplicative reasoning we want them to do. And, as you mentioned, the place values are taken out of the process so students don't really have a sense of the reasonableness of their answers. I saw Jennifer Bay-Williams speak recently and she said that our goal is not to get every student to do the algorithm, but instead to help students know when they need to use an algorithm and when they don't need it. Exploring strategies like double and halving (45x16=90x8), using a friendly number and adjusting (99x5 = 100x5 - 5) , 5 groups are half of 10 groups (10 x 4.8 = 48 so 5 x 4.8 - 24), etc are other ways our students need to have experience with so that they can efficiently arrive at their answers. We can use the area model to show all these flexible strategies as well.

Always such great insights Brittany!! For your question at the end… I’m undecided! But I do think everyone should learn the area model for multiplication.

I absolutely love the area model. I always tell my students that I wish I would have been taught this when I was a student.

the extension of the area model is the expansion of brackets in algebra, particularly quadratics. This makes sense of the FOIL method and relates it to abstractions of bases and place value. It is the bridge to making sense of the algorithm which can be extended into the multiplication of three sets of brackets. The area method can also be related to Japanese multiplication, counting intercepting lines and both lake sense of finding the roots of a quadratic, using alpha and beta, multiplied to give c and added to give b in ax^2 + bx +c.

I personally do not think we should emphasize standard algorithm multiplication or division until students have consistently demonstrated an in depth understanding of partial products/quotients. As a student I had to learn it without any other strategies and had no idea of what I was actually doing regarding place value. I simply memorized a series of steps and some cute little saying to help me remember. I struggled my way through it and totally hated math all through college until I started teaching 4th grade. Now math is my favorite thing to teach and it has been very eye opening.

If the students know their multiplication tables then both area model and algorithm. If not, area model.

I do not think that we should focus on the algorithm. It does not save that much time and it does not lend any help to allow the students to explain what they have done.

The area model is naturally limited to 2D, the algorithm is not. First Area model, then the algorithm.