Here’s an activity that lets students practice multiplication facts. It’s basically a flashcard application.

The issue here is that if a student does not know their multiplication facts, they have no way of figuring them out. The feedback is extremely simple, too simple to be useful.

Here’s another activity.

This activity is slightly better. Students still do not directly know what the correct answer is, but they at least now have a mechanism for determining it. Unintuitively, feedback that requires a student to think is more effective than simpler feedback, so it’s possible that this level of feedback is just right — students may not actually benefit from just being given the answer.

This next activity is more introductory by design.

The benefit of this activity is that students are more able to connect something they know to something they may not know. For example, if students know that 2 × 4 = 8, they may be able to use this fact to derive that 3 × 4 = 12.

This activity is an extremely traditional activity where students fill in an entire hundreds chart.

The key benefit to this hundreds chart versus a pencil and paper activity is that students can request feedback at any time by clicking submit. This means that students who are using patterns to complete the table are able to double-check those patterns before having a mistake propagate into other rows and columns of the chart.

What none of these applications do is give feedback to a student based on their thinking. How would one go about designing such an application?

In this 2016 paper presented at ICCM 2016, the authors analyzed common errors made by students to single-digit arithmetic problems to see if they could rationales for these errors. Knowing why a child might write that 3 × 5 = 12 would be useful when engineering feedback for that child! In this case, the authors note that the child may have been skip-counting or using repeated addition and lost track of how many times they added 3 together. So instead of 3, 6, 9, 12, 15, the child thought 3, 6, 9, 12. The feedback here might be to skip count with the child and count the number of counts with them. If a child enters 3 × 5 = 14, then it might not be the number of skip counts that is the issue, but the addition from 12 to 14 instead of 12 to 15. Our feedback would have to be different!

Unfortunately, I have not yet designed an activity that does this, but the idea of feedback matching the thinking students do is a key component of the work I do.

The point is that with some small design decisions, we can modify activities that essentially only assess learning or build recall and turn them into activities that students can learn from.