Module 5-M Content Revisions

[StevenClontz]

Part 1 (lecture)

I received some natural questions about when compositions and matrix multiplication may be reversed. It'd be natural to have activities to answer those questions (rather than just an aside in 21.11).

We also could use some images/examples to illustrate 21.10 and 21.11.

[StevenClontz]

Part 1 (TBL)

It was a dud for me. Full disclosure: my top students in 3/4 teams were absent, but even still, I'm unhappy with the scaffolding.

In particular, I had to hold a lot of hands to start 21.4 as they didn't know how to compute S(T(e_1)). I think the big issues is that A1 should be expanded to have them do more than blindly convert function to matrix. We already discussed having them do the other direction, but I think their major blind spot is the lack of ability to compute linear transformations of vectors, particularly when only given a matrix. One student actually called me out on this: "which standard was this again?" I mumbled something about A1, but we really should broaden it to cover this important skill explicitly.

I ended up skipping 21.5-21.9 and 21.13 due to the slow pace of the first few exercises.

I noted again during this class the need for images/examples to illustrate 21.10 and 21.11.

[siwelwerd]

They had trouble doing S(T(e_1)) in the first part of 21.4, but once we talked through that they were on their way. We didn't have time for 21.9, but they did work through two complete examples. They successfully did 21.12, but I hand-waved 21.13 due to time constraints.

My gut feeling is that while my top lecture students were ready for today, on average the TBL students did better with the lesson today.

I note that the RAT specifically asked a question about computing the image of a vector under a linear transformation given the standard matrix; maybe we need to hit this harder in the readiness assurance process. It actually never occurred to me to have a standard for that because it is so basic.

[StevenClontz]

It actually never occurred to me to have a standard for that because it is so basic.

You'd think, but I had to hold a lot of hands yesterday to get them to do this. Generally speaking, I think anything non-prerequisite that we want to cover in the readiness assurance process should be represented by a standard.

[StevenClontz]

Part 2

This went very quickly (30min?) in lecture, but I ended up tweaking things on the fly. Since it ended early enough, I actually made some changes that I think were for the better before the TBL section. I didn't merge them into master (see 2bd175c1fc708d5627e889114768726df0e31106 ) because it was so last-minute, and I neglected to offer the revisions to @siwelwerd as an option (sorry).

Even with those changes, we finished in ~35 minutes. I actually made up an extra activity on the fly mimicking an M1 exercise to start the class, because I felt that they didn't really follow the previous lesson. I think it was a good idea anyway, since it helped warm them up for the first planned exercise (coming up with the identity matrix).

TBL class actually called me out on two things that I noticed during lecture. (1) We didn't cover a new standard today. We don't have to cover a new standard every day, but I think a standard on defining matrices R such that RA is the result of a row operation wouldn't be a bad thing for next semester (and gets a total of four M standards to match the others). (2) Activities 22.6-22.9 are unmotivated: it's not clear what they have to do with matrix multiplication (the content of all three M standards). They have more to do with A3, injections/surjections. I don't really think they are needed for Part 3, so I'd chuck them. In particular, it was noted that 22.9 was already covered in the 22.6 activity (bijections must be square, actually, 22.6 went further and said bijections have RREF(A)=I).

[siwelwerd]

Today started out as a bit of a train wreck that I guided into (what I think was) a real valuable learning opportunity. They did the first activity okay, but then ran into all sorts of trouble with the first row operation. The root of the problem is that they didn't understand matrix multiplication yet. We spent 15 or 20 minutes in some quality productive struggle just on part 1 of 22.3. After 10 minutes, they had all written down the correct matrix S_1, but were doubting themselves, and one team told me they were certain it was wrong. So on the fly I had them all multiply the correct S_1 times A. They struggled with this, but with team-by-team coaching all ended up okay with it. Part two went better, and by the time we got to part 3, they made short work of the permutation (which is in my opinion, the hardest).

Activity 22.5 is a good activity that probably belongs on a different day (I think the reason it is here is so that we have the notation AX=B). My printed out notes didn't match the slides, which had a trick question for (f)--I believe you corrected this, but somehow it only made it into the notes and not my slides. It did reveal that they still don't quite grasp injectivity and surjectivity,

I didn't have time for 22.6-22.9 due to spending at least 30 minutes on 22.3.

Looking ahead to part 3, can we put an example in for 23.2? I am 99% sure it is too abstract for them.

And go ahead and merge those minor changes in to the master branch.

[StevenClontz]

merged in 2bd175c

[StevenClontz]

new 23.2 is in cd9f91f (built all files in c8e64fc)

I really like this new version, thanks for asking to change it. It now works on two levels: students can plod through the numbers, or simply use the definitions to knock things out trivially. I expect teams to take different paths, producing nice discussion.

[StevenClontz]

The root of the problem is that they didn't understand matrix multiplication yet.

Yeah, we need to spend 1.5 days on this, not just 1 day.

[siwelwerd]

Finished over 15 minutes early today (nearly 20 minutes early). They had all sorts of trouble getting started on Part 1 over 23.4 I think the notation was the biggest sticking point. However, once we got through that, they blitzed through the next 3 parts, as well as activities 23.6 and 23.7. This might be a better spot for activity 22.5. The extra time can be dedicated to more matrix multiplication next semester.

[StevenClontz]

Yeah I set them up to do 23.4 Part 1 by giving them the augmented matrix from the start. (Could have alternatively added a Part 0: rewrite the equation T(X)=e_1 as a linear system of equations.) Assuming you meant Activity 22.6 (sort all the statements about inj/sur/bij linear maps) rather than 22.5, then I agree putting that on this day is ideal. I also agree we can spend more time on matrix multiplication next semester.

[StevenClontz]

As for my update: I also finished early, and my experience seemed similar to yours.

[siwelwerd]

I updated all 3 days of module M; I tweaked the first day based on the fact that I ran out of time in 2 of your 3 classes (and to remove all this language about dot products since our students have not all had Calc 3 and don't necessarily know what a dot product is), and I went ahead and changed days 2 and 3 based on our above comments. The (currently numbered) M.2.2 was on your day 1 slides, but your first two classes didn't have time for it, so I bumped it into day 2 for my classes.

[StevenClontz]

dot product

If that's a concern, wouldn't it make more sense to take 20 seconds to define it (which I did on the chalkboard, but should have done on the slide). I think removing the reminder of how to calculate T(v) using A and v is an error, as is removing the observation of how to quickly compute a cell of AB using rows/columns: I'm not sure students will be able to tackle the construction of row operation matrices without being able to quickly get up and running with matrix calculations.

On the other hand, thanks for revising these... I was about to start editing at the 11th hour now that I'm back in town, but I'll jump in with teaching your revised M.2 tomorrow. I did tweak a couple things in M.3, e.g. I think it's best for these sorting activities to start students with one item in each group.

[StevenClontz]

I'll stand by using a dot product description, but I'll walk back saying we need to be explicit about how to compute individual cells of a matrix product. Instead, I had a lot of success with M.2 today after I emphasized with everyone how to find columns individually. This will also help us get better M1 responses that show work rather than just writing a final answer (that could be given by a calculator).

[siwelwerd]

Instead, I had a lot of success with M.2 today after I emphasized with everyone how to find columns individually.

Yes, I think this is the key point, emphasizing that the columns of the standard matrix are the images of the basis vector; and to find a column of a product, you chase the basis vector through two maps, which amounts to doing a linear combination of the columns of the second map.

[siwelwerd]

M1 went fantastic in my classes; very sharp contrast with yours. I attribute this to the readiness assurance process. The scaffolding in the first multiplication activity matched up perfectly with their readiness level. Lots of them thought the RAT was easy, but the contrast with your classes in their ability to get started on the activities was stark. Clearly the RAT cannot be simply dropped; if you are not going to do it, you need to do something else instead.

Interestingly, every team all day long missed the first question (although one person had the right idea, just mixed up domain and codomain). But I think that's almost a feature rather than a bug, it set up some really good inter-team discussions that helped set everyone up for the subsequent activities that they had no trouble with.

Looking back at our comments above, I think the modification of the A2 standard, along with asking 2 specific questions on it on the RAT really made the difference in this sequence of activities.

[StevenClontz]

I agree about needing something for readiness assurance. Although I'll note that it feels like they rallied and did pretty well on the M1 standard on Friday (based on my first section's responses, compared with my recollection of last semester).

Something I'm considering experimenting with this summer: prerequisite standards. Only one M/checkmark earnable for each, which may be earned via the iRAT/tRAT process if you get it correct individually and if the team gets it right on the first try. Otherwise they show up on the quizzes like everything else. This might make the iRAT/tRAT process more satisfying for all parties involved.

[siwelwerd]

Really happy with how this module is working out this time around. Part 2 finished 5-10 minutes early and thus could use one more activity, but otherwise went really well.