After this year’s TMCNYC, we sent out a survey to everyone who attended. We asked, *“What was your favorite part of TMCNYC18?”*

Here are most of the responses:

- Doing Math and hanging out with people.
- Probably Doing Math with Ben, Cici and Melvin. Really interesting series
- Being able to do interesting math with other participants, math that I could see how it directly related to high school math, without being high school math.
- Doing Math sessions & just chatting with people
- Doing math. I always think the most beneficial PD for math teachers is being engaged in math. I wish that session after lunch was longer.
- Probably the “doing math” sessions! All three were fun, engaging, thought-provoking, and well-run.
- Doing Math
- The Doing Math sessions were fantastic!
- Doing math – I loved each of the math sessions.
- The expanding math strand!!!!! I also enjoyed the doing math sessions.

It seems that pretty much everyone’s favorite part of our conference was the hour we spent in the middle of each day, doing math together.

In light of this feedback, we’re going to engage in some public reflection. What made the three Doing Math sessions so popular? What makes a successful Doing Math session? How can we (and others) replicate the fun that we had at TMCNYC18?

We (the organizers of TMCNYC) asked each of the Doing Math presenters to reflect a bit on their session this past August. The first post comes from Benjamin Dickman, whose Doing Math session was about problem posing and the multiplication table, a context that we had a ton of fun with at TMCNYC and that some have already brought into their own classrooms:

**Facilitating a Doing Math session on Problem Posing [****Benjamin Dickman****]**

I see a lot of mathematics education through the lens of problem posing. I see problem *solving* as asking oneself a lot of questions – posing problems, in some sense – like, “Does this remind me of a related, or simpler, problem?” or “Can I represent this problem using a picture or a diagram?” or “Who is awake right now and might have an idea about how to solve this problem?” But, what is the actual source of the problems with which we are engaged?

I worry that too few of us have had the opportunity to create a problem that was our own. In my anecdotal experience, people get excited when they make something novel that engages others, and I want to see more of this – the excitement from being creative, and the engagement around new problems – in the vast, ever-expanding world of math and math education.

Two ideas that are central to my decision to present on problem posing:

**(A) **Problem posing is a skill that can be introduced and practiced during a short session; and

**(B)** Problem posing is a skill that can be developed over a lifetime, and brought back to the classroom – or other learning sites – in many different ways.

I think these are themes. Specifically, can you answer the following?

**(A*) **What is a topic that you can broach meaningfully during a short session?

**(B*)** What is a topic that participants can continue to explore after your session ends?

I presented at TMCNYC18 on problem posing with the multiplication table. I picked the topic of the multiplication table because:

**(1)** Problem posing is so wide-open that I felt it needed some constraints to keep us on the same page, or at least in the same book;

**(2)** I expected most participants would be familiar with the times table, and might even harbor negative conceptions about it based on perceived proximity to rote memorization or drilling; and

**(3)** I have thought a lot about the multiplication table in the past: enough to feel that I had something original to contribute, enough to feel that I could respond effectively to ideas or questions that arose as relates to this topic, and enough to feel that new ideas could provide opportunities for more careful thought later on.

I think these are themes, too. Specifically, can you answer the following?

**(1*)** What is a big topic that you can home in on, whether through focused questions, a sequence of problems, or some other approach to structuring?

**(2*)** What is a subtopic, or object, or idea that will be accessible to a wide range of participants – and how might you think about differentiating instruction so that rich mathematics can be surfaced?

**(3*)** Related to the previous two questions, and perhaps worth asking yourself beforehand: What do you know a lot about – meaning that you have given it thought in the past, and are excited to give it further thought in the future?

I hope that what I have written above can be helpful to someone who might facilitate a future Doing Math session, or something like it. The **only** prerequisite that I can think of for facilitating such a session is that you genuinely want to; even then, you might wish to run a session, but be concerned that it will not go well. I have certainly experienced this type of worry or anxiety or dread, and my experience has been that it subsides only with experience. But, I **know** that you have something worth sharing – even if you have not yet identified what it is – and it would be a shame to let your self-doubt deprive others! If you want to contact me about planning for a potential Doing Math session – whether initial brainstorming or finishing touches – then email will be a great choice: firstnamelastname[at]gmail[dot]com

**Notes on Numbered Themes for my TMCNYC18 Doing Math Session:**

I begin with **(1)** and **(3)** and then return to **(2)** to investigate just one specific problem posed by a participant, Brian Palacios, during this Doing Math session at TMCNYC18.

**(1):** Perhaps the best I can do is link to a write-up of the actual talk! **Problem Posing with the Multiplication Table**. (Many thanks to Andrea Kung for typing out the problems, and her solutions, for that which was presented!)

**(3):** A couple of participants (Sam Shah and Dan Anderson) used the numbers adjacent to the main diagonal of perfect squares to connect the times table – a square display of multiplicative, number theoretical information – to Pascal’s Triangle – a triangular display of additive, combinatorial information. This provided me with new fodder for thought: What other connections exist between these two representations? I have not [yet] given this much thought, and I invite the reader to marvel at will!

**Also (3):** I was pleasantly surprised that Grace Chen, in her closing session on the last day of TMCNYC18 (Structured Reflection on “Expanding Mathematics”) re-mentioned an aspect of the definition that I gave for a **problem**. This definition can be found in the link above, but here it is:

Problem: A question for which the method of solution is unknown at the outset.

This was followed immediately in my talk, and write-up, by a remark that begins:

There is an implicit human in the definition above: the word “unknown” entails the

existence of a living entity…

I think that *the [implicit] human in mathematical problems* is a topic to which more attention should be directed, and look forward to thinking it through to greater depths.

**(2)** By applying the approach to problem posing that was discussed during the beginning of this Doing Math session, Brian formulated the following problem:

Consider a table in which the entry of row a column b is ab-(a+b). What is the sum of all

entries in that table?

I do not recall dimensions being given, so I will solve it below for a 10×10 table; the generalization to an nxn table or nxk table is left to the reader!

**My solution to Brian’s problem****:** Add 1 to each of the 100 entries in the 10×10 table; this adds a total of 100, so when we find the modified table sum, we will go back and subtract that 100 off. Each entry is now ab-(a+b)+1, which factors as (a-1)(b-1). In a standard times table, we would have row a column b containing ab; here, we have simply reduced each of those factors by 1, which results in a topmost row and leftmost column of zeros, and the rest of the table looking just like the 9×9 table. To compute the sum of that 9×9 table, we use the same method described at the start of the talk (and in the link to its write-up) to find a total of 45^2. But, we still need to subtract off the extra 100; so, our answer is: 45^2 – 10^2 = (45-10)(45+10) = 35(55) = 1925.

I feel fairly certain that the reader has known about the multiplication table for many years, but has never seen Brian’s problem before. (I had never seen this particular question, but my experience thinking about the topic of multiplication tables, factoring, and summing table entries definitely prepared me to broach it effectively!) This question, for me, provided a wonderful opportunity to explore a familiar mathematical context in a novel way.

Read over the presentation if/when you get a chance, and let me know: What multiplication table problems can you create?