I first started doing number talks (also called math talks) to start class when I taught 8th grade. If you’re unfamiliar with number talks, this site by Fawn Nguyen has some great stuff to get started. My first year teaching high school (Algebra II, Precalculus, and Calculus) I stopped, opting for Visual Patterns, Open Middle, Which One Doesn’t Belong?, and a few other rotating warm-up routines. I thought that the skills involved in number talks, while useful for middle school students, were less relevant for upper high school.

I came back to number talks at the start of this school year, and I’ve been happy with the results. When I wrote a problem up on the board one day toward the end of the year, a student blurted out, “oh, I love these”. That’s just one student, but engagement was usually high. Efficient strategies for these problems often did not come easily to students, which suggests that there’s potential for learning from them. More importantly, as they became comfortable with the routine, many students who were rarely willing to share started to speak up and take more risks.

Here’s a selection of my favorite problems, many courtesy of Fawn’s site.

Picking two numbers and an operation is often insufficient for a great number talk; I’ve found that careful selection of the numbers involved to ensure a variety of strategies is worth the effort every time.

I have one lingering question for next year. Engagement during number talks seems high, and seems to engage both high-performing and low-performing students. There is clearly a need for the skills that number talks are targeting. At the same time, I don’t have any real evidence that students are learning these mental math skills. I think they are, but that’s based on my intuition and a few anecdotes. One challenge is that I tend to cycle through a variety of types of number talk problems that require different strategies. One goal for next year is to reorder the number talks I use so that I expose students to one type of problem 2 or 3 times, lead a discussion that attempts to consolidate understanding of relevant strategies and when they may be useful in the future, and then revisit that problem a few weeks later. Hopefully this sequencing will provide more robust evidence as to whether or not students are actually learning.

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Michael PershanI’m also a fan of using number talks in high school classes! I like your thoughts about how to assess whether kids are developing their mental arithmetic skills.

One thought I had, while reading this post, is that the mental math doesn’t actually need to be about arithmetic. Almost none of the talks that I ran this year with Algebra 1 were about arithmetic. My favorite talk of the year was some mental math solving systems of equations. I forget the details, but it looked something like this:

x + y = 10, y = x + 2

y + x = 20, y = x + 2,

x + y = 20.5, y = x + 2,

etc.

Once we’re able to do mental math with all sorts of non-arithmetic topics, I think that mental math can more specifically address the core content of our high school classes. My favorite thing to do with mental math is use it to respond to student work i.e. a replacement for feedback.

dkane47Post authorThis is awesome! Will definitely try to squeeze in some stuff like this. I like that sequence as well — I’ve heard those called number strings, where you do a few in a row. Seems much more purposeful, only issue is time and one thing I like about number talks is that doing one can take <5 minutes some days.

Benjamin LeisI’m not sure I understand why efficiency is very interesting here. It seems like the only real goal would be a reasonable estimate or accurate results without taking “too” long and anything beyond that is kind of pointless.

Does computational weakness show up elsewhere?

dkane47Post authorOoh ok. Here’s an example of what I was trying to get at:

When finding 3.75*32, very few students multiply by 4 and then subtract a quarter; most either try to multiply the decimals, multiply by three and add three quarters, or convert to 11/4 and try to work it out from there. When the subtracting strategy is shared, many other students think it is a better strategy than their own. And that type of thinking is useful in lots of contexts outside of these mental math exercises. My point was just that, since students are hearing about methods that they a) haven’t thought of and b) think are useful, there’s potential for some useful learning.

Does that answer your question?

Benjamin LeisHopefully this isn’t coming off as too contrarian. That gives me a better sense yes of the aim. If that’s the case though, you’re really just hoping for a transfer of skills and I’m not sure your current strategy of rotating tasks needs tinkering. I.e. if you want to learn flexibility and focus on problem solving then you need to continually introduce new problems and shift contexts. If you want to focus on getting great at mental math then you need to sequence and repeat mental math problems to build enough practice.

I’d add that there is a fair amount of talk that the more generalized problem solving doesn’t transfer that well between tasks. If you could take that 5-10 minutes and do 1 more whiteboard problem within Alg II etc. and discuss student solutions would that be more powerful? For that matter could you find examples of computation to discuss within the natural flow of the classroom? I’ve been wrestling with the place for warm-ups this year albeit in a much more limited context. At any rate, it was another thought provoking post.

Ben

dkane47Post authorDefinitely not too contrarian! I appreciate the chance to push my thinking.

I think connecting some of these skills with course content — different estimation strategies or decompositions of numbers — would be an awesome next step. It also sounds ambitious in terms of the effort but possibly also the payoff.