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      I'm a retired math teacher. I spent 42 years in the classroom, grades K-12. Now I work mostly with teachers. I blog about math education, and I maintain a very large math education website. Over the decades, I have developed much curriculum involving manipulatives, but my interests are quite a bit broader than that: when it comes to teaching math, "there is no one way".



      In my other life, I am a member of the National Puzzlers' League, and the co-creator of the cryptic crossword that appears in every issue of The Nation.



      I am married to an epidemiologist and live in Berkeley, not far from my two adult children.

      You mentioned your math education website and blog. I imagine for a new teacher that all of the resources and insights shared on them could be useful. What are some things that a veteran teacher would find useful on your website, which they may not have seen before?

      I don't actually know the answer to this question. On the one hand, the site is huge, with a large variety of materials. On the other hand, veteran teachers are very different from each other. What I can say is that the site is somewhat unusual. Many very helpful sites are dedicated to a single idea, such as Visual Patterns. Mine has tons of stuff, and as a result it's hard to find things in it. What I hear from experienced teachers who use the site is that it's a good source of student-centered, group-worthy, discussion-generating activities, mostly for grades 7-12.

      If I was a veteran teacher and I was looking for discussion-generating activities on logarithms, what would you recommend from your site?

      Actually, I only have one activity about logarithms, but it’s a good example of my sometimes offbeat approach to curriculum. What I often do when introducing something new is start with the concept, and only name it later. In the case of logs, I remind students about scientific notation. “For example, 300 is 3 x 10^2. Today, we're going to use super-scientific notation, where it's just a power of 10.”

      

“10 to what power = 300?” We can figure this out by graphing y=10^x, and y=300 in Desmos or GeoGebra, and looking for their intersection.

      It turns out that 10^2.477 is very close to 300. We could also have found this answer by trial and error, or by using a computer algebra system, say the one in GeoGebra or in some calculators.



      So I ask the students to make a super-scientific notation table, and follow up with questions about it, all of which can be answered by using laws of exponents. 


      Of course, what is going on here is that the exponent of 10 for 300 in super-scientific notation is actually the log of 300 in base 10 logarithms. So students find that they already know the laws of logs: they're a version of the laws of exponents. 



      Here's the link:

      In your experience, is there a minimum class size required to avoid the sound of crickets during student discussions?

      Maybe a dozen kids? There's a different issue if the class is too large: it makes it difficult to call on all students in one class period. 



      The main issue is not so much class size, but the sort of questions you ask, and what precedes an all-class discussion. If the students have had a chance to grapple with a question, they are more likely to get involved in the discussion, so in my view, whole-class discussions should alternate with group work. 



      If students worry about being wrong, they are more likely to hold back. So it is important to make it comfortable to make sure there is no stigma to giving a wrong answer. One way to do that is to take down multiple answers with a poker face. Another is to praise participation rather than correctness. Over time, praising correct answers discourages participation, by limiting it to those students who are sure their answer is correct. On the other hand courageously volunteering an answer one is not sure of deserves explicit appreciation from the teacher. Giving a correct answer is its own reward and there is no need for the teacher to dwell on it.

      Any general tips to foster student engagement during group activities?

      Group work should be the default setup. Students need not collaborate on every question, but they should be able to ask each other for help. The groups should be selected randomly (I use playing cards) every other week or even every day. The way to get engagement, as well as any desirable behavior, is to ask for it. In other words, instead of speeches to the class ("It is nice to help one's neighbor"), intervene directly: "Lucy, can you help Tom with #3?" "Aisha, move closer to the group, you can't participate if you sit this far out."

      

If the work is too easy, students do not have any reason to participate in group work. It is better to aim a little high, and make sure the work is challenging. If it's too hard, you can always give hints to get things rolling.



      One very powerful technique to improve group work, which I learned from Carlos Cabana, is the "participation quiz". You give students work for which you're pretty sure they won't need your help and you project your computer screen. On your screen, you describe every desirable behavior. "Jose took out his calculator" "Joanie is sketching the graph", and so on. Pretty soon, your class turns into a surreal zone and you hear things like "Are you having trouble with #3? I'm happy to help you!" (at which point the student looks up to make sure you caught that.) Students gradually realize that those behaviors actually pay off in their ability to learn, and you reap the benefits long after the quiz.

      In June and August, you are doing a Visual Algebra Workshop and I know you’ve been a workshop presenter at NCTM conferences. For math teachers who have never presented at a conference before, can you share how you become a presenter?

      If you have some ideas about curriculum and pedagogy to share, and especially if you've tested those ideas in your classroom, you should definitely consider sharing them with others. The details of how to submit your proposal depend on the conference, and are generally well explained on the sponsoring organization's website. Still, it is helpful to attend the conference to get a more concrete sense of what is expected of a presenter. If your proposal is accepted, you should start preparing early, so as to have time to collect data from your classroom, such as student work, or anecdotes about what happened when you taught this material. Speaking at a conference is a great way to clarify your ideas about teaching. You come back to school invigorated.

      “In my other life, I am a member of the National Puzzlers' League (NPL)”. For those unaware of this organization, could you talk about it and your participation in their activities?

      In spite of English not being my native language (or because of that?) I developed an acute interest in word puzzles, especially cryptic crosswords. Many years ago, I read about the NPL in a book about anagrams and palindromes, and I decided to join. This was one of the best decisions I ever made: it felt like I had found "my people". Many members are crossword constructors, and all of us enjoy solving word puzzles of all types.

      We meet yearly at a convention in July, and create puzzles for each other which are published in a small monthly magazine, The Enigma.

      My contributions, for years, were cryptic crosswords, which led to a volunteer position as the cryptic crossword co-editor, and later to an actual (low-) paying job as the cryptic crossword co-constructor for The Nation. You can find out more about all this on my puzzle page

      and on the NPL website.

      I believe that my involvement in puzzles has had a huge influence on me as a mathematics curriculum developer. I've written about that here.

      How can we best stay up to date with you?

      Now that I'm retired from the classroom, I stay involved in math education largely through my online presence. I'm on Twitter as @hpicciotto.

      I blog somewhat irregularly at https://blog.mathed.page. I also use that site to announce upcoming presentations and workshops, as well as any updates to my math education website (https://www.mathed.page). On that site I share curricular materials (middle school and high school math), plus some philosophizing about math teaching (often combining some blog posts into a more permanent form.)

      Finally, you can subscribe to my newsletter, which I e-mail to about 3000 people every two or three months.