In Chapter 5, Wendy Ward Hoffer writes about the importance of time spent discussing mathematical content. She once again reminds us that "students benefit more from solving and then discussing a few problems in depth, rather than completing numerous practice problems." While this idea was discussed earlier in the book, it was a good reminder during this chapter because discourse, for obvious reasons, takes time and doesn't fit with the "drill and kill" mindset.

As always, I love Ward Hoffer's lists and there were several in this chapter. On page 75 she lists ways we can invite students to share. While I have used some of these in my classroom, the one that intrigued me was "explain your vigilance". Without Ward Hoffer's example (What were some of the speed bumps you encountered when solving this problem?), I might have glossed over this list completely. What I like about this is that it validates students' struggles. It says that it's not only okay to struggle, but it's also normal to struggle. This is important for all students, because on some level, I think if there's no struggle, there's no gain (the no pain, no gain mantra that coaches use). As an aside, one group of students who I think need to experience this more is the honors students. (Mine frequently say that math is too hard (mainly because they've never been challenged before). I explain to the students and their parents throughout the year that if the student get through the year without feeling a little uncomfortable, I haven't done my job well because they haven't been challenged.) Working through challenges, whether in the math classroom or in sports or in a career, is an important life skill.

Throughout the book, Ward Hoffer shares the importance of withholding correct answers in order to discuss errors. One of the things I occasionally do with my students when we are looking at homework answers is post an incorrect answer. Students often assume if it came from the book, the internet, the teacher, or their parents, it must be correct. They assume that if I'm giving them an answer, it must be correct and so they mark their answer wrong and never question why they got it wrong. Students who not only catch the error, but justify their thinking are rewarded in some way. By working with errors, Ward Hoffer is validating student thinking by recognizing that errors aren't something to be afraid of because

__we learn from them__.

An ah-ha for me was Ward Hoffer's suggestion of allowing students to revise their thinking. I think this is a natural extension of discussing errors, but one that I think I have missed. Ward Hoffer writes, "

*an incorrect answer is the beginning of learning*; when we maximize the opportunities revealed by our own and our students' mistakes, we model the

*growth mindset*, the belief that we are all capable of overcoming our difficulties and achieving." (emphasis added) Allowing students to revise their thinking tells them that our classroom isn't a "three strikes and you're out" kind of place. Just because they have had some struggles, doesn't mean they're out of the game and unable to recover. When we allow students to revise their thinking we validate the idea that learning is a process.

I'm off to make classroom posters to help facilitate student discourse.

For the kids,