I am currently helping twenty-seven K-8 teachers brush up on their math skillz. The teachers I am working with are honestly a JOY to work with. The first few days we had comments in the daily feedback that revealed discomfort and frustration with math and with the idea that we were asking the participants to NOT FOCUS ON THEIR TEACHING but to FOCUS ON THEIR OWN UNDERSTANDING. Boy have they come around!!!!

**Teacher participant (after seeing a colleague at the board):**"Are you going to ask for more people to share whether they did the problem differently?"

**Me:**"Why yes, I was now that you mention it."

**Teacher participant (in the process of standing up to come to the board):**"Oh, good because I did it a totally different way, and I want to show what I did!"

**Another paraphrase of a student talking from her seat:**

"Can I share something I thought of about the area model? It's kind of like when we add 0 a bunch of times so that we could subtract negative numbers....because you can add 0 as many times as you want and still get 0. It's like that, except... really I should just show you."

Marie's "1 and 3/4 times 1 and 2/3" |

"we had 1 and 2/3 going on to the right forever and 1 and 3/4 going down forever, and the product is just the overlap...So we don't need to think ahead of time of how big a rectangle we need". So since you can see 35 squares, each of which is 1/12, you know that the product of 1+3/4 and 1+2/3 is 35/12.

I thought that her invention was really slick. She noticed that adding extraneous area actually made the concept shine through. We had just finished talking about how "adding 0" and "multiplying by 1" could be useful.....She really internalized that..

Marie asks LOTS of questions. And as the workshop has progressed, the number of excellent questions that are brought up AND answered by the participants themselves just keeps growing. Granted, these teacher participants are being paid to attend, but I don't think that's all that is motivating them to ask such wonderful questions. I think one reason is that they are not being graded.

Many mathematicians (including myself) are appalled at the low level of understanding and lack of interest in understanding shown by many elementary school teachers as well as those who aspire to be elementary school teachers. But being appalled doesn't accomplish anything.

**LEARNERS' RIGHTS and MY RESPONSIBILITIES AS AN INSTRUCTOR:**

Many people talk about the "Rights of the Learner". But we (my teaching partner and I) REALLY took that seriously. Clearly it's important to create a classroom environment that is comfortable, but that doesn't mean "feel-good" or "dumbed-down".

**I now attempt to do the following:**

**1) meet anger about and frustration with the material with a sense of humor as opposed to dismay or disapproval.**

**2) acknowledge that learners may have previous experiences that we can learn from.**

**So**

**we should solicit anonymous feedback, show that we have read it, and publicly address or discuss it.**

**3) encourage learners to reconcile old ideas (whether correct or incorrect) with new ones Asking them to "just start from scratch" or "just forget about the algorithm" is unfair.**

**4) keep in mind that learners want to please the teacher (especially if a grade is involved). When there is no grade involved, students are more likely to express their frustrations honestly, but still hesitate for fear of "upsetting" the instructor.**

**5) be consistent in "walking the walk". In other words, if I talk about the philosophy of "why something should be discovered or learned independently", I have to try to avoid things like writing on students' papers or directing them on what to write. I have to invest in their thought process whole-heartedly regardless of my concern for "covering material".**

For instance, after actually discussing and debating about the abhorred-by-mathematicians "FOIL method" of multiplying binomials and how it was related to the distributive property, one student jokingly said "Well I prefer the SARAN WRAP method." After we had discussed the method (its advantages and disadvantages) heatedly AND with humor, even though they knew I "didn't approve", students felt comfortable saying that they "used FOIL" to solve the problem. To me, this is much more preferable because they are being HONEST about how they thought about the problem rather than trying to make me feel good.

That's all for now...So many thoughts are running through my head. But here's a link about Finland's lack of grading which has means that in Finland The differences between weakest and strongest students are the smallest in the world!!!