1) Students ask mathematical questions (that in turn drive their learning)
2) Students and teachers seem happy to be in class and proud of their contributions
3) Students and teachers relate what they learn to what they already know (in math and in their lived experience) and to what they want to know more about/do better (in math and in their lived experience)
4) Students and teachers collaborate to learn and make sure everyone is learning
5) Students and teachers communicate their ideas fluently out loud and in writing
6) Students solve novel problems with a variety of strategies, tools, and representations
7) Students and teachers ask not just “am I right?” but “how do I know?”
8) Students and teachers ask, “what other math/patterns/generalizations can I discover from this?”
9) Students and teachers ask, “what are we learning about? what’s the big idea? what do we need to practice to get better?”
What’s missing? What’s superfluous/not fundamental? How do you know that your classroom looks like you want it to? How does this relate to what you assess?
Wonderful post! And I can tell you these are the exact questions/ideas I discuss with kindergarteners, so even (and perhaps especially) the youngest of mathematicians can access these questions.
I’d add:
Students and teachers can articulate how they’ve changed as a mathematician/learner over time or as a result of new learning.
Students and teachers create “original works of math”–they solve problems they have imagined/created/pondered, not just those presented to them.
Great Prompt – agreed.
Here are some ones I thought about:
– The students barely notice me, because they are working deeply together on important math.
- They say, “Wait… don’t tell us. We’re almost there. Give us a few minutes.”
- They keep working even after the bell has rung.
- They don’t need me to tell them if they are right. They verify and check their work, and convince themselves (and others) based on correct mathematics.
- They detect errors in the lists of “solutions.”
Exactly!
When I first read this blog, those were my exact thoughts William. When a students says, “wait, wait, don’t tell me” you know they are deepening their mathematical understanding. And I might add, I suggest this to them by saying those words early in the year when someone solves something and wants to be the first to tell me. I say, “oh, wait, don’t tell me. I have not had time to solve this yet and I am still thinking about it.” I tell the students I like to solve it with them because because everyone learns from the sharing of mathematical thinking along the way to the solution.
The fact that they don’t need me and keep working after the bell rings is such a great moment in teaching! :)
- All students feel comfortable contributing, not just the ones who “know” the answer.
- Students know that asking a good question is just as smart as knowing an answer.
- Different kinds of smartness are being valued, not just quickness: insight, systematicity, clever representations, and important connections.
- Students make judgments about each other’s contributions based on mathematical reasoning, not based on their sense of how “smart” the person saying it is.
Good ideas! I would add:
Students and teachers both use technology to make math “real”
These are great!
I also love it when the enthusiasm and ownership of the problem solving that happens in class begins to creep into students’ conversations outside of class. I would add…
* Student learning becomes dinner table conversation. They share their discoveries with their families and want to share those subsequent conversations with the class.
Great thoughts on what would make a wonderful mathematic classroom for learning. It would be great in any subject area. How the learning will be assess is the element which inspires my thoughts. Can the assessment lead to further growth or discouragement? Students need to begin the exploration knowing what outcome is expected. Examplars of these outcomes can help the learner know the direction to focus. If the students and teachers clearly understand expectations, the assessment would hold no surprizes and the learner should be able to assess their own learning. I know this would be the ultimate assessment, however time and the real world interfer. I have saw this modeled and would like to collaboratively develop this for my grade six curriculum. My goal is to do one unit this year and evaluate the effectiveness with my students.
This past year I had a group of students with multiple behaviour problems, and weak independent work skills. When surveyed on what was the most helpful in a math, they responded that being able to understand what was being assessed and being provided multiple opportunities to be successful on the assessment.
“Students and teachers collaborate to learn and make sure everyone is learning.”
This is probably the statement that stands out the most for me. If all members of a learning community are concerned about each others understanding the result would be powerful. In our culture there seems to be a “me” mentality, once I understand it I am done. The statements that put “we” into the learning help build that type of environment. Those would be great ideas to list: How can I develop a community of “we” with my learners.
Great synopsis, Max (and others!). I want to share this with my colleagues.
I know when I’m doing it right…
- when students invoke their own creativity to show me understanding in ways that I would never have imagined
- when teachers in other disciplines report that students are talking excitedly about math problems in their classes or write about math in their journals (I’m not talking about the teachers complaining about Johnny working on his Algebra homework instead of paying attention to the lecture in Science)
- when I get a postcard from the Grand Canyon from a student telling me about all the applications of Statistics she observed on her drive out west with her family
When students show teachers a new way of seeing/solving a problem we are doing something very right.
A great post, and good responses! Let me add (or second):
I know I’m doing it right when …
-Students ask questions that I don’t know the answer to.
-Students ask each other for help before they ask me.
-Students aren’t afraid hypothesize and be wrong.
-Students treat each other’s ideas with the respect they deserve.
Great ideas and comments.
I’d add
I know I’m doing it right when …
Students experience the joy of doing mathematics and the self confidence that that builds.
I think in the olden days,the method of teaching made students think, wow that teacher is smart. Look how she/he can stand up there and know, say, and do all that mathematics. But enabling *them* to do the thinking, wondering, and figuring out gives them that confidence and joy.
I love the discussion. Thanks!
Well done, a great list…superb.
Maybe something could be added along the lines of “students and teachers discuss errors and misunderstandings “
A good list, Max.
I’d like to add that I think #2
2) Students and teachers seem happy to be in class and proud of their contributions
is really important. I’d replace “seem” with “are”. If we focus only on content, we are missing a huge part of what’s important.
Great list. It would be wonderful if all classes shared these traits.
Another one might be
A student who is not taking math this semester (or who is taking math but with another teacher) comes looking for you to share a new math idea or problem with you because they knew you would be excited to hear about it!
That would be a student that truely appreciates what the joy of learning is, knowing that we too share that joy.
All said perfect!
What great affirmation as we begin a new year to remind us that we have all seen this in our classrooms!
I would add—
I know I am doing it right when–
students start telling me what they notice and wonder about everything in math class, not just when we are problem solving.
Applications should naturally follow from the big ideas. The points you’ve made here are key to giving teachers and students confidence to let that happen in the learning process. Teaching problem solving using a learning standard is fundamentally different than presenting a learning standard. Making a real effort to have assessment not be a regurgitation of information and instead an application(s) of mathematical concepts, is in my opinion, a true indication of learning.
Your comments are right on and should be the focus of lesson planning…. the real shift in education will begin when students feel compelled and empowered to learn and place a high value on their own contributions and thinking.
All great ideas.
I know I’m doing it right when I hear students talking about a PoW in the hallways between classes and during their lunch break. It is so nice to know that something triggered them to wonder! I also get a thrill when other teachers tell me that my kids are talking about a problem as they enter their class. Just to know that their conversations/discussions aren’t finished in math just because the bell rang is quite a reward for me.
From the number and quality of the comments, you have obviously hit on a topic which is very important to Maths teachers and their pedagogy. If we achieve a class with these attributes, then teaching Mathematics becomes a work of love and passion indeed! Keep the great posts coming Max!