Elicit Student Thinking

In Michigan, at least, the high leverage practices are dominating the teacher preparation conversation. For our new certification programs, the state waaants to know where are we doing it. Nothing new there, just... enumerated. Some are beyond what we can do in math classes. But where it all starts for us, I think, is eliciting student thinking. I have an interview project coming up, in an elementary teaching class I'm teaching for the first time, and thought to ask on social media what teachers do or think about this core process. (Used to just be Twitter, now I'm trying Mastodon and BlueSky.)

#classroommath #mtechat one of the main objectives with preservice teachers is to work on the practice of eliciting student thinking. What advice do you have for them? How did you get better at it? What are you working on now?

Lani Horn had a quick, impactful response (bsky). "I find that the work of eliciting needs to be followed with some work on listening and interpreting. i have seen some folks stop at just eliciting, and it becomes "what do you think? what do you think?" without any connections built."

Elizabeth continued: "OMG yes. Learning to listen is a challenge for many pre-service & new teachers, and I often wonder if this is because they feel so rushed/anxious themselves.

Listen, swallow, take a breath -- just because a student has spoken in response to a prompt doesn't mean I've heard them yet.

Another thought -- could we also stipulate that asking a clarifying question can also be also an essential part of teacher listening?"

Shelby Strong also responded to Lani: "YES. It's not enough just to hear a bunch of different ideas; what is similar and different about those ideas?"

Learners are definitely interested when they know you are interested. How many times have they had a teacher gloss over their answer while really just looking for what they want to hear. "Tell me more" is a phrase I try to use a lot.

Mike Steele said: "When you catch yourself thinking about what to say next when students are talking… don’t. Focus on listening. Take a beat before speaking."

Nick Smith noted: "I like the other replies here and I'll add, "Never say anything a student can say." 

I think too often I'm doing the thinking for them. The less I talk, the more I hear their thinking instead of my own." 

Good indicator. Especially tough as a novice, maybe, when you have to think more about what's next.

Shelby also said: "Get students to turn and talk and prep them that you are going to ask them to share what their partner said. It lowers the stakes because it's just one other person listening to their ideas, and it takes the pressure off of sharing their own ideas."

Karen Campe responded: "Oooh I like that... encourages careful listening to your partner!"

Coutney Flessner added: "I LOVE using this strategy. I also don’t have students share their work. They already know it! The class analyzes it and we ask the author if we missed anything. Kids are significantly more engaged with each other and math with both these instructional routines. Ryan Flessner introduced both to me!"

Your students who always talk will still try to say what they said, but I think this is a moment for them to hear what others see them as saying. 

Karen was also thinking about wait time: "An important step is to give Ss individual think time before talking to partners/groups or sharing out to class -- the T shouldn't solicit any responses until that essential time has happened. 

That way, everyone engages, & no priority to fast thinkers or those who can do it in their head. 

Then you can elicit their thinking.

I was so bad at wait time early in my career that I had to actual count on my fingers (behind my back) to be sure I gave thinking time before discussing. 

Tierney Kennedy also has a teacher hack: "My advice: take a drink bottle with you to groups. When kids ask a question or when you ask them one, take a mouthful. It builds in an automatic thinking moment. Plus sometimes they end up answering their own question." 

Trey Goesh thinking similarly: "I like to have students take 60 seconds to record their ideas silently before having them talk to a partner.

You can feel the tension ratcheting up as they collect the ideas they want to share."

Wait time is such an amazing tool. Really lets learners know you are really asking, not just checking 'any questions.' 

Dee Crescitelli has the objective in mind: "Listen to student thinking with an ear for the mathematical goal of the lesson… we should be thinking about the math story the classroom discussion is telling. How do student responses & representations connect to tell that story?"

Peg Cagle also thinking about what you're asking: "Make sure that you ask about their ideas/thinking not their “answer”, and make sure they know you are genuinely curious about & interested in what they tell you. Answers w/o thinking-worthless. Ideas w/o answers, immensely valuable…& to everyone in the room!" 

Tara Maynard: "Try to always find a positive in their thinking and then find the misconception. Ask students to write, draw, sketch how they found their solution, not just verbal interaction. Always trying to provide feedback that is helpful yet doesn’t take hours." 

This might violate Mike's advice to keep your mind where it's at, but I do this a lot, thinking about the summary/reflection for the lesson. I probably open floor question too much to summarize, but if there is an idea missing, I like knowing whom to call. 

Arika Byman said: "Model genuine curiosity every chance you get. Provide sentence/question stems to help students organize and articulate their thinking. Be patient and persistent!" 

The stems idea is another idea of which I don't do enough.  The curiosity is crucial. There are so many things I genuinely want to know about learner thinking, why not ask?

Another few people were thinking about the math about which you're asking:

Rose said: "# talks and visual patterns tell Ss you want to know what they 💭 esp bc everyone has dif ideas. Shifted my mindset too!

Working on: design small group tasks/materials that encourage Ss to share their ideas w/each other. Generally ⬆️ S talk. S talk = window into their thinking."

Sian Zelbo said: "One aspect of eliciting student thinking is asking questions that are open-ended enough that you get a range of answers.  If you ask something that is essentially procedural students can't share their thinking bc they have none."

Lastly, is sharing thinking a part of your class culture?

Susan Russo said: "One thing that helps is to model your own thought process out loud. Not: this is what I’m doing but: First I notice this, and that leads me to think it might be good to go this way so I’ll try that and see where it leads. But now I wonder… 

If you are also sharing genuine thinking it is a great model for when you ask for theirs. 

What did you notice about these responses? What would you add or emphasize? There's a comment section below just waiting for you.

I'm really grateful to all who responded. Whatever media site we wind up on, I'll be there, because talking to teachers is the best way to teach better.

 

 

A Sorted Beginning

First day of Geometry and Measurement for K-8 Teachers today.  I did some improvising that turned out well, and wanted to think about that a bit. So, quick blogpost.

We were starting with Piece of Me, an activity I've stolen or modified from David Coffey. (I called it Piece of Mind today. I have a pun problem.)  The idea is that instead of the instructor droning on, not looking at you, students find out what they're interested in. One modification I do sometimes is to have them start in their groups. Develop two questions for your tablemates, then ask the person on your right. When they were done, I asked for the questions. I often write down student responses on the whiteboard, just from the principle that it helps them feel listened to. If I'm doing it, I write down them all. Just on impulse, I decided to sort them as they came in.

Now what? I said that I had sorted them. Each group should come up one more question for each list. A few minutes to discuss, then everyone stands up. After you give another question you can sit down. I don't call on people, just first to speak. The only rule was that we had to have one for each column before another one in a used column. When one was suggested, I just asked the class "Agree or disagree?" and we put it where the majority agreed. Sign one of a good semester: no one asked me if they were right! Actually that's sign two. Sign one was that they started on questions without a single person asking me what the columns were. Not that there's anything wrong with that, but the willingness to just give it a go is great.

Then they picked one question to all answer at their table. After all this (about 25 minutes) I asked: were we doing math when we did this? Some yes and no, but when a yes argued that we were noticng, sorting and analyzing by characteristics, plus looking for patterns she crushed the opposition.  I made a point that I want class to be free for people to speak, even if they are the only ones with an opinion.

Finally we did the teacher piece. Lots of why am I teaching, why I am a prof questions, with too long of stories from me. Questions about the course were about working with students, how are they being assessed, what does homework look like.

The next activity is one of my favorites for attributes, and I have used this with all ages.

Game: In or Out?

Set up: draw a circle-ish shape, or lay down a rope, or divide the room in half... two regions is what we're looking for. Players standing around a circle works best in my experience.

One player comes up with a rule that can be determined to be true or false for each player. True, they're in, false they're out. Starting with the player to the rule maker's left, they guess if they're in or out. If they are correct, they can try to guess the rule.

If you need winners, coming up with a rule that no one guesses is a win or a point.

I started with are you wearing sandals. 5 or 6 and they got it. We had rules about shorts, shirts, hair color. One fellow who's rule was at least as tall as me when he was the tallest in the class. I had an every other rule, that went about 15 deep. One great rule was whether you were standing in the shade or not. The rule maker was just at the edge, so we had 15 no's, then yes's until someone got it. We talked about the math we were doing, and I was able to connect their comments to the importance of non-examples. I also talked about the activity being accessible to many different learners and free for different kinds of participation.

When we came back inside, they talked about these questions in their groups:
1)    What were some of the rules used? 
2)    Was there a rule that was easy to guess?  Why?
3)    Was there a rule that was difficult to guess?  Why?
4)    What is a rule that would divide our class into two groups of roughly the same size?
5)    What are two rules that would divide our class into 4 groups of roughly the same size?
6)    Why do two rules divide a population into 4 groups?  Give an example.
Extension: Into how many groups would 5 rules divide a population? N Rules?

To reflect we tried for number 6 as a class. They came up with three ways to visualize in the classroom:

• hand up, stand up. One question you stand for a yes, the other you raise your hand for a yes. (New to me!) It was neat to be able to look at an individual and interpret, but not a good display to get a sense of the group.
• end, middle, end, double no's elsewhere. Worked okay, but not as well as ...
• four corners for four groups. Once we tried it, they divided by a Cartesian scheme, with one direction the first question and the othe question the perpendicular direction. This they liked, and appreciated how there was a dividing line for each question.

We tried:

• sibling & more than 1 sibling: It turned out we all had siblings, and someone noticed that even if someone didn't, they wouldn't be divided on the other question.
• wearing shorts & wearing denim: four groups, but it's hot so not many non-shorts.
• curly hair & shoulder length or longer hair: not many short and curly. This prompted the question - do they have to be linked? No? Well, then...
• pet & eat a good breakfast: yes fish count as pets. Not many non pets, but good division on breakfast. I mean bad because people, it is the most important meal of the day.
• TV in the bedroom & eat a good breakfast: computer in the bedroom doesn't count, even if you watch TV on it. (Hmmm.) This was almost perfect, 6,6,5,5.

They jotted down their take aways, then shared in group. Good stuff about nature of the activity, how much of the problem solving they did,  how engaging the sorting questions being about themselves were. I shared how a teacher had students write down sorting ideas, which she could then screen for sensitivity.

I am looking forward to a semester with these people!

Why Questioning?

Found at From-Student-to-Teacher Tumblr
From Joy of Literacy

Last week became questioning week with student teachers. It came up in two action plans and we were able to have some really interesting discussions about it.

A lot of what I'm writing about here is from work with David Coffey, inspired by Kathy Coffey, and processed from Mosaic of Thought (link includes Chap.1 as a sample) among other books. Neither Dave nor I can remember the actual origin... which is sometimes symptomatic of having done it yourself.

In my own growth as a teacher questioning is definitely one of the places where effort and reflection have helped me improve. The first level was just asking better math problems. More open-ended, that required more problem solving. When the problems are better, there's more interesting things to ask the students about later. The next was to ask more appropriate questions. Some of those excellent problems I gave to students were too much for students. This is a zone of proximal development idea. When it was too much, I needed to scaffold. Later I learned about rephrasing the question instead of narrowing it. Later still I learned about demonstrations of how I think about a problem, sometimes more appropriate than guiding students through it. Then you ask 'what did you notice?', which is - of course - one of the all time great math questions.

One of the best things I ever learned was to stop being the authority. I don't say what is right or wrong. I ask the students if they 'agree or disagree?' That might start an actual conversation. 

One of the first things I really noticed about teaching was that students would tell me that they couldn't do it (whatever it was at the time) but when I asked them questions they could get from beginning to end with no difficulty. So, obviously, I needed to teach them to ask those questions of themselves.   It was difficult. Really difficult. Shifting from asking them 'how to do (next step)?' to 'now what?' and 'how do you know?'

Literacy learning experts are good about this idea of questioning as a process, with the idea that questions are how we move ourselves forward. I like this framework to help me think about the kinds of questions that I'm asking.


Obviously, we have had a tendency to ask too many literal and application questions in math class.  I think about inference questions being predictions, reading between the lines, hypothetical questions and the like. Analysis questions are reflections, synthesis, connections, recommendations and so on.

What I shared that seemed to tie it together for the student teachers is a simple idea: ask to find out what I want to know about. I don't need to ask for answers - I know those. I don't need to ask right or wrong. I need to ask about what they are thinking. Students know when a question is genuine, and this simple idea has improved my assessment more than anything else.  I'm more persistent in getting answers when I really want to know them, also.

From A Softer World

Some good student teacher writing on questioning:

• Caitlin - Open Ended Questions
• Stephanie - Questioning Techniques

Skemp Discussed

This semester we had the opportunity to discuss Richard Skemp's great article on Instrumental and Relational Understanding in class. (“Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006; link goes to a pdf hosted at Portland State) Students read the article, with the following home workshop. When we came to class, they discussed at their table and then made one 'slide' presentations to the class on 2 ideas.  The questions are the ones I used for an online discussion before, recorded in this blog post.

Home Workshop 16 - Learning Math
Instrumental Understanding
How do I understand?
“Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006
Discuss in class Wednesday
Objective:  TLW use specific questions to better focus on and understand relational understanding.

Schema Activation: How do you multiply fractions?  How well do you understand the multiplication of fractions?

Focus:  Directed reading.
Below in the activity are a dozen discussion questions. As you read, keep notes on your thoughts about the questions.Read through the questions before reading the article.

Even though this article was written for teachers, Dr Skemp wrote mostly for researchers, and at times the language is a wee thick. Press on!

Activity: Read the article, jotting notes on the 12 questions below.  These are just notes, and you may find you have no thoughts on a couple of them. In the reflection you will expand on your thoughts for two of them.

1) What is the point of starting off with the Faux Amis story? 
(A faux amis are two words in different languages that sound similar but mean differently.  Sopa (soup) and soap (jabón) are my favorite from Spanish.  Skemp says that the ways we use "understanding" are as different as if they were faux amis.)

2) What is your favorite example of “rule without reason”? Why?

3) Does the author’s idea of looking for your own examples and his three reasons for it make sense? Why?

4) Explain Skemp’s two kinds of mismatches (in the classroom) in your own words.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

6) What are Skemp’s faux amis in mathematics teaching? Is either one an issue in your math major classes here in GVSU?

7) Would you add any advantages to his list for instrumental mathematics?

8) Would you add any advantages to the list for relational mathematics?

9) Do you agree with the advantages that he lists for the two types?

10) What’s an example of relational understanding in your non-math life?

11) What’s an example of relational mathematics understanding for you? How do you know?

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Reflection: Pick 2 questions you would like to talk about in class, and write a thoughtful response to each.

• Look over your notes/highlights/work from reading.  What did you take from it?
• In your own words describe the ideas of instrumental and relational understanding.

The other thing that we've been developing in class is the idea of questioning, both as a teacher, and the benefits of student to student questions.  This class struggles with being quiet, but by the fourth group, they've hit full class discussion mode.  I really think that conversation is the only way to work towards understanding of big ideas. I filmed the first two groups and then handed off the iPod for recording.

In their groups I asked them to share their reflection from the workshop, they discussed a bit, and then to decide on two points to present to the class. Mostly they used the questions to frame their points. We talked a bit about presentation zen, and I asked them to make a 'slide' for their two points on the board, with the idea to not have a lot of text, but to support their idea with a succinct statement or even better, a visual. They did an excellent job, and I hope you enjoy sharing in their discussion.

Group 1 focused on questions 5 and 8.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


8) Would you add any advantages to the list for relational mathematics?


Group 2 

2) What is your favorite example of “rule without reason”? Why?


 5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


Group 3

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?


11) What’s an example of relational mathematics understanding for you? How do you know?


Group 4

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Group 5

1) What is the point of starting off with the Faux Amis story?

10) What’s an example of relational understanding in your non-math life?
  (Nice because it was ambiguous whether their example was relational or instrumental.)

As I listened to their discussion, I was struck by how many of the concerns of inservice teachers they already have, which is a real testament to the idea from The Teaching Gap that teaching is a cultural activity.  If we do not do something to resolve the tension between what teachers feel they are expected to do (the job) and what they want to and should do (the vocation), we're not going to make any progress.  It's almost what Skemp is talking about with the faux amis about our two ideas of learning. The same two ideas are competing and confusing us when we use the word teaching.

Glyphs to Data to Display

Me.

Story of a 2 day lesson.

In the preassessment for my 6 week math for elementary class, it turns out that the students were pretty strong on the basics of statistics measurements and displays.  So instead of spending a lot of time telling them what they know, we went right to data collection.  One of my favorite ways to collect data is a glyph.  The first teacher I saw use one was Char Beckmann, but I think there's an old Teaching Children Mathematics (er, Arithmetic Teacher) article about them.  [Found it:  Cartland, Patricia E., What's in a Glyph?, Feb 1996, 324-28] Definitely one of my main faults as a teacher is trying to get too much out of a lesson, so beware that here.

Objectives:  TLW

• collect statistical information
• formulate questions
• organize and analyze data
• display data to address a question
• consider what features of displays are effective
• consider types of questions teachers ask
• review how the next day's test will be evaluated

Day 1: (40 min)
Schema Activation:  what would you be interested in knowing about your classmates?
They brought up music, sports, food, family background, and the like.  (I think of these as cultural identifiers.)

Focus:  introduce the idea of glyphs.  The handout has this information on it:

Glyphs

The circle on the other side of this page will be your face – but not the face you see in the mirror!  A face that tells much more about you…

Hair:  Put a hair on your head for each person living in your residence.  Curly if they are 18 or younger, straight if they are 19 or older.

Eyes: I purposely leave left and right ambiguous (like a mirror or a mask) because I want there to be issues in data collection, but

• left eye - favorite pet :   circle-dog, oval-cat, triangle-bird, spiral-fish, X-exotic or other, crescent    don’t like pets       
• right eye - favorite TV show:  rectangle-reality/game show, square-police/mystery, trapezoid-doctor/medical, rhombus-historical/documentary, kite-sports, Y-other (should be a chevron), crescent-don’t watch TV

Nose:  Make a shape with as many sides as books you read for fun last month.

Mouth  :If from outside the state, add a tooth for each year you’ve lived in Michigan.

• Smooth line: from Grand Rapids area
• Crooked line: from Michigan but not GR
• Rectangle: from another state in US
• Circle: from another country

1)    What other characteristics of a face could we use to ‘store’ information?

2)    What are some other categories of information that would be good or interesting to represent on a glyph?

Activity:

As a whole class they added categories for music (country, hip hop, chill, rock, classical, show tunes) as the left ear, food (italian/pizza, asian (incl. sushi), mexican, chocolate) as the right ear, and hobbies (sports, outdoors, shopping, games, etc.) as the eyebrows and came up with symbols -usually very pictographically, for each.  (It's okay for me that my starting questions are a bit boring,  as they ask about what they care about.  I don't have to do it for them.)  They completed their glyphs.

Reflection:  look over the glyphs of the whole class and share their reactions.
They remarked on how much they enjoyed making them, their interest in each others, and how cool it was to see them together.

Day 2: (2 hours)
Schema Activation: polish up your glyph (or make one if you were absent), move it to the back table, see what you notice about them as a group.  (Most people from Michigan, lots of pizza, music types remarked on, questions about living situation...)

Focus:
First we reviewed the types of elementary displays and covered any questions.  They asked about pictographs and boxplots.

Teaching note:  I like talking about questioning with statistics anyway, but they were really interested in what Jo Boaler brought up about teachers' questions at the last book club, so the timing was perfect.

Activity:
Pick a column.  Pick questions in that column so their numbers add up to four or more.  Answer the questions. (Display) means that a display is required.  Make a poster of your answer that includes your justification.

They collected data and dealt with interpretation issues, what to do with people who gave more than one answer, left/right, figured out pretty efficient ways to record their data, and started discussing display type.

After the posters were mostly complete they were passed around, and each poster was evaluated by each other group using our communication rubric: 0, 1/2, or 1 each for clear, coherent, complete, consolidated and content.  (Created with Coffey so the consonants all alliterate appropriately.)  This is how their exams will be graded the next day, also.

We came together as a group and discussed what makes for effective displays.  This is what they thought.

We also discussed the question types and made connections with reading.

• Literal.    Literal questions have answers that are found directly in the text or are answered by factual recall.  Examples:  How many people come from Michigan?  What was the main character's brother's name?
• Application. Application questions require computation or processing to determine the answer from the information at hand.  This usually is considered more mechanical than involving conceptual reasoning.  Example:  What's the average number of books read this month?  How long was Bilbo's journey?  
• Inferential.    Inferential questions require the answerer to create something unique, something that is implied by the information at hand.  Sometimes this is by combining prior knowledge and experience with literal information.  Inference may require students to imply, guess with support, or deduce.  Can be forward-looking.  Example:  How far would the average drive be from where people are from to Grand Rapids?  Why did the character do that?
• Analysis.    Analytical questions may require synthesis of literal information with information from other sources.  They typically require justification.  Sometimes analysis is referred to as synthesis.  It revolves around examining the information in and from the problem and solution.  Reflective in nature.  Example: Is there a correlation between favortie food and favorite music?  Why did the character do that?

Reflection:  Pick two of the following to address as a group.  Turn in your group response.

• What issues came up in data collection?
• How did you go from glyph to data?
• What was a useful form for recording data?
• How did you decide what graph to use?
• Any decisions you would make differently if doing it again?
• What do you see that was effective in other groups work?

Typically people thought about the teaching implications of what we did, but also thought a bit about the effective display idea.

Bonus:  better classroom decorations.