Category Archives: Questioning

I Still Have a Question About…

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We did not get through all I intended today to allow some students who wanted to watch the inauguration that opportunity.  But we did address a couple of more questions from the 2-minute reflection students had completed.  You can look back to the previous post to see the original task.

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We addressed the two blue questions in the after lunch class.  Why can’t you multiply the numbers by each other?  Well, lets see.  Again, as I did with another class, I asked them to add two numbers that would give us 18.  We graphed our responses, then graphed the equations x+y=18.  And likewise, give me two numbers that would multiply to give us 18.  We graphed our responses along with xy=18.

When we added the equation to the product set, students were caught off guard with what they saw.  WHY is there a graph in the third quadrant?  Will that red curve ever cross the y-axis?  Doesn’t it eventually get to the axis?  Again, just attempting to address their question, by looking at a couple of horizontal translations and introducing them to that boundary line called an asymptote, led to even more wonderings.  Which is what math class should be about.  As long as they were on task, I continued to go with their questions.  Only 3 students were not interested, who would likely have been off task no matter what I chose to do, so I made the decision to keep going with the majority’s curiosities.

Another student asked about our statement “x cannot be in the denominator” but yet when we find rate of change with a table of values, we compare y over x. Hmmmm.  Good question.  So I gave a table of values, asked the student to talk me through finding rate of change.  When we wrote our ratios, what values did we use?  Not the actual y and x values, but the change in y over the change in x.

The entire class really reminded me that we can say something with an intent, but what our students hear is something else…how important communication is, how important is it that we allow some time to process and clarify their misunderstandings.

Finally, we addressed the question, we’ve been told x’s exponent must be 1 in the linear function.  We’ve seen greater than 1, but if it is less than 1, can it still be linear?  Let’s see.  Go to y=, type in x and choose an exponent less than 1.  What do you see.  Share with your neighbors.  So, how would you respond to this question, students?

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To me, this was one of the most productive two days I’ve had in this class.  Students were engaged because we were addressing their questions.  I’m not sure I actually answered their questions, but I provided them with some examples that allowed them to answer their own questions.

Better Questions Week 3 #MTBoS

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betterquestions

I’ve pondered this challenge for a couple of days.  So many options!  But a tweet from @mathymeg07 led me to a post from @MrAKHaines blog Math Pun Pending.

The post was celebrating a variety of strategies his students had use to answer the question:

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He wrote:  When I wrote the question, I had anticipated that students would use a couple of different strategies. What I didn’t know was that my 25 students would use a combined seven correct solution strategies to solve this problem.

Two parts to my post:  1. How can I make this an open question and 2. How can I use student samples to develop a better lesson in the future?

How can I make this an open question?

A.  Name a point that is NOT on this line.

B.  Name a point that this line passes through.

Thanks to @PIspeak‘s TMC14 session in Jenks, I urge students to “Support your claim with evidence/reasoning.  I want to see your thinking!”

How can I use student samples to develop a better lesson in the future for my classroom?

I appreciated the fact that he never explicitly taught “the teacher’s efficient strategy” but allowed group discussions and support to drive the lesson.  Students shared ideas.  The last paragraph  in his post says, “My students are acting like mathematicians, y’all. They’re using their toolkit of math ideas to solve problems flexibly. I couldn’t be happier.”

In the end, that’s what we all want – students thinking on their own, making sense and being confident enough to explore a problem with their own ideas.  So, how does this tie in with the Better Questions prompt?  My outline of the lesson feels a bit like those I’ve used from Formative Assessment Lessons, but I feel it lends itself to students doing the thinking, talking – I only provide the materials and support to make desired connections that will lead to the learning goal.

I’ve been following the #T3Learns chat from Wiliam’s book.  In chapter 3 of Embedding Formative Assessment, it suggests using student sample work. How might I structure a lesson, utilizing student samples of this question?  In Principles to Actions, MTP3 states Effective teaching engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

  1. Begin with the same question. Allow students to write a response. (3-5 min)
  2. Place students into small groups and allow them to share their approaches. (5-7)
  3. Allow groups to explore student samples, making note of different ideas, what they like/ways to improve, questions they’d like to ask the student. *maybe as a gallery walk? (15-20)
  4. Discuss their findings as a whole group. (10 in)
  5. Connections to/between the different mathematical representations. (5-10 min)
  6. Written reflection:  (3-5 min. possibly use as a start up / bell ringer to begin class with following day – providing an opportunity for retrieval of previous days information?)
    • my strategy was most like:____
    • the strategy I liked most was ____ because ___
    • the strategy I found most difficult to understand was ___ because ___
    • Which approach was most efficient?  Why?
    • What do you think was the BIG IDEA your teacher intended for you to learn/understand?
  7. Transfer…provide a few, different contextual problems that allow students to connect the mathematics to something tangible, maybe in a problem posing situation (should this be small group?  individual? ) (5-15, would this be better as follow-up the following day?)

Timing is often an issue for me.  I want to provide students with enough time to make sense/discuss, but not so much time it feels long and drawn out.  Are the times I have listed appropriate?

Please offer suggestions.  How have you used a similar approach successfully in your own classroom?

Students Making Sense of Quadratics

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I realize some folks will bash me for sharing this from an Algebra 2 class, but based on benchmarks, most of my students have major gaps in quadratics. 

I began with reviewing multiplying 2 binomials on our whiteboards.  I shared the box/area model and several smiles celebrated because they “saw it” and were doing it correctly!

Last week, I pulled out a box of Algebra Tiles.  We literally explored building squares.  I wish I had taken pictures because some of their squares were like a grandmother’s beautiful quilt blocks.  I began tying it back to our box/area models -I’d rather think of it as leading (not forcing) their thinking – but they were quickly picking up the patterns. 

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We then began looking at the algebraic equivalents, again, with a sketch along side allowing them to “see” the process.

Our next step was to find the missing value without tiles/picture models…and then I asked them to review their multuplying with 5 expressions alongside.

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“What? You think they’re the same thing?!?” I asked,  “Prove it to me. Well, by-golly-jee. You are on to something!”

The following day in class, I made a HUGE ordeal of different ways to write zero.

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I explained our next few minutes were a process. But we talked about it, step by step, completing the square, adding ‘that zero’ in our expression, the separating the trinomial and 2 constants.  Rewriting our trinomial as a binomial squared.

Ok. Why in the world would anyone want to do this?  I told them we were finding hidden information.

As they arrived at this form (x+4)^2 – 9, I paused, reminding them to think back on our function transformations before Christmas break.  How would this function y= (x+4)^2 – 9 move on our graph from this one y=x^2?  Quiet. “Move left 4 and down 9!” Someone exclaimed.  Really? Are you sure? We graphed the two and yes, it did just that.  So what does this tell me about my parabola?  They didn’t say vertex. Or minimum.  They said it shows us how the graph was transformed. 

I will take that.

I then asked them to move left 4 and down 9 from the origin.  What have you found? The lowest point.  The vertex. The minimum. All their responses, not my statements.

We set our expression equal to zero and solved the equation, using our inverse operations.  They made the connections with the x-value of the vertex being the “center line” of the parabola.  They realized the +- 5 were steps in either direction from the center line.

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I most appreciated the questions they asked on #3, 4 and 7.  Several chose #7 thinking it was shorter, thus less work. Snafoo. No middle term. What happens? 

I suggested they look at it from a transformations point of view.  Someone shared-It doesn’t slide left or right, only down.  Another student said-well, that’s the easiest equation to solve! (Yep.)

Why did #4 bother some? The middle term had an odd coefficient.  But once they shared their thinking, ok. Got that one too!

#3 was what we math folks recognize as perfect square trinomial.  But for the students, it was an a-ha.  Again, using the transformations context, we moved right 5, but not up or down.

L: But I thought all quadratics intersected x-axis twice?  I asked – did this one? No.

What about y=x^2 + 3?  It moves up 3. Ok. How many times did it intersect the x-axis? It doesn’t.   Hmmm.

A student who is rarely engaged then asked, if you can make a parabola that doesn’t intersect the x-axis, can you find one that doesn’t intersect either axis?  Me: Can we? What would it look like? S: Noooo. As its going up, increasing, it would be increasing outward, too!  More discussion, between them. Me not included. I was smiling.

And their questions were what drove our lesson today.   And I was so excited, telling them their questions make me think! And when they’re asking questions, their brains are processing the information – making it their own.

It was a good day.

SMART goals #MTBoS30 Day 4

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So, in Kentucky we are piloting PGES – Professional Growth and Effectiveness System.  Component are based on the Danielson Framework for Teaching.  I, personally believe it is an opportunity for major impact on student learning…

Back in the fall these are 2 goals I wrote for my classroom…  Wondering what feedback anyone could offer (please and thank you).  Do these follow the SMART goal format? What adjustments do I need to make?

I completely understand these should have been revised earlier in the year but it’s been a learning process for me. 

Goal 1:  Student Growth

Student Growth

My goal is for all Algebra 2 Learners and will be measured using Discovery Education Benchmarking Assessments.

My goal is for all students to improve their overall score by 15%  on the next Discovery Education Benchmark assessment.  If this is accomplished, 90% of my students would move up at least one proficiency level.  This is an achievable goal considering an average of 8% will move the class averages up one proficiency level.

Looking over commonly missed questions, I found there were gaps in the areas of Quantity and Functions for my students.  I plan to address these by:
• Using resources like Estimation180, 101 Questions, Visual Patterns as bell ringers for class discussions to build learner confidence and numeracy reasoning.
• Our most current unit is an Overview of Functions – through interactive, engaging instructional activities, learners will have an opportunity to talk about and discuss things they notice about different types of functions.  This will allow for conceptual development of learning targets.  Intentional formative assessments will allow me to adjust my plans daily. 

Goal 2: Professional Growth…

The Classroom Environment

I want to provide more meaningful problem solving opportunities for students to engage in discussion with their peers through activities that highlight and allow for students actively use the 8 Standards of Mathematical Practices.

3 goals for my administration/ peer observations:
Better questioning to draw out student ideas / strategies;
Provide quality tasks and structure the class time in a way that allows ALL students to be engaged in learning and discussion.
Develop a better culture of listening by lessening the amount of times I repeat what a student says – encouraging students to listen closely as their peers are talking.

I will read Powerful Problem Solving by Max Ray and participate in an online chat – then implement strategies discussed, reflecting, adjusting and sharing either through chat or blogging.
I will use suggestions from 5 Practices for Orchestrating Productive Mathematical Discussions by Smith & Stein to plan learning sequences that will impact student engagement and learning. 

No jumping in, silent/listening, no repeating & my win for the day #ppschat

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Our #ppschats the past few weeks have brought some a-has and good reminders for me.  Here are a few adjustments I am trying to mindful of:

1. When students are working in small groups, I have often jumped in to their conversation when I heard them going in the wrong direction.  Of course, my intentions may have been to redirect them. 
 
Needed adjustment:  Be silent and listen.  Giving them space to muddle through their own thinking without jumping in and telling them what I think they should think.  A key for me may be to keep my tablet, clipboard, post-its to jot down notes of their conversation.  Key points to reference back to later.  Jot down questions I might like to ask.  Not sure when to ask /share, maybe as a quick revisit before the end of class?

2. In my efforts to “value” what a student shares, I often find myself repeating. Afterall, those softspoken students need to be heard, so I repeat it so classmates across the room “hear” them.  Oh, no.

Needed adjustment: Ask student to speak up so others can hear.  If they are intimidated, offer an encouraging word, let them know you like it, find it interesting or you want others to hear it.  When I repeat, I am causing others to not listen because they know I’m going to repeat.  Oh, my.  Guilty.  Who knew?  What are ways you create a listening community of students? 

3.  I may ask for volunteers and the same 8 people are sharing.  Spread the love.

Needed adjustment:  I tried to be very purposeful in sharing this week.  In geometry, I picked a problem several seemed to have trouble with.  I structured the task with Know: what information is given, Notice: what do I notice about the diagram? Other information I can use to move me further?  Wonder: What other measures can I determine? How can I justify my reasoning? 

Students took couple of minutes individually, to jot down a couple of things in each bullet, then in their groups of 3 to discuss.  I asked each group to pick 1 thing they felt was important to share.  Yep.  Good ol’ Think-Pair-Share.  As I went around to groups, I arrived at one who said…they already shared ours, so I used the suggestion from PPS to +1 on the board.  It seemed that others were really listening to what was being said.

A win in class today as I gave students a diagram with no questions.  They noticed/wondered and it was a statement from a student that 2 chords were congruent.  When I asked them to convinece me…their statement was quite fuzzy ending with “it just seems like they would be.” I challenged the others to prove or dispute the statement… 

“Oh yea” high fives, “we got it!” & “you’re genius!”  Students celebrating something they hadn’t seen before. 

What was even better, the way they justified their reasoning…all different, not one that I had seen myself.  And that is why I don’t care for the answer key as the answer key. Students sharing their strategy and each confirming the others.  Hearing them say ‘that’s cool’ to another student’s strategy.  Engaged while looking at a different approach.  I truly feel the take aways from a single problem approached this way is valuable.  It was a productive day.

Student Reflection on HW

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When I get back from a conference, I have the best intentions of sharing, but its nearly 3 weeks later and I am just starting to get caught up…only to realize there are less than 3 weeks of instructional time before Christmas break. 

Starting to stress in my Geometry blocks classes…similarity (although I tied in some with our congruence unit and they used dilations in our transformations unit…) right triangles and circles…then a super dooper quick approach to modeling via 3-d problems.  Anyone have an amazing project that ties circles and right triangles together?  Anyway, a bit off topic, because the stress causes me not to focus.

  I attended a session led by @ottensam sharing different approaches to ensure we are integrating the SMPs in our instruction.  He was very engaging and shared some simple, research-based strategies.

A great idea he shared was to change up the way we approach homework.  One simple suggestion was to ask students to eflect on the problems…which were most alike? Most different?  Why? Which one did you think was easiest? Most difficult, why?  I had students to do a quick write using this idea this past week.  Once they were finished, they had to meet with someone they did not sit next to and share their responses.  Finally, I called on students, asking them to share -not what they had written- but something they had heard. 

I am always amazed at student responses when I use startegies similar to this and could kick myself for not being more intentional, more often.  Several shared exact similar/different pairings but for totally different reasons.  I love it, being able to see and hear their ideas and thinking. 

Happy Birthday #Made4Math !!! Formative Assessmemt Reminder Cards

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First, just let me say a big THANK YOU to @druinok for beginning #made4math and to all of the generous folks who have openingly shared their classroom ideas, lessons, tips over the past year.  I was overwhelmed with how quickly it took off!  Still, today, I am amazed at the generosity of this community.  I have learned so much and my classroom was definitelh impacted by your awesome ideas!

My share for today was initially a result of a convo with @rachelrosales and @druinok, brainstorming ways to organize reminders for the numerous formative assessment techniques…something simple, at your finger tips. 

I loved @druinok’s post today and her Student Engagement Flipchart.  Very.Nice.  It will definitely be on my to-do list for a future project.  However, I am choosing to share a similar idea, just a bit different format.  I cut down index cards to fit sports card pages… pack of 10 for $1.  I am able to display up to 90 of these reminders ranging from formative assessment techniques to various strategies for student engagement, reflection, etc. 

Front side of card has title, with some information…

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Back side of cards has description, suggestions, reminders…

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I have placed the pages in a small 3 ring binder which can easily hold more pages.  Currently, I am trying to include summaries/reminders of techniques I have used or see being easily modified for math class.

Looking forward to learning and sharing more FA techniques with my amazing PLN!!!

Pam Wilson, NBCT
Currently Reading
5 Practices for Orchestrating Productive Mathematical Discussions, Smith & Stein
Teach Like  a Pirate, Dave Burgess
From Ashes to Honor, Loree Lough