Fostering “Mathusiasm” with Jo Boaler’s Task “How Close to a 100?”


Good to have you back math nerds!  And if you’re just checking in for the first time, welcome!

I had the opportunity to spend a day working with 10 elementary instructional coaches and reflecting on our work this year in the math classroom.  At one point, a colleague called another colleague a “rat bastard!”

Now that’s a successful PD, am I right?

Let me back up a step.  I’m a strong believer that all professional development workshops for math educators need to foster their “mathusiasm” (math + enthusiasm).  We all need opportunities to do math, to explore and play with content in an effort to develop a fuller, deeper, more flexible understanding of concepts.  We must be enthusiastic as we think more deeply about simple ideas and practice engaging in thoughtful mathematical discourse.  If we want to foster math nerds in our classrooms, we must practice being them ourselves outside of the classroom.

So when I came across Jo Boaler’s wonderful game “How close to 100?” in her book Mathematical Mindsets, I was very eager to try it out with this group of coaches to see what “mathusiasm” could be generated.  What transpired was one of the mathusiastic dialogues I’ve had in a PD, and I’d like to tell you about it.


You can read the rules at the link above.  Here’s a brief explanation using language from Boaler’s site:

  • This game is played in partners. Two players share a blank 100 grid.
  • The first partner rolls two number dice.
  • The numbers that come up are the numbers the player uses to make an array on the 100 grid.
  • They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible.
  • After the player draws the array on the grid, she writes in the number sentence that describes the grid.
  • The second player then rolls the dice, draws the number grid and records their number sentence.
  • The game ends when both players have rolled the dice and cannot put any more arrays on the grid.
  • How close to 100 can you get?

We played this version of the game once before playing a variation (also on Boaler’s site) where each player has their own blank 100 grid.   This creates the opportunity for the players to choose to be collaborative (help each other) or competitive (beat each other).

It was during this moment that a player called her opponent a “rat bastard!” for rolling the perfect roll to win the game.  I chalk that up to evidence of vested engagement and mathusism!  After playing several rounds, the coaches started to think of variations that would add nuances to both the game play and the mathematical thinking.

Here are some things that I heard/recorded while they played:

  • Laughter.
  • What makes a roll good?  bad?
  • I’m wondering if there’s a time limit.
  • How do we decide where to put shapes?
  • Probability…
  • Twenty four!  Sh*t!
  • Are bigger numbers better at the beginning?
  • Where should I put squares?  Where’s the worst place to put them?
  • How that we see our list of number sentences, how could we have done it better?

Profanity aside, those are some things we want to hear our students say and ask in the classroom.  They are evidence of having a mathematical mindset and provide a wealth of opportunities for teachers to structure the flow and sequence of discussion.

Here were some key takeaways that came out during our reflection conversation about the educational value of playing “How close to 100?”

  • The game promotes social interactions and gets kids engaged with each other.
  • It has a wonderful balance between giving students freedom of choice (where to put the array) and freedom from choice (have to put it somewhere).
  • The game incorporated a layer of strategy and anticipation that lays an excellent foundation for talking about probability.
  • The strongest math student doesn’t always win.
  • All students can feel successful and develop math confidence.
  • It’s a puzzle!  Everyone loves puzzles!
  • It reinforces a spatial understanding of multiplication.
  • It is fertile ground for using peer-to-peer academic language.

And here were some variations and wonderings that they had about the game moving forward:

  • Allow students to be able to have some passes for rolls.
  • Modify the rules for ending.  When is the game over?
  • How could we do a collaborative whole class version?
  • What would a 3-D version look like with three dice?
  • You could roll several times (like 3 times) and decide where you want to put them all.
  • You could choose if you want to put it on your grid or your opponents grid.  And where to put it.
  • We play until we’re bored.  When bored, we change the rules.
  • How could we rework the number sentences to incorporate the distributive property.  For example, 6(4) = 6(3 + 1) = 6(3) + 6(1).  Could we allow students to then make their 6 x 4 array a 6 x 3 and a 6 x 1 array and allow them to graph them separately.

Thanks for checking in math nerds.  I invite you to experience the game in your PDs or classrooms and share any mathusiams with us here in the comments section.  Please be in touch.

Teaching Geometry with Ms. Pac-Man

I support an 8th grade math teacher in Inglewood, CA.  Fabian is a first-year teacher with a lot of professional responsibilities on his plate (8th grade science classes, an environmental science class, advisory, and 8th grade math classes).

He is also a former student of mine.  I taught him AP Calculus about 10 years ago.  It’s been fun closing the circle (pulling back the curtain? bridging the gap?) between student and instructor.

About a month ago we were discussing the 8th grade geometry standards (transformations) and his teaching for the next unit.  We looked at how MathLinks 8 sequences the geometry concepts for students and guides instruction for teachers.  (Disclosure:  I’m an author of the program.)  While the MathLinks curriculum is great for conceptual development, it still felt too formal as a starting point for his students.  We wanted to invite students into a conversation that would require them to use math.  We didn’t want to start the unit with a conversation about math explicitly because we knew some students would be turned off.

So we turned to Robert Kaplinsky inviting, “low floor” lesson about the movements of Ms. Pac-Man.  I used to work for a non-profit that used a free software program called GameMaker to teach students mathematical concepts through video game design.  (I’m not sure where the project is at now, but it was called MathMaker and you can find more about it here.)  Kaplinsky’s lesson is an example of the main philosophical thrusts of MathMaker:  students can learn math not just by playing video games, but also by talking about video game design and making their own games using mathematical reasoning and simple drag-and-drop computer code.  (I miss working on the project and have more to say about it’s value; geek out with me about it sometime in person!)

The goal of our intro-to-the-unit lesson was to get students engaged quickly and talking to each other about describing motion.  We followed Robert’s write-up pretty closely.  We played the first video a few times and chose the prompt:  “Describe Ms. Pac-Man’s movements.”  We let them talk in groups first and then started making a class list.  To keep engagement and honor participation in thinking, we repeatedly used the statement: “Raise your hand if your group said a similar thing.  High five each other!”  To attend to precision, we followed up slightly imprecise statements with “Did anyone say the same thing in a different way?”  We did not introduce “translate,” “reflect,” and “rotate” language.  But we did unpack what they meant by move, flip, and spin (and the like).  The definitions of their informal language ended up being recorded and we attached the formal vocabulary to their informal definitions in the next lesson.  We did our best (but may have missed that mark wth some students) at only formalizing the math after conceptual understanding had been established.

Robert’s sequence of videos accurately predicted our student misconceptions.  (The initial reflection was the hardest for students to see.)  The videos allow students to see their mistake without feeling they are wrong or “bad at math.”  The follow up questions are great.  “How far did she move?  Which way did she turn?  What do you mean ‘flipped’?” were some questions we created too.  If you use this lesson, I encourage teachers to go slow in the discussion and to take the time to unpack student thinking.

We concluded the exploring part of the lesson after we had flushed out all of their language about Ms. Pac-Man’s motion.  Then we asked them what math they saw in their words.  After a brief discussion, they wrote their responses on exit slips.  We wanted to save Robert’s larger task for later when they had a stronger grasp of the language and math.

The rest of the lessons in the unit started with some sort of connection to video games whenever possible before launching into more formal conceptual and structural “patty-paper” type lessons for each rigid transformation.

Here’s a quick snap shot of some of the Warmups:

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Note the addition of numbers.  We were slowly adding in more structure.

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Then we put in more structure using coordinates.  (Not sure why Ms. Pac-Man is stretched in the image above.  Her distortion was not intentional.)

And we used more formal Warmups as we progressed:

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Eventually, Fabian created an assessment that was very similar to Robert’s task.  In addition, he had 4 more formal problems asking students to perform or analyze rigid transformations on object.

Here are some of his statements in our reflection conversation today:

  • Students of all levels got on board with the opening lesson.  It’s such an easy thing to get kids talking about.
  • Ms. Pac-Man makes the math so conceptually obvious.  She needs to look where she’s going.  Orientation matters!
  • There was much less student push back when the test came.  They all wanted to do it and every student completed the test.  That was the first time this year that every student made a legitimate attempt.
  • Only stronger students could use abstract notation and coordinates to describe motion in the coordinate plane.
  • All students have room to improve their precision.

Some things we’re pondering moving forward:

  • Most students still struggle to describe reflections mathematically.  All students did well describing rotations and translations.  We wondered how to leverage the Ms. Pac-Man task to do more reflections.  Could we get a clip of here moving turning around vertically when she moved horizontally and horizontally when she moved vertically?  We wondered about making one for next year.
  • Also, how do we get her to reflect across an axis?
  • How do we get students to attend to precision by checking to see if their answers are right?
  • How do we get them to keep using the tools (namely, patty paper) when their use of mathematical structure isn’t accurate?
  • How do we get them to embrace the notation more?
  • A sequel idea was to have students pick a ghost in the image and describe how that ghost could get Ms. Pac-Man if she wasn’t moving.  This would allow them to practice describing transformations in a more “open middle” way because there are several answers that yield similar results.  Letting students pick their ghost also invites them to choose their own level of complexity.  Some ghosts are a lot further than others.



Thanks Robert for making such a great resource!  Kudos to Fabian for finding ways to make math more inviting and to keep all of his students engaged in math as a first-year teacher!

Teaching is an amazing journey.

The Freakonomics of Professional Development: How Do We Put the “D” Back in “PD”?

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Yes, Allen, I am talking about practice.

On my drive back from NorCal to Venice, I listened to some Freakonomics podcasts.  My head exploded.  I’d like to tell you about it.

They’ve been doing a series on the research about self-improvement, productivity, growth mindsets, and grit.  In a sense, what does it take to expand and maximize your potential talent?  Since I spend a bulk of time in the professional development field, I’m always interested in the science and research about adult learning, especially as it relates to perfecting a skill or talent.  How can we maximize our talents as math educators, speakers, facilitators, coaches, and support providers?

Teaching (or more specifically:  the act of instructing) is one of the most cognitively demanding tasks to master.  The sheer amount of meaningful data a teacher needs to absorb, sort, and evaluate in an hour of instruction is staggering.  I would love to see a chart about careers and the number of meaningful, consequential decisions a professional needs to make in a day.  How high on the chart would teaching be?

As a result, teachers need a significant amount of professional coaching, collaborating, and calibrating in order to improve their craftsmanship at the art of instructing.  Likewise, teachers must continue to have a growth mindset and gritty view of their practice if this professional support is going to lead actual, measurable, and enduring professional growth.

There are obviously many obstacles to navigate on this battlefield (school boards, unions, funding cuts, local/regional/state/federal politics, crappy textbooks and a crappy adoption process, standardized tests, shortsighted leadership decisions, poverty, unequal access and equity to good math instruction…).  I don’t intend to address them here in this post, and I’m intentionally setting them aside.

I’m just asking:  How do we best maximize the professional development for teachers (and our own development as PD providers)?

What I would like to share are a few excerpts from the podcasts and my take on how this relates to creating successful professional development opportunities.  My analysis is at the bottom.  As per usual, it’s a rambling journey until then.

I want to start with Nigel Richards.  Maybe you saw this pic floating around Twittersphere:

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What Nigel did was very impressive.  I don’t mean to take anything away from him or come across as diminishing his accomplishment.  I am suggesting that what he has accomplished is a tremendously difficult DOK1 task.  The task (memorize the French dictionary) was not complex, but it was very difficult.  (Check out Robert Kaplinsky’s own thoughts on DOK and complexity here.)

Many math teachers commented on Nigel’s performance by suggesting that we have a national history of teaching math with the same mindset to our students.  We don’t need to teach kids the art and utility of the French language; we just need to get them to memorize a bunch of French words without attention to context or meaning.  We could get students to “win” against a DOK1 standardized test, but they couldn’t speak the language or use it in any meaningful way.  Here in California, the old standardized test was so simple and so low on the DOK scale, teachers could (sort of) successfully game the system and get mathematically illiterate students to show a “false positive” on the California test.  (Check out David Foster’s work on the topic.)


This isn’t news to many people in the math education world.

What I’m suggesting is that much of the PD that math teachers have received for years has taken the same shallow approach to the skill development of teachers.  In other words, PD DOK has been at level 1, and we need to shift it to levels 3 and 4.

So how do we create meaningful and purposeful DOK3/4 PD for math teachers that leads to growth in how we teach math?  How do we help teachers create new meaning and new understanding about the art and craft of teaching math students?

That’s where the Freakonomics podcasts come in.  You can find the full transcripts here, here, and here.

In an interview with professor/author/researcher Anders Ericsson, Stephen Dubner (host) dives into what the research says about how people become exceptionally good at something.

Basically, it comes down to practice…purposeful, deliberate practice.  And a lot of it.

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Ericcson continues later:  “There are several components to deliberate practice, but generally, it’s about using good feedback to focus on specific techniques that will lead to real improvement.”

My key takeaways:

  • Deliberate practice requires focus on a goal.
  • It requires a teacher/coach/colleague to help collect and provide good feedback on progress to that goal.
  • Most importantly, deliberate practice requires maximum effort and being outside your comfort zone.
  • Sometimes deliberate practice is genuinely not enjoyable.
  • The last comment by Ericcson sounds like the backbone of all good professional development whether it is lesson inquiry, co-teaching, videotaping, etc.

Every day, we ask our students to commit to deliberate practice.  How do we create professional spaces for the adults that work in schools to do the same thing?  Because it’s really important that we include all adults, at all levels, in all roles, regardless of experience in the culture of deliberate practice.  Just because we’ve been doing something for years, doesn’t mean that we can excuse ourselves from extending our comfort zones.

I hope my doctor agrees with me because this research shocked me:

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My key takeaways:

  • How do we know if we’re making mistakes?  How do we know if we’re not?  How do we know what types of refinements we need to make?
  • Improvement only comes from processing feedback and data through reflection and the help of others.
  • Experience is not the same as practice.
  • Once a level of automaticity is reached, skills can actually decrease.
  • Teachers who say “Observing others or being observed is a waste of my instructional time; there’s nothing new for me to learn” are correct, sadly.  They have nothing new to learn because…
  • …deliberate practice also requires a growth mindset.


In what ways are teacher “evaluation tools” like the old DOK1 standardized tests?  How can we create more meaningful and more formative opportunities for teachers at DOK3/4?  Because if we’re setting the bar at some walkthrough checklist once a year, it’s very easy for a teacher to “pass the test” but never come closer to maximizing their talent.

Let me be very clear.  I’m not suggesting we need longer or more frequent checklists for teacher evaluations and walkthrough tools.

I’m suggesting that if we want to move math instruction, we need to shift the dialogue about teacher professional growth as something entirely different and separate from the machinery of teacher evaluation (and for what it’s worth, SBAC data).  We don’t need a more robust summative assessment for teacher effectiveness.  We need a 1000 more formative assessment opportunities rooted in the elements of deliberate practice and embedded in a cycle of reflection so teachers and leaders can calibrate and improve how to maximize their potential as educators.

At least that’s my interpretation of what Ericsson’s research suggests and what Malcolm Gladwell suggests below (in a different Freakonomics podcast):

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My key takeaway:

  • Deliberate practice without reflection is useless.

Without reflection, there is no professional development or growth; effort and practice will bear no meaningful fruit.  Reflection requires positive school culture, growth mindsets, and trust.  The depth and value of the reflection is limited by these factors.

Creating positive culture requires deliberate practice for school leaders; they need to create a culture of what Angela Duckworth calls “grit.”

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My key takeaways:

  • Nobody is born with a quota of “grit.” That’s fixed mindset thinking.
  • Grit is something that we can intentionally cultivate in ourselves and in others that we care about.  That’s growth mindset thinking.
  • Grit can be taught.

According to Duckworth, there are four key components to building grit:

  1. Interest and passion for something.
  2. Focus on deliberate practice.
  3. Purpose.
  4. Hope.

You can read all about it here.

My analysis:

If we want to create successful, gritty math students, we need to cultivate passion with interesting math tasks and compelling problems.  Deliberate practice gets tedious and boring; humans crave novelty.  To maintain and deepen student interest and passion, we need to leverage nuances in our classroom.  We need to put students in situations where they are encouraged to find another level/dimension/application to their thinking.  In other words, we need compelling and interesting sequels as Michael Fenton suggests in this great Ignite! talk.  Furthermore, the math work we ask students to do must have a purpose beyond “you need to know this for next year…”  The purpose of math assessments is to make math authentic and meaningful.  If the purpose of a math class is to “pass the test,” then most students, even the ones with the highest grades, will not develop a sense of grit and perseverance.  Lastly, students need to have hope that no matter where they are in their math journey, there are going to be problems and mistakes that they need to bounce back from.  Hope of future success is why we get up from the current failure.  Mistakes cannot erode hope.  Mistakes should build optimism, not pessimism.

If we want to create successful, gritty math teachers and school leaders, we need to cultivate passion with interesting and worthwhile professional tasks like lesson inquiry, co-teaching, coaching cycles, filming, observation and reflection etc.  Deliberate practice gets tedious and boring; teachers and leaders crave novelty even if they say they don’t.  It’s nice to be on auto-pilot because we’ve taught the lesson 40 times over the years, but that permits us to be bored.  And our boredom will be conveyed to our students.  It’s nice to have our PD plan “dialed in” after being a leader for a decade, but teachers will feel our boredom and flatness and will most likely imitate our lead.  We need to put ourselves in situations where we can find nuances in our work with others and our students.  (And this means venturing beyond the boundaries of our comfort.)  We must work together to create our own sequels.  The purpose of professional assessments should make our work more authentic and meaningful.  If the purpose of evaluation is to make sure we “get those boxes checked,” then we will stunt our own growth and never fulfill our potential.  We will lose our sense of purpose to our work.  Lastly, we need a culture of where professional struggles and instructional mistakes are reflected on and celebrated as evidence of learning and professional growth.

We all need to remain optimistic and hopeful in the face of our many daily failures.  We must cheerfully engage in our own productive struggles because students deserve inspiring instruction from passionate teachers.  Similarly, teachers deserve professional development that is rooted in deliberate practice and fosters grit and rewards growth mindsets because teaching math is really friggin’ hard.  How else are we going to get students to go beyond their comfort zone and take risks and celebrate mistakes if we don’t model it for them?

Thank you for listening to my thoughts and rambles.  I appreciate you.  Feel free to share your thoughts in the comments and help me further the dialogue.



The Power of a Mathematical Mindsets and Rethinking the #ClassroomClock

Hello Math Fans!  Welcome back.

I’m currently reading Jo Boaler’s new book Mathematical Mindsets and I’m looking for bookclub buddies, so I’d love to know if you’re reading it and how it’s supporting your thinking and work.  I invite you to share your thoughts in the comment section or reach me on Twitter (@mathgeek76).

If you’re not reading it, I encourage you to get on it.  It’s amazing.  Leave the blog now, and read more about the book and buy a copy by clicking here.

Seriously.  I’ll wait.

You back?  Good to see you again.

So I had the chance to work with 11 amazing elementary academic coaches last week for a one-day PD.  We delved into Boaler’s book, explored some new teaching strategies from the #MTBoS world, and then used Andrew Stadel’s #ClassroomClocks as a tool to think about how we spend our instructional time in the math classroom.

We learned a lot, had a load of fun, and I’d like to tell you about it.  Also, I think that there will be a lot of PD delivered to teachers and leaders about the powerful ideas and impactful research in Mathematical Mindsets and I am hoping that this post (and future blog posts) can add to the discourse about how best to maximize professional learning for teachers and support them with these instructional shifts.  Please feel free to use any of these resources if you think they are helpful.  Better yet, help me understand how to make them better!

There’s so much rich content in MM that I had difficulty choosing which reading to focus on and I also didn’t want to get too bogged down in unpacking the finer nuances of Carol Dweck’s elucidating model about growth and fixed mindsets. So I started with the following prompt:

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I had them find an eye contact partner from another table and spend some time talking about their response to their chosen challenge.  I did not have them share with the whole group; I just wanted them to have a chance to personalize their thinking to prepare them for this slide:

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I think there’s an assumption (misconception?) out there that a person is either one mindset or the other.  We are usually some combination of both and it often depends on the context.  A teacher or student could have a fixed mindset about their math ability but a growth mindset about learning foreign language.  What we need to focus on is monitoring our own mindsets in different contexts and developing growth mindsets for students and teachers in all areas of learning.

[Aside:  As another example, I confessed in the PD that I battle a very fixed mindset about learning to play the guitar.  I was born with clumsy fingers and a lack of physical rhythm and it’s easy to use that as the excuse instead of focusing on deliberately (and joyfully) improving my practice through commitment and focus.  Sigh.]

We delved into pages 4-9 in MM.  To support their learning and their thinking, I used the following strategy with the coaches.

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For those of you who just ordered the book online and don’t have your copy yet, here’s my choice for one of most important messages from the reading:


The coaches spent a solid 60 minutes on the reading and really chewing through the material.  I had some hesitations dedicating this much time when planning the PD, but in hindsight, it was time well spent.  I think it’s essential to support the learning of adults using similar instructional practices that support the learning for their students.  This means framing the content well and creating time and space for learners to explore their thinking with each other.  Good learning takes time and discourse.  If we want our teachers to embody this ethos, then we need to provide PD that embodies it too.

To help them organize and integrate their new learning, I invited them to go through this playful opportunity.


Here’s what they created and presented to the whole group:

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After a break, we explored two wonderful instructional strategies Which One Doesn’t Belong and Notice/Wonder.  These resources connected nicely to our work from earlier in the morning because they are both low-floor/high-ceiling and low-risk/high-reward learning opportunities for all students to experience meaningful brain growth.  Check ’em out.

The bulk of our afternoon work was framed around Andrew Stadel’s invitation during his NCTM Ignite! talk to rethink how we make the most of our instructional time.  In case you missed it, a super brief recap of his presentation and my takeaways:

  • Here’s an image of how NPR divided up an hour of programming:Screen Shot 2016-05-03 at 4.18.08 PM
  • Key takeaways:  It’s tight, efficient, and leads to great radio.
  • Here’s how Andrew’s evolved his thinking about how best to use instructional time:
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    Key Takeaways:  Like radio, classroom time needs to be tight, efficient, and lead to great learning.  Closure is essential; students must have time to reflect on learning and teachers need to capture data.  Instructing to the bell isn’t a productive habit.  If the primary purpose of math class is to do math and learn how to question and reason mathematically, then the primary focus of classtime should be carved out to let students do that.  A very effective way to do that is to offer an interesting and compelling task that is accessible to all learners and using student work to inform and guide instruction.  Whenever possible, evidence of student thinking should be used to structure discourse.

Before revealing Andrew’s thinking to the academic couaches, I shared the NPR slide with teachers and then gave them this prompt:

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And here’s a snapshot that three of them came up with:

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I then shared Andrew’s slides about his own clock(s) and we had an organic and inspiring conversation about the interesting differences and overlap between their thinking and Andrew’s.  Almost all the clocks talk about a warm-up, an application problem, some time for conceptual development, and some time for closure.

The coaches also incorporated some language around “fluency practice” and “concept development” into their lessons.  Their district has adopted Eureka for their curriculum and this language is used in their own lesson clocks.  Here is Eureka’s snapshot of classroom clocks for a whole module:

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Five things that I noticed and wondered about classroom clocks:

  1. Yikes!  Elementary students do math for a long time!  I had put two clocks on the page because I’d thought we’d do a “thinking before/thinking after” activity with the second clock.  But they obviously needed the second clock because their classes go more than 60 minutes and some as long as 120 minutes!  Five days a week!  That’s a lot of time for rich exploration.
  2. So, why are middle and high school math classes closer to 45 minutes?  Seriously?  Anyone know?  And why are we surprised when we don’t see enough thoughtful exploration/investigation/discourse in the secondary classroom?
  3. There’s a lot of ways to interpret and execute “concept development.”  It might be an appropriate label for the blue sections in Andrew’s last two clock images.  I think it’s a term that needs to be unpacked because it contains a certain blend and sequence of student discourse, problem solving/investigation, direct instruction as Andrew suggests.  What’s the best blend?  What’s the best sequence?  What resources/structures should we be using in PD’s with teachers to help them figure this out?
  4. I’m doing a lesson inquiry with some 2nd grade teachers next week using Eureka.  I’m eager to see more about the choices the authors make around “concept development” on a more granular level.  I’ll keep you posted.  But I’d be interested to hear from any Eureka users out there that may have something to contribute to the dialogue.
  5. I might like Eureka’s “Student Debrief” more than the standard “Closure.”  I may steal it.

So, my professional takeaway questions moving forward:

  • How best to incorporate the brain research and instructional strategies described in MM into productive PD for teachers?
  • How best to navigate the “concept development” chunk of our classroom clocks?  What ingredients (and in what order) make the best recipe for fostering growth mindsets?

If they are questions that resonate with you, let me know!

Stay groovy, math nerds.


#MTBoS Life: Ramblings and Musings from an Inspirational NCTM

I am long winded.  Most of my friends and colleagues know this.  And their true compassion shows as they listen as patiently and intently as they can and ask me questions to help my thinking.  Sometimes that question ends up as “So how does this connect to what we’re talking about?”  And I usually don’t know.  Perhaps my long-windedness is my way of talking out all the thoughts in my head, and talking it all out leads to me making connections and deeper meaning.

In that spirit, this post is me talking out loud.  There’s so much to think about after the NCTM conference this week and I don’t know how it all connects just yet.  So what follows is a stream of thoughts without much thought to coherence or order.  As always, I’m interested in your thoughts and insights and questions.

  1.  #MTBoS:  The best faculty lounge around.
    • I joined Twitter and became a member of the #MTBoS community in November at CMC South 2015.  (Thank you @Math-m-Addicts and @robertkaplinsky!) It has revolutionized not just the quality of my work with teachers but how I orient myself to my work in general.  I have these amazing colleagues in my pocket at all times available for support, guidance, inspiration, and coaching.  It’s an empowering feeling to know that I have access to all this collaborative potential from such talented and giving colleagues.  And I feel so proud to offer what I can to such an amazing group of collaborators.
    • While Twitter and #MTBoS have improved the quality of my work in so many ways, my interactions at NCTM reminded me that the building of human relationships is at the core of our work and a measure of our health as a professional community.  There is obviously tremendous potential power and efficacy in our ability to tweet back and forth about resources, strategies, ideas.  But the conference was a chance to meet the tweeters behind the tweets.  So at NCTM, I intentionally sought out to meet my #MTBoS collaborators in the 3-dimensional analog world of human to human contact.  And on that journey, I interacted with so many more people than I ever had at a conference before and my #MTBoS community grew and grew.
    • Every single one of those interactions was warm and inspiring.  Here are just a few that are sticking out right now.
    • I got to finally play with the wooden tessellations of @Tranglemancsd that I had heard so much about.  And during that play, got to meet and chat with him and others while also getting my geek on with shapes.  See for yourself:FullSizeRender 2
    • I met @MFAnnie and had the opportunity to thank her in person for her inspirationally pithy talk at an Ignite! event about the power of getting students to notice and wonder.  I’ve used that strategy hundreds of times in PD with teachers and have shown that video dozens of times over. And I’ve had the joy and pleasure of witnessing first-hand how notice/wonder engages the thinking and wonderings of thousands of students.
    • I got to meet @bstockus and thank him for his inspiring work about Numberless Word Problems that I’ve been able to use to empower teachers to make their textbooks more engaging and inquiry based.   Keep banging on your drum beat Brian!  We’re all better for it.
    • I met @gfletchy in the middle of one his talks (literally) at the NCTM Central booth in the Exhibit Hall.  He was inspiring me and other teachers about the power and joy of using estimation strategies as a way to inspire elementary students in one of his 3-Act Math lessons called Array-bow of Colors.  He recognized me in middle of his talk, stopped his presentation, came over and said hello and shook my hand, then seamlessly returned to his presentation.  His energy and passion is infectious.
    • There were so many more interactions than I could possibly write about.  Thank you to all who shared time with me to talk and build the human relationship behind the tweets.  You’ve filled me up with inspiration and wonder.
  2. #MTBoS:  The most open and inviting faculty lounge around.
    • I had the pleasure of volunteering at the #MTBoS booth.  Here was a common beginning to my interactions with with people walking by:
      • Them:  “Hey, what’s #MTBoS?”
      • Me:  “The best faculty lounge around.”
      • Them:  “How do I join?”
      • Me:  “You just did.  Welcome aboard.”
    • It was wonderful to extend the same open and warm invitation to others that I had received from the #MTBoS community.  And my reward was seeing folks feel happy and excited to be a part of a community of excellence and curiosity, to feel included in the struggle to find ways to engage students, to know that there were other collaborators just like them who are striving to find solutions to the challenges of math education…and that there was a place where that could happen freely and openly.
    • And that’s perhaps the amazing thing about #MTBoS:  It’s a digital world that builds the opportunities for us to strengthen the human relationships that will create the courage to continue our work.  Shared experiences like NCTM create the meaningful connections, not the tweets.  Tweets sustain the relationships between those experiences.
  3. Matt Larson’s Ignite! talk left me thinking about many things.  One thing I’m wondering about:  Our work is often held back or derailed (at worst) by the politics of the “math wars.”  Are there science wars?  English wars?  Do teachers of other disciplines face this challenge?
  4. Andrew Stadel has asked us to think about our #classroomclocks and thinking about how we divide up time in our classrooms.  I’m eager to use his clock template as an activity with teachers and coaches in future PDs to start a conversation about student engagement and the lesson planning process.
  5. Michael Fenton has me thinking about the value of sequels and leveraging students’ familiarity with “characters” in a math problem to bring new light to old understandings.
  6. Robert Kaplinsky moved many souls with his talk at ShadowCon16 about empowerment and sharing his personal stories.  He has me pondering power versus influence and the significance of context in determining which of the two is being wielded.  How do we foster and use our influence as teachers to create a classroom culture where students feel empowered and compelled to take ownership over their learning?
  7. I wear many hats in my work.  One of them is as an author for the Center for Mathematics and Teaching, a non-profit publisher of some amazing middle school math curriculum.  So many teachers came through our booth to talk about math, curriculum, and geek out on math education.  It was so rewarding to meet so many hard-working and inspired educators.  I’m grateful for CMAT for the opportunity to attend NCTM and experience the rich opportunities for professional development.
  8. Apparently I don’t look like my Twitter profile pic.  Huh.
  9. I heard this somewhere and scribbled it in my journal:  The best questions make students more curious about the answer.  Thank you somebody.

More to come.

When you reflect on your NCTM experience, what might be your most meaningful learning that you would like to incorporate moving forward?

The Powerful Phrase: “…in a way that makes sense to you.” (Part 2)

Goodness!  Time flies when you’re having fun.  It’s been some busy times here at Undercover Calculus; I need to buckle down and write more.  Thanks for being a patient fan!

In my last post, I started talking about a clothesline math activity I facilitated with several elementary math coaches during a recent PD.  And now that I have a moment, I’d like to share more about my professional learning from the experience.  And just because I love acronyms, we’ll call these PLMs for Professional Learning Moments.

If you missed part 1, scroll down to the next post or click here.  Just to recap, I had asked participants to make a “too low,” “too high,” and “just right” estimate for the number of Skittles in a bag.  Then, in two groups of four, they were asked to put their numbers on a clothesline “in a way that makes sense to you.”  While they worked, I tried to facilitate two conversations by popping back and forth in an effort to set up a compare and contrast reflecting conversation as a whole group.


I had asked the group that made the clothesline (pictured above) to think about ways to make their clotheslines more accurate.  I was trying to get them to think about scale and attending to precision.  (I wish I had known about Andrew Stadel’s “Place, then Space” language to frame the question.  But now I’m ready for next time!  Thank you Chris Shore!)

Then I approached the other group who made the number line pictured below, took a peek at what they had done, and started to move some of their numbers around in an effort to simplify their clothesline for our debrief conversation.

“Whoa whoa whoa!!  What are you doing?” they decried!  Clearly, they had become protective of their clothesline and they reminded me that I had asked them to put them in order “that made sense to them” and that their order does indeed make sense if I would get out of the way to let them explain.

PLM #1:  When asking open questions to learners, I need to keep open perspectives about their answers…and I need to do this intentionally and mindfully at all times.  I had judged to0 quickly because I didn’t see something that made sense to me.  And I almost missed a chance to learn.

PLM #2:  Individuals and groups naturally become protective over their work if given the opportunities to take pride and ownership over the space in which their work occurs.  This protective pride can happen with the clothesline activity.  I’ve seen elementary students show more care and motivation in their learning because they come to see the clothesline as their space to show their work and something worth being proud of.  The same was true with these educators.  They liked the way they represented their reasoning and they were eager to defend their arguments from critique.


Like the other group, they clumped their too lows, just rights, and too highs into clusters on the clothesline.  But if you notice, their numbers don’t go in order.  They had discussed changing it, but the more they talked the more they liked what they had created.  They wondered aloud, “Does putting them in number order make our reasoning any clearer?  In fact, isn’t our chosen way clearer?”  Fair point.

Eventually both groups merged together in front of this number line and we continued to reflect about our learning and implications for our work teaching students.  Some key pieces of learning and wondering from the discussion:

  • They liked that each group member had a different colored marker.  And that you could see the order they went in for each category of estimates.  This allowed them to see patterns in their thinking.
  • They also asserted the notion that the clothesline wasn’t necessarily a number line.  It was physical model that they could manipulate how they wanted.
  • That this representation was more useful than just putting them all in number order and calling it finished.  It was about making choices that made sense and explaining them.
  • We wondered how best to ask for the low and high estimates to avoid answers like 1 and 1,000,000,000.  The best we came up with:  “Give a low/high estimate that you’re pretty sure is too low/high.”  If you have some input, please let us know in the comments.
  • This activity is great for students who need to move in order to learn.
  • Clotheslines are great for fostering student discourse in the classroom for learners at all levels.

And penultimately, PLM #3:  I need to be aware of my own lens as a facilitator and an instructor if I want to focus on people’s reasoning, not just their answers.

And finally, PLM #4:  Finding ways to add “…in a way that makes sense to you…” to questions or directions opens up learning opportunities for all learners and puts the focus on explaining sense making rather than just getting answers.


The Powerful Phrase: “…in a way that makes sense to you.” (Part 1)

The Powerful Phrase: “…in a way that makes sense to you.” (Part 1)

I had the opportunity to work with several K-5 instructional coaches last week during a one-day workshop on increasing student engagement and discourse in the math classroom.  (My favorite workshop to lead!)  Leadership in the district is making a focused effort to support teachers in their practice of creating classrooms where Math Practice 3 thrives and students are asked to share their thinking and reasoning with others.  Creating such lively classrooms often means escaping the confines of a textbook, opening up math lessons to allow for more student choice and voice, anticipating student thinking, and embracing the surprise when they do something unexpectedly brilliant!

In our workshop, we explored ways to use estimation strategies to get all students participating in a conversation using mathematical reasoning.  We then graphed our estimations on clotheslines, and that’s when the real magic started to happen!  It was an enriching professional experience for all of us, and I’d like to tell you what we learned (and I will in Part 2 of this post).  More importantly, I’d love to hear your thoughts on the questions below.

The workshop outcomes that framed the language and the focus for our day:

Screen Shot 2016-03-16 at 1.14.45 PM

After grounding the group and framing our work for the day, we launched into an activity that required participants to make an estimate about how many Skittles were in a regular-sized bag.  (I had a bag on hand to use as a visual.)  Here were the prompts:

Screen Shot 2016-03-16 at 12.21.28 PM

Participants wrote each of their estimates on folded sheets of paper.  They were then divided into two groups of 4 and asked to do the following:

Screen Shot 2016-03-16 at 12.21.40 PM

(A quick aside to say thank you to Andrew Stadel and Chris Shore for sharing their valuable and useful work around estimation and using clotheslines as a way to make number lines more engaging for learners.)

“…in a way that makes sense to you” is a phrase that I’m learning to add to many of my questions/prompts, and I’m encouraging teachers and leaders to do the same.  The phrase creates an invitation that there is no “right way” but that there might be many sensible ways that could be worth exploring.  This phrase invites students of all levels to make meaning for themselves.  Compare that to the prompt:  “Accurately scale your clothesline and locate your estimates appropriately.”  This closed question has only one right answer and is riddled with language and ideas that are important to learn, but in this case, may turn off struggling learners from engaging in thinking and showing what they can do  Again, the focus of the workshop was on finding ways to encourage discourse for all learners, and I’m discovering that “in a way that makes sense to you” is a great phrase to use when asking students to explore their thinking with each other.

Here’s what the groups (of 4) did with their estimates.

Group A:


(Note:  The group above placed their two 20s and their two 100s on top of each other.)

Group B:


(Note:  A “too high” estimate of 300 was written in green and got cut off on the right side.)

I invite you to take a moment to ponder each clothesline because I’m curious to know your thoughts.

What do you notice?  What do you wonder?

How do they compare?

What questions could you ask to promote thinking?  What further directions would you give the learners?  Where would you take this with your students?

In Part 2, I’ll share our insights, wonderings, and the key pieces of learning and growth we made for ourselves as teachers and learners.




Numberless Word Problems 1

I had the pleasure of working with 2 amazing 3rd grade teachers last week.  They had the usual concerns and frustrations about teaching word problems and were eager to find a better way to get their kids enthused and engaged.  So, we chose to do something different.  And I’d like to tell you about it.

About a week or two ago, I came across Brian Bushart’s (@bstockus) posts about using/writing Numberless Word Problems (NWPs) that you can read about here.  (Regina Payne (@reginarocks) has also been involved in launching the work around NWPs as well.)

I showed Brian’s post to the teachers.  After a quick read, they were hooked and we set to work.  (Reason 1 why NWPs are awesome:  They trigger a “I can do that!” response for teachers.)

We dove into their curriculum and found the usual dry word problem that sometimes leads off a lesson:  Yasmin has 24 large apples and she makes 8 loaves of apple bread.  Each loaf contains the same number of apples.  How many apples did Yazmin put into each loaf?

The textbook heavily scaffolded the solution process with the usual stuff that sucks the joy out of learning and exploring.

So this is what we did and how we did it.

The first thing we did was change “bread” to “pie”.  Partly because pie is more delicious and also partly because we thought the loaf/loaves represented an unnecessary hurdle for English language learners.

Then the teachers decided to help bring in some roleplay and story telling to set the scene and frame the work for the day by displaying the following slide.

Screen Shot 2016-03-08 at 11.10.45 AM

Then they showed “Clue 1”.

Screen Shot 2016-03-08 at 11.13.56 AM

They asked:  “What math do you see in this sentence?”  Students paused and thought and then started talking with their tablemates.  All students were engaged and discussing what “several” and “some” meant.  One student ponders aloud, “How many apples does she have?”  (Reason 2 why NWPs are awesome:  Open questions mean all students can play.)

Student responses were then charted.  One student mentioned that large apples means less apples were needed than if she used smaller apples.  Classmates listen and nod in agreement.  Another noted that “some” pies meant that there had to be more than one pie.   (See image below for a capture of their responses.)

On to Clue 2:

Screen Shot 2016-03-08 at 11.24.41 AM

Teachers asked:  “How does this new clue change what you already now?”  Full discussion starts to unfold at their tables.  Some students hollered out:  “It’s a multiplication problem!”  “No!  It’s a division problem!”  We asked for evidence and they talked about counting equal groups or making equal groups.  Another student suggested that there had to be more apples than pies which led to an interesting discussion.  Some students wanted to know how the apples were sliced.  (Reason 3 why NWPs are awesome:  Students drive the conversation!)

Clue 3:

Screen Shot 2016-03-08 at 11.30.03 AM

Teachers asked:  “What do you notice?”  Then after student conversation:  “What do you wonder now?”  Student responses…  How may pies did she make?  How many pies can you make with 24 apples?  How many applies will she use for each pie?  I bet she made 6 pies with 4 apples in each pie.

Teachers also asked students to make an estimate.  “How many pies do you think she made with 24 apples?  And why?”  (Reason 4 why NWPs are awesome:  Students can make estimations to their own questions.)

Finally, the 4th Clue:

Screen Shot 2016-03-08 at 11.43.51 AM

Teachers:  “What questions can we answer now?”  More student discussion and exploration.  How many apples will she use for each pie?  How many slices of apple for each apple?  each pie?  How many slices of apples are in 8 pies?

Students set to work answering their questions.  A few interesting things to note:

After 45 minutes, all students are still into the problem.  Butts are out of seats.  Heads are together.  Students talking to students.  Introverted students scratching their heads as they silently look over their own work.

Two stronger students started wondering how many apples would she need for 10 pies?  They started to list some ideas and started to make the rudimentary beginnings of a ratio table and informally talking about 3 apples a pie as a rate!  I was blown away.

And then…. the lunch bell and students groaned!  (Reason 5 why NWPs are awesome:  Students are bummed out by the lunch bell.)

Closing thoughts:

  1. I can’t wait to do this again.  Neither can the teachers or the students.  They want to do it weekly.  Creating an NWP was simple and easy, and they were able to create them using their existing text.
  2. At one point during the lesson, the teacher asked a “closed” question during the lesson.  The shift in the energy in the room was drastic and sudden:  students got quiet, discussion got stifled, and the teacher became the only math-doer in the classroom for that moment.  The students were afraid of giving the wrong answer to a right/wrong closed question.  The teacher realized the shift in energy; we (the observers) noticed it too.  After a moment, the teacher backtracked and said “Let’s come back to that question later, OK.  Who else has something interesting to say about this problem?”  BAM!  Students are back on track and thinking and exploring and doing.
  3. But that moment was my key piece of learning.  Open questions allow for all students to play.  It demands that stronger students be more than algorithm followers and actually do math.  It makes discussion and sharing of ideas easier.  It reduces anxiety. Obviously, closed questions are necessary too!  But if dialogue, discussion, and full student engagement are goals to target while tackling word problems, then open questions are the way to go!  And NWPs allow for that to happen in a structured, scaffolded approach that teachers can use to guide learning.
  4. I’ll never look at a math textbook the same way again.  I’m looking forward to seeing and hearing about what this looks like in middle school classrooms and in high school.

So:  What do you think?  How could you make it better?  What learning has this created for you?


Two remaining tidbits because I’m long-winded like that.

Here are the student charts if they interest you.

Screen Shot 2016-03-08 at 11.32.27 AM


Here is a cool extension that they created to help reinforce learning.  They created a clue booklet for another NWP.  The goal was that students would be able to write and show they’re thinking at each step of the way.  I tried to show that in this video below so you could see the problem, but it’s a comical disaster.  Who knew that stapled paper was so hard to work with??  Anyway, you might find it useful.

Finding Common Factors: Making Over A Textbook Problem

I had the opportunity to work with a team of three 4th grade teachers the other week.  We were exploring a lesson about finding factors for numbers less than 100 (4.OA.4).  I’d love to hear your input and comments and ways to make this better.  I’m always eager to learn and grow.  My hope is that you also find this post useful in your own work.

We spent 4 hours on Tuesday discussing student engagement and examining how their current curriculum frames the learning experience for students around factors.  (Spoiler alert:  the curriculum sucks life and wonder out of learning.)  It was immediately apparent that the structure of the lesson in the curriculum was incongruent with our vision of a classroom of students engaged in mathematics and producing knowledge (and allowing us to understand and identify their own misconceptions).  So we gave the lesson a makeover and then spent 4 hours on Thursday each teaching the lesson to different groups of 4th graders, observing student learning, and reflecting on our work.

There are a lot of layers to our learning but I want to share (and get feedback) about making over the textbook problem.  If you’re reading this, you are probably already aware of the significant body of work around finding ways to restructure the lessons found in traditional math curriculum and “opening them up” so that students can explore, inquire, and make meaning for themselves.  (Thank you Dan Meyer, Robert Kaplinsky, and Nanette Johnson among many many others!)

The textbook presents the following “essential question” to students:  How can you use the “make a list” strategy to solve problems with common factors?  Ironically (tragically?) the publisher claims that this lesson addresses Math Practice 5:  Use appropriate tools strategically.  That practice asks that “mathematically proficient students consider the available tools when solving a mathematical problem.”  In the EQ, that tool is given; there is nothing for the students to consider.  Can student use tools strategically if we’re telling them what tool to use?  

The text then offers this problem as an entry point into common factors:

Version 2

Exciting?  Compelling?  Engaging? 

We didn’t think so either.  We discussed many of the obstacles to student learning that were inherent in this problem.

Second, this problem is, for a lack of a better word, lame.  We discussed giving coin manipulative to groups of students so they could create the problem and act it out for themselves.  But we were struck by how arbitrary the problem was.  What’s the point?  Why does Chuck care to do this?  Why would a student want to help Chuck out?

Lastly, the graphic organizer was dauntingly tedious.  The curriculum says that this lesson was also about Math Practice 1.  Presumably the graphic organizer was to teach students how to make sense of the problem and demonstrate perseverance.  We looked at it and asked ourselves, “How are we going to get students to make meaning of this without being a ‘sage on a stage’ leading our students through it?”

Here’s the graphic organizer:

Version 3

There were a lot of things we didn’t like about the organizer.  We felt like it was a really long journey to find common factors and there wasn’t much room for student exploration and discussion.  And the structure of the answer states that there are exactly four answers.  And at the end of the problem, we are still left with a stale “So what?” taste in our mouths.  “We did all that work, for this answer?”  Furthermore, this problem has a closed beginning (every student solves the same problem), a closed middle (every student uses the same organizer and fills in the same blanks), and a closed middle (every student should have the same answers).  

The lesson continued in the text with the same similar structure of problems.

Common Factors.001

It’s important to note here that students are not asked to group different things into equal sized groups that have the same ratios of unlike things.  We wanted to believe that’s what was being asked because it’s useful to do such things.  For example, if a food bank asked students to make food boxes containing equal amounts of water, bread, and cans, that’s a worthwhile problem.  But #4 on the right above doesn’t!  It’s asking students to make equal sized boxes where each box contains only water, bread, OR cans.  Why would we want to do that?  And why would we want to spend an hour of math time figuring out problems like this?

So, we refocused our work on the following goals.  We wanted students to solidify their understanding of factors (and how they are related to products).  We wanted students to discover and identify common factors of numbers.  We wanted to see students interacting and cooperating with each other as they do the math and the thinking.  And we wanted to make sure that students of all math levels could participate, learn, and grow.  Low floor + high ceiling = full student engagement.

We created the following pathway.  Each of the three teachers made modifications to this lesson to suit their own student needs.  (If you want to see the lesson plan:  Lesson Inquiry for 4th Grade Factors.)

We started the lesson by asking students:  “Write as many numbers as you can that have 6 as a factor.” 

We gave them about 2-3 minutes to do this.  We like the question because it was open.  There are an infinite number of numbers with a factor of 6, albeit only 16 less than a 100.  It was both accessible and challenging because the measure of success was “as many as you can”.

Two amazing things happened. 

First, some students wrote something like this:


We anticipated this mistake and celebrated it because it answers a related question.  We borrowed Dan Meyer’s question “What other question does this solution answer?”  This allowed the class to celebrate the incorrect answer by showing us what it does teach us while also reviewing factors and products.  It’s also a reminder that constantly asking students the same question in the same format (“List the factors of …”) conditions students to follow a structure at the expense of thinking flexibly.

Second, some kids started writing stuff like this:


We saw students developing different strategies for finding numbers that have 6 as a factor.  The students doing similar work above started to make connections between factors and multiples.  Other students started doing skip counting.  Other students dug out their times tables.  Students were able to share strategies and reasoning.

One student suggested that 32 had 6 as a factor.  The discussion around this error was rewarding and enriching.

The teacher recorded student responses on the board and then chose 4 numbers to work with (such as 12, 36, 42, and 96).

Then we asked students to choose a number and list all the factors.  We allowed students to choose the number that challenged them but they also felt like they could struggle productively.  In one class, we had something like this on the board:

12: 1, 2, 3, 4, 6, 12;

36: 1, 2, 3, 4, 6, 9, 12, 18, 36;

42: 1, 2, 3, 6, 7, 14, 21, 42

96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

In small groups, students were asked to make statements about what they noticed.  We recorded all the responses on the board and unpacked what we could in the time remaining.  Students identified that they not only all had 6 as a common factor, but also 3 and 2.  We asked why that was and students had a rich discussion.  One student asked:  “If two numbers have 6 as a factor, will they always have 2 and 3 as factors?”  Sweet.  We asked students to make other statements that they thought were true and recorded them for use in the next class.

Below is a thing outline/pathway that we created and use to guide instruction.  Every lesson looked differently in reality, but had similar threads that I’ve described above.

Lesson Inquiry for 4th Grade Factors

Where did we miss?  How could we have done it better?  What might be useful to you?

Interested in your thoughts.  More to come soon.