Why Does Math Look Different?

The most common question I am asked related to my profession (by adults mostly outside of education) is, ” Why does math look different than when I was in school?”  If I was speaking to you in person, I would share my thinking with lots of gestures and excitement because I love math that much.  In lieu of a personal conversation, I will do my best to paint the picture with words.

The short answer: It’s you, it’s not me.  The long answer: It’s all about perspective, and here’s why… Constructivism is rooted in the work of Jean Piaget from nearly 90 years ago.  This theory of learning best describes the phenomena (to many) that is modern mathematics teaching and learning.

At any given point in a student’s learning journey, he/she has some accumulation of mathematical understanding.  Then, for every learning experience, opportunity to do math and reflect on the thinking behind the calculations and representations, the student’s networks of understanding are added to, changed, or somehow extended.  This is how you know learning has occurred – it has impacted the student’s thinking either through verifying beliefs or challenging understandings.

Now, when the students move along this continuum of mathematical understandings, they strategically modify or verify their existing knowledge or they abandon it for more efficient, accurate methodologies or connections.  This is powerful – strategic abandonment of ideas for new, more relevant ones.

Back to the perspective claim.  As adults, we see the methodologies of teaching and learning mathematics from our current state.  We are efficient, accurate mathematicians.  We know what tools to select to solve problems, appropriate processes to use to solve problems, how to verify accuracy of solutions, and means to explain our thinking.  We stand very far down the mathematical journey, relative to the position of children in elementary or middle school, for example.  If we were to walk ourselves back down this path, past when we first used a calculator to balance our bank accounts, past when we first learned Algebra, beyond our days of fractions and long division, we would find ourselves in a world of inefficient mathematics.  But this is all relative.  This is about perspective.  Using repeated addition, such as 4 + 4 + 4 + 4 + 4 + 4 + 4 to arrive at 28 is less efficient than using multiplication, such as 4 x 7 instead.  And, counting 1, 2, 3, 4 and 1, 2, 3, 4 and 1, 2, 3, 4 (seven times) is even less efficient.  Once we truly understand that multiplication is a more efficient method to find the total number of objects in a given number of equal groups, we strategically abandon the method of repeated addition.

So, math does look different.  This is true.  However, it is all about perspective.  You should change your perspective and put yourself in the shoes of the children.  You were once there, but you have strategically abandoned those memories for ones that are more efficient!

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Change your perspective and you may understand why math looks different.


Want to learn more?  I recommend Elementary and Middle School Mathematics: Teaching Developmentally by Van de Walle, Karp, and Bay-Williams.

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The Power of Constraints: Binary Questions

One strategy I often employ in professional learning with my educators is to rate themselves on a scale of X to XX.  Similar to estimation on a number line, they are to reflect and consider where they stand with relation to a given prompt.  Though this line of thinking supports “correct” responses anywhere along the continuum, it is missing one key ingredient: a constraint.  There is no true decision making required, no absolute committment.  There is no one or the other.  No A or B.  No 1 or 0.

Enter: Binary Questions.

Recently, I introduced the concept of Binary Questions to my educators in a professional learning experience.  This small act prompted many, many conversations centered around possible prompts for their own classroom.  (Success!).

The constraint of decision making adds a twist to this reflection in a new way.  This is not a popularity contest or true/false question.  Rather, it is a visible sign of individual voice.

In the future I will not abandon the number line, approximation form of questioning by any means, but I will certainly consider this structure as well.

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Stay the Course

I worry about teachers this time of year.  I worry that they hear two messages from administrators: one of accountability and state assessment and one of high quality teaching practices.  I worry that they fear poor ratings and in response make poor teaching and learning decisions that are contrary to what they know is best for kids.  I am not referring to all teachers here, but I am confident this applies to more than one and one is too many.

So, I share 7 tips to help you stay the course.  You can do it.  I know you can.

1.  Refer to your curriculum as your guide.  A guaranteed and viable curriculum is reassuring for you and for your students.  It helps you sleep at night knowing you have provided opportunity for your students to access the learning and that no student drew the short straw and ended up in your room.  If you don’t have a curriculum to depend upon, call me.

2.  Consider what you are measuring and what matters.  Keep one eye on profound, long-term learning.  If the work you are providing your students cannot be ultimately mapped to this, reconsider your plan.  I challenge my teachers to design the best in the world learning experiences for their students.  Loosely paraphrased, this means you should do the best you can with what you have, and we have a lot.  This also means that   we have lessons that bomb, that hindsight is 20/20, and that’s ok, too.  When we know better, we do better.  Period.

3.  Focus on the quality teaching practices you have come to know.  Remember the goal is LEARNING.  That’s it.  Your job is to remove all barriers possible to provide access to the content for your students.  Sometimes you are able to help a student make the smallest little inch down the developmental progression of learning.  Celebrate this, as this is learning and movement in the forward direction.  Think about what your students will learn today – not what they will do today.  Let’s have less doing and more learning.

4.  Reflect on past lessons, units, and years.  Focus on what has worked.  Do more of that.  If you don’t know, ask your students.  They know what helps them learn.  Find one thing that you did that led to learning.  This might be the way you designed your classroom learning environment, it might be a tool that helped your students see an abstract concept, it might be one good, solid question that got them thinking.  Whatever it was, embrace the positive and replicate it.

5.  Reach to your team and your PLN.  Do not be afraid to reach out to others with questions.  Advocate for the students in your classroom.  You are not alone in needing help.  The absolute strongest, most confident teachers did not get there alone and they have not arrived at perfection.  Keep growing.  

My PLN amazes me.  Some days on my commute home I reflect on the contacts I’ve made that day and think about the great minds in education to whom I refer as my friend.  Here’s a little backstory: If I have met you in person before and we’ve talked math, you’re my friend.  This may not be reciprocal (yet) and I am ok with that.  If I have not met you in person before, but I know we’d hit it off, you are my “friend” and I look forward to dropping the quotations.

6.  Give yourself a moment to take a deep breath.  Don’t panic.  Don’t revert to short-term, superficial learning experiences that you know deep down are not good because you feel rushed.  Find one good thing and do that.  Then find another.  Until you feel like you can breathe again, when you are able to make bigger plans.

7.  Avoid the countdown.  (This may ruffle some feathers.)  If we are focused on learning, then why are we counting down to a time that we will no longer be together to learn?  This message may say to kids, there are only X more days I have to see you in my class (that’s terrible, by the way).  Let’s focus on today and tomorrow, not next year when I may not get to see your smiling face everyday.

I encourage you to do the right thing and do it all the time.  You know what that right thing is and your students deserve it.

The Writing on My Wall

Two months ago I began reading The Writing on the Classroom Wall by Steve Wyborney.

Wait.  Let me back up a bit… Two and a half months ago I had the opportunity to witness Steve Wyborney communicate via FaceTime with one of my 3rd grade educators and her class.  Later that day I ordered TWOTCW because of how he interacted with those kids.  The conversation was supposed to be via Skype, but due to technological difficulties, we migrated to the educator’s cell phone.  Steve was so patient.  At one point he wrote notes and held them up to the camera when we couldn’t hear him.  The content of the conversation that day with the kids was important (it was about SPLAT Math), but the message he sent in his kind voice, welcoming language, and genuine love of learning was so memorable I knew I wanted to learn more from him.

As I read TWOTCW I made notes (sketchnotes) of the Big Ideas.  My sketchnotes represent what images come to mind when I read.  What words stand out.  What colors inspire thought.

On every page I included the same central image – inspired by the book’s introduction.  Steve challenges us to connect with others, to take risks, to share ideas.  As I completed each page of notes, I shared them via Twitter.  Write(sketch), reflect, share.  Wash, rinse, repeat.

On every page I also included a lightening bolt.  More to come on that.

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Now for my first Big Idea.

It hit me like a slow bolt of lightening.  (There’s the image.)

Learning is not paced.

Sometimes moments of clarity come in rapid succession and sometimes they come few and far between.  Sometimes we seek understanding as the answer to a question or the solution to a problem.  Sometimes understanding leaps out of the page in front of us without prompting, like a pleasant surprise we didn’t expect.  But usually, understanding comes after much contemplation.

This is why we should write, reflect, and share.  If we are in the habit of capturing our learning and documenting it, we will be on the lookout for new understanding as content to write about – whether this happens once per day (as if we can be that lucky!) or once in a while.

So, learning does not occur like clockwork.  Wake up, it’s 12:00, time to have a profound thought.  No.  Learning is not paced.  It is not a box to be checked every 400 meters as you pass the starting line.   It is, though, something to celebrate as forward progress occurs – maybe inches at a time, but inches no less.

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Let the journey begin.

 

Creating Digital Breakouts with iWork: What I have learned so far

I challenged myself to create Digital Breakouts using the iWork Suite because I know the power of the features embedded within the programs.  I chose to use Numbers as the platform for the content and Pages for the final certificate.  Two of my Digital Breakouts are linked below.

Why Numbers?

Numbers is so much more than a spreadsheet program!  I use Numbers as a combination of a graphic editor, word processor, and spreadsheet all in one.  Without the constraints of page size or the need to scroll through a document, students can toggle between sheets easily using the tabs at the top, like bookmarks.

The use of iCloud with Numbers is valuable!  I use view only and share the link through email and social media.  Then, students can download a copy and complete the Digital Breakout without altering the original.  In addition, teachers can download a copy and make edits, if they wish, before sharing with their own classes.  I also appreciate the ability to make edits to the document and not worry if students have access to the latest version.

How to Use the Sheets

I create one sheet for each lock or step in the process, in addition, I use the first sheet as a welcome/introduction and the last one for the final lock.

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I take the opportunity to embed clues within the titles of the sheets as well as tab numbers as an indication to move through them sequentially.

How to Use Shapes

The new shapes in iWork are powerful!  They are clean, clear, and editable.  To edit the shapes, choose a shape to include on a sheet, then select the shape and choose Break Apart.  This will separate the shape into component parts.  Select the shape again and choose Make Editable.  This will allow you to change the shape, such as toggling between straight and curved lines and adding new lines.  Adjusting the color is also an option.

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How to Use Cells for Responses

I create a small table on each sheet to capture responses.  I layer these cells on top of an image related to the theme of the Digital Breakout.  For example, if the purpose was to determine a 5-digit password, then I would add 5 cells across the top of an image that resembles spaces to enter a password.  In this case, the Digital Breakout includes 5 locks, one per cell in the password.

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How to Use Conditional Highlighting for Feedback

I use Conditional Highlighting to provide positive feedback for correct responses.  Select the cell, then Format, Cell, and Show Highlighting Rules.  I choose to use Green Fill as the indicator of a successful response.

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The Final Lock

I layer cells over a shape for a final lock.  I insert a formula to reference the value in the cells on the other sheets.  This way, the solution for (or a clue for) the final lock will be displayed on the last sheet once they are determined.

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Certificate of Success

I use Pages to create the certificate of success.  Then, I share it through iCloud and set up a password for access.  The password is the key to open the final lock.

In Pages, I use the Kids Certificate template, alter the center image and text and in a few short minutes, I have a certificate of success ready to go.  Students can add their name and date to this Pages document and save it to indicate they have successfully completed this Digital Breakout.

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The first Digital Breakout I created involved a lot of problem solving every step of the way.  The second one was a bit easier, with the effort spent mostly on the content – as it should be.

Anticipation Guides

*Note: This post is a work in progress.

Students need to listen, speak, read, and write about mathematics as well as archive their learning process.  Teachers need windows into their students’ minds, to see their thinking. 

Using Numbers I have created Anticipation Guides for students to assess their own understandings of mathematics content (before and after learning experiences).  

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With the integration of apps such as Explain Everything or SeeSaw, learners can annotate their thinking with text, language, and images/sketches.  Through the process of design, delivery, and reflection, students and teachers will have the opportunity to access their current level of understanding in relation to the developmental progression of learning mathematics.  

The button below provides access to Elementary, Middle School, and High School appropriate templates.

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The buttons below provide access to grade-specific Anticipation Guides in iCloud.

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#NoOfficeDay

Inspired by Matt Arend’s No Office Day plan to devote about one day per month out of his office and into the classrooms with learners, I spent my first No Office Day today, the first day of school.

My Why

I choose to intentionally schedule time out of my office and onto campuses in order to connect (and re-connect) with educators and learners.  I choose to not lose touch with the daily responsibilities, challenges, and celebrations that come with being a teacher.  I choose to recognize the impacts that my daily decisions make on teaching and learning.  I choose to know why I serve children and adults in my district through my role as Math Director.

My Day

As liaison to Wilson Elementary, my day began there with the awesome opportunity to welcome parents and students to the campus on this, the first day of school!  I offered to take family photos and opened car doors before the school day began.

Then, I headed to Cottonwood Creek Elementary.  A fifth grade educator had reached out to me and invite me to his classroom the first week of school for a visit.  I wouldn’t miss it!  Invitations from educators like this take priority on my calendar.

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Then, I headed to Coppell High School for the very same reason – an invitation from an educator that I would not miss for the world.

I had the opportunity to read responses that Algebra II learners contributed to the prompt: What are the qualities that you look for in a teacher?  The most common response I noted: kindness.  This will come full circle later in the day.

The next two stops were Wilson (again) and Cottonwood Creek (again) to deliver lunches to the campus administrators and counselors.  This wonderful tradition provides us one opportunity to show these campus leaders how much we appreciate them.

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Next, I headed to Richard J. Lee Elementary and did not make it past the media center where Blue House learners were exploring with Legos.  They invited me in and I had a lengthy conversation about how their day had gone (not great but wonderful!) and how much they love the Blue House!  This was a highlight of the day, for sure!

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My No Office Day ended back at Wilson with dismissal.  This is when I asked kindergartener after kindergartener about his/her day.  Every one of them mentioned how his/her teacher was nice, or kind, or smiled.  From 5 years old to 17 years old, that is what is important: kindness.  And, back to dismissal – Of course I would brave the Texas heat and sun to open car doors and wave parents through the car rider line.  I wish I could do more.

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I am looking forward to my next No Office Day and I am so very thankful for Matt Arend for opening my eyes to this valuable way to invest in my teachers and students.  I am not sure when or how often I will be able to spend a day in this way, but I know when I do, it will be worth every minute!

Building a Campus Mission

I was extended the opportunity to lead my liaison campus, Wilson Elementary, through a process to develop their campus Mission.  The voice of each staff member was welcomed and the outcome included a statement that captured their purpose, their why.

First, we focused the work on the existing Wilson Elementary Educator Creed.  This common language sets the tone for the campus by announcing the responsibilities, expectations, and motivation for the educators.  Those educators who were present at Wilson when the Creed was written shared stories of the development process as well as the impact it makes on their classrooms with the new hires.

I am a Wilson Ranger Educator.
I have great expectations for my learners and myself.
I accept the challenge to be the best I can be.
It is my responsibility to create a learning environment
conducive to optimum academic, social, and emotional growth.
I provide a model of decorum and respect
that guides my learners as well as honors them.
I cherish every learner.
I change the world one learner at a time.
I am a Wilson Ranger Educator.

Then, we used an Affinity Map to collect ideas to use to build the Mission.  The educators were given 3 minutes to brainstorm thoughts – one per sticky note.

Looking for common topics, the educators then collected ideas and grouped them with those at their table group.

As a whole campus, the educators gathered in groups, one representative per common theme, and wrote them on sentence strips.

The sentence strips were collected and organized with a pocket chart.  The Mission was arranged (and re-arranged) until it captured their fundamental purpose.

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Finally, the table groups indicated their support in a non-verbal way with plastic cups.  Green: We are good to go.  Red: We have a question or concern.

After some editing, the final Mission reads:

At Wilson Elementary, we provide a safe environment where each learner is loved, valued, and accepted.  We embrace and address the needs of the whole child, encourage academic, social and emotional growth, and develop character to the highest level.  We promote wonder and build a community of empowered, curious minds.  We inspire lifelong learners who positively impact the world.


This process was efficient and effective in developing the campus Mission.  I modeled classroom facilitation strategies and emphasized the necessity to use this Mission as a lens through which to consider all children.  I am looking forward to seeing this Mission in action at Wilson!

 

 

 

What Your Students are Thinking

How do you know that your students are thinking?  How do you know what EACH ONE of your students is thinking?  I encourage the valuable implementation of mathematical discourse communities to model problem solving, promote higher depth of knowledge and the necessity for justification of mathematical arguments, promote the use of accurate academic vocabulary coupled with growth of language for English Language Learners, allow students to build their confidence in mathematics, and provide opportunity to formatively assess students throughout.  This formative assessment is what your students are thinking.

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mathematical discourse community (MDC) includes productive mathematics discussions, centered around common problem situations in which the environment is safe enough for all students to contribute in multiple means.  MDCs are accessible for all learners, including those exhibiting gaps in prerequisite knowledge or academic vocabulary.

Though the benefits are great, building a mathematical discourse community in a classroom is a challenging feat for educators.  Quality should trump quantity and conceptual understanding is a goal worth seeking.

Lynn Liao Hodge and Ashley Walther describe 4 initial practices to build a foundation of productive discourse in mathematics classrooms in their article Building a Discourse Community: Initial Practices:

  • Use a more open task
  • Support think, pair, and revoice/compare
  • Offer three ways to participate
  • Define a contribution

In my work to design and support curriculum for PK-12 mathematics, I recognize the need for a balance between scalability and fidelity.  That is, what structures I put into place in my curriculum that can be utilized horizontally (throughout the school year) and vertically (throughout the courses) by educators, and in doing so, can be done well – accurately, efficiently, and effectively.  I choose to implement a few structures in order to support teachers to do them well.  3-Act Tasks, Notice and Wonder, and Which One Doesn’t Belong? are three such structures I work to embed in my curriculum.

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Hodge and Walther refer to open tasks as those with embedded student choice, multiple paths to solve, the necessity to justify solutions, all within a real-world scenario.  Related to this, I strive to embed inquiry-based learning experiences, in the structure of 3-Act Tasks.  I focus on 3-Act Tasks as a means for clarity, consistency, and opportunity for professional learning related to the implementation in the classroom.

When not deep within a 3-Act Task, it is important to continue to support the discourse community in the mathematics classroom.  I argue Notice and Wonder and Which One Doesn’t Belong? are two structures that continue this work to allow conversation, comparison and justification by students about mathematics concepts.

When teachers use the Notice and Wonder structure, it causes them to pause.  It also pushes the ownership of the knowledge to the students.  The conversation begins when a prompt of some sort is shared with the students.  This prompt may be an image, an object, a graph, a situation, or any other representation that leads to interpretation.  The simple question, “What do you notice?” is open enough to encourage contributions.   Based on the work of Hodge and Walther, it is advised to encourage participation by more than a single student through think, pair, revoice/compare.  I challenge you to avoid choral responses as a go-to with Notice and Wonder.  This excludes nearly all students in the room, allows students to hide behind their classmates, and narrows the scope of the conversation.  To move beyond verbal-only contributions, I challenge you to support sketches, gestures, and other non-spoken notices and wonderings.  Then, when the conversation lends itself to transition to the wonderings, the question, “What do you wonder?” is used.  Read more about Notice and Wonder here.

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When teachers use the Which One Doesn’t Belong (WODB) structure, it also causes them to pause.  Wait time is a valuable commodity in the mathematics classroom!  WODB also pushes the ownership of the knowledge to the students.  Similar to Notice and Wonder, a prompt is used, but this time it includes multiple items of some sort.  This prompt is usually a set of images, but does not have to be.  Any set of images, objects, graphs, situations, or other representations that lead to interpretation can be used for WODB.  The simple question, “Which one doesn’t belong?” is open enough to encourage contributions – with justification.   Seek participation by many students (ALL) through think, pair, revoice/compare.  I challenge you to avoid choral responses as a go-to with Which One Doesn’t Belong?.   Hodge and Walther say to offer three ways to participate.  What three ways will you support in your classroom?  What tools will you encourage your students to use to communicate about mathematics?  Sketches, gestures, and other non-spoken ways to identify and justify Which One Doesn’t Belong? builds the mathematical discourse community in your classroom.  Need a place to start with Which One Doesn’t Belong?  Try this or this.

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Focusing on the means to contribute, I encourage reflection on our English Language Proficiency Standards.  That is, opportunities for students to listen, speak, read, and write about mathematics during the learning process as well as while they are demonstrating their understanding.  These four modalities should not be seen as a checklist, because listening is not hearing, speaking is not repeating, reading is not looking, and writing is not copying.  I bring you back to the initial questions on this post:

How do you know that your students are thinking?

How do you know what EACH ONE of your students is thinking?

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Balanced Mathematics

As the leader of mathematics curriculum and instruction in my district, it is imperative that I clearly articulate the vision for teaching and learning mathematics and support it with a guaranteed, viable curriculum that includes practical, actionable items for the classroom.  As I work to complete a new curriculum rooted in the beliefs outlined in the Visioning Document and in response to the results of a recent, deficit model audit, I recognize the need for a succinct balanced mathematics plan.  This model of lesson design, instruction, assessment, and learner support will be evidenced throughout curriculum documents and professional learning opportunities for educators and campus administrators.  Successful implementation will include district-wide common understanding of the purpose and implementation of the balanced mathematics model.

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Within a balanced mathematics program, classroom structure supports intentional, responsive lesson design and facilitation of learning experiences in order to guide all students to success in mathematics.  

The components of the CISD Balanced Mathematics Plan include:

  • Conceptual Understanding: Learners understand mathematical ideas, make connections to other topics, and are able to transfer thinking to new situations in order to solve problems. Conceptual understanding builds to procedural fluency.
  • Inquiry:  Formulated by educators and learners, compelling questions are developed and subsequently guide inquiries into concepts and problems related to specific learning outcomes. (Supporting English Language Learners: Inquiry)
  • Fluency:  Procedural fluency is demonstrated by students as they carry out procedures flexibly, accurately, efficiently, and appropriately.  Fluency in the mathematics classroom is built on a foundation of conceptual understanding, strategic reasoning, and problem solving.  Mental math and estimation are aspects of fluency developed within balanced mathematics. (link to NCTM position statement)
  • Discourse:  Mathematical Discourse Communities are fostered to support students to make and test conjectures, question, and extend concepts in a welcoming classroom environment with student-created norms aimed for conceptual understanding.  (Link to article, Supporting English Language Learners: Academic Talk)
  • Intervention & Acceleration: In order to respond to the academic needs of all learners, mathematics content is intentionally designed and delivered in small group instruction with progress monitoring toward specified goals. (Supporting English Language Learners: Student Ownership of Learning)
  • Evidence of Learning: Learners demonstrate conceptual understanding or skill development through multiple modalities.  This evidence is created for learners to track their own progress and for educators to formatively monitor learner progress throughout the content.  In addition, the accumulation of evidence of learning builds toward a learner’s portfolio in a summative manner.

Components of this Balanced Mathematics Plan are evidenced in curriculum documents.

  • Conceptual Understanding: Curricular resources are organized into the Concrete/Pictorial-Representational-Abstract learning continuum.  This sequence of instruction builds a thorough understanding of mathematical concepts as learners progress developmentally, building upon prerequisite knowledge and extending to subsequent ideas.  Conceptual understanding is built through low-floor, high ceiling learning experiences.  That is, the content is accessible to all learners, including those who may demonstrate gaps in procedural skills, as the problem may be solved accurately through less efficient methods.  These less efficient methods would be used by a learner who has not yet attained procedural fluency in that given process.  High ceiling tasks allow for extensions for learners ready to make connections deeper in the content or beyond the scope of the course.  These tasks do not limit thinking with one-step, simplified processes.
  • Inquiry: Whole group or small group learning experiences following the 3-Act Task format. (link to post by Dan Meyer, link to post by Mary Kemper) The structure of 3-Act Mathematical Tasks provide built-in opportunities for learners to build background knowledge, connect to prior learning experiences, access the mathematics without the barrier of language, and extend thinking, as appropriate.
  • Fluency: Station-based or other individual learning experiences intentionally designed and accessed by learners to reinforce or extend conceptual fluency within the zone of proximal development.  Learners archive evidence of work and are held accountable for their fluency growth.  Specific practice to attain fluency should be intentionally designed, based on the current level of understanding of the learner and the next developmentally appropriate goal.
  • Discourse: Prompts such as What do you notice? and What do you wonder? as well as Which one Doesn’t Belong? are used to promote academic discourse and specific talk moves such as I agree with ___ because ___ are used to promote accountable talk related to the concepts within the unit of study.  Structures such as Number Talks provide opportunities for mathematical discourse by all learners during a whole group or small group experience.
  • Intervention & Acceleration: All students set SMART goals related to concepts as appropriately aligned within the developmental progression of mathematics.  For those learners identified as At-Risk, this small group instruction as guided mathematics fits within the CISD Mathematics RtI Program.  For those learners not identified as At-Risk, setting goals, receiving guided mathematics instruction related to the goal, and monitoring progress toward the goal, supports growth as well toward acceleration.
  • Evidence of Learning: Learners create evidence of understanding as aligned to the CISD Mathematics Transfer Goals as they display, explain, justify, and communicate mathematical ideas and arguments using multiple representations, including symbols, diagrams, graphs, and language as appropriate with precise mathematical language.  Learners use technology, as appropriate, to demonstrate understanding using verbal and visual articulation with annotation apps (such as See Saw or Explain Everything).

The components of the CISD Balanced Mathematics Plan are implemented in a synchronous manner.  Inquiry-based experiences include opportunities for mathematical discourse and result in evidence of learning, for example.  A mutually exclusive mathematics plan would not maximize the potential to balance the intentional, responsive design and facilitation of learning experiences to guide all learners to success in mathematics.