Making the Shift to Standards-Based Grading

Recent conversations with my teammates at my school have gotten me thinking about my own grading practices. While my district takes a “standards-based grading” stance, the grading practices I see happening at my school do not reflect that stance. Most of the teachers at my school still employ very traditional practices when it comes to grading – marking each question right or wrong, and converting the number correct out of the total number of questions to a percentage. The percentage grades are then translated to a scale of 1 to 4, 1 meaning “Below Standard” to 4 meaning “Beyond Standard”.

The Problem With Traditional Grading

Education is about learning. Everything that we do as teachers should be for the purposes of developing student thinking, reasoning, and sense-making. Traditional grading, however, does the complete opposite. Collecting student work in the form of a test or quiz and assigning it one grade based on correctness of the work does not help develop learning – it sends a clear message that performance is valued over learning. It  provides no valuable information to teachers, parents, or students about the learning that has taken place.

Consider this example: At the beginning of the school year, my teammates and I decided to give a baseline test for math. After a lengthy discussion, I managed to talk them out of giving a comprehensive test, instead focusing on the big ideas in our first math unit – Meanings of Multiplication and Division. We decided on several tasks that incorporated big ideas students were to have learned in Grade 2, such as recognizing rectangular arrays and using repeated addition to find the total number of objects in them, as well as the big ideas in Grade 3. Our goal was to elicit student thinking with regards to making sense of, representing, and solving problems involving multiplication and division. We all gave the test in the same way – silent classroom, little or no support from us or from each other, completing the tasks independently. Afterward, we all graded them separately before reconvening to analyze the data.

My colleagues came to that meeting with a single grade for each student, a ranking of 1, 2, 3, or 4 based on the percentage of the questions that had been answered correctly. Only answers had been considered; drawings and other models, even if partially accurate, had been ignored. I, on the other hand, came prepared with a scale score for each task, and the rubric I had created based on the standard each task was assessing. The scale for each task included descriptors for students who demonstrated little or no understanding of the task, some understanding, good understanding, and deep understanding. I was able to show my teammates, for each task, which students needed more support and practice and which students needed more challenge and extension. I also had created a spreadsheet for the upcoming unit with each standard to be addressed listed, and students’ scores from the baseline assessment listed and color coded.

I don’t intend to make it sound like I have standards-based grading completely figured out – I most certainly do not. My point is that the traditional method of grading that my colleagues had used did not provide the depth of data we were looking for. It provided a broad snapshot of their students’ performance, but gave little in terms of their thinking.

What’s Needed to Make the Shift to Standards-Based Grading

1. Changing the Mindset

Making the shift to standards-based grading is not an easy one. It first and foremost requires a paradigm shift in which teachers, parents, administrators, and even students differentiate between learning and performance and placing more emphasis on learning. In the current culture, learning is defined as knowing, evidenced by repeating back what has been taught. In a standards-based culture, learning is defined as thinking, evidenced by how students use skills in new situations (Vatterott, 2015). Making this shift requires more opportunities to learn skills through problem solving, to engage in the process of not knowing, of reasoning, of modeling, of sense-making, and of argumentation in order to understand concepts, make connections, and solve complex problems.

Grades are interpreted by students as definitions of who they are. It’s not uncommon to hear terms like “C-student” and conversations about grade point averages. When this mindset is encouraged, it creates a cycle of unnecessary low or unrealistic high expectations. If we work to change our grading mindset to one that values feedback and continuous improvement, then the grade assigned becomes all but meaningless compared to the power of the descriptive feedback about how students can improve and keep learning.

2. Changing the Way We Assess

To most, “assessment” is synonymous with “testing”. While the two are related, they aren’t the same. “Assessment” is the action of observing student work and comparing that work against a standard measure. “Tests” are things, one type of tool that can be used to help us with the act of assessing. Tests themselves are not necessarily a problem, but the way we use them definitely is. As I described in the situation with my teammates, using a test as a single defining grade doesn’t provide adequate information about student learning, especially if that test has been designed with low-level, rote knowledge tasks.

When designing assessment tools, including tests, that maximize the data we want to collect, we first need to consider the standards we are striving toward. I’ve often heard educators debate about whether standards are a “floor” or a “ceiling”; they represent neither the starting nor ending point of instruction. In my opinion, standards are stair steps, interconnected and developmental, building off previous experiences and contributing new perspectives and complexities.

During the assessment process, it is not enough to seek out what students can “do” or “not do.” Instead, we need to seek out evidence of the understanding, skill, and application embedded in each standard or set of connected standards. We can do this by creating a rubric for each question, treating each one as its own entity.

A great example of this assessment method comes from the Eureka Math Mid- and End-of-Module Assessments. These summative tests include multi-part questions and rubrics for each question, describing the increasing complexity of reasoning needed to achieve each level. One question and corresponding rubric from a Grade 3 Test is shown:

1. Mr. Lewis arranges all of the desks in how classroom into 6 equal groups of 4. How many desks are there in his classroom? Show a picture and multiplication sentence in your work.
   • What does the product in your multiplication sentence represent?
   • Fill in the blanks below to complete a related division sentence.
     • _____ ÷ 4 = ______
   • What does the quotient in the division sentence represent?

The rubric shows the standards addressed in the question, as well as a 4-point progression toward mastery of these ideas. By matching student work to the descriptors in the rubric, we get a much clearer idea of their learning, and therefore can make better decisions about how to move their learning forward.

3. Changing What and When We Grade

Part of the reason students interpret their grades as a reflection of themselves has to do with the frequency with which their work is graded. In the traditional culture, that means that students are almost constantly getting messages about their ability through their grades, exacerbating the “I’m a C-student” mentality.

Another problem with the traditional system of grading is that, in order to obtain grades, teachers often give tests, quizzes, and other interruptive assessment tools. In a standards-based grading system, students know the learning goals, the progression to reach mastery of the goals, and receive constructive feedback about their learning on a daily basis. This doesn’t mean that students receive grades every day, rather their work is formatively assessed using the established goals and learning scales, and teachers and students keep track of their progress in a variety of ways. A great book by Skip Fennell, Beth Korbett, and Jonathan Wray provide some great strategies for using formative assessment. Other books by Robert Marzano, Tim Westerberg, and Susan Brookhart provide guidance to helping teachers make the shifts needed to improve the current paradigm.

 

No matter where teachers are in the shift to standards-based grading, its important to keep making small steps to lasting changes. Otherwise, we risk overwhelming ourselves and undermining our own efforts. I believe we can and will make a difference.

The (Mis)Perception of Student Ability – and What We Can Do About It

Ability – the capacity for doing something or achieving something 

The word “ability” is tossed around a lot in educational circles. It often refers to the talents and skills students demonstrate. In the performance-based culture that defines the education system in the U.S., teacher focus on student abilities has led to wide-spread beliefs that sorting students according to their abilities and providing instruction based on what we think they can do is the most effective way to help students learn.

Wrong.

Someone’s ability to do something can be determined only by that person. No one has the right to tell you what you are “able” to do.

Everyone Can Learn Math

When I first took Jo Boaler’s How to Learn Math for Teacher and Parents course several years ago, this was one of the overarching themes. It is such a simple statement, and one I intuitively knew to be true, but I had never heard anyone actually say it out loud before.

Everyone can learn math. Everyone. No one is born with mathematical prowess. Our learning of mathematics is just as much psychological as it is physical. We are a product of our opportunities and our experiences, our cognitive growth resulting from an interaction between basic human capabilities and “culturally invented technologies that serve as amplifiers of these capabilities” (Bruner, 1957).

Bruner elaborates on the notion of “cultural technologies” to include not only obvious devices, but also more abstract ideas such as how we categorize phenomena. In the case of mathematics education, the cultural technology in question is how we assess, perceive, and categorize students’ “abilities”, thereby sorting them into like-ability groups and, in my opinion, suppressing (as opposed to amplifying) their basic human capabilities.

Mathematics has long been treated as a performance subject, thought of as a list of isolated skills that students either are able to learn or are not. Jo Boaler (2015) refers to this as an elitist culture, as the practices often seen in mathematics classes fit Bruner’s cultural technology of a “sorting mechanism”, separating students into those who can and those who cannot do mathematics. In order to, as Dr. Boaler states, “achieve higher and equitable outcomes” in mathematics, we must not only recognize this elitist culture, we must begin doing something to change it.

Thinking Mathematically vs. Doing Mathematics

Equity in mathematics education is a topic that I have become increasingly passionate about over the past several years. Since moving from the classroom and into a coaching role, and then back into the classroom again, I have encountered scores of teachers who hold strong to their beliefs that some students can do math and the rest cannot. They teach as though the “rest” aren’t even there, only calling on those who raise their hands first, or who they know will have a “right” answer so they can just move on.

To make a shift in my own classroom, I try to create a culture of “mathematical thinking” instead of “doing mathematics.” This means that the forefront of work during our math block (as well as in other content areas) needs to be centered around problem solving, reasoning and proof, representation, communication, and connections (NCTM, 2000). No matter what content we are working with during our class time, my ultimate goal is to make sure students are engaging in one or more of these five processes.

The Launch Unit I created this past summer was a positive start to this shift, and the strategies I learned from Dr. Boaler in keeping these ideas alive really only scratch the surface of the kinds of changes that I feel need to happen in classrooms. I recognize that by making these changes in my classroom I can impact the lives of my students for this one year, but what happens next year? And the year(s) after that? Chances are, they will likely encounter more of the same performance-based, elitist culture that they experienced before my classroom.

Equitable Practices for Mathematics

Recently, our staff participated in a professional development session focused on culturally responsive teaching. One of the biggest takeaways from that session for me was the idea that we want to move all students toward being independent learners (Hammond, 2015). The excerpt that we read and discussed was reminiscent of Jo Boaler’s recent article Mathematical Freedom (2017), as it discussed the need to think beyond surface-level culture (music, food, clothing, customs, etc.) and instead dig deeper into the culture of learning within our classrooms and schools.

Hammond (2015) defines culturally responsive teaching as “an educator’s ability to recognize students’ cultural displays of learning and meaning making and respond positively and constructively with teaching moves that use cultural knowledge as a scaffold to connect what the student knows to new concepts and content in order to promote effective information processing” (p. 15).

To understand the reach of this definition, I connect back to Bruner’s ideas about culturally invented technologies, and instead of only thinking about “students’ cultural displays of learning” as an extension of their ethnic background, I think about these displays in terms of how students have learned to “do school.” The culturally invented technology that Bruner refers to is the culture of schools to present mathematics as a performance-based subject. Therefore, it is the educator’s responsibility to use such knowledge of school culture and strive toward more productive and equitable instruction.

In their article for the Journal for Research in Mathematics Education, Bartell, Wager, Edwards, Battey, Foote, and Spencer (2017) created a framework for linking equitable mathematics teaching practices to the Standards for Mathematical Practice (SMPs). They began this framework with a table describing equitable mathematics teaching practices.

Equitable Mathematics Teaching Practices

This table includes may of the same recurring ideas about necessary shifts needed to make mathematics instruction more equitable. The authors then suggest a process for making links between the equitable practices and the SMPs, as shown below.

Achieving Equity through Facilitating Thinking and Problem Solving

Bruner, in his book The Process of Education (1960), presented a counter-argument to Piaget’s ideas about readiness. “He argued that school waste time trying to match the complexity of subject material to a child’s cognitive stage of development. This means that students are held back by teachers as certain topics are deemed too difficult to understand and must be taught when the teacher believes the child has reached the appropriate state of cognitive maturity” (McLeod, 2008).

By framing our instruction around facilitating thinking and problem solving, we can impact all students’ learning and development. Jo Boaler’s ideas about low-floor, high-ceiling tasks are a great resource to teachers in making equitable shifts. These tasks help to shift the focus of traditional mathematics instruction from that of answer-getting and learning “how to do the math” to thinking, processing, and problem solving. Students can approach these tasks in a variety of ways, using their current mathematical knowledge. The teacher can then guide their mathematical learning by helping them think about reasoning, representing, and communicating their understanding while connecting that understanding to new concepts, structures, or patterns.

These scales created by Jo Boaler and her team at Youcubed help to show the kinds of specific shifts required in order to make this new framework a reality. The first one, focused on organizational freedom, shows the continuum of the norms that make up mathematics classrooms.

The second one, focused on mathematical thinking, shows the continuum of student approaches to mathematical work in classrooms.

Changing the Way We Work

In order to combat the culture of labeling students based on our perceptions of their abilities, we need to do more than provide surface-level lessons about grit, mindset, and other social-emotional topics. We need to begin taking a hard look at how we do things in our classrooms, identify practices that enable a culture of performance and elitism, and take strides together to change the way we work. The research and resources are out there; the ideas here are not new. But the changes in our language, practices, and treatment of “less-abled” students depend on teachers taking action in our classrooms, schools, and districts to disrupt “business as usual.”

Keeping the Mindsets Strong Post-Launch

It’s been a week since completing Unit 0 – Launching Our Mathematical Mindset Community (all four weeks) and my Math Block is in full swing. I shared in my last reflection that I am using a combination of Eureka Math, Zearn, and a few choice other supplemental materials to help my students uncover the mathematics expected in 3rd Grade.

In sharing my work through social media, I’ve gotten a few questions regarding how I plan to maintain the mindsets established in Unit 0 throughout the rest of the school year when most curricular materials are not designed with those ideas at the forefront. In this post, I want to offer how I plan units and lessons aligned to the standards while also keeping the Mathematical Mindset norms alive and well.

Anchoring Our Learning

First, I have this poster, which we built over the course of the four weeks, in a prominent place in the classroom. We refer to it often, especially to help connect what we are doing to why we are doing it. This poster is an anchor to the learning that is done on a daily basis.

Digging Beneath the Surface

I have read lots of posts and blogs about teachers having their students use growth mindset language in response to  classwork and homework, which I believe is a great start to helping students develop a more positive disposition toward mathematics. Using this kind of language in conjunction with old ways of working (closed tasks, procedural-based instruction, skill and drill-based class- and homework, etc.) merely scratches the surface of the kinds of shifts that we need to be making.

I think back to the first class Dr. Jo Boaler offered – How to Learn Math for Teachers and Parents. This was the course that really started to shift my thinking not only about mathematics instruction, but about what developing a growth mindset really means. At its very core, mindsets are psychological. To me, this means that observable actions, spoken words, and student attitudes are only part of the equation (no pun intended). There is also the matter of what we do in a classroom that affects students’ subconscious thinking, and I think that this matters more than what they are encouraged to say or how they are encouraged to behave.

According to Dr. Boaler, there are four categories of teacher actions that can encourage positive mindsets – Norms, Tasks, Assessment and Feedback, Grouping Students.

Norms and Expectations

This element of Mathematical Mindset classrooms was established during Unit 0, and is dependent upon the other three to continue to be effective throughout the school year. This is why my Math Mindset Charter is a constant anchor to my students’ learning – I want them making explicit connections to what we are doing in class to the norms we have already established. If we spent time talking about making mistakes, de-emphasizing speed, and the importance of collaboration, but then grade work based on mistakes made, give timed tests, and deny students opportunities to talk to each other about their math work, then the words on the poster are just words. They have no meaning. I believe that teacher decisions with regards to the next three categories are what give the norms we established their meaning.

Tasks, Struggle, and Discourse

The heart of mathematics teaching is the tasks that we give to students. Rich tasks are those that have a degree of openness (open-middle or open-ended) and help students to learn procedures with connections or to do mathematics (not just calculate answers). I always refer to Levels of Cognitive Demand (Smith & Stein, 1998) when selecting or designing tasks for my instruction as well as my assessment. The higher the cognitive demand, the more students will experience productive struggle. The more they struggle, the more I can facilitate discourse about their ideas. In the lower levels of cognitive demand, there is little to nothing for students to discuss, and so their mindsets about mathematics are not developed.

Games: Accessible and Higher-Cognitive Demand Tasks

One of my favorite tasks to use in my lessons are math games. Games, when designed well, often fall into the higher-levels of cognitive demand while also providing a non-threatening, low-stakes environment for learning for all students, regardless of their perceived “level”. Games also provide students with opportunities to practice using mathematical language, helping to reinforce conceptual understanding while developing procedural fluency.

Here are pictures of my class playing games involving multiplication (equal groups and arrays) and division (finding the size of a group).

Through these games, I was able to differentiate for students using questioning and use of tools (you can see some students using whiteboards to draw while others are using counters). I was also able to assess student understanding of the main focus of the lesson by designing the games with the lesson’s success criteria in mind.

My favorite places to find quality games are:

Open-Middle and Open-Ended Tasks

Since most curricular materials are primarily closed tasks, I spend my planning time working to open them up. Any task can be made an Open-Middle task merely by allowing students to make sense of, represent, and solve the problem in any way that makes sense to them. Open-Ended tasks are those tasks that have multiple possible solutions, and looking at the different solutions students find provides opportunities for uncovering the underlying mathematics within the task.

I try to use a mixture of Open-Middle and Open-Ended tasks to begin each lesson, where students work with a partner or their Math Team. I call this the Team Problem Solving portion of the lesson, and I try to keep it to a 10-minute time frame. I do this after my Number Talks Warm Up, which is also a 10-minute time frame. So, my “whole group” teaching consists of two 10-minute activities – Number Talks and Team Problem Solving.

Assessment & Feedback

I use the same approach to assessment tasks as I do for instructional and learning tasks. I try to keep the cognitive demand high, and I allow students opportunities to represent and solve the problems in ways in which they are comfortable. On select tasks, I ask them to include a specific representation in their solution process so that I can get a sense of how they are developing their fluency with important structures, such as using place value or properties of operations.

According to Dr. Boaler, assessment for learning is much more effective in helping students develop mathematical understanding that assessment of learning. Our educational system is so overrun with tests and equally poor measure of student achievement that assessment of learning (tests, quizzes, final reports, grades, etc.) need to phased out as much as possible.

One of my main goals this year is to incorporate formative assessment practices into my pedagogical repertoire. My main reason for doing this is the evidence suggesting that students learn more when assessment is woven into the fabric of instruction and feedback is frequent, accurate, specific, and timely. I recently purchased Francis “Skip” Fennell’s The Formative Five: Everyday Assessment Techniques for Every Math Classroom and plan on using it for my own professional growth goals this year.

Valuing Mistakes and Responding to Student Needs

We all make mistakes, especially when we are learning new things. Students not only need to hear us tell them that mistakes are a valuable part of learning, they also need to feel that mistakes are valued within the learning process.

In connection to designing rich tasks, teachers must also consider where students might falter and how to most effectively respond to their needs. Questioning is one of the most effective ways I have found to differentiate tasks for students in their varying responses.

Here is an observation form that I created with varying levels of questions based on my observations of students during their work time. I am able to jot down their name and check off questions I ask. It also helps me keep track of what questions I am asking to whom and make sure I strive to ask all students the highest levels of thinking questions.

Differentiated Questioning Checklist

Exit Tickets

Exit tickets provide me with a quick snapshot of how each student has responded to the instruction during each lesson. I use them at the end of my lesson time (the last 5 minutes). I have found that the Exit Tickets provided by Eureka Math and Zearn are sufficient for the purposes of analyzing student thinking and planning future tasks.

Grouping Students

With the evidence presented by Dr. Boaler about the counterproductive effects of ability grouping, I have made it my mission to keep my math groups as heterogeneous as I possibly can. While there may be times when I need to group students based on similar needs, these groups will be merely temporary, kind of a catch-and-release approach. When I “catch” students who have similar struggles, misconceptions, or confusions, I will bring them together for some guided problem solving, then “release” them back into the mixed group.

Collaboration: A Constant Expectation

To keep the various groupings of students in my classroom heterogeneous and mutually supportive, I have my students form and change Math Partners and Math Teams several times throughout the year. Part of the expectation of daily math work, like games and problem sets, is that students work on problems together, sharing their ideas, acting as skeptics, and justifying their thinking to each other through multiple representations. I often provide sentence stems to guide students in using precise mathematical language. For example, when playing the Go Team Division game (from Math Games: Skill-Based Practice for Third Grade by Hull, Balka, and Harbin Miles), one expectation for the game that I added was when students solved a division problem such as 20 divided by 4, they were to say to their partner, “Twenty objects shared into four groups is five objects in each group.” This was meant to emphasize the partitioning (sharing) meaning of division.

Putting it All Together

By focusing my planning time on rich tasks, invisible assessment practices, constructive feedback, and attention to how I am grouping students, I believe that I can maintain a focus on the Mathematical Mindset ideas established during Unit 0. Striving to meet these goals is beyond any set curricular materials or scope and sequence. I can use any of the materials at my disposal to meet these goals (with some modification). The materials I choose to use on a regular basis (Eureka Math, Zearn, Math in Practice, etc.) are due to the fact that they require the least amount of modification, therefore increasing the efficiency of my plan and prep time.

Unit 0 Reflections – Week 4

Of all of the lessons I wrote for the Launch unit, Week 4 was the part I was most apprehensive about. Looking back, I felt uneasy about how much was packed into a single week of instruction, knowing full well that students often need more time than we are willing to spend to learn routines and procedures. Still, the extending the Launch beyond four weeks made me uncomfortable, so I ignored my inner idealistic voice and jammed all of the components of a functional Math Workshop into a single week.

Last weekend, I decided to abandon that plan.

One reason had to do with school scheduling – our PTA had scheduled our Fun Run fundraiser event on Friday afternoon smack in the middle of my math time. So that took me from an already full five-day plan down to four days.

Another reason had to do with technology. I wanted my students to begin using Zearn, an awesome new (and free!) digital curriculum (details about this to come). I knew my students would need extended time learning how to log in to our district computers, and then learning how to log in and navigate the Zearn program on their own.

A third reason had to do with something all teachers stress over – meeting the unreasonable high demands of pacing during the school year. I had not yet started working on any of the major emphasis topics (we’d touched on some place value, addition, and subtraction ideas in previous weeks), and I knew at some point my principal was going to be asking questions. In addition, I began to consider how much time it would take on a weekly or even daily basis to prepare all of the games, activities, and lessons a fully-functional Math Workshop would require. A priority goal for myself is to lighten my workload, not exacerbate it.

So, I changed the plan. I started on Monday with an idea for student Math Notebooks.

Here’s my class working on decorating the cover of the notebook ( you can see my example up on the screen).

My idea was to have students begin each topic with an “I can” statement and essential vocabulary. Here’s what that page looks like:

Whenever possible, I plan on including visuals to connect the vocabulary terms to what we are learning in class. I also began making vocabulary cards for a Math Word Wall.

An Interlude about Zearn (this is NOT a sales pitch!)

Since I knew I was going to have my students use Zearn during their independent work time, I needed to restructure my math block. Zearn recommends students completing four digital lessons each week, meaning they would need to be using it just about every day. The typical Zearn classroom is divided into two groups: one that is working on their digital lessons and the other that is working with teacher guidance. Halfway through the session, the two groups switch. The math block is still started with a whole group warm up and some Team Problem Solving (about 15 minutes total).

I wanted my students to keep a record of their Team Problem Solving in their notebooks, so I decided that the next pages of the notebook would be devoted to this goal.

Students glue the problem into their notebook, and them work together to try to solve it using the RD2W Process (see Week 2). It was amazing to listen to their collaborative groups discuss ideas, ask each other what they should draw, and debate what the equations should look like. It made me really feel that spending the time during Week 2 was totally worth it!

Next, I wanted to start at least getting students comfortable with playing Math Games. I use a wide variety of resources for games (or I make them up based on the focus lesson). I think that games provide students with low-stakes practice time during which they can talk with each other using mathematical language, and apply the concepts we are learning during the lesson. Below are some of the resources I go to regularly for Math Games:

 

 

 

 

 

 

 

All of these are available to purchase – just click on the image.

The first game I used with my students was from the Partner Games kit called Groups Galore. You can see my students playing, and download the files below.

Groups Galore Game Board (Word)

Groups Galore Game Board (PDF)

Groups Galore Game Rules (Word)

Groups Galore Game Rules (PDF)

Groups Galore Recording Shee1 (Word)

For the purposes of introduction, I limited the game to factors of 2-5 (friendly numbers). In the game, students roll two numbers cubes numbered 2-2-3-3-4-5. They place the cubes on the game board to create a multiplication expression. They then build the expression using counters (making, say, 3 groups of 4). They then talk to their partner, saying, “I made 3 groups of four. That would be like if I had 3 bags, and each bag had four books in it.” They then record the factors (3 and 4), the product (12), an addition equations (4 + 4 + 4 = 12) and a multiplication equation (3 x 4 = 12) on their recording sheet. Partners take turns doing this and after 5 rounds, they each add up their products (some addition practice thrown in for good measure!). The player with the smallest total wins the game.

I find games to be much more engaging and a better use of practice time than any workbook or worksheet could ever be. I also find them very easily differentiated, as you can switch out number cubes or playing cards, add in or remove different layers of rigor, and even use the game itself as the focus of the teaching point when working with struggling students. Students are then invested in learning the mathematics because the problematic aspect of each game design is the mathematics students are expected to learn.

Launching Zearn

Zearn is a digital curriculum that mirrors the Eureka Math curriculum. The number sense activities, lesson topics and sequence, and representations included in Zearn are the same as are presented in the Eureka Math materials. This makes it an easy choice for my classroom, as I am a big fan of the coherent sequence of the Eureka Math. Students work through digital lessons independently for half of the math block, then work with the teacher for the other half. Best of all, Zearn Math is a non-profit organization whose mission is to help all students access quality curricula in order to foster a love of learning math, so it is free to sign up and use!

When students work on digital lessons, they work with virtual maniuplatives to help link the concrete, pictorial, and abstract levels of representation. They also have a Student Notes page for each lesson to allow for transfer of learning through writing and to provide teachers with documentation of their work. In addition, every lesson concludes with an Exit Ticket (the same ones as included in the Eureka Math curriculum). I am still figuring out what is going to work for me, but right now I am leaning toward having them complete the Exit Ticket after participating in the Small Group Lesson with me. Digital lessons are set up like this:

To be honest, the only part of the lesson sequence of which I am not a fan is the Sprint (because they are timed). They DO follow distinct patterns and are designed to have students notice and use those patterns to increase efficiency, but the timing is unnecessary (in my opinion). I just tell my students to do their best, and if they would like to redo a Sprint without the timing, they can let me know and I can print them out for them.

Students need their digital device (my school has laptops), headphones (to follow along with the digital lessons), and their Zearn Student Notes (I print out all of the notes pages for the Mission we are working in) in order to start the digital lessons.

The first day I launched Zearn (Tuesday), I displayed the main lesson (called Math Chat) on the interactive whiteboard. The next two days, I set up each child with a laptop so they could practice logging in and getting familiar with the digital tools.

I have to say, I have never seen such engagement during a math block (possibly due to the novelty of the program – time will tell), nor have I seen such retention of conceptual understanding from day to day. I designed my Team Problem Solving tasks for Wednesday and Thursday to review the concept of equal groups in multiplication in real-world scenarios (asking students to name objects that we could replace the word groups with, such as baskets or hands, and then asking students to find an example of equal groups in the classroom and using a multiplication equation to represent it, such as our desks or titles of posters). My students even told me that Zearn was “way better than IXL” (a program our school uses), because it was all timed and, according to them, “kind of boring.” Even better, students kept asking me all day on Thursday if they were going to work in Zearn that day!

Balancing the Digital and Small Group Lessons

Since the lesson part of Zearn leans toward more of a direct instruction approach, I have decided that my Small Group Lessons will be 100% problem- and game-based. It will be during the Small Group Lessons that I will engage students in collaborative problem solving, complex tasks, and discourse. I think this will be even more successful that when my original plan was because I will only be working with 10 students at a time, instead of trying to balance my time among the whole class. Zearn will make it much easier to differentiate for my small groups, while they work at their own pace through the digital lessons. I truly feel that I will be able to promote reasoning and problem solving even more with this new structure.

Looking Back and Thinking Forward

Overall, I feel the Launch Unit was a complete success (even though I completely changed Week 4). Students understand my expectations for their math work, including collaboration, and as long as I strive to keep those ideals at the forefront of my planning, I will be able to keep those expectations relevant all year long. Zearn will make my prep time even easier, since I no longer have to prepare anything for their independent work time. I can devote my planning and preparation time on Small Group Lessons – choosing good tasks, making games, and checking the Zearn progress reports to ensure that every one of my students is learning and loving mathematics.

Unit 0 Reflection – Week 3

This week, the focus of our math class was to establish the expectations for group work. I have to admit, based on observations of my students working cooperatively on activities over the past two weeks, I was a little apprehensive about how the week would go. Deciding to put a little trust in them (and myself) resulted in a tremendous week of learning for all of us.

The week began with some activities from Elizabeth Cohen’s Designing Groupwork (2014) and NCTM’s Smarter Together (2011), both of which focus on Complex Instruction, an instructional approach that helps students develop collaboration skills as they learn. You can also learn more about Complex Instruction by visiting the San Francisco Unified School District’s webpage.

One of the activities we did was the Broken Circles challenge (although I used Broken Squares). Students were challenged to silently exchange puzzle pieces so everyone on the team could make their own complete square. No one was done unless everyone was done. they needed to rely on each other to notice and pass the pieces needed, but they couldn’t gesture or point in any way.

This led to a discussion about what we like and don’t like to experience when workng in groups.

Which finally led to the creation of our expectations chart.

Next, I wanted the students to begin developing some collaboration habits, especially around sharing ideas, questioning each other, and making sure that everyone in the group gets a chance to participate. We worked on the Number Visuals task from YouCubed for two days, during which students felt some struggle (as several of them did not know what was meant by the word “pattern”). Still, we persevered. On the first day of the task, I just asked them to work together with their team to see what they notice and begin making connections among the visuals.

While there was some discord within the teams, that was to be expected on their first venture into collaboration. So, on the second day, I did a quick mini-lesson about patterns (growing and repeating) and scaffolded their conversations with the introduction to our Discussion Language (or Talk Moves).

During this time, I also made it an expectation that every member of the team use a different color to represent their work on a new, blank Number Visuals page (so I could see all of their contributions) and that each team would be responsible for sharing their results under the document camera.

Finally, we worked as a class to reflect on our group work and developed a rubric together that would help us get better in specific areas. This discussion was rich, and the students had many great ideas for what each level of performance looked like to them. Here is our final, compiled scale.

What was surprising to me that, although my students freely admitted that they were “probably at a Level 1 because they were just beginning with this” (their words), I actually felt that they were operating at the Level 2 descriptors – room to grow, but on the way.

What Worked During Week 3

Again, lots of ideas worked during this week. I am really glad that I dedicated a full week to laying this groundwork, as there aren’t many opportunities to explicitly focus on collaboration once we begin focusing on standards. Collaboration will still be an important part of our daily work, and I am glad that I took the opportunity to let students feel what being a part of a collaborative team felt like before taxing their brains with new math content.

The “Who Are We” task on Day 2 was a different, but good, starter to their new teams. I liked how it made me mix up the class (which they enjoyed as well). I plan to keep Teams heterogeneous, and pull students with similar needs when necessary. In fact, the whole week moved smoothly, as though the cohesiveness of the lessons had an inherent flow to them. This was by design, but it still surprises me when my grandiose plans actually pan out.

What I Might Change

During the “Who Are We” task, I felt that the student interactions were a bit superficial, so I might try to change the questions a little so that their conversations are more math-centered. I also might think about doing some pre-work with patterns and relationships that they might be familiar with before giving them the Number Visuals task. I might take Marilyn Burns’ advice to set up a task with a similar, but simpler task so that students activate some prior learning before tackling the real task.

Overall, I am deeply proud of my students this week, and extremely grateful that my plans have been working (so far). Next week, we finish our launch with an introduction to the Math Workshop stations and thinking about how I can modify our outdated and traditional curricular materials to continue building on the work we have done so far.

Unit 0 Reflections – Week 2

This week, my focus was laying the foundations for problem solving in my classroom. It represents the culmination of an idea I have had for several years now – setting these expectations with students early in the year so we can focus on the important aspects of problem solving (sense-making, reasoning, and representation) rather than realizing too late that students’ previous learning involved only answer-getting with no real understanding of the problems they were solving.

Setting the Stage for Problem Solving

In Week 2 of my Launch Unit, I borrowed a problem solving process from Eureka Math (a.k.a. EngageNY) called the Read, Draw, Write Process (RDW). In it, the W part refers to two types of written work, writing an equation and writing to answer the question. Just to make the “equation” part and explicit expectation, I modified it to be the RD2W Process. The chart below shows what I expect from students in each part of the process.

What Worked During Week 2

This week, many things I had originally planned worked really well. It also helped that we focused on perseverance during our Social-Emotional Learning Unit (using ClassDojo videos). In Day 1, we discussed what effective problem solvers do, using our discussions from Week 1 as an anchor. Students completed a sorting activity involving descriptors of Problem Solvers and Problem Performers. This idea originally came from an article from Teaching Children Mathematics.

Students also received a copy of this chart to keep in their Math folder.

Once we had worked to define the behaviors of problem solvers, I introduced the RD2W Process. This day was rather challenging, as I knew students had never been expected to represent problems using math drawings. I was also planning on using a type of problem they had never encountered before – numberless word problems. I chose to use numberless problems because I wanted students to focus on making sense of the situation and looking for mathematical relationships (add to, take from, put together, take apart, and compare) without being distracted by numbers. With all of this in mind, I prepared myself to be extra patient with their resistance. As predicted, they struggled, but I was ready to shower them with praise of their effort and keeping at it even though it was challenging. Students used whiteboards to record their work. I even had a couple of students volunteer to present their work to the class, and I praised them highly for showing courage and taking a risk.

By Day 3, I realized that my students needed more work with drawing math pictures and being able to identify the structure of a problem. These structures can be seen in the Glossary of the Common Core State Standards for Mathematics (CCSSM) and are based on the Cognitively Guided Instruction (CGI) structures.

The Math Pictures I chose to introduce are those that I knew students should have had some experience with in previous grades. Later in the year, we will add graphs, area models, and other models pertinent to Grade 3 content.

Students then spent two days practicing drawing representations of problems cooperatively using a Scoot activity I found on Teachers Pay Teachers. I chose to use 2nd Grade level problems so that students could focus their energy on drawing the representations and writing equations, rather than the complexity of the problems. On the first day, I blacked out the numbers, making them “numberless” problems once again. On Friday, I gave them the same problems with the numbers showing, and their job was to focus on identifying and labeling the unknown amount in each problem using a drawing and an equation.

What I Might Change

There really isn’t much I would change about this week. Although students felt the struggle early in the week, it was really a blessing in disguise as we continued to have conversations about ways to work through that struggle the rest of the week. They will remember this experience for the remainder of the year, especially whenever anything we are doing feels “hard.” I had considered including some two-step problems, but I think that might have presented too much frustration for them at this point in time. We will be working on solving word problems every day during the rest of the school year, so they will have plenty of opportunities to work through more complex problems.

Overall, I am really pleased with how this week went. Using the cooperative activity actually helped present students with a need to identify productive behaviors for group work (which happens to be next week’s focus). Halfway through the launch, and my students are well on their way to a productive math learning year!

Unit 0 Reflections – Week 1

With Week 1 of my launch unit in the books, I thought it would be a good idea to reflect on the week and share what worked, what didn’t, what I had to change, and what I learned. This past week was full of excitement for me and some new ways of thinking for my students. The videos from YouCubed’s Week of Inspirational Math were a big hit, and definitely did a great job launching student thinking about the different norms we will be using throughout the school year.

Some of you who follow me on Facebook have been asking how I organize my classroom for math. Being an elementary teacher with a relatively small classroom presented me with some challenges in my organization this year, but here is what I came up with:

These pictures show how my classroom is arranged and organized. My interactive whiteboard and presentation station is located to the left as I walk in the classroom door. My students’ desks have been rearranged already, but the space is very narrow, which presents challenges for making sure everyone has a good vantage point. Most of my walls are windows, making hanging up anchor charts a challenge. The accordion walls stay where they are, and they are pretty much all the space I have. The current display (Problem Solving with Math Practices) will eventually be replaced with strategy charts that correspond with our current unit. Toward the back of the room are the “student cubbies” I was provided that are too small to hold any of their belongings, so I made them my math thinking tool center. I have a variety of manipulative materials collected over the years, and they are organized into tubs and ready for students to learn how to use them.

What Worked During Week 1

The Math Attitude Survey I asked my students to fill out on Day 1 was especially enlightening. I learned a lot about my students’ mindsets, which helped me to prepare for the rest of the week. Many of them understand that mistakes help them learn, but are still uncomfortable when they don’t know something in in math. That makes it my goal this year to help them learn to be a little more comfortable in “not knowing” and “being wrong.”

I also felt that my launch into Number Talks went well, although I am going to make some tweaks to my format. This week I used PowerPoint to display the dot images (and quickly take them away), and then I recorded students’ thinking on chart paper. At this rate, I will end up using all of the forests in the Pacific Northwest for my Number Talks charts, so I have decided to start using Microsoft OneNote instead. You can watch a video of me doing a Number Talk with my class here.

Another success this week was with the tasks that I asked students to engage in, including the Squares to Stairs task and the Types of Mistakes task.

While many students experienced productive struggle (probably for the first time!) on the Squares to Stairs task, many were excited to share the patterns they saw and their ideas for extending the pattern without actually drawing the stairs. They also continued using the language of mistakes the rest of the week, particularly noticing their A-Ha Moment Mistakes and Stretch Mistakes.

My favorite lesson of the week was centered on Speed Isn’t Important (Week 1, Day 4). When I asked students to describe how they felt when they took timed fact tests, they responded (with my principal present – how perfect!) that they felt “nervous,” “uncomfortable,” “rushed,” and “pressured.” Even my one student who receives pull-out Gifted services admitted that he didn’t like them. They were really excited when we created the following chart, especially when I told them that they would never take a timed test in my class.

What Didn’t Work (or What I Changed)

Duirng the Types of Mistakes lesson (Week 1, Day 2), I included an article from Mindset Works about different types of mistakes. I had fully intended to rewrite this article over the summer into more student-friendly language, but time got away from me. So, instead of doing the jigsaw, I created the recording chart (something I probably should have done to begin with) and read the article aloud to my students, asking them to listen and jot down notes about each type of mistake. This turned out to be a good change, as most of my class admitted they’d never been expected to listen and take notes before. So, for many of them, this proved to be a great opportunity to stretch their listening brains.

I also changed the activity for Day 4, as we had begin working on some growth mindset coloring pages earlier in the day. I challenged them to make sure no two touching shapes were the same color. While this activity is not necessarily mathematical, it did reinforce the message that speed isn’t important; taking time to be precise with our work is. Students complained a bit (at first), but after a while they really seemed to enjoy it.

Finally, at the request of my team, I changed the activity for Day 5. The rest of my team had their students create “Number Posters” depicting different numbers that relate to their lives. I wasn’t going to have my students do it at first, but it was insisted that we all have something similar to display for Open House. They turned out pretty nice, nevertheless. I also think it helped to reinforce the overall message that numbers and mathematics are a part of everyone’s life, if we look hard enough.

Number Poster

Overall, I think the first week went very well, and we now have the norms established for a productive year. This week coming up will be focused on my expectations for “showing their work” when they are solving problems. I have a good feeling about the rest of this year!

Strategies for Developing Multi-Dimensional Fluency

In my last post, I shared my thoughts about broadening our definition of fluency in  mathematics from the one-dimensional goal of getting correct answers quickly to a more multi-dimensional approach. I described six goals, according to evidence, change what it means to be fluent with mathematics. These goals are:

1. Accuracy – getting correct answers, and understanding why they are correct
2. Efficiency – looking for and using shortcuts, and understanding why those shortcuts work
3. Flexibility – having a variety of strategies from which to choose, and understanding why the strategies work
4. Appropriateness – being able to defend a strategy as the most appropriate for a specific problem
5. Knowing From Memory – using a fact or procedure so often that it becomes a part of our working memory for easy retrieval during more complex problem solving
6. Application and Transfer – using known facts and procedures in creative and inventive ways when solving unfamiliar, non-algorithmic problems

Understanding the importance of each of these goals requires a shift in how we do things in our classrooms. No longer can fluency be achieved through timed tests and speed drills alone (as I have previously shared). Instead, fluency must be continuously developed and maintained through a variety of visual, low-stakes activities.

Building Fluency from Understanding

It is interesting to note that the many of the dimensions of fluency include a component of understanding. The National Council of Teachers of Mathematics (NCTM, 2014) considers conceptual understanding, strategic reasoning, and  problem solving to be the foundation of true fluency. Students need to understand why the answers they are getting are correct, why the strategies they are using work (based on place value, properties of operations, and number relationships), and why the shortcuts they are creating are possible. They also need to use their mathematical understanding to be able to defend the appropriateness of the strategies they choose.

Understanding, and therefore fluency, take time to develop. It is unrealistic to expect children to commit facts and procedures to memory before they have had ample time to explore the underlying meaning of the concepts involved. Listed below are tables showing the whole number and decimal computational fluencies expected by the Common Core State Standards for Mathematics (CCSSM) in Grades K-6, as well as some of the possible connections within the standards designed to help build the understanding necessary for fluency to develop.

It is clear (at least to me) that by speeding up or skipping work related to the conceptual prerequisites, students are not provided enough opportunities to develop and maintain true fluency. This happens most often when we depend on the textbook for our interpretation of the standards. I have found that most textbook resources treat conceptual standards in a very procedural manner, inadvertently missing important concepts in lieu of having students follow prescribed steps of a process.

Classroom Activities for Developing Fluency

In writing this, I have begun to realize that the best way to develop fluency is to give students plenty of time and space to work with the conceptual prerequisites shown in the tables above. The prerequisites involve students working visually, making sense of strategies and related written methods, and applying their previous understandings of place value and properties of operations to develop new, more sophisticated and (possibly) efficient strategies. Instead of just giving students time to practice, we need to think about activities that will help students practice productively.

Number Talks

One of my favorite activities to help students develop understanding and strategic thinking is Number Talks. In a Number Talk, students are presented with a visual, expression, or equation, and are asked to interpret the problem in some way. Developed by Ruth Parker and Kathy Richardson in the early 1990s. Originally designed to engage students in mathematical discourse and meaningful practice with computation through dot cards and mental math, Number Talks have expanded since then to include reasoning tasks such as Which One Doesn’t Belong?, Would You Rather?, 101 Questions, Estimation 180, and many others.

During my own classroom Number Talks, I have made it a personal goal to include a visual component to all of the tasks that I choose, because I believe that students learn mathematics better when they are able to make visual connections. Number Lines in particular are among my favorite visuals to use.

If you are new to Number Talks, or are looking for more guidance on how to improve them, you might consider consulting one or more of the resources listed below.

• Math Perspectives – resources and advice from the creators of Number Talks
• Mathematical Number Talks – consultation from a passionate Number Talks teacher

Books:

Fluency Games and Finger Activities

In her article “Fluency Without Fear”, Jo Boaler of Stanford University argued for a better way to help students reach fluency. The article included an Appendix with a variety of games that students can play to help build and maintain fluency without the pressures of timed tests and speed drills.

In another article by Dr. Boaler and Lang Chen, titled “Why Kids Should Use Their Fingers in Math Class”, the argument is made for students using fingers in strategic ways helps increase math achievement. Dr. Boaler also provides some ideas for activities that can help children use their fingers strategically.

Student Tracking

In an interesting blog post, lead CCSSM writer Jason Zimba describes the way he helps his own daughter keep track of her single-digit facts at home. This has inspired me to design ways, either using a similar “flash card bank” or using the addition and multiplication tables to have students shade in the facts that they “know from memory” (of course, after they provide ample evidence of their knowledge and understanding).

Classroom Assessment

In my opinion, it is not enough to just change our practices in terms of learning activities. We must also change our practices when assessing fluency as well. Using learning activities that send a clearer message of what fluency means is useless if we continue to resort to timed testing.

In their article “Assessing Basic Fact Fluency”, authors Gina Kling and Jennifer Bay-Williams provide suggestions for formative assessment strategies that can be used to assess the dimensions of fluency, including interviews, observations, journaling, and (untimed) quizzes.

Another free resources that teachers could use is Marilyn Burns’ Math Reasoning Inventory, which can hep guide interviews and help teachers keep track of student thinking strategies when solving computation problems.

Additional Resources

Here are a few more resources that I have found particularly helpful in my quest to improve my teaching and assessment of fluency:

Mastering the Basic Math Facts Series

Mastering the Basic Math Facts in Addition and Subtraction by Susan O’Connell and John SanGiovanni

Mastering the Basic Math Facts in Multiplication and Division by Susan O’Connell and John SanGiovanni

Time’s Up on Timed Tests

Final Thoughts

I am in no way suggesting that everyone needs to rush out and purchase every single one of the many resources I listed here. Like the fluency we are striving to develop in our students, making shifts in our classroom practices takes time and patience. I suggest setting a goal for yourself; maybe you want to begin implementing Number Talks as a daily routine, or you want to find some easy alternatives to timed testing. Then, seek out one or two resources that can help you along your way, and join some Teacher Support groups via social media (Jo Boaler’s How to Learn Math, Math in Practice, and YouCubed are three that I frequent often through Facebook). Also, don’t be afraid to ask these driving questions with your grade level teams and school leaders. Together, we can all make math classrooms a better place for students to truly learn (and develop fluency)!

 

Multi-Dimensional Fluency

Mathematical fluency tends to be a hot topic among teachers. To clarify, how to help students reach fluency is a hot topic. The idea that students need to reach fluency with mathematics is not generally contested. Teachers rarely engage in discussions about whether students should be expected to reach fluency, but rather which strategies work the best to get them there.

Historically One-Dimensional

When is comes to working on and assessing fluency in mathematics, teachers typically use one activity – timed tests. Timed tests involve a collection of single-digit fact problems, during which students have a finite amount of time to answer as many of the problems correctly as possible.

Teachers have used timed tests for decades, without question, thinking that the data collected from them was effectively helping them determine whether or not a student had achieved fluency. If so, the child was rewarded; if not, one type of learning activity was used to remedy the deficits. Speed drills, such as flash cards, mad minutes, and other timed activities have been the popular go-to choice of teachers when it comes to math fluency.

However, the practice of using timed tests and speed drills in the attempt to help students achieve fluency only provides a one-dimensional view of what mathematical fluency really is.

Broadening Our Definition

The National Council of Teachers of Mathematics (NCTM) describes procedural fluency as the ability to apply procedures accurately, efficiently, and flexibly; to transfer procedures to different problems and contexts; to build or modify procedures from other procedures; and to recognize when one strategy or procedure is more appropriate to apply than another (2014). This definition highlights four major dimensions of fluency:

1. Accuracy – the result of the calculation must be correct, and students must understand why the result is correct.
2. Efficiency – the pathway to achieving an accurate result should be done with the fewest number of steps.
3. Flexibility – the student has a variety of strategies from which to choose (based on place value, properties of operations, and number relationships) and does so regularly.
4. Appropriateness – the student is able to defend their strategy choices as the best choice for the problem at hand and recognizes that the one particular strategy is not the most appropriate for every problem.

In addition, the Common Core State Standards for Mathematics (CCSSM) include the expectation to know from memory all sums and products of two one-digit number. Although this is an important goal on its own, knowing basic facts from memory is only one step toward developing and maintaining fluency. Baroody (2006) describes basic fact fluency as “the efficient, appropriate, and flexible application of single-digit calculation skills” (p. 22). Notice the dimensions of efficiency, appropriateness, and flexibility are echoed in Baroody’s work, combined with the key word application – meaning that students should work toward knowing the basic facts from memory in order to apply them efficiently, appropriately, and flexibly when solving complex problems.

One of the major misconceptions about fluency is the “basics come first” mentality – that students cannot engage in complex problem solving until they know the basic facts. With this mentality, students come to learn that mathematics is about getting right answers fast. For those that struggle with memorization (which is a large majority of students – think about how many students in your own class resort to counting on their fingers!), this ideas causes math anxiety and a complete dislike of the subject (Ramirez et al. 2013; Young, Wu, and Menon 2012). In fact, Boaler (2015) found that memorizers were the lowest math achievers.

Multi-Dimensional Fluency

Based on the evidence, teachers need to start making shifts in the way that they approach fluency development. I deliberately use the word development, because I think that fluency is not an end goal. Students do not achieve fluency, it is developed and maintained through application of current knowledge in unfamiliar situations and constantly challenging and questioning what we think we know.

When approaching fluency in our math classrooms, we need to move beyond the one-dimensional view of getting right answers fast. Teacher need to look at the multiple dimensions of fluency and implement activities and assessments that encompass all dimensions.

1. Accuracy

Classroom activities and assessments should involve making sure students are calculating accurately. This could include asking follow-up questions such as “How do you know that is right?” and “What does that mean?” Students who can compute accurately understand the underlying meaning of the operations being used.

2. Efficiency

Activities and assessments should include tasks that involve students in thinking about the efficiency of their strategies. Even if they already know the fact from memory, they should still be asked to think about multiple ways to achieve the answer and to make connections between the variety of strategies at their disposal. This could include questions such as “Can you combine any of your thinking steps?” or “How might you use (insert known fact or number concept) to find (insert unknown fact)?”

3. Flexibility

Activities and assessments should include tasks that involve students in thinking flexibly with a variety of strategies. Similar to efficiency, they should be asked to think about several different ways to perform computations. Flexibility also involves students in making connections between strategies to see how they are related. This could include questions such as “Can you solve that in a different way?” and “How is (known strategy name) similar to or different from (unknown strategy name)?”

4. Appropriateness of Stratgey

Activities and assessments should include tasks that involve students in thinking about the appropriateness of the strategies they ultimately choose. Discussions around appropriateness should strive to move students away from resorting to counting one-by-one or even skip-counting in favor of more sophisticated (and efficient) strategies. This could include questions such as “Why did you choose that strategy?” and “Why did you think that strategy would work the best?”

5. Knowing From Memory

Activities and assessments should include a component where students may indicate that they knew an answer from memory. We’ve all encountered students, when asked how they know if an answer is correct (see Accuracy), who respond with, “I don’t know, I just knew it.” This response can no longer be accepted at face value, and students should be pressed to support their claimed knowledge with evidence. This could be as simple as providing a check box option next to each fact on handout labeled “I knew this from memory,” and then asking the student to explain how they would help another student who didn’t know the fact from memory yet.

6. Application and Transfer

Activities and assessments should include tasks that require students to use their computational thinking in order to solve more open or complex problems. This might include looking for arithmetic patterns, demonstrating creativity in inventing strategies, or even making their own puzzles and games that utilize flexible, strategic thinking. These kinds of activities should not be taught to students ahead of time, however, as that would defeat the purpose of assessing their ability to transfer their knowledge.

It goes without saying that these shifts won’t all be easy to make. To many curricular resources still depend on speed drills and memorization techniques in conjunction with fluency tasks. However, I believe that these shifts are imperative if we are to achieve the larger goal of developing mathematical thinkers. Mathematics is not about rote memorization or racing through timed tests. Mathematics is an open, creative, and visual subject in which all students are capable of excelling. As math teachers, we can take small steps toward making these shifts a reality, starting with resisting the urge to give timed tests or use speed drills and misinterpreting those data as fluency.

References

• Boaler, J. (2015). Fluency without fear. Retrieved from www.youcubed.org 
• Gojak, L. (2012). Fluency: simply fast and accurate? I think not! NCTM Summing Up.
• Kling, G., and Bay-Williams, J.M. (2014). Assessing basic fact fluency. Teaching Children Mathematics, 20(8), 488-497.
• National Council of Teachers of Mathematics. (2014). Procedural fluency in mathematics. Position statement.
• Ramirez, G. et al. (2013). Math anxiety, working memory and math achievement in early elementary school. Journal of Cognition and Development, 14 (2), 187–202.
• Young, C.B., Wu, S.,  and Menon, V. (2012). The Neurodevelopmental
  basis of math anxiety. Psychological Science Online First, March 20, 2012.

Launching Our Mathematical Community – A Unit for the Beginning of the School Year (Grades 3+): Week 4

The first three weeks of this launch unit have focused on helping students make some major shifts in their thinking about how mathematics is learned. In Week 1, we established several positive classroom ideas that would help guide our math work regardless of what it is. In Week 2, we specifically focused on problem-solving and recording our thinking about problems. In Week 3, we worked to become a collaborative community of mathematicians by learning the expectations for group work and class discussions.

Week 4 focuses on another aspect of math instruction that I feel is critical to student success – the Math Workshop. I define Math Workshop as times when we have done some learning as a class through problem solving, collaboration, and discussion, and we need some time to process and practice that new learning in a variety of ways. This week of lessons was particularly challenging to write as my first goal was to write lessons for my own classroom use. However, classroom management structures can vary greatly from teacher to teacher, and much of what I included will not be applicable to every situation. I did my best to include notes in the body of each lesson that explain my decision-making process. Feel free to make adjustments as you see fit.

My first exposure to the ideas of Math Workshop came from Laney Sammons’ Guided Math (2009). The concept of Math Workshop is a variation on the Reading and Writing Workshop model, in which students work on small-group tasks every day. I am a believer in the power of whole-class problem solving via collaborative group work and discussion, so I do not conduct a Math Workshop every day. I try to plan at least one Workshop each week, depending on the topic.

Shown below is how I structure my Math Workshop times:

Warm-Up – I start every class, regardless of the structure, with a Warm-Up. So far in this unit, I have only used Number Talks as the Warm-Up. As the year progresses, we will also use more Which One Doesn’t Belong tasks or Notice and Wonder tasks, again depending on the topic we are warming up for.

Status of the Class – Next, I do a quick check-in with students to remind them of responsibilities, expectations, and assign particular workstations (Computer Stations and Conferences or Clinics). I then give students an opportunity to choose their workstation, announcing to the class in which workstation they will be working and why (they state what their personal goal is).

Independent Work Time – This is when the majority of the class is working independently toward their personal math goals. Independently does not always mean they are working alone (as this would contradict our important message that mathematicians work better together). Some workstations require students to work alone, while others provide opportunities to work with partners. I choose to not have students work with more than 1 other person during Math Workshop to help keep the noise level appropriate. The Workstations I offer are inspired by Laney Sammons’ companion books in the Guided Math program – Guided Math Conferences and Guided Math Workshop:

• Conferences/Clinics – Math Conferences are brief, 1-on-1 meetings between the teacher and a student, in which the teacher tries to get to know each student individually as a learner and collect information about their specific learning needs. Math Clinics, on the other hand, are “Guided Math Groups” where I pull small groups of students with similar needs to provide more targeted instruction. In this introductory week, Math Conferences are launched, whereas I don’t start Math Clinics until we get into our first Unit of Study and I have collected enough individual student data through pre-assessments, initial interviews, and exit tickets linked to the first few lessons of Unit 1.
• Games for Mathematicians – These are partner games focused on specific math content, but more importantly they require students to use Talk Moves to defend their moves during the game. I love using games as engaging ways for students to practice content, rather than endless worksheets. Shown below are some of the resources where I find good games.
  • Math Games for Number and Operations and Algebraic Thinking Grades K-5 by Jamee Petersen
  • Well Played: Building Mathematical Thinking through Number Games and Puzzles Grades 3-5 by Linda Dacey
  • Every Day Counts Partner Games (these are older resources still sold by Houghton-Mifflin Harcourt, albeit pricey; I have reproducible books for Grades 2, 3, and 4)
• Using What We Know – One of the most challenging components of effective math instruction to include on a regular basis is application and transfer, in which students use what they have been learning to work on nonroutine, project-based problems. In this station, students will have those opportunities to work on unsolved problems, real-world projects, or even connect their math work to other subject areas.
• Independent Work – In this Workstation, students will work by themselves to finish incomplete lesson work, work at the Computer Stations, or work on anything I may assign them as an extension of a Conference or Clinic. Independent Work is split into MUST DO work (any of the above) and MAY DO work (additional individual games or activities that students may choose from if they complete the Must Do work before time is up).
• Developing Fluency – Fluency (accurate, efficient, flexible, and meaningful application of strategies) is an important part of math learning, but timed tests, flash card drills, and other speed-based activities can be counterproductive. In this Workstation, students may play individual or partner games that are designed to help develop fact fluency, or they may work on activities that help to develop fluency with problem solving. Either way, students will have to blend their understanding of math concepts into their work here to develop fluency without relying on rote memorization.
• Expressing Math Ideas – Writing is another component of learning mathematics that students rarely get adequate time for. In this Workstation, students will keep a Math Journal to express their ideas in writing about a variety of math topics and task cards, including communicating problem solving strategies, expressing mathematical relationships, or describing patterns.

Listed below are the lessons included in Week 4 – Establish Routines and Procedures for Math Workshop. I have provided links for a Word document and a PDF file for each lesson, as well as Word documents and PDF files for additional materials that I created. Any other resources I have referred to in the lessons are noted in the Resources section of each lesson. Feel free to download, read, and leave feedback. I would love to hear what others think!

Week 4 – Establish Routines and Procedures for Math Workshop

Lesson 16 – Introduction to Math Workshop and Preparing for Math Conferences

Lesson 0.4.16 – Introducing Math Workshop and Preparing for Math Conferences (Word)

Lesson 0.4.16 – Introducing Math Workshop and Preparing for Math Conferences (PDF)

Math Conference Notes (Word)

Math Conference Notes (PDF)

Lesson 17 – Fact Fluency & Using the Computer Stations

Lesson 0.4.17 – Fact Fluency Games and Using the Computer Stations (Word)

Lesson 0.4.17 – Fact Fluency Games and Using the Computer Stations (PDF)

Lesson 18 – Using Task Cards & Math Journals

Lesson 0.4.18 – Using Task Cards and Math Journals (Word)

Lesson 0.4.18 – Using Task Cards and Math Journals (PDF)

Lesson 19 – Playing Partner Games

Lesson 0.4.19 – Playing Partner Games (Word)

Lesson 0.4.19 – Playing Partner Games (PDF)

Lesson 20 – Celebrating Our Mathematical Mindsets

Lesson 0.4.20 – Celebrating Our Mathematical Mindsets (Word)

Lesson 0.4.20 – Celebrating Our Mathematical Mindsets (PDF)