I created this task for Kindergarten MLS: K.DS.A.1-2 and CCSS: K.MD.B.3. Here is the activity in Desmos.
What do you notice? What do you wonder?
Are there more tigers or zebras?
How many more?
How do you know?
I am just obsessed with these tiny Tic Tac boxes and had to create a task so that I had an excuse to buy them! This task is well suited for several grade level standards ranging from 2nd grade to 4th.
The grade 2-3 option is in orange below and the 4th is in blue.
CCSS 2-3 Options: 2.NBT.B.6, 3.OA.A.3
CCSS 4 Options: 4.NBT.B.5 or 4.OA.A.3 (with write an equation)
Grades 2-3 Act 1
Grades 4-5 Act 1
What do you notice? What do you wonder?
Grades 2-3 question: How many little Tic Tac containers are in the big one?
Grades 4: How many Tic Tacs are there in all of the containers?
Write an estimate that is too low, one that is too high and one that is reasonable.
Grade 2-3 Act 2:
There are 15 of each flavor.
There are 4 flavors.
Grade 4 Act 2:
Write an equation with a letter standing for the unknown quantity and solve.
Act 3 Grades 2-3
Act 3 Grade 4
This task isn’t as straight forward as some, but I was making a pie and only needed one crust. This recipe makes 2…so I had to figure out how to adjust my ingredients. I think this could be a good one for 5.NF.B.7. If you have a way to make it better, please share!
How much of each ingredient is needed to make half the servings?
What information do you need to solve this problem?
I finally got outside for a bit today just before the sun set and wanted to get my vegetables started. I had saved some pots from last year, but I ran out and remembered I had empty cardboard egg cartons! When I started to plant my left over beans, I realized this could be a really fun task. I love this task because it reminds us that even though we are currently stuck at home, we can still do the things we love and watch life bloom all around us! This could fit several OA and NBT standards in grade 1 and 2, but I like 1.NBT.C.4.
How many beans did she plant?
What’s a number that is too high? Too low?
What information do you need to solve this problem?
There have been multiple studies over the past few years about detracking math pathways to provide equitable experiences for all students. This excerpt from the National Council of Teachers of Mathematics article, Linking Catalyzing Change in MS Mathematics illustrates NCTM’s stance on inequitable pathways:
“The course pathways that a high school offers are most often a continuation of the course pathways of its feeder middle schools. Although de-tracking might seem like an impossible feat, Catalyzing Change offers evidence of it being done successfully (NCTM 2018, pp. 18–19). In the middle, we must find a way to rebuild our structures to dismantle tracking and instead implement effective intentional and targeted interventions for students who need additional support. Further, there is no race to calculus, and schools should not succumb to perceived parental or district pressure to accelerate. In the limited cases in which acceleration is warranted, it should be done appropriately. In other words, “care must be taken to ensure that opportunities are available to each and every prepared student and that no critical concepts are rushed or skipped” (NCTM 2016, p. 1)
I am a mother of two boys. They were both identified for our district gifted program at a young age. I remember being so excited that my child was gifted. I felt a sense of pride and elitism that he was given “the gifted gene.” I cringe as I think back to my narrow view of intelligence and the conversations I have had with him in the past. As a parent and an educator, I have learned much over the years as my first son journeyed through school. I watched as his attitude towards mathematics changed drastically as he began to struggle with mountains of homework and procedural mathematics.
He graduated last year and despite scoring the highest on his mathematics section of the ACT, is terrified to take a math class in college…to the point that he feels his dream of becoming a marine biologist might not be achieved because he is so “bad at math.” Let me assure anyone reading this that he is in fact, not bad at math. He is like so many like him who have had bad math experiences that were perpetuated by a broken system. Experiences I as a parent encouraged and despite being a part of the educational system, did not understand. He has been taught that mathematics is about rules and procedures, memorization and test taking. I write this post in the hopes that other parents who might be uneducated (as I was) in the potential perils of acceleration, will read it and think twice about the pathway they choose for their child.
Please know that my son has had fantastic teachers over the years. I do not blame a single teacher for my son’s math anxiety. I blame the educational system that is wrapped in red tape and far removed from research.
I have talked to district coordinators across the state and the country who are frustrated that expert voices are not used in policy making. I recently attended an NCTM workshop on Catalyzing Change where we dedicated over half of our time to how we might make small strides in helping our communities and policy makers understand the research recommendations. I heard frustrated voices and saw defeat on the faces of math teachers who seek to change pathways for students as they looked to the battle ahead. My hope is that some fraction of those teachers and leaders will continue fighting for equity in education and for instruction that allows for deep understanding of mathematical concepts. I hope that they continue to advocate for kids even when their proposals are rejected time and time again…because that is why we have chosen this profession.
My second son was quite a different story; Different in the fact that I was more knowledgeable and less naive. When he tested for the gifted program (a choice I still struggle with), I talked to him about it first. I explained that this was a choice program that he could participate in and shared with him some of the classes that were offered. I explained that it would be a day away from his home school each week. I also explained that some people thought it meant you were smarter than other kids. I assured him that was not the case, his intelligence is directly related to his work ethic and curiosity for learning. It has always been his choice to go or remain at his home school.
He is now in 6th grade, the year that math pathways are chosen. Our district offers a pathway that splits 7th grade math into two tracks, 7th grade math or 7th Grade Accelerated. Students in the accelerated pathway continue to 8th grade algebra and then to Geometry in 9th. Students must score advanced on MAP and receive a minimum iReady score of 531 at middle of the year to be qualified for advanced math courses.
I have known for years that I would be requesting he be placed in 7th Grade math and not accelerated. My opposition to this has only amplified as more studies have come out and my understanding of the standards has grown. Our district has also adopted an inquiry-based mathematics curriculum resource that I believe offers students a much deeper understanding of the standards.
He qualified for 7th grade accelerated on both measures. I have attached his reports below so that you can gain a better understanding of what I am trying to illustrate. The first figure shows his score on the Missouri standardized test last year. Note the narrative under his performance level.
“Advanced demonstrate advanced proficiency in the knowledge and skills necessary at this grade level/course of learning as specified in the content expectations. The students are well prepared for the next grade level or course and are well prepared for college and career readiness.”
Anything stand out? Nowhere in the narrative does it suggest that he should skip a grade because he is advanced. It states that he is well prepared for the next grade level course; 7th grade math.
The next figure shows a breakdown of his score by domain. Notice that he has two domains that although he still scored advanced, are not maxed out. Notice that Geometry, Measurement and Data and Statistics was his lowest category. I say this because his greatest growth this year according to district testing is in Geometry. In talking with teachers who teach accelerated courses, in order to accommodate the compacting of the curriculum, they are forced to leave out many standards in these domains. Math is a web that is interconnected. If students lose out on experiences with standards that help build on others, they will have gaps in understanding.
The next figure shows a report from his MOY iReady placement. Note that he is 27 points above the minimum score for placement in accelerated math yet his placement is Late 6 in all domains; he is on grade level.
Below are the unit dependencies for the standards aligned curriculum resource we have adopted. As you can see, the pathways for 6th, 7th and 8th grade math are interconnected. This resource builds procedural fluency through conceptual understanding. Students make sense of rich tasks through productive struggle and collaborative problem solving. I don’t want my son to miss out on any of that as I believe (and NCTM recommendations agree) that a strong foundation in the 6-8 learning standards will set him up for success in Algebra in high school.
It is important that we, as educators, but also as parents think deeply about the messages we are sending to our kids (and students). Students are given a very clear message when they are or are not placed into advanced pathways.
I am a math person.
I am not a math person.
I have chosen to discuss this with my son and explain to him why I have requested he be placed in 7th grade math. I want him to know that I have learned from my mistakes and I care about his understanding and attitude towards mathematics. I am confident that as a parent, I will continue to make mistakes, but hopefully I can remedy this one and his love for mathematics will remain intact.
Anyone who knows me knows that my greatest passion besides teaching and learning is ocean preservation and environmental awareness. Tonight while checking my Facebook feed, I came upon a cause that I support and realized that it would be a perfect springboard to a conversation with students that could have an impact on their math reasoning as well as spread awareness about the global marine debris problem.
Here is the scenario along with a few questions to spark conversation. This is an opportunity to remind us that math is a powerful vehicle for many things – including some of the world’s most pressing problems. I think these could turn into an even bigger research project that involves looking at other areas that are affected or at the costs of cleaning up this area including labor, equipment, disposal, etc. There are so many possibilities…
What do you notice? What do you wonder?
What question could you ask about this scenario?
What is the average amount of money donated by one person?
Here is the link to the donation page so you can look at the new totals and ask some additional questions. (and if you feel the need to donate to the cause while you are there…the turtles thank you)
The past few years have been an exciting time in math instruction. Research on brain plasticity and mindset have caused a shift in the idea of what it means to know and do mathematics. Consequently, our district has seen a downhill trend in standardized test scores in mathematics over the last few years. This has forced us, as educators, to take an intentional look at our teaching practices.
The problems students are being asked to solve require interpretation, analysis, and resilience. We knew we had to change our instructional practice to reflect the rigor of the tasks they would be asked to solve. Yes, on the test, but ultimately as future employees and citizens of our world.
Gone are the days of I do, we do, you do. That model never allowed for enduring understanding of conceptual mathematics. It was designed to promote recall and computation and it succeeded in that. However, we now know that we are preparing students for a world that no longer requires such skills. Our charge is to develop thinkers. Problem solvers. Grit.
This offers a docatomy for many educators, one that faces off the way they were taught to teach and what research now shows as best practice.
As we began our pilot of IM K-5, our teachers were struggling with that very issue. For years, we had implemented a workshop model that followed the I do, we do, you do model and used ability grouping to differentiate content for students. Based on recent research, we had modified that model to also include whole class problem solving days, but we had lots of work to do in supporting teachers in their new roles as facilitators.
The IM curriculum was designed for much of the time to be spent with students working in heterogeneous collaborative groups and partnerships while the teacher monitored and facilitated discussions. When the pilot teachers met after implementing a few lessons in Unit 1, they discussed their struggle with what differentiation looked like in this model.
It became clear that we had had a very narrow vision of differentiation. This required us to look at differentiation in a new light; one that empowered students to be the mathematical experts in the room.
We began looking deeper at The 5 Practices for Orchestrating Mathematical Discussions to begin that shift.
Through the careful planning and monitoring of student interactions, teachers could utilize student work to connect strategies to support students at each level of their learning.
Illustrative Mathematics allowed this to become common practice due to the structure of the lessons and the ample opportunities for formative assessment.
Anticipate
IM has structured the teacher support materials to offer possible student solutions for each of the warm ups and activities as a means for teachers to begin planning for what strategies they might want to call attention to. This was an invaluable support for teachers as many might not have been able to generate all possible solutions on their own.
Monitor, Select, Sequence, Connect
During the warm up and activities, students are encouraged to approach the concept in multiple ways. The teacher monitors by watching students approach the problem and recording strategies.
The teacher can then select which strategies she/he feels need to be highlighted based on instructional goals. These goals might differ from day to day.
For instance, one day the teacher might choose student work to highlight because it provides other students a structure for keeping track of their work.
For example, in a first grade classroom where students are working on combining two collections, many students might be recreating the collections by drawing strawberries when they are part of the scenario. A teacher might have a student show a piece of work that has drawn a literal interpretation of strawberries and then after show a piece of work that a student drew circles to represent strawberries (Sequence). She/he might ask connecting questions like, “what is the same about these two representations? What is different? Which do you think was more efficient?” (Connect)
Or in the same scenario, a student has drawn random circles on the paper while another student draws five on top and three on the bottom to show the two collections. Or perhaps, a student used different colors for the two collections. A teacher might say, “show me the five in the first collection. How about the three in the second collection. How do I know which collection I am looking at? How does that support me in finding a solution?”
One of the greatest struggles voiced by teachers in shifting to a role of facilitator is having the right questions to ask. We use this monitoring sheet that includes a place to note possible questions you might ask to connect strategies or dig deeper into student understanding.
I truly believe that problem based resources, like Illustrative Mathematics, have the potential to not only support students in developing enduring understanding of mathematical ideas, but also to develop the capacity of teachers in becoming expert facilitators of student learning.
Last year, we began reviewing curriculum resources for middle school math. When we put out the proposal asking for submissions, we asked for some very specific things based on recommendations from the National Council of Teachers of Mathematics (NCTM).
Some of the recommendations and research we used in identifying high quality instructional materials are listed below.
We used 8 Effective Teaching Practices from NCTM’s publication Principles to Actions.
According to Hattie’s research, Teacher Estimates of Achievement had an effect size of 1.62 and Collective Teacher Efficacy had an effect size of 1.57.
We utilized the research presented by Jo Boaler on the brain’s ability to grow and change to set the stage for holding high expectations for all students. We knew that quality tasks for students was a must have. Rigorous tasks with multiple entry points.
However, we were also looking for resources that would build teacher capacity. We weren’t looking for another resource that would have teachers reading out of a textbook. We wanted to shift instruction and pedagogy in a way that would develop students as critical thinkers and problem solvers and teachers as facilitators of learning, not keepers of information.
This graphic has showed up in many places, but I first saw a similar one in Boaler’s book, Mathematical Mindsets.
On first look, it made me think of the book, Hidden Figures. The women in the retelling were hired by NASA to be computers. Their job was to crunch numbers, and to do it fast. However, with the arrival of the computer, the women were faced with becoming obsolete to a machine. So they adapted by becoming what NASA needed most at that time, problem solvers.
It was clear that if continue to teach math in the way we always have, we will produce the same types of thinkers; students who are able to follow directions and procedures; computers. What the world’s largest companies are asking for are thinkers. Students who have been taught how to think about problems in new ways and approach them from multiple pathways. Students who can work collaboratively to come to a solution.
This is what we knew we needed – instructional materials that were student-centered; Resources that allowed kids to make sense of problems and persevere in solving them.
Which leads me to another set of standards that played a huge part in selecting a 
resource; The Standards for Mathematical Practice.
The content standards get a lot of attention across the country, but many people (even some math teachers) are unaware that there are also a set of standards for mathematical practice.
These are the employable skills that are essential for our students to leave our systems with. Many of us have heard the quote mentioned by Richard Riley, former US Secretary of Education,
“We are currently preparing students for jobs that don’t yet exist, using technologies that haven’t been invented, in order to solve problems we don’t even know are problems yet.”
So the question remains, how DO we prepare students for such a world? If we use employable skills as a model, all signs point to developing students who can think, reason, and solve problems as part of a team.
In our search, we reviewed several resources, one of which was Illustrative Mathematics. Illustrative Mathematics was the highest ranked curriculum by http://www.EdReports.org, an independent non-profit designed to support schools and educators in evaluating instructional materials.
We took a team of teachers to a 4-day training in Atlanta, GA to evaluate it further. The teachers were hooked and requested to pilot it the following year. We opened the pilot up to all of our middle school teachers and many participated.
After piloting the resource, we collected feedback from teachers and heard things like, “Illustrative Mathematics has made me a better (more effective) teacher.”
“My students are talking about math in ways I never imagined they could.”
“One of my students who had never spoke in class before, came up and explained their strategy for a number talk today.”
They reported increases in demonstration of understanding on summative assessments, district benchmarks, and time spent on productive struggle.
The pilot experience was not easy. Teachers struggled with the change in pedagogy. They struggled with beliefs and practices that they had held and used for years being turned on their end. They worried that students wouldn’t formalize concepts by the time standardized testing came along. However, at the end of the pilot, teachers overwhelmingly wished to continue using the resource due to the growth they had seen in students thinking and reasoning skills.
We knew that in order to shift instruction in this way, we needed to support teachers with upfront and job-embedded professional learning opportunities. So we arranged for Illustrative Mathematics trainers to come in to do a 2-day training for each of our teachers, SPED teachers supporting math, ELL coaches, Learning Specialists, Title 1 Support and any administrators who wished to attend.
We then planned four quarterly half day collaboration days for teachers at each grade band. Tomorrow is our first collaborative day together and I am excited to continue our journey on refining our practice.
I’d love to hear stories from other districts and ideas of how to best support teachers in the continued implementation of problem-based instructional materials.
After falling in love with teaching mathematics in the classroom, I knew that I wanted to eventually end up in a learning support role for mathematics instruction. Shortly after attending a Math Add+Vantage training offered by our district Title 1 department, I accepted a position as a K-5 Numeracy Coach. I was trained shortly after as a Math Recovery Intervention Specialist.
I spent six years working with teachers on intervention and instruction. The schools I worked in had some of the highest (over 85%) free and reduced lunch rates in our district. Students came to us with varied educational backgrounds, many with very few experiences in numeracy. In Title 1 Math, we utilized an instructional configuration for Math Workshop that the team had created based on the Comprehensive Literacy Model. I worked with teachers to get math workshop and guided math set up in their rooms for those six years. We looked at data, we visited other classrooms, we pre-assessed and re-assessed. Our students were showing growth, but they weren’t making huge gains in proficiency. The question was, why?
I learned many lessons in my years as a coach, but probably the one that has influenced my philosophy on education the most is that it is possible to intervene too much; and we were doing it. I had set my expectations for students too low and was not exposing them to tasks that allowed them enough time and space for productive struggle. We were so concerned with meeting students “where they are” that we had completely shifted instruction to intervention and removed the rigor. I’d like to say that this realization came overnight, but the truth is it came after years of reading, learning, teaching, reflecting, and examining my practice. To be honest, I’m still on the journey.
Two years ago, I accepted the curriculum coordinator position in our district for K-12 Mathematics. The position had been re-imagined and was posted as Coordinator of 21st Century Numeracy. When I accepted my current position, I was charged immediately with putting in place a math workshop model for all of our elementary buildings. I was excited about this because I knew it would help teachers be more purposeful about instruction and this created opportunities to open dialogue about student understandings and next steps for instruction.
I told myself that we were doing workshop right because we had flexible groups that changed regularly. But the truth was, in practice, that wasn’t the case in every classroom and even with flexible grouping, we were still tracking students into ability groups that stayed fairly stagnant. Even though our practices were perhaps the best case scenario for math workshop, we were missing the boat on many other aspects of high quality instruction.
I also struggled with where and when to use high quality tasks such as 3 Act math tasks. Our current structure did not allow for those opportunities.
It was really a perfect storm that led me to my current understandings and beliefs.
Jo Boaler‘s book, Mathematical Mindsets, spoke to my heart. The findings that she proposed based on brain research that a) everyone can learn math at high levels, b) mistakes are really important in the learning process, and c) we must value depth over speed, made me look at our practices through a new lens.
It was Dan Meyer‘s Ted Talk that introduced me to 3 Act Math years and years ago that helped me develop my belief in problem based learning (and become obsessed with creating 3 Act math tasks for elementary). I’ve had the opportunity to attend a few of Dan’s sessions at conferences since then and his words stuck with me, “Be Less Helpful.”
I couldn’t reconcile those words against some of our instructional practices. Everything we had designed to support students was about being more helpful – which had the effect of making them helpless. We noticed that when students faced something difficult, they would immediately ask for help instead of persevering. (For my thoughts on praise and learned helplessness, check out my prior blog post, How Do I Change Math Class Tomorrow?)
It was Robert Kaplinsky‘s Open Middle problems that provided the perfect model of how a task with multiple entry points could allow all students to engage in high quality math tasks.
It was Christopher Danielson‘s books, Which One Doesn’t Belong and How Many?, that tugged at the importance of sense making.
It was Sherry Parrish‘s book, Number Talks, that allowed me to look at fact fluency and number sense in a whole new light.
It was Brian Bushart‘s Numberless Word Problems, Steve Wyborney‘s Estimation Clipboard, Fawn Nguyen‘s Visual Patterns…the list goes on and on.
It was Graham Fletcher‘s link between 3 Act Math and the 5 Practices that had me looking at the power of combining high quality tasks and intentional teacher moves.
In 2017, at the NCSM National Convention, I was given a copy of Jennifer Lempp‘s 
new book, Math Workshop. In her book, Jennifer proposed a mix of guided math days and what she calls task and share days. Task and share days were essentially days in which students work in heterogeneous groups to solve problems and share their solutions with the class.
It was after reading her book that I realized it was time to modify our instructional model. The following year we introduced two instructional models: Math Workshop days and Problem Solving days.
That was our first step in a journey toward equitable teaching practices.
I truly believe that the lack of quality tasks (with multiple entry points) is the reason we have struggled with mathematics instruction for so long and why we have not been able to shift our pedagogy from teacher-centered to student-centered instruction. As the list above clearly shows, that time is over. In my future blog posts, I will share other open source resources for high quality curriculum that puts students in the drivers seat of enduring understanding.
I happened to be at Staples and I glanced down the Post-it aisle and noticed that not only was there a giant pad of Post-it notes but now they had come out with an even bigger pack of Post-it notes! I bought them not knowing exactly how I would use them. I remembered some other 3 Act tasks I had seen using Post-its and an article I read last year in an NCTM journal about using Post-its for area and perimeter. I thought about how I could use them in a task for area in a way that makes it a little more engaging for kids and that’s how Stick Up Robot came about.
Estimate. Write a too low and a too high estimate.
Robot Blackline (can print as a scaffold for organizing work)
mscastillosmath 