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"If we are serious about encouraging students to develop growth mindsets we need to provide open tasks that have the space within them for learning, not short tasks that students are meant to get right or wrong. Tasks are made more open when they have or encourage:
• Multiple entry points
• Multiple ways of seeing
• Multiple pathways and strategies for solutions"

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"Low floor, high ceiling stretching helps engage young students with interesting and challenging mathematical ideas that may serve as context for learning existing content that may be less interesting and challenging mathematically"

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Gadanidis (2012)_Why Can't I Be a Mathematician?.pdf

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7. Average Score on Each Unit's Summative Assessment

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7. Average Score on Each Unit's Summative Assessment

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While both problems led to student engagement and interest, students spent more time working on the LF/HS task. Additionally, students wrote more on their boards for the LF/HS task, with both greater organization and dilligence. After the class in which the students engaged with the LF/HS task, about half the class remained late to check answers and compare procedures. This implies that students were more engaged with the LF/HS task than the non-LF/HS task.

Furthermore, there is a significant increase in average performance on summative assessment for the unit that featured LF/HS problems. If the summative assessments truly measured understanding of the material, then this would imply that the integration of LF/HS problems, at least in some small way, increased students' overall understanding of the unit's material.

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This problem, also original, is LF/HS because any student can enter. Even if a student's strategy is to brute-force the problem, as long as the student understands the very basics of factoring, he can dig into the problem and make progress toward a solution.

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While scaffolded and challenging, this problem is not LF/HS because of the challenge of finding a foothold. There are only a few right answers, and they are hard to progress toward without having an idea of how to solve the entire problem.

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To answer questions 1 and 2, I examined student work and assessment scores from two units. The first unit covers the simplification of single-variable first-degree rational equations. The class worked through this unit in October. The second unit, covered in May, introduced students to factoring.

I selected these units because they were the two most challenging units of the year. In both units I presented students with challenging problems, yet only in one did I integrate the Low-Floor/High-Ceiling approach. Since the same students did the problems in both units, I concluded that these two units would present the most meaningful data from which I could draw conclusions.

In the examples that follow, it should be noted that the data sets are somewhat small and skewed. The topics in the units are slightly different, and my teaching style changed throughout the year. The students' experiences in the class, and in their lives, may have changed throughout the year, potentially contributing to the changes observed. That said, the data still provide a glimpse into the effect of integrating a LF/HS approach to math problems.

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In addition to their research on mathematics education, Jo Boaler and her team have a repository of LF/HS problems.

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.
— Dan Meyer (@ddmeyer) April 27, 2018

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.

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I've heard "low floor high ceiling tasks" too many times. They're "roach motel tasks" now. Easy for students to check in but hard to check out. This is canon! 🐛💯🐛.

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10. How has this study impacted my teaching philosophy?

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10. How has this study impacted my teaching philosophy?

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Click below to find out!

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Boaler, J. "Unlocking Children's Math Potential: 5 Research Results to Transform Learning,” You Cubed. Retrieved from https://www.youcubed.org/resource/short-impact-papers/

Gadanidis, G. (2012). Why Can't I Be a Mathematician? For the Learning of Mathematics, 32(2), 20-26. Retrieved from http://www.jstor.org/stable/23391959

(As described in the paper to the left, the above two articles (along with more resources on YouCubed) opened my eyes to the credibility and power of LF/HS problems. In addition to these, contextual and overarching information came from the sources below:)

Leinwand, S. (2014). Principles to Actions: Ensuring mathematical success for all. National Council of Teachers of Mathematics, Incorporated.

Stein, M. K., & Smith, M. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. National Council of Teachers of Mathematics, Incorporated.

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Synthesizing personal research and design with existing scholarship on the subject, it seems that the answer to both Critical Questions #1 and #2 is “Yes.” Students demonstrated harder work and deeper understanding on assessments, both formative and summative. While anecdotal and deriving from a small data set, this conclusion motivates deeper engagement with LF/HS problems throughout mathematics curricula.

From this conclusion, I began to work on different approaches to designing original LF/HS problems and modifying existing problems to become LF/HS. While I continue to work on this process, and will only improve as I practice, I have laid out a framework for this process, which is linked below. I hope to engage with this template as I create more LF/HS problems in the school years to come.

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6. Student Work Example #2: LF/HS Problems in Unit 8 (Factoring)

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6. Student Work Example #2: LF/HS Problems in Unit 8 (Factoring)

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In this unit, I assigned the original problem below to my students in class as a no-stakes formative assessment. Pictures of their written work are attached.

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5. Student Work Example #1: Non LF/HS Problems in Unit 3 (Rational Equations)

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5. Student Work Example #1: Non LF/HS Problems in Unit 3 (Rational Equations)

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In this unit, I assigned the problem below to my students in class as a no-stakes formative assessment. Pictures of their written work are attached.

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1. Compared to non-LF/HC problems, do LF/HC problems motivate students to work with greater focus for longer periods of time?

2. Compared to non-LF/HC problems, do LF/HC problems help students develop deeper understanding of the relevant topics?

3. How can a math teacher create original LF/HC problems or modify existing problems to be LF/HC?

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To understand, categorize, and create the problems that best fulfill the goals of a mathematics class.

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1. My Topic and its Significance: Low-Floor/High-Ceiling Problems

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1. My Topic and its Significance: Low-Floor/High-Ceiling Problems

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During my first year as a high school math teacher, I taught an algebra class that can only be described as stratified. Ten students, all freshmen, showed up with wildly different backgrounds and experiences. Most were new to the school. Some were arithmetically strong yet totally new to algebra. Others had taken most topics but lacked conceptual mastery. Two were retaking the class.

In the face of such different levels, I longed to find a way to differentiate the class. However, I did not want to fragment the class into different groups, for I value the community of the class. Furthermore, as a new teacher, I lacked the resources or experience to plan multiple lessons for different students.

In the face of this challenge, I turned to Low-Floor/High-Ceiling problems. Low-Floor/High-Ceiling problems—or LF/HC problems, as I will abbreviate onward—build in differentiation to discrete tasks. If properly designed, they allow all students to enter the Zone of Proximal Development. The challenge is neither too nuanced for lower students to find a foothold, nor is it too simple for a higher student to feel bored. Math blogger Dan Meyer humorously calls these “Roach Motel” problems, for all students can check in but struggle to check out.

While theoretically perfect for my class, I was curious to see how my students would take to LF/HC problems. Specifically, I wanted to know if my students would find LF/HC problems more engaging. Would they work harder and longer on LF/HC problems? Would students ultimately gain a deeper understanding of the topics tested by LF/HC problems? If so, I wanted to learn how to integrate more LF/HC into my curriculum. How could I create original LF/HC problems? How could I modify existing problems to become LF/HC? This Padlet documents an examination of these questions, synthesizing personal research and design with existing scholarship on the subject.

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North Shore Country Day School | Winnetka, IL
Algebra 1 — all freshmen
Honors Precalculus — mostly juniors

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A Chicago Private School | North Shore Country Day School (NSCDS)

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Welcome to NSCDS, a coed JK-12 day school in Winnetka, Illinois. Our college-prep program balances rigorous academics, arts, and athletics.