Problem Design – Summary

I’m wrapping up my series of posts on problem design. It was fun; sorting worthwhile problems by the Standards for Mathematical Practice has been enormously helpful in writing problems for my students every day. This idea was inspired in part by Ben Blum-Smith’s post on a language for problem design, but also from my frustration with conversations about the Common Core centering on “conceptual understanding” and “deeper learning” without defining what those terms meant. Conceptual understanding and deeper learning are certainly important however they are defined, but they’re defined differently by every math teacher who uses them. The Standards for Mathematical Practice give teachers a common language for the specific habits of mind and types of understanding students need to be proficient at mathematics.

There are tons of great resources that have been built all over the MathTwitterBlogosphere around the Practice standards, but many of them focus on rich tasks, inquiry-based instruction, 3-acts, and similar materials that teach full lessons based on these practices. While I love many of these resources (see this page for what I use in my classroom, and here for even more references), I’m more interested in how smaller problems, things that take students 2-10 minutes to complete, can teach, reinforce, and assess the Practice standards. This was a part of my goal in developing this language for problem design, and it became more explicit as I found more and more ways to think about each of the Practices.

To summarize my thinking, I’m categorizing both my taxonomy of problem design and a number of other resources available around the internet by specific Standard for Mathematical Practice, with the idea that this can serve as a guide to thinking about the different types of understanding students need in order to be proficient mathematicians.

MP.1 – Make sense of problems and persevere in solving them:
Low floor problems, hard problems that are accessible at a range of levels allowing students to fail productively on the way to an answer
One-way problems, problems that are extremely difficult and require thinking beyond what’s expected in a standard, and allow students to synthesize concepts and apply them in a more rigorous way

MP.2 – Reason abstractly and quantitatively:
Reversing, asking students to solve a problem both forwards and backwards to build algebraic thinking
And Fawn’s visual patterns

MP.3 – Construct viable arguments and critique the reasoning of others:
Splitting provoking student discussion by asking a question with several clear possible answers that are in sharp disagreement

MP.4 – Model with mathematics:
Withholding, asking students to identify the information necessary to solve a problem
3-act tasks

MP.5 – Use appropriate tools strategically: I have no answers for this one.

MP.6 – Attend to precision:
Thwarting, asking students a question where application of the standard procedure is likely to lead to a wrong answer
Disguised problems, problems that ask students to apply a concept in a format they do not usually see that concept

MP.7 – Look for and make use of structure: I have no answers of my one here, but number talks and counting circles are the two best ideas around

MP.8 – Look for and express regularity in repeated reasoning:
Baiting, providing students an opportunity to see a pattern and use it to solve a problem strategically

Finally, the last two problem types I’ve explored don’t fit neatly into a Practice standard, but are incredibly valuable tools in formative assessment and building understanding of specific concepts –
Jamming, asking a student a question on a concept where the usual procedure cannot be applied
Non-examples, juxtaposing questions that are similar to the standard being taught or assessed but require a different concept to see if students can differentiate between them

That’s what I got.

Problem Design – Withholding

Withholding comes from my struggle with figuring out how to teach mathematical modeling. There are an enormous number of resources out there for modeling lessons (see here, here, here, and here for a ton of examples), but what I’ve struggled with more specifically is teaching the skills that allow students to be successful in modeling. I don’t have any good solutions for this, but one piece that I think is critical, not only for mathematical modeling (MP.4), but also for reasoning abstractly and quantitatively (MP.2) and algebraic thinking in general.

Withholding is taking a problem, then asking what information is needed to solve it. Withholding is absolutely useful at a basic level–like giving students one side of a rectangle and asking what they need to know in order to find the area, or which angles are needed to identify the third angle in a triangle. However. it has much more value as a problem solving question asking students to analyze a number of possibilities and determine the easiest route to a solution.

Some examples:

What would you need to know to know in order to find out how much printer paper is on the shipping crate?

(from Andrew Stadel)

What do you need to know in order to determine if the polygon below is a square?

A function goes through the points (2, 5) and (3, 7). What do you need to know to determine the y-coordinate where x = 4?

Which dimensions are needed to find the area of the figure?

Withholding is a pretty simple idea that is mathematically rich when it is applied to an open middle problem. Being able to identify necessary information, in particular in divergent problem-solving situations, is a critical skill in modeling with mathematics. However, I’m much more pessimistic on the value of withholding relative to other problem types I’ve talked about. Mathematical modeling is a complex skill that requires sustained attention and practice, and withholding is only one very small piece of that.

Problem Design and Conceptual Understanding

Three things I believe:

1. Students need to understand math conceptually. They need to make sense of problems, be able to explain why the processes and procedures they use are correct, solve problems in and out of context and apply concepts in novel situations.
2. Successful inquiry-based instruction (coupled with appropriate practice) leads to greater conceptual understanding and a productive mindset around mathematics.
3. Inquiry-based instruction often fails  for many students due to insufficient background knowledge, imperfect facilitation, or a lack of a desire to learn the material.

I believe in direct instruction. I believe that, under certain conditions, telling students . There’s a reason teachers don’t pretend in any other discipline that all necessary knowledge can be discovered if only we structure an ideal inquiry environment and inspire students to want to learn. This is not to say that direct instruction is ideal for every lesson, or every moment of any lesson, and in particular I’m pretty put off by teaching that looks like this, but I don’t shy away from telling my students key information when I feel they are engaged, the material is appropriately motivated, and they will have a chance to apply it independently and in novel contexts right away.

The classroom I believe in (and my classroom is moving in that direction, but is far from there) is one where students are inspired to answer mathematical questions every day through perplexing tasks (whether or not they come from the real world), and then find them the tools they need to answer those questions. This often means motivating a new topic through a rigorous task, leading to direct instruction on the topic, some combination of guided practice, partner work, and whole-class problem solving/analysis, and independent practice for students to stamp their understanding. It also often means using inquiry to allow students to talk about math, explore new topics, and construct their own understanding. One is not inherently more valuable than the other. Either way, I believe that the most valuable mathematics comes from students wanting to answer problems that require them to make sense of both mathematics and the world around them, and to pause and think about the mathematical concepts that they are applying.

Anyway, while I’ve spilled lots of digital ink on the idea of problem design here, and in particular thought a lot about the value of a taxonomy and language of problem design, I think it can be effectively distilled into two key ideas that channel the Standards for Mathematical Practice:
1. Great problems give students reason to pause, be thoughtful about mathematics, and consider their solution path.
2. Great problems provoke solution paths that are divergent and different from conventional procedures applied to the concept.

These are imperfect, but the more I write about problem design the more I see these two ideas at the center of problems that promote high-quality mathematical thinking.

Problem Design – Baiting

Baiting is giving students an opportunity to notice a pattern in similar problems or repeated problems and apply it to infer a rule or shortcut. Baiting is what great teachers do in inquiry-based instruction, allowing students to construct their own mathematical understanding of a topic through patterns, sustained reasoning, and careful definition. However, baiting can also be applied to MP.8: Look for and express regularity in repeated reasoning in more innocent contexts. Baiting is valuable in constructing student understanding of a topic, but inquiry risks wasting valuable instructional time if the lesson is not successful, and facilitating successful inquiry is challenging. Baiting can be used in applying a concept to explore its mathematical structure and reason to deepen understanding, rather than to create it.

Some examples:

I got this example from a visualizing math post referencing this

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How many triangles are in step 1?
How many triangles are in step 2?
How many triangles are in step 3?
How many triangles are in step 4?
How many triangles are in step 10?

I’m giving a new name to an old idea, but I think the twist is the value in baiting a concept after it’s been taught. Inquiry is intimidating for teachers not experienced with it, but students believing in the power of their reasoning is critical and possible no matter what style their teacher chooses. Every classroom can focus on giving students opportunities to reason about mathematics and draw inferences about quantitative relationships, and build their capacity to see mathematics as a logical system of rules and relationships that are connected and full of meaning, rather than procedures to be memorized, applied, then forgotten.

Problem Design – Reversing

Reversing is the act of taking a problem and reversing a given and an unknown. It’s simple, but critical in building intuition and intellectual need for algebraic structure. One trap I’ve fallen into is teaching opposite directions of the same concept on separate days. While that can lead to superficial, short term success, it doesn’t promote understanding. Instead, introducing the Pythagorean Theorem and having kids solve for both the hypotenuse and the side the same day promotes productive struggle and more flexible understanding.

For me, reversing falls under MP.2: Reason abstractly and quantitiatively. It’s critical in building students’ connection between arithmetic problems and the value and power students gain in using variables and algebraic structure to give math meaning.

Some examples:

Find the area of a rectangle given two sides; find a second side of a rectangle given the area and one side

Find the mean of a data set; find the missing data point given the other data points and the mean

Simplify an algebraic expression; identify a missing coefficient given the simplified expression

Find an interior angle measure of a regular polygon; identify a regular polygon given an interior angle measure

Simplify an exponential expression; identify a missing exponent given the simplified expression

Use an equation to generate a table; generate an equation from a table

Manipulate logarithms; manipulate exponents

Calculate percentages; apply percentages to identify parts and wholes

I think the biggest pitfall to building abstract reasoning through reversing is dividing these objectives into separate parts. It’s tempting for teachers to try to isolate skills. One day, teach finding the area of a rectangle, have kids practice a bunch, they do well with it. The next day, have kids find the missing side length of a rectangle, have kids practice a bunch, they do well with it. It’s gratifying as a teacher for lessons to go well, but the conceptual understanding is quickly revealed as illusory, and students struggle to think flexibly about the big idea — the multiplicative structure of a rectangle.

While students will struggle with questions reversing a concept, their struggle is a struggle with rigorous, broadly applicable algebraic concepts. Cutting out those concepts reduces math to a set of discrete series of steps that must be memorized and consist largely of arithmetic, which is the opposite of MP.2.

Problem Design – Context

One of my favorite phrases in the Standards for Mathematical Practice is this

the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved

from MP.2. Anyway, after manipulating away with my taxonomy of problem design, I want to pause to think about the context of where I began and where I’m going.

I’ve hit on three of the Standards for Mathematical Practice:

MP.1: One-way problems, low-floor problems
MP.3: Splitting
MP.6: Disguised problems

Each of these types of problems is meant to teach the mathematical practices.

The final two problem types, non-examples and jamming, could fit in a number of the mathematical practice standards. However, I would categorize them differently, as problem types more useful in formative assessment than in learning. I don’t love drawing a distinction between the two because I want them to be one and the same, but in everyday practice I think that distinction is valuable. Jamming and non-examples give extremely valuable information to the teacher, and less value in learning to the student. One-way problems, low-floor problems and splitting give less information to the teacher. Disguised problems are somewhere in between, and I think any problem that address Attend to Precision will do that, but I think disguising a concept is one of the most rigorous and valuable ways to teach students to engage in MP.6.

3 practices down, and a ton of room to grow . I have some ideas for MP.2, MP.7 and MP.8. I’ll leave MP.4 to the 3-act experts, although I’m working on a one-pager with my principal that I might share along those lines. Writing a problem for MP.5 (Use appropriate tools strategically) is a mystery to me, beyond the basics of using tools in the math classroom. Definitely an area of growth for my practice.

Plan:

I’m really enjoying writing this series on problem design. It’s been really valuable for me in articulating and sharpening many of the half-formed ideas I have as I plan lessons. I have a half-dozen more ideas in the pipeline for problem design, and I’d love to think more about how they fit with the standards for mathematical practice and formative assessment as i write about them. In particular categorizing problem types, and thinking about how these categorizations are useful or not useful, has been a particularly valuable piece of thinking for me.

After that…I’m not sure what’s next after problem types. Engagement strategies, types of context, or tackling that monster of modeling with mathematics. We’ll see.

 

Problem Design – Low Floor Problems

Low floor problems have been talked about plenty before. I’m borrowing ideas in this post liberally from work by Dan Meyer and many others on 3-act tasks. However, I think there’s something worthwhile in the idea of a low floor problem beyond the scope of modeling and 3-act tasks. Dan Meyer:

Set a low floor for entry, a high ceiling for exit. Write problems that require a simple first step but which stretch for miles. Consider asking students to evaluate a model for a simple case before generalizing. Once they’ve generalized, considered reversing the question and answer of the problem.

So there’s the idea. But I want to go deeper into what a low floor problem does for learning, in particular in the context of MP.1: Make sense of problems and persevere in solving them, as well as for lower-skilled math students.

One of the challenges in teaching students who struggle with math to persevere in problem solving is the extremely short, negative feedback loop they often experience. Teacher presents challenging problem. Student doesn’t have a clear path to the solution. Teacher prompts student to think about it and apply what they know. Student gets nowhere. Other students solve it, student tries to listen to the answer, and internalizes their own powerlessness in the math classroom.

While a low floor is critical for a successful modeling task, low floor problems can also be valuable tools in teaching students their own power as problem solvers as stand-alone questions.

Some examples:

This question from Math Arguments is deceptively easy to begin. Try it:
2² ends in a 4
12² ends with 44
Find a square that ends with 444.
Find another that ends with 4444.

Plot all of the points on the coordinate plane whose x- and y-coordinates add up to 7. What is the equation for these points?

This question from Five Triangles shows a cube on which adjacent midpoints have been connected by line segments, and the corners cut off along those line segments. However, unload some of that literacy load, and ask students how many edges there are in the polyhedron. See what they come up with.

If a > 0,  b < 0, label each of the following as always, sometimes, or never true.

For the lower grades, try something like this, and see what they do with that blank:

Fill in the blank to make the addition true.

  4 9 7 1
+3,_ 8 9
  8 3 6 0

The 5 key on your calculator is broken. How can you add 458 + 548 + 345 without using the 5 key?

I would be remiss without mentioning Visual Patterns and the wealth of learning bound up in simple patterns. The low floor here is drawing the next step, and then the next, and then the next. Here is a favorite of mine:

 

All of these questions have a few things in common.  First, they have an easy first step. They offer potential for exploration in several directions. Answers are possible both through guess and check, logic, or application of a mathematical concept.

Finally, all of these questions, and many rigorous low floor questions, have an algebraic component. Each problem has underlying algebraic structure that allows further insight and provides opportunities for rigorous extension. More than that, an intuition for the meaning of an unknown, and the power that unknowns can give mathematicians, is a central bridge between elementary and high school mathematics, and many of these questions effectively scaffold that understanding.

 

Problem Design – Non-Examples

Non-examples aren’t revolutionary to problem design, and would maybe more appropriately fall under problem set design, but I’ve found them to be a critical and incredibly useful way to see where student thinking is and challenge students to push their understanding.

Non-examples are usually routine questions — different from the conceptual thinking that other elements of problem design try to push, non-examples are innocuous but well-placed questions to assess whether a student can differentiate one concept from another. Non-examples require students to attend to precision (MP.6) by discerning between applicable concepts. These are most useful in student practice, after they have been introduced to new material and before they are pushed to deepen their understanding.

Some examples:

Geometry:

While students are practicing area of quadrilaterals, give them a triangle question.
While students are practicing area, give them a perimeter question.
While students are practicing reflections, give them a translation.
While students are practicing the volume of cylinders, give them a rectangular prism question.

Number & Operations:

While students are practicing one operation, give them a word problem requiring a different operation.
While practicing rounding, ask students to write a number in expanded form.

Fractions:

While students are practicing converting between mixed numbers and improper fractions, ask them to round.
While students are practicing simplifying fractions ask them to convert to a decimal.

Proportions:

While students are practicing finding a part of a whole, have them use the part to find the whole.
While students are practicing proportions given as percents, give them a problem using ratios.
While students are practicing finding unit rates, give them a problem requiring a percentage.
While students are practicing finding ratios, give them a problem asking for a probability.

Expressions & Equations:

While students are simplifying exponential expressions (), have them combine like terms with exponents.
While students are solving word problems requiring equations, give them a problem requiring only arithmetic.
While students are practicing square and cube roots, ask them to divide common perfect squares and cubes by other factors.
While students are practicing writing equations from a graph, ask them for a unit rate.

Statistics & Probability:

While students are practicing finding the mean, ask them to find the median.
While students are practicing interpreting measures of variability, ask them to interpret a measure of central tendency.

 

The idea of a non-example is not to ask random questions interspersed with questions aligned to a daily objective, but to place well-chosen questions at several deliberate places in a lesson that assess whether students can distinguish between similar concepts. If a student answers a non-example correctly, it doesn’t mean they’re necessarily mastering the material. However, if a student struggles on a non-example, and in particular if they apply the skill they’ve been practicing to the wrong concept, it provides valuable information about their fluency with that concept.

This is where choice of non-examples become critical. Random, unrelated questions are unlikely to provoke misconceptions or push students to think critically about the math that they are doing. A non-example should be structurally similar so that a student mindlessly applying a procedure will continue to provide that procedure without pausing to examine the new context. This doesn’t mean trying to trick students — throwing one subtraction question into a mad minute of multiplication — but choosing deliberate questions that push students to be thoughtful and deliberate about the processes they apply to math problems.

At worst, non-examples reveal students with the mindset “the teacher is showing me how to do this thing; I just need to do it over and over to make the teacher happy”. I was floored one time when, in an exercise working with circles, a student encountered a problem asking about a triangle and started to write area equals pi times radius squared, and identified a number as the radius of the circle before I stopped them to think for a moment about what they were working on.

At best, non-examples assess a challenging skill — having enough working memory capacity to think about both the execution of a new concept and the characteristics that allow that concept to be applied, as well as creating a habit in students to attend to precision in applying mathematics to problems they encounter.

Non-examples don’t teach students in and of themselves, and they don’t give valuable information on higher-order thinking, but they can be incredibly useful as an early-stage formative assessment tool to learn whether students are ready to push their understanding to a greater depth.

Problem Design – Splitting

While many of the elements of problem design are useful tools in formative assessment or pushing students to think deeper about a concept they have had some experience with, splitting is most useful when a concept is first introduced. Splitting a concept takes students whose understanding is immature and gives them two (or more) options to argue about, with the goal of provoking 1) rigorous, engaged mathematical discussion and 2) a student-centered understanding of the concept. Splitting is an incredibly useful tool for application of CC.MP.3 Construct viable arguments and critique the reasoning of others.

Some examples:

I once saw a master teacher ask a group of elementary students who had just been introduced to decimals which was greater:  .9, or 1.1. She was an absolute master — half the students thought .9 was greater because the first digit was larger, and an excellent discussion ensued where all of the students became convinced that 1.1 was actually greater and had the chance to articulate their understanding of place value.

When introducing the idea of association in scatter plots, students catch on quickly to the idea that negative association corresponds with a negative slope, and positive association with a positive slope. Then, throw this at them:
A scatter plot has negative association. Which statement is true?
As one variable decreases, the second variable decreases.
As one variable decreases, the second variable increases.

Fraction comparisons: Which is greater,

Does this scatter plot have an outlier?

Show students this student work, and ask if it’s correct and why:

Splitting is a key step in CC.MP.3: Construct viable arguments and critique the reasoning of others. Students can best construct and critique arguments if the mathematics are worth arguing about. This requires well-chosen arguments that provoke genuine disagreement–and disagreements that can be facilitated so that as many students as possible come out with a deeper and more flexible understanding of the topic.

Problem Design – Jamming

Jamming is taking a concept that is connected with a common procedure, and asking a question that cannot be solved using the procedure. The question jams that concept, because it assesses whether a student understands the concept well enough to apply it without the procedure.

The word jamming comes from Ben Blum-Smith’s blog, who was referencing Cody Patterson, so no credit goes to me for this idea, but I’ve adapted it in my classroom and want to share it in my series of posts on problem design.

Some examples:

There are tons of tricks out there for factoring quadratics. Jam the concept with a question like this one, and see if students can solve it:
What integer c will make the polynomial below a perfect square?

Two great questions for elementary or lower middle schools folks–students may successfully apply the standard subtraction or mixed number multiplication algorithm to these questions, but for each what we really want is the number sense that allows a student to do these in her head.

This one is a bit more advanced, and effectively jams fraction operations (and might fall under Ben Blum-Smith’s thwarting as well).

The distance formula on the coordinate plane is a fascinating application of the Pythagorean Theorem, and a common victim to memorization without understanding. See what students can do with this:
Which points with integer coordinates are exactly 5 units away from (-3, 4)?

More fun with geometry:
What is the length of the side of a cube which has a volume numerically equal to its surface area?

Jamming is a critical step in formative assessment after students have been introduced to and had a chance to explore key aspects of a new concept. This is both valuable information for the teacher on student’s depth of understanding, and a chance for students to stretch their knowledge through non-routine problems and critical reasoning.