The Mathematical Practices that are in the College and Career Readiness Standards for Adult Education define what it means to be a mathematically proficient student. As adult education instructors, our job is to help our students navigate over the swells in the tempest of their angst until they get to the point where they have enough confidence in their own abilities to weather the sea of mathematics.

Consider the first standard, MP1: Make sense of problems and persevere in solving them. It explains that our students should be able to approach a math problem or situation — something a bit challenging that they might never have seen before — and find a way into solving it. They should be able to develop flexibility with numbers and the ability to approach problems with the understanding that there are multiple pathways that can lead to a solution. They should have confidence that if they stick with it long enough, they can find an appropriate strategy that will solve the problem. That means that we, as their teachers, need to provide intriguing math problems that catch our students’ attention and then nourish and celebrate the different methods of problem solving that our students intuitively gravitate towards. * (**Mathematical Mindsets, Jo Boaler, 2016)*

One resource I strongly recommend for these kinds of math problems is **Open Middle**. As of this writing, there are over 300 high-interest problems housed at the website. They are organized by grade level and topic that make it easy to zero in on a particular skill level and area of focus that you want to bring into your classroom.

So what makes a problem open-middle?

The website explains:

- They have a “closed beginning” meaning that they all start with the same initial problem.
- They have a “closed end” meaning that they all end with the same answer.
- They have an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem.

###### Problems with Multiple Solution Methods

As an example, consider the following problem. At first glance this may seem pretty straightforward, requiring only a knowledge of the rules of subtraction in order to solve it

Each problem comes with a hint. For this problem the hint is: Remember that zeros are not allowed. Because you can only use each digit once, the digit in the hundreds place can’t be the same.

Try solving it yourself before looking at the student solutions listed below. Keep in mind that there is one optimal answer to this problem albeit multiple ways to arrive at this solution.

*(To find this and other subtraction problems – Open Middle Subtraction)*

This is listed under Grade 2, so keep this in mind when navigating the website. Don’t necessarily start with high-school level if your students need the basics in algebraic thinking and are just beginning to learn how to be flexible with numbers. If you’ve already worked out a solution to the problem, read on to see examples of how my students approached this.

Notice that one student did not arrive at the optimal answer on his first try. He was a tad disappointed when he saw that a classmate had found a smaller difference until I asked him if there might be another combination of numbers, besides what his peer had already found, that might lead to the optimal answer of 14. This led to a flurry of renewed interest in the problem as all students, even those whose calculations had already led to 14, busied themselves in trying to discover what other number combinations would work. Two of those solutions are pictured above, which clearly indicate a pattern that can lead to additional solutions.

To wrap up this activity, have each student who discovered the optimal answer of 14 explain how they arrived at their answer. When all solutions that students found are placed on the board, ask students what they noticed about the pattern that is evidenced in the solutions. If they have not already found and commented on the pattern in the combinations of numbers that work, ask students if they see any patterns in the solutions and discuss how the pattern changes with each equation. Recognizing patterns and developing flexibility with numbers go a long way in helping our students to become mathematically proficient.

###### Problems with Multiple Solutions

Some of the problems on the website are also open-ended in that there are multiple, sometimes infinite, answers to the problem. Students may still arrive at their answer via multiple solution pathways, but now everyone in the class has the ability to create and defend their own unique math situation. Table of Values: Function is one such example, which simply asks students to create a table of values that represents a function.

Similarly, there is a problem called Table of Values: Not a Function.

These activities can reinforce the basic definition of a function. Many times we focus our instruction on finding the rule of a function when we present a table of values to our students, but it’s also important that they are able to look at an input/output table, a graph, a range and domain set with arrow notation, or sets of ordered pairs and correctly identify which ones represent a function and which ones do not.

*(To find these and other function problems – Open Middle Functions)*

There may be no readily apparent rule for a set of values that our students might see on an HSE exam, other than it is or is not a function based on the definition of a function: a relationship between two values where each input value has only one unique output. While having an understanding of this rule is a procedural skill, requiring students to create their own set of values using their choice of an input/output table, graph, range and domain set with arrow notation, or sets of ordered pairs, requires a higher depth of knowledge, especially if students are subsequently tasked to create a real-life scenario that their function represents. Students are more apt to remember the basic definition of a function if they have engaged in this kind of an open-ended activity.

As a follow up to this, you can assess student’s understanding of a function with the TASC/ORT Style Problem Assessment Packet, (see examples below) aligned to Units 1-4 of the CUNY HSE Mathematics Curriculum Framework.

Whenever you are planning a unit on a particular math topic, it is well worth visiting Open Middle. There is such a wide range of problems, you are sure to find something to engage your students and teach them that they are capable of creative problem-solving.

Explore the site and then take a minute or two to add a comment below.

Let us know what problems you use and how they go with your students.

This review of Open Middle originally appeared in the Spring 2017 issue of *The Math Practitioner* which can be downloaded here. This issue includes a printable version of an Open-Middle problem, Create Squares. The review has been modified to reflect the style of the reviews at CollectEdNY.

The Math Practitioner is a newsletter published by The Adult Numeracy Network (ANN) for teachers of adult numeracy and high school equivalency math.

If you are interested in receiving this newsletter on a regular basis, please join our national community of adult education math practitioners at the ANN website.

ANN is a community dedicated to quality mathematics instruction at the adult level. We encourage collaboration and leadership, and would like to hear about what you are doing in your classroom to help your students reach their goals in mathematics. We’re also interested in your “aha” moments as an adult education instructor. To that end, we encourage submissions of articles, activities, and other items of interest related to math for adult learners for consideration for our newsletter. Please direct all correspondence regarding *The Math Practitioner* to: Patricia Helmuth, mathpractitioner@gmail.com.

Patricia,

Thank you so much for introducing this fantastic resource.

I really love how the Open Middle problems engage students, especially in a mixed-level math class.

Consider an alternative to the subtraction example you wrote about, where a teacher gives their students a handout with a bunch of problems requiring the subtraction of three digit numbers.

Faster students who are comfortable with the procedure will race through the handout, some completing it before you’ve even finished handing out the sheet. And then you have to give them something else to do. This problem gives students a lot of practice with calculation (think about how many different subtraction problems they’ll do on their way to solving this problem) but it requires more than that. Students have an opportunity to reason, look for patterns and structures, which deepens their learning and extends it beyond just this one problem. There is a puzzle-like quality to the problems on Open Middle. Students persevere because they are not just doing all of these one-off calculations. They are doing a series of calculations that connect to each other and are moving towards the overall goal of finding the smallest difference. It is also very easy to extend the problems here for students who do finish early. If a students finds what they think is the smallest difference you can ask them to prove that their difference is the smallest. You could also ask them for the largest possible difference between two 3-digit numbers, given the same conditions.

I love that this site has challenging problems for exploring number sense and operations and challenging problems in functions, algebra and geometry. One thing I really appreciate about the algebra problems is that because they are “open middle” – meaning they have more than one way to solve them – they are accessible to a wider range of students. When students work on problems where there is only one solution method, it often curtails their sense-making and perseverance because for those kinds of problems, you either know how to solve them or you might as well put your pencil down. These open middle problems keep students in the game and give them an opportunity to do a lot of great mathematical reasoning.

Also, I had a lot of fun working on this great Open Middle function problem posted on Twitter by Graham Fletcher (@gfletchy) – https://pbs.twimg.com/media/C9VkpFwVwAAV5BH.jpg:large