Three Good Problems

“Problem solving is hunting. It is savage pleasure and we are born to it,” said Thomas Harris.

A good problem engages the learner is a pleasure that feeds our desire to do more. As Thomas Harris suggests, we are born to it and yet our learners so often drudge into our math classes like an imaginary, sinister puppet master is forcing them to their seats in our classrooms, as if he couldn’t think of a better torture. There is some light though, when learners engage in a good problem, something magical happens, and it is truly beautiful to see how that transforms their learning. This entry is about three problems that I have had success with, three problems that engage learners and cut the strings of this imaginary puppeteer, allowing math to come alive for both me as the facilitator and for the students as learners.

Let’s define our terms, before we start a discussion about them and we all end up confused. When I say a good problem, I don’t mean just a problem that made me think or was a posed a challenging solution, though these are part of what a good problem should have, it is by no means all. A good problem should be accessible to a variety of learners, across various age levels and across various skill levels. A good problem should be able to scale to various levels, having multiple extensions and avenues to approach, extend, and explore. A good problem should engage the learner at a gut level and scale up from there to meet the needs of the learners. A good problem should have a guiding question of interest, from which the mathematics flows, being made visible in the process of the solution and not the only goal of the problem.  A good problem provides a framework, a context in which the mathematics makes sense, and the mathematics is a part of the solution, not the end goal. A good problem should also provide learners with the opportunity to make conjectures, generalize their thinking, and find exceptions to break their rules.

A good problem is a rare thing to encounter, which is why, when found, I hold on to them with baited breath. I am enthralled to find good problems, to explore good problems, both from the learner’s perspective and from the facilitator’s perspective. To this end, I wish to share three good problems that I have had the opportunity to explore in both roles, and in doing so I hope this sharing will start a conversation to find, create, or share more like problems.

Number One – The Locker Problem

This is a well known problem in many math geek circles, this problem has many forms and many purposes, I will share a brief bit about how I typically broach this problem – if there is interest in an extended coverage of this problem, I will gladly elaborate in a later entry.

The version I typically use is to either model the approach of what is happening with a mock-up of the situation, having students watch, think about what they saw, and write a guess as to what is happening. Then I have students share out one observation at a time with a partner, which immediately provides me with formative feedback on where the learners are in their understanding. Depending on the grade level and ability of the learners, I usually start with either twenty or one hundred lockers and students, each taking a turn changing the state of the locker.

The processes by which we communicate the problem setup varies from having the learners demonstrate to using literacy tools and having learners interact with text first, but in all cases a careful exploration of the problem is necessary before learners are launched on solving the problem. The scaffolding for the problem may come in the learning that I want from the students, or in giving them an organizational tool to aid in making their thinking visible, or using various learner interactions to expedite learners’ thinking.

All this setup to say the problem is basic terms is: A school has a tradition where 100 students number off from 1 to 100, each student will go down a row of lockers also numbered 1 to 100, and change the state of every locker. All the lockers are closed, when the first student goes by and opens every locker, the second student goes by and closes every even locker, the third student will close locker 3, but open locker 6, close locker 9, open locker 12, and so on, this pattern continues for all 100 students. Which lockers are still open when all 100 students have taken their turn?

There are many ways to change this problem from the number of lockers to the depth of the questions that you ask, which opens it up to a wide range of learners. I have had success with grade levels from first grade to AP calculus students, the depth and complexity of questions, the scaffolding, the discussions were very different, but the problem was the same.

Number Two – Building Triangles (also works with Squares)

This problem is extremely fun and if you provide any equally sized objects, like toothpicks or straws, learners are allowed to actually build the problem and gives some kinesthetic learners an opportunity to attach to learning. This problem is great because learners are creating an experience, if learners are able to experience the mathematics, they are much more likely to access the desired learning. The question has a similar depth of complexity of questions that you are able to ask from simple arithmetic to analysis of quadratic functions to area models to pattern recognition (generalizations) to hypothesis breaking. Like the Locker Problem, the way in which I facilitate the problem varies on the learners and the learning target, but much of it is similar to that which I discussed in the Locker Problem.

This problem is stated something like:

Consider this sequence of diagrams:

The first is made up of a line of three line segments of equal length. You can think of the second diagram as made up of nine segments of that same length, and so on.

  1.        Find the number of segments of the given length that would be needed for the tenth diagram in this sequence.
  2.        Explain how you would find the number of segments of the given length that would be needed for the 100th diagram in this sequence.
  3.        How would you find the number of segments of the given length for any diagram in this sequence?

Problem Three – Pattern Building

Originally, I had a specific problem in mind (I will save that problem as a discussion for a later post) because this is more general and may be more rewarding. Identifying patterns is hardwired in our brains, creating problems where learners are asked to identify any patterns they find, recording each of these patterns, and discussing them with the learners does a world of wonder for learners. For example, each student will be able to identify some pattern, this rewards that intuitive nature learners need to foster and develop as they become mathematically proficient. Pattern recognition problems are easy to create, there are patterns all around us, and is a great way to jump into many problems as a warm up.

An example of this kind of lesson could be something like displaying the following sequence of numbers, and asking learners to write down anything they notice, giving them like two minutes to look at the string of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ….

Students share their observations with a partner, back and forth, then with their pod. Students vote on the top (one or two) observations that they want to share with the class. As each group shares outs, I would record their results, asking how many other learners noticed the same pattern. We will discuss the various relationships, and I will have the learners describe how they might be able to get the next term, or the 1000th term. Learners will then try to derive a rule (a function) that describes how to relate the various terms. Learners will work together to produce their rule, and then we will come back together to see if we have a rule. Once we have a few candidates, I will ask groups to try to find exceptions or see if they can break their rule. Once we are sure we have something, I may show them something like

I will ask how might this fit into the rule(s) they’ve created? This lesson may go in a variety of ways, and it could be used to teach a variety of the Mathematical Practices that are embedded into the CCSSM. Using the idea of pattern recognition to move into the mathematical content, I facilitate the discussion and the learning without revealing the structures the learners ‘discover’ on their own, this allows for learners to feel like the learning is directed by them, that the ideas discussed were their own, which typically translate to higher retention and a deeper understanding  – not to mention I always learn something new from students too.

With these three problems, I would love to hear how you facilitated cutting the strings of the puppeteer for your students. How might you be able to facilitate opportunities to enjoy the savage pleasure of solving good problems?

If there is any interest on more details on how I have implemented any of these in the past, please let me know. Happy mathematics.

Literacy in Mathematics

The role of literacy is a life skill we often assume is relegated to English class or we confuse our learners reading a text with comprehension, and we need to play a much larger part in creating opportunities for learners to practice and develop their literary skills. Mathematical texts tend to be very dense, meaning the text is both rich in academic language and conceptual or procedural knowledge that is very difficult for learners to access, let alone understand. Giving learners the opportunity to access, decode, interpret, and understand mathematical texts prepares them to be self-directed, self-motivated learners. If learners are able to read a mathematical text, interpret the context of the text, and be able to write and speak about the text then imagine how much more likely our learners will be successful. Literacy is difficult to teach it is a process where slow growth in learners may be challenging on the nerves, and the dealing with some of the learners’ push back may be challenging, but it is a worthwhile endeavor for our learners. To this end, this post will be about three ways you may incorporate literacy into mathematics (really any subject). Number One: Anticipation Guide The anticipation guide is the only one of the three that requires you to do a little work ahead of time. The anticipation guide at its heart is just a way to get learners thinking about their own knowledge and provides learners with a buy in and access to the text. There are many ways that this may look like in your classroom, but I like to have around five questions, which I like to write addressing learners’ misunderstandings. The form usually is looks something like what you see below: You may notice that some of the statements are untrue and vary in level of difficulty, this is to illustrate that untruthful statements are useful, especially when they are a misconception the learners may have and the varying degree of difficulty is meant to illustrate the variety of ranges in classes you may use this structure in. In the classroom, there are numerous ways to use this, like as a warm-up for the “Me” column and then give learners the opportunity to read the text and finish the second column, then your lesson begins with a discussion of the text and you are able to address misconceptions or elaborate on ideas presented in the discussion. Imagine how much more buy-in you get and the lesson is learner-directed, not a lecture of information the learners may not wish to hear about. Number Two: Frayer Model The Frayer Model is an organizational structure for learners to access academic vocabulary and many, many other things if you wish to be a little clever. This structure has many representations you may find online or in most ELD texts, that you may print off and hand to your students, but I prefer learners make their own. The structure has the form shown to the left. For learners to make their own, they need a blank sheet of paper and then they can fold their paper “hamburger” and “hotdog” style to create four even sized squares, and then write the word in the middle. They can also turn it over to write another word. This is one of my favorite tools to use in the classroom because of the variable activities that I have my learners interact with text with this. As a classic example, I may pull four academic words from a lesson or unit and give direct instruction on these. Then I give learners the opportunity to create their own assigning pairs for two words and another pair with another two words. Each student is responsible for their word, they create their Frayer model, they share their model with their pod, and they must choose one of the four to be the model of a word they want to present. Then each member of the group will go to other groups with their best representation, they must describe the other representations and then detail why they think this is the best one. That same learner will then hear similar stories three times, and they will return to their original group to share out. The next day there are four posters in my room, each area laminated square meter of instructional gold. Learners will see some partially filled out squares from the day before, as pairs they have a few minutes to fill out the remaining squares however they see fit, and return to their seats (this is the warm-up). Within five minutes they should have 20 answers ready for me, the result of which is like a “quiz.”  This same routine is also used with classification schemes and detailing steps to various problem types, and now with multiple representations. Number Three: SQRQCQ This is another approach at getting learners to understand that academic texts are terse materials and usually require more than one read to obtain the important information. The power in this approach is that it costs nothing more than having learners all be able to see the text, and the persistence of having learners be able to feel free to share out. The letters and what you do are shown on the right. When I employ this strategy, it is usually with a particularly dense and rich text that I really want the students to understand. I usually project the text and give a copy of the text for the learners to annotate. If the text is about a paragraph, I will give them a minute to read it quickly, and we talk about what words stuck out or what they noticed the text is about. I usually do a think-pair-share here, and I record some of their responses. On the next pass, I give about three minutes to read and encourage learners to annotate the text, write down thoughts, details, or perform necessary computations. Learners then share what they highlighted, details, or what computations they made with a peer, then we will share out whole class and I record these details like the first pass. We then give it one more pass and learners make final calculations, incorporate final details, and write a sentence describing why they believe their solution is correct. Peers will share with a partner one more time, make any adjustments from their sharing, and then random selection will determine who gets to share with the whole class – the results of this discussion are recorded. The last thing we do is talk about how our thinking changed with each pass, what became clearer, how our understanding grew with the processing and multiple approaches to the text, all of which is brought out from asking the learners about how our thinking changed over the analysis. Conclusion Well that’s three of my favorite structures, combined with in-class activities that allow learners access to the literary components of the dense texts that they often encounter in mathematics. The structures provide a natural way to incorporate them into your lessons without much additional preparation and being driven by the learners, the interactions and engagement are much deeper and meaningful than by purely direct instruction. I hope you enjoyed this discussion and you will find some ways to incorporate these (or any other) structures into your lessons to increase learners’ literacy. If you do have any success with literacy in mathematics or if you have additional suggestions, please share I am always interested in learning new or seeing that wish is already known in new ways. Please enjoy making your learners’ thinking visible.