Cheers to Better Deals – The Inequality Story

downloadAlthough I have attempted to teach this topic in many ways at a variety of levels, I always find it interesting to see how learners attempt to grapple with the concept and understanding. Learners who struggle with understanding linear graphs typically have an even more difficult, and when you add-on this is the first time the learners have literally ever met the material I am at the source of the stream that I  see in higher education. The topic is linear inequalities, the grade level is sixth grade, and I am attempting to teach these young minds this topic in 50 minutes, or less, and it is their first time (Does anyone else hear the Mission Impossible theme song here?) Well, I accepted the mission, I would attempt this process, twice actually, as I would teach this lesson back-to-back in two different group of 6th grade learners. As you are aware, I am a huge fan of 3 Act math lessons, with one of my favorite resources coming from La Cucina Matematica, where they shared Estimation 180 and found this amazing 3 Act lesson. As with any lesson, I have to adapt it a little to make it match my teaching style, and went in feeling ready to give it a tackle. In my career, this is the first time I have attempted to teach this lesson with a 3 Act component, and it is the first time I felt compelled to not give any instruction, but let the students conversation and discovery lead the way. One of the big pieces I was looking forward to was having students see their responses projected in front of them with Desmos, using their group responses to guide the inquiry. One note on this topic, I would prefer to have assigned each student their own opportunity to play on Desmos and then share out whole class, but I would need some additional front loading days to teach students how to interact and use Desmos, which I am unfortunate in that I didn’t have that opportunity, so we did that part whole class. My lesson followed this the PPT I made here or you can see below, which went off overall pretty well.

Immediately, I must say that the kids in the first class thought that Woody had cussed in the first act, so the second time I gave the lesson I had to front load that the bleep was covering up the answer to a question we were looking to answer. Another thing I picked up on right away, is that students struggled with figuring out what the humor was, they needed the mathematical background to understand what the reason for the humor was, which is an avenue to pursue with them later. The way that the various groups interacted, the way in which the conversations went within each group depended a lot on what they noticed from the clip. I was surprised by the number of kids who actually knew the show, a big thanks to reruns and older parents, as those students were more interested in the problem since they had more background knowledge of the show. The beautiful thing about this lesson, is that the discussion and interactions gave the students the opportunity to discuss and find the break even point, with about half the students understanding why the deal was better or worse. Quantitatively speaking more than 90% of the learners were able to understand that $25 per week was the break even point, and they were able to understand that certain values like $30 a week was greater than while $20 a week was less than the value of the original deal. We started to talk about how $24.99 per week was still less than the original $100 a month raise, while $25.01 per week was greater than the original $100 a month raise; however, there were only about six kids per class that were able to articulate this point and why in their exit slips. The next steps in this lesson would be to flush out the idea mentioned above, that any value greater that then $25 per week is a better deal (more money) for Woody, while any deal of $24.99 per week or less is a worse deal (less money) for Woody. The discreteness of money helps make this more concrete which was one reason I thought it might be very valuable for these learners, not to mention money is something they are familiar with. Overall, I would say the lesson went well and promoted a lot of great discussions among the students, it removed any prior need for mathematical knowledge so all kids had an entry-level assessment into the game, and the students discussing the break even point was a huge win. The use of Desmos really helped visualize it for students, and will give them some place to start building on. I feel confident this was by far the best attempt at securing an understanding of this difficult concept and I imagine that with follow-up and repeated visits this topic will be mastered by most of these young learners.

Not ODEs – The Variation of Parameters Method

VersatilityWhen one attempts to define versatile, something like “able to be adapted to many functions or activities” may come to mind. As a classroom teacher (and now working K12) this idea comes even more naturally as I know have the opportunity to see how successful and adaptable an idea or lesson may be with learners of various levels and ages.

Recently, I had such an opportunity to try marry a couple of ideas I have brewing in the back of my mind with four classes, two 6th grade classes and two 8th grade classes. The original idea stems from two separate sources a random search and from Mark Driscoll’s amazing book Fostering Algebraic Thinking in which the heart of the lesson follows. The other idea is my love of working a 3 Act lesson into anything that I can teach. So I will discuss both lessons and what I learned in what follows, but I’ll first set the stage for the lessons.

This year, I have had the pleasure of working a lot in the sixth grade, a grade I felt comfortable with my background (as I have described before) doesn’t lend itself naturally to the younger learners. I am working on getting some practice with many folks favorite age group, the infamous eighth graders. So I had this opportunity lineup, within a day of each other, I had the time to teach the 6th graders on an introduction on Equations and Expressions, and an introduction for the 8th graders on slope.

Enter my normal teacher brain, whose thoughts in no particular order went something like this: 1. I don’t have much time to prepare two very different lessons, and make them meaningful, 2. Students need hands on connection, an anchor, that we will be able to refer to throughout the unit,Teacher Brain

3. Students like to build things and I like to facilitate through questions and setup versus talking to myself in a lecture format, 4. Using 3 Act format gets a lot of buy in early, prompts students’ thinking in the direction I need, and makes for a natural direction to move forward, 5. What do I have around me that would allow me to use the same format, but change the questions (or purpose) of the lesson with the same manipulatives?….

Enter the work setup by Driscoll, he posed a Matchstick problem that extends, this would work perfectly, and the dollar store has popsicles by the hundreds for, well….

My manipulatives clearly in hand, my thinking went to the idea that I wanted the sixth graders to build the triangles, as shown in the video, counting the number of triangles formed and relating that to the number of sticks required to make the triangles. There is a natural differentiation here too, advanced learners may recognize that there are more triangles depending on how they build their triangles together. The relationships that start to form here, are well beyond the scope of this lesson, but what a great learning experience for those kids ready to go beyond. As the students build the triangles, they record the data, and then the extension question(s) go like: 1. How many sticks would we need if we were building 100, 1000, or 1,000,000 triangles? 2. How many triangles could we build if we have 100, 1000, or 1,000,000 sticks?

Good ThingsThe math nerd in me love these questions because it extends to the idea and in light of actually building it, this is a natural way for me to introduce the idea of generalizing, enter expressions and equations. In addition, this also shows the doing and undoing process of the relationship between the number of triangles and the number of sticks, an important piece that we sometime overlook.

Using a similar idea in eighth grade, the idea there was to tackle the misconception that these learners had in regards to scaling, the idea that the ratio of two similar triangles is the same though the lengths of the respective sides may be much different. The learning that I wanted the students to see was that as they build the right triangles, make measurements (by counting sticks) and comparing the ratios of “rise” to “run” they would discover that the ratio was the same. I was expecting some errors, like how the students build and counted the right triangles, which I was hoping would be a great “teachable moment.” Students would get a lot of collaboration and would work with a small data set of 9 other ratios calculated from other student groups, reinforcing their understanding.

What was used is shared is here.

What I learned through this experience is the importance of modeling and being very clear with what was required of the learners. After reflecting on these lessons for a while, and thanks to invaluable input from #MTBoS and #mathchat folks, the lessons need some tweaks here and there. Specifically, the amount of scaffolding and teacher direction are not calibrated for optimum performance. Having giving them a test run, I like the core ideas and I think the experience was very valid, but I know some major revamping I will be doing for next year.

What did go well was the conversation and engagement when students were actually building and working with the manipulatives. I am forever amazed that something as simple as small, wooden sticks could get even the most ardent stalwart working through the lesson. The classroom was buzzing, that’s the moment I love to hear, kids actively involved with their learning. I will update this and other thoughts as I move forward, getting back to blogging and catching up has been a hard road to toll.

Happy Mathing!

A Hard Act to Follow

After attending one of the most powerful mathematics professional developments with Matt Vaudrey and John Stevens as the chefs in La Cucina Matematica they shared a wonderful site that fits right up my teaching alley, Estimation 180. The site offers a day-to-day estimation scheme, which helps students increase their mathematical thinking and sharpens their estimating skills. I love the idea getting students to use their instinct and to reason through their thinking on figuring out these problems. I always find it amazing how even the most reluctant learner will throw their hat in the ring with a compelling prompt, taking a risk when there is no wrong or right answer. The learners are always so passionate about their reasoning and it sparks some great on topic discussions.

That aside, Estimation 180 offers another nugget of gold, which are pre-made lessons, based on the 3 Act strategy implemented by Dan Meyer (to the best of my knowledge he is the first person I know to coin this term). On the teaching front, I often adopt this model, I find it is a natural way to increase students buy in and a fun way to facilitate learning. My background is based heavily in secondary and college curriculum, so I have spent this year getting my feet wet with the elementary grades. Of the number of lessons I have done this year in lower grades, I was extremely excited to see an awesome lesson prepared at Estimation 180, ready to go, which I saw as the second graders I was planning the lesson for are just entering addends with three digit numbers. The lesson, Bester Egg Tower, was a perfect starting point, I built a presentation around it, and went to try out the lesson. A preview of the materials I downloaded and my presentation are here.

The lesson went off very well, the kids were so excited, we had them sharing in groups of three, building the trays of eggs with ice cube trays, and Styrofoam colored eggs (colored for each group). The pace was pretty quick, as I tend to be very high energy in the classroom (a behavior management and buy in tactic as much as my personality), and I like to keep things moving according to my quick formative assessments, I adjust as necessary.

Although I felt great about the lesson, there were many things I learned that I will be more aware of with more practice. First of all, when students wrote what they noticed and what they wondered, I was surprised that about half the class didn’t know what to write. As a teacher, I am concerned because the 6 year-olds in my life are suffering from the same condition, they are not as excited about school, and don’t ask nearly as many questions as I would imagine they should be at this age. If kids don’t wonder about anything, then learning doesn’t follow as readily. So front loading what it is to wonder, and having question stems (and sentence frames) will be great additions to this lesson.

That being said, the next thing I was surprised to have students share out as their questions (which are recorded in the notepad document here) were either about why the chickens moved so fast or why the teacher was stacking the eggs. I was anticipating the latter, but reflecting on the lesson the former makes a lot of sense, as kids wouldn’t be used to seeing chickens moving in high speeds….leave it to kids to point out the obvious and make the teacher wonder.

The biggest moment for me came, when I asked the kids about the ten frame the carton of eggs, I asked the kids to compare and contrast the two. One student shared he contrasted that one was even and one was odd.

Ten Frame vs Egg Carton

As you can imagine, my teacher brain was thinking but they are both even, and I was thinking I needed to ask for clarification. My normal inclination may have been to ‘jump in’ to ‘save this poor kid’, but I didn’t and I asked for clarification. The student said that the ten frame as five in a column and the egg carton had six, so one is odd and one is even (granted those weren’t his exact words, but that’s what he meant). Do I really need to say more here? Wow, I know.

One adjustment I would make for next time is to have a picture of just one row of the stack of eggs, I think having the kids get to the point where they are understanding just one row (3 dozen eggs, so 36 eggs in a row) is a great starting point to move forward. We finished on getting kids to the “doubles” plus 1 number of 12, and we talked about the strategies the kids used to get their result.

All in all, it was a tremendous amount of fun, and I can’t wait to do this lesson, in another second grade class. I can also tell the students loved the time we spent, because it took literally 3 full class hugs and several high fives for me to be able to get out the door. Two weeks later, I see a group of those kids on campus, they run over to me, hug me with unison cries of “When are you coming back….” Needless to say, second graders may be my new favorite grade level, at least until I teach Kindergarten again.

Here are some students’ work from the day, I tried to capture the range of responses that the kids produced, the gradient of abilities.

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A huge thanks to the chefs at  La Cucina Matematica and for the folks at  Estimation 180 for making these great resources available and for inspiring some alternative fun. I have loved the 3 Acts theme for a while, but seeing that it works with second grade was amazing. The last thing I forgot to mention is that the kids had a very hard time estimating how many eggs they thought would be in the tower, this shows that the kids may not have a strong sense of the size and quantity of these numbers. As they are just now learning about 3 digit addends, I think this reflects in their difficulties with estimation, which is another reason why the estimation practice may be helpful – especially in terms of thinking about the reasonableness of their answers in their own work (some SMPs) in the mix as well.

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.