Recently I’ve become aware of the reality that as math teachers, we often talk about what problems we will be using and even how we will implement them but frequently we miss opportunities to discuss why we’re using them. Consider Graham Fletcher’s amazing sugar cube task on volume of a rectangular prism. I can think of three different reasons why you would do this task, and all three of them greatly impact the decisions you make during the problem and consequently how the students learn.

Reason #1: Use the problem to introduce a new concept
A problem like the sugar cube 3-act task would be an amazing way to introduce 5th graders to volume of a rectangular prism. You start the unit with this problem, realizing that theoretically students do not have the skills to complete it because they have never learned about rectangular prism volume. They may be able to make the necessary connections and complete the task, but that is not critical. It would be just as wonderful (and maybe even more so) for students to get to a place where they want to figure out the answer but realize that they don’t have the skills yet. Students now have a headache and crave a mathematical aspirin to make it go away. Once they realize that they are missing this knowledge you can then pause the problem, tell them that this is what the class will be learning about for the next few days/weeks, then build conceptual understanding and procedural skills from the context. Finally, you can come back to the same problem at a later date to complete it. It will be a great litmus test for students to demonstrate their newly acquired skills.

Consider though how this could potentially look to two observers: one who understands the reason why you are using this problem and one who does not.

Observer Perspective

• An observer who understands that you are using this problem to introduce a new concept expects that students will be curious and possibly baffled by the problem. They’ll see how students will be eager to learn more about how to solve the problem. Whether or not you finish the problem is inconsequential because you can come back to it at the end of the unit. Either way, you invested time in developing a marvelous context that will pay dividends throughout the unit.
• An observer who doesn’t understand that you are using this problem to introduce a new concept may wonder:
• What was the purpose of this problem?
• Why didn’t you finish it?
• Why didn’t you let students struggle through it?
• Did the teacher end the problem because he or she was confused and gave up?

Reason #2: Productive struggle
Sometimes the goal is just to let kids productively struggle and build their critical thinking skills and perseverance. It may be an especially challenging problem or perhaps it addresses standards from a different grade level. In Graham’s problem specifically, you want students to discover that cubes laid out in this structure make counting them easier.  Either way, your goal is to let kids make their own connections and rely on each other as much as possible. Accordingly, to prepare, you may come up with a set of least helpful questions and hints that will give students just enough of a nudge to keep them going but not so much that it takes away their opportunities for critical thinking. With the goal of productive struggle in mind, finishing the problem during the class is nice, but not necessary. Wherever students get to in the problem is wherever they get to, as long as they are learning along the way.

Observer Perspective

• An observer who understands that you are using this problem to let students productively struggle is looking to see that kids are working hard to make connections on their own. The observer realizes that your choice to give minor hints and not tell students what to do is not laziness or being unprepared, but rather trying to give students the least amount of help needed for them to keep going. This is similar to a bench presser and a spotter. A spotter’s job is to give the bench presser the least amount of help needed so that he or she can lift the weight. Too much help and it’s the spotter who gets stronger. Too little help and the bench presser dies. As part of this process, students will inevitably demonstrate many of the Common Core Standards for Mathematical Practice including 1, 3, and 6 as they try to make sense of the problem and convince their partners why they are right or wrong.
• An observer who doesn’t understand that you are using this problem to let students productively struggle may wonder:
• Why did the teacher let the students sit there confused instead of telling them what to do?
• Did the students even learn anything because they never figured out the answer?
• Why didn’t the teacher finish the problem?  Did she lose track of time?

Reason #3: Problem completion
There are times where we just want to start and end a problem within a given class period. Maybe it’s because you have colleagues who have never tried a problem-based lesson like this one and they want to see how students progress throughout the entire problem. Maybe you are being evaluated by an administrator and want to show how a complete problem goes. Maybe it’s because you have very limited time and are trying to keep up with your colleagues. The reality is that when you have to finish a problem within a specific window of time, you may have to limit students’ discussions or other opportunities for making connections so that you have enough time to progress towards the end.  In this sugar cube problem, students may start to notice that you can add up all the layers to figure out the number of cubes, but may have trouble generalizing it into multiplication. With the goal of completing the problem in mind, you may decide to just tell them that mathematicians have noticed this pattern too and have come up with a formula.

Observer Perspective

• An observer who understands that you are using this task to show a completed problem realizes that you had to make some sacrifices to complete it in a single class period. The observer is better prepared to imagine how different parts could be expanded, given more time, and is appreciative of having experienced the problem in Cliff Notes form.
• An observer who doesn’t understand that you are trying to complete the problem in one period may wonder:
• Who really did the work today: the students or the teacher?
• Why did the teacher not see all those great opportunities for students to make their own connections and take advantage of them?
• Why did the teacher give such obvious hints and tell them what to do?

My recommendation is to have conversations with your colleagues about why you are using specific problems.  You may find that your “why” is different from theirs.  Not having that conversation before the problem usually sounds like this after implementing the problem:

• Teacher 1: “Man, that problem was challenging for students.  They spent the whole period working on it and didn’t even finish it.” (Goal is productive struggle)
• Teacher 2: “Yeah, it was challenging but I gave them some hints and we worked our way through it because we are beginning a project tomorrow.” (Goal is problem completion)
• Teacher 3: “We didn’t actually finish the problem.  We started it but we’ll come back to it later once they’ve learned more about the concept.” (Goal is introducing a new concept)

Update (4/27/2016)
Recently I observed a classroom where the students began the period reviewing homework and then spent the rest of the period reviewing for an upcoming assessment using Kahoot. I found myself wondering if the problems being used fell into any of these three categories, and it didn’t feel like they did.

– is using a problem for assessment its own category?
– does this categorization only apply to non-procedural problems?

1. I really enjoyed reading this, Robert. This year, I feel that I’ve gotten much better at asking myself, “What do I / we want students to learn and be able to do by the end of the lesson?” when lesson planning, which has helped me craft my learning objective or target and how I want the students to engage in the SMPs and content.

But…now that I reflect on it, I’ve only accomplished two or the three parts — the “what” and the “how.” I need to be better at reminding myself about the “why” — why am I using this activity, why am I asking students to do this, why is it important for student learning.

I’ll be honest…I’ve definitely used certain lessons or activities multiple times because I simply enjoy them or students enjoy them. I would like to think that there are many other teachers that are in the same boat. I will say that I’ve gotten better at choosing lessons or activities that are congruent to my learning objective, but I am (and I think other teachers are) still missing that third and critical piece to the puzzle.

I appreciate you reminding me and others that we need to start with the “why” with everything that we do in our classrooms, conducting professional development, and our conversations with students, parents, and administration. The more we communicate our intentions and have these conversations, the better we will all be — teachers and students together.

• Robert Kaplinsky says:

Thanks Daniel. This whole post is very post-dictable to me in the sense that it is all so obvious to read now that it’s been written, but yet it had never occurred to me until very recently. Definitely worth thinking about personally but also worth communicating to others when we do lessons in their classes.

• I completely agree. It makes total sense when I read (and re-read) this post. I appreciate you taking the time to formalize this for us. I’ve been guilty of trying to accomplish all three of the reasons and I know it could have gone better if I just focused on one.

One question…do you think it is possible or appropriate to use a math problem with the purpose of accomplishing more than one or all of the reasons you listed? For example, can a 5th grade teacher use Graham’s sugar cube task to accomplish productive struggle and problem completion?

• Robert Kaplinsky says:

Great question Daniel. Short version is that I don’t know for sure. Two thoughts though:

– First, I am far from certain that this is a comprehensive list of why people use problems. There are likely others.

– Second, my gut tells me that one reason is should be prioritized above the other. For example, in your scenario, problem completion is likely the priority. So, that would look like, “I’ll let the kids struggle as much as possible, but the problem still needs to get done so I might cut them off if we start to run out of time.”

It seems though that some reasons are more at odds with one another. For example, problem completion and introducing a topic don’t go well together. If you use a problem to introduce a topic, you have time to come back to it, so you don’t necessarily want to give them formulas needed to complete the problem that they will discover soon.

Thanks for pushing my thinking.

2. This post really resonates with me, both as a teacher and an instructional coach. Additionally, you know I’m a big fan of starting with “why” and the purpose (or learning objective). Your offering of possible uses and conversations can really help support a collaborative math department, or initiate a more collaborative department.

One thing I would advocate (and add to this post) is the need for teachers to solve the task. It might be assumed while reading your post, but assumptions can be detrimental. I think if a teacher understands the importance of solving the task ahead of time (and not the night before), it will be extremely beneficial. In solving the task, the teacher can anticipate as many students solutions and misconceptions as possible, making it easier to determine where they want to use the task: new concept, productive struggle, or problem completion.

Tagged under my “Sticky Math Blog Posts – 2016”

• Robert Kaplinsky says:

Yes, I completely agree with you. I guess I would classify it this way:

What = Choosing the problem

How = Using the 5 practices, which includes doing the problem of course

Why = One of the three reasons above or any of the other reasons I will realize exist over the coming years.

3. Michelle says:

I would like to see PB questions for Algebra 2 for polynomial or rational problems. I’ve been following you and reading your materials and have used some of your videos. But I would like to see some examples for these functions

• Hi Michelle,

This is a common question I hear too. I’m curious if you’d be able to share any tasks you’ve created or have come across in a textbook that would represent polynomial or rational functions.

If not, what are some of your favorite problems from the math textbook, curriculum, or resources you are currently using with polynomial and rational functions? I’m curious because sometimes textbooks have decent contexts related to higher level mathematics and we can use them as a starting point in finding PD questions for Algebra 2 like you requested.

Thanks.

• I don’t have a problem for you, exactly, but I think James Tanton’s “Personal Polynomial” task might be the start of something good:
I, of course, would cheat. My full name is PATRICK, which means I’d need to find a function f(x) such that:
f(1) = P
f(2) = A
f(3) = T
f(4) = R
f(5) = I
f(6) = C
f(7) = K

That’s too much. It’s been a while since anyone has shortened my name to the androgynous PAT, but since it’ll make my life simpler, I’d try to find a function f(x) with:
f(1) = P
f(2) = A
f(3) = T

Shouldn’t be too hard; some simple quadratic should do the trick.

4. I love how you’ve broken this down so more people can understand it! I used to do this a lot when I was in the classroom and some people would think I was crazy until they saw that my students could actually problem solve through some things better than other teachers. Plus I loved engaging them and getting them to the point of begging me to teach them the little piece they were missing. Keep up the great work!

• Robert Kaplinsky says:

Thanks Sarah. The more we can all get on the same page in terms of goals and expectations, the better. I appreciate the support.

5. Great post, Robert. I just engaged teachers in a the what and how and why today. The why got cut short due to time. Your post makes me rethink conversations I have with teachers and the order in which we address these questions. I’ll definitely be more deliberate about beginning with the “why” when planning with some 8th grade teachers on Monday.

As you continue wondering about problems used as assessment being in their own category, do you think that may depend on the reason for the assessment? Is it formative or summative? What is the teacher’s reasoning for giving the assessment. Is it just to get a grade or is there a more pedagogical reason?

Lots to think about here. Thanks making me waking my brain up… again!

• Robert Kaplinsky says:

Thanks Mike. Regarding your questions about using problems for assessments, in short: I don’t know. The lesson was just quizzing kids with no reflection whether kids got the answer right or wrong and obvious purpose. Let’s be clear, I have done the exact same thing when I was in the classroom.

Now with more experience and perspective, I try to have a specific goal for everything I do. Certainly I don’t hit it every time, but now I have something to measure against to see how effective what I did was.

6. It is interesting that the “hows” of doing the problem should also be nested in the “whys”. Eg.” I used pairs or groups of 3 because I know in their productive struggle there will be a lot of mathematical reasoning and communication happening. They were on vertical surfaces so their work was exposed to others making it more likely that the key conceptual understanding could be mobilized in the class and aid problem completion. We “assessed=discussed” the clarity of each group’s communication so that students would acquire problem solving skills for other problems. I summarized what I saw in the student solutions to draw out the key conceptual understanding. . .” This process is mirrored in the comment discussions after your initial post. You have utilized the blog as our problem to extend our thinking in our (& your) planning of, observing of and reacting to lessons.
Thank you for a rich time.

• Robert Kaplinsky says:

What a nice comment! Thanks Fred. I didn’t reflect on how this discussion may relate to implementing a problem. Thanks again for chiming in.

7. I am getting observed today, so this is perfect timing! Truly a bit nervous because I am introducing a problem today that I know students will not be ready to solve. This goes along the Dan Meyer, Jo Boaler, yourself and other philosophies about creating that desire to learn the math. Making the math interesting for right now, today, so there is a want from the students to learn the procedural that compliments the problem solving. My principal is pretty great, I think once I explain this, he will be on board. I thank you for writing this so I can share it with him today.

I noticed your last comment about observing a teacher who reviewed homework and then spent a class preparing for an assessment. We still live in two worlds. The reality of needing grades, and the hope for change to move towards standards based instruction. I struggle living in both worlds and try endlessly to somehow bring the two in sync. No, it is not a perfect situation, but we can’t give up!

One of the new additions to my culture this year comes from Jo Boaler. I am incorporating a Participation Quiz. I have been tweaking day after day as I can’t seem to achieve the right balance quite yet. Today, I will be attempting version 6. This grade will actually come from student working together, having conversations, etc. Perhaps it is one way we can bridge the two schools of thought. It might be one way to get the standards and grades on the same page (until we can get rid of grades!!!). Do you have any other ideas?

• Maybe I’m just missing the point here, but I don’t see what’s wrong with practicing for an assessment. Once you’ve done the productive struggle and developed the concepts, you need some practice with them to develop procedural fluency. Then on the assessment you can demonstrate that procedural fluency as you solve smaller problems in context. Also, you can use those skills to solve more complex problems in future lessons.

• Robert Kaplinsky says:

Hi Kathy. Perhaps some of the misunderstanding comes from what type of assessment you are talking about. If you are doing formative assessment, you want to see where your kids are at so you can adjust accordingly. You would not practice especially for that because you’re just trying to figure out where they are. For a summative assessment, then at the end of some period of time you’d like to measure how well they understood the concepts. So, I guess all that you teach them goes towards practicing for that assessment.

I’d also be careful to separate practice from being exclusively connected to procedural fluency. You can also practice applications and developing conceptual understanding.

• Cheryl Corral says:

Hi Kathy, I think the problem with the test review using Kahoot is that there was no way of knowing why students would get an answer wrong or knowing who did. It mostly benefited those who were already well versed in the procedural and conceptual understanding. Those who weren’t got wrong answers bit no clarification of their understanding. This type of review simply creates a feeling of failure and frustration for the students who are getting wrong answers. I would NOT want my students going into a test with that cloud of negativity hanging over them. Rather, I want them coming into the test feeling confident and eager to do their best. Kahoot is fun and engaging but maybe not the best way to prepare students for a test.

• Robert Kaplinsky says:

Hi Meryl. I’m glad that this post was useful for you. It sounds like there is a lot of potential with your Participation Quiz. I hope it went well. I don’t have any other ideas for you at the moment, but if I do, I’ll add them to this post. Thanks!

8. I have been thinking about the role of imaginative education in developing problems solving. We (most) often ask questions that students work on to find a solution path. There is a large focus on ‘what the answer is’ and less on the process. What if we were to reverse this; show sample student solutions and have students imagine how the other students solved the problem? This would create much more emphasis on the process. Students would be looking at what the ‘mathematics’ ‘speaks’ about their process. A simple example is array counting with a picture of dots and students asked to show how they might determine the number of dots. Instead, here is a picture of some dots. One student said it was 4(3) – 2 = 12 – 2 = 10 dots. Show how you think they were seeing the picture. Self-assessments of participation and peer comments will be a big help but ‘learning work’ should be in process and not summatively graded. This comes when the learning has processed.

Just my nickel worth of thoughts as Canadians have done away with the penny and inflation has made my two cents, a nickel.

On the quizzes, I don’t think we truly have the balance until students are assessing on the criteria and not marking on them to arrive a grade in a formative setting. Checkups are key to seeing where each student is at and where their next steps might be. Grades interfere with the analysis of criteria.

• Robert Kaplinsky says:

Very neat idea Fred. It reminds me of the work I did with teachers this week around picking a selection strategy when implementing the five practices. Instead of going from concrete to abstract, one group picked abstract to concrete and tried to get people to see a formula and be able to explain where it came from. It worked out pretty well.

Thanks for adding to the conversation!

9. Shelley says:

I think this applies to more than just colleagues observing us. Parents too wonder what is going on in math class when they get their child’s version of what happened in class. They don’t know your why at all! I’ve been a math co-op teacher for homeschooled kids and I am really sensitive to the judgement of parents when I try to do things differently from the “old traditional way.” Your post helped me realize why that is! It does look dumb to those who don’t understand your goal. (And especially to those that don’t agree with your goal.) Now I know what I can do about it for myself. I thought I just wasn’t confident in my math teaching skills; but now I think it’s more that it was about knowing in the back of my mind that those “observing me” (and in online classes that are live the parents are literally observing you without any feedback) were passing judgement without the full picture. Thanks for this post!!

• Robert Kaplinsky says:

Hi Shelly. I hadn’t thought about the perspective of parents’ perspective but I totally see what you mean now. I’d love to hear about how this went if you tried sharing this.

10. Linda says:

Thank you for the many thought provoking posts! In considering the above update regarding assessment as a reason for working through a problem, I think it’s more than simply assessment prep. Some problems allow us to target the current level of mastery for a specific standard. Whether it’s test prep, classwork, homework, or an actual assessment, sometimes we need to see that a student can recognize a related efficient procedure and apply it effectively in order to independently solve a problem. Have they reached mastery for a specific standard? Complex problem solving is excellent, but it involves a great deal of standard and practices overlap as well as collaboration. That makes it difficult to look for individual weak areas that require further targeted instruction. Besides, not all problems in life are complex. Sometimes the goal is to recognize and use the most efficient method to get something done.

• Robert Kaplinsky says:

Thanks Linda. Your comment really made me think. I agree that there exist problems used for that type of purpose. In this particular situation, it didn’t seem like the problems were really being used in such a way that complex problem solving was being assessed. Truly it is a challenging thing to measure, but in the situation I was in, it seemed like it was more like a summative assessment being paraded as formative, and with no particular purpose in mind.