Tag Archives: problem-based learning

Word Problems and the Problems with Words

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Yesterday, I posted a new 3-act task on the blog. In the tradition of digital mentors like Graham Fletcher, Andrew Stadel, and Dane Ehlert, I will rarely post an activity on the blog that I don’t intend to use in my own class with students. Today, we did Make It Rain.

Here is what my students noticed during Act 1…

  • There’s a lot of money
  • There are 20’s, 10’s, 5’s, and 1’s
  • There are more 20’s than 10’s

And here’s what they wondered…

  • How much money is there?
  • Why did it go from greatest to least?
  • Why was it being spread out?
  • What kind of bills were in the pile?
  • How many of each bill is there?

My students are used to analyzing their questions collaboratively. Some of the students noted that we couldn’t answer the “why?” questions without asking the person in the video, who we did not have access to (even though it was me!)

So, then our wonderings looked more like this…

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A Snapshot

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First, a little background.

The theme of our spring unit is always financial literacy. As teachers of students with varying degrees of need, strength, and interest this means different things for different groups of students. One of my groups is working on selling tickets for our school play, Alice in Wonderland.

We sell tickets at two price points. An adult ticket costs $10 and a child/student ticket costs $8. This is partly my doing, because having two different prices sometimes allows my students to investigate more interesting mathematical questions. Today was one of those days.

Show-goers are also able to purchase play tickets in one of three ways: cash, check, or online with a credit card. My students record the type of ticket and the method of purchase for each order in a table. Students then represent this information visually using graphs. We will use these tables and graphs later on to reflect on the trends and patterns in the ticket sales to make suggestions to our play directors for future ticket sales initiatives. But that’s the bigger picture and I promised you a snapshot. So here it is.

I realized I had been giving my students too much information. As they recorded the total amounts of cash, checks, and credit, I was also telling them the type of ticket. Today we began our routine of using math to figure out the type of tickets using our knowledge of the ticket prices and total amount of money. I gave them this problem as a warm-up:

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Productive Struggle vs. Frustration

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This past Friday, I gave an introductory presentation on the educational ramifications of new brain and psychological research, specifically, Carol Dweck’s Mindset. What came out of the discussion during the session, was that our school already does a fairly good job of inherently implementing most of the underlying themes in Dweck’s research.

What we realized we still needed to work on as a school, was allowing students to struggle productively. Robert Kaplinsky recently posted his ignite talk from the Northwest Mathematics Conference. Kaplinsky gives a very accessible account of the differences between productive struggle and what he calls, “unproductive struggle.” In our school, “unproductive struggle” is frustration.

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Contemplate then Calculate: Dominoes

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The math department at my school is working to implement instructional routines in our classes for several reasons. The first reason is to increasingly apply the standards for mathematical practice on a daily basis. The second reason is to improve our communication as a department by creating common language centered around shared routines and activities.

One of the instructional routines we plan to implement is Contemplate then Calculate (#CthenC on twitter). Created by Amy Lucenta and Grace Kelemanik at the Boston Teacher Residency, #CthenC is a highly structured routine that simultaneously allows for open mathematical thinking and problem solving. Contemplate then Calculate highlights looking for and making use of mathematical structure in problem solving.

In an effort to assist the implementation of Contemplate then Calculate at our school, I created this activity to do with my class as a model for implementation and discussion.

Here is the task:

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How many dots?

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3-act Task: Trail Mix

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Student engagement is a funny thing.

On twitter I’ve been pretty critical about using extrinsic rewards to increase student engagement.

Today was our 100th day of school (as calculated by our students!) To celebrate we made 100 piece trail mix. Our trail mix included: cheerios, chex, raisins, and M&Ms. Candy! Talk about extrinsic student engagement! Before we dove in to the rewards food, I gave my class the following problem:

We have 4 ingredients to make trail mix. How many different combinations of ingredients can we have if our trail mix only has 100 total pieces?

The students persisted through their work on this word problem, until they arrived at various solutions based on their calculations and personal taste. For instance, one student is allergic to nuts and could only eat the cheerios and raisins, so that impacted his work on the problem. The students worked diligently and happily ate the trail mix once they had arrived at a reasonable solution.

However, after class I channeled Graham Fletcher and Dan Meyer to try to make this mathematical experience a more rich one for the students. So, here is a preview of the 3-act task we will be doing tomorrow in class…

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Scaffolding For Executive Functioning

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Over at Reason and Wonder, Michael Fenton is exploring the possibilities for using Alex Gendler video puzzles in the classroom. Michael’s wonderful take on these rich resources, reminded me of one of the main goals of this blog, to show how students with disabilities can access rich mathematics instruction.

As we began this school year, my goal was to model how our class valued perseverance and sense-making over answer-getting. I did this for a couple of my classes by using Gendler’s Zombie Bridge Problem video. The video is long and there are a lot of details to account for before you can come to a reasonable solution. This requires quite a bit of what is called executive functioning. Executive functioning includes (but is not limited to) the abilities to initiate a task, make a plan, prioritize information, organize information, think flexibly about strategies, and self-monitor (i.e. check your work). Sound familiar? My students tend to struggle with executive functioning skills and this is often where my scaffolding is targeted.

To help scaffold my student’s executive functioning while solving the Zombie Bridge Problem, I used EDpuzzle. EDpuzzle allows a teacher to modify an already existing youtube or uploaded video by cropping it, including voiceovers and adding questions. Here is how I used EDpuzzle to scaffold the Zombie Bridge Problem.

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First, I cropped the video to exclude the solution. As anyone familiar with 3-acts knows, the solution is vital, but should come after students have had time to explore first! So it was gone.

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Unscaffolding Math Problems

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Several months ago, the new NPR show Invisibilia did a broadcast about expectations.  The main theme of the program was that the expectations others hold of an individual can effect the outcomes of that individual, either positively or negatively.  If you’d like to know more about this idea please listen to the radio show, its great!

I wanted to incorporate the show’s theme into a blog post about special education math classes, but was unsure how until Alex Overwijk, a teacher from Ottawa, sent me the following comment about my post about scaffolding…

Expectations...

Expectations…

This led me to consider how the over-scaffolding of mathematical tasks and problems for special education students creates an atmosphere of lowered expectations.  Both Alex and I agreed that students with disabilities need a certain amount of scaffolding to be successful. What we didn’t know was to what degree and when this scaffolding should be provided.

Thinking more deeply about this question, I believe the degree to which scaffolding is provided to students with disabilities is a very individual, personalized process.  Great special ed teachers who understand their student’s learning pathways will be able to determine the appropriate level of scaffolding for them.  But the timing of when scaffolding is provided can show students what a teacher’s expectations are for them in math class.  If scaffolding is implemented too early in a lesson or unit, students may feel a sense of lowered expectations which according to Invisibilia would result in lowered outcomes as well.  You can’t get much earlier in a lesson or unit than the pre-assessment, so let’s start there.

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Noticing and Wondering About Scaffolding

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Often I wonder how much to explain or define for my students before engaging in the problem solving process.  Proponents of sense-making in mathematics classes like Dan Meyer and The Math Forum encourage presenting a perplexing scenario to students and letting them develop the questions to be answered using math.  This is a very enticing proposition.  Who wouldn’t want a math class which uses the Socratic method to solve problems as a community.  I do!  Professor Ilana Horn recently wrote a piece investigating the merits of this pedagogical philosophy with other popular options like Doug Lemov’s Teach Like a Champion.

Some students, however, need more scaffolding, language support, cultural background, or skill reinforcement before they are ready to grapple with a truly perplexing situation.

Vygotsky's Zone of Proximal Development

Vygotsky’s Zone of Proximal Development

For instance, what if your students view their zone of proximal development much differently than you, as the educator, do?  What if the student views every problem as lying in the outer ring, but it truly lies in the middle or inner ring according to your professional opinion?  Which leads into my question about problem-based learning.  How much do you scaffold for students who need it before you set them free to make sense of a great, perplexing mathematical scenario?

This is a major question for special education math teachers.  How much scaffolding is too much so that the process of solving the problem is taken out of the hands of the student?  One area where this comes up is when teachers are deciding what order in which to present information to students during the problem solving process.  As an example, here is a problem I have been developing in which there are two components.  Which of these components should go first in a truly problem-based classroom?  Maybe you can help me figure it out!

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A Student’s Eye View

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Ashli Black, over at Learning to Fold, recently posted this little bit of wit and whimsy.  The post essentially recounts her experience in algebra classes and compares it to the experience of contestants on an extremely confusing, quite vague, and thus hilarious math game show.  Ashli makes the point that, “As that kid without conceptual understanding in algebra, this skit is pretty much exactly what it was like in class for me. Confusing, almost no stated rules I understood, and at any moment the scene might change or I might be shoved in a box for not achieving Wangernumb.”  Ashli considers the difference between teaching for conceptual understanding and teaching for procedural understanding in her post, but it got me thinking about my own students.  I often think my students are holding their breath, waiting for me to tell them their answer was in fact “Numberwang.”

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My goal as a special educator is to communicate the day’s lesson or task so the students will be able to access, understand, and apply the mathematical content.  This often leads to accommodation, modification, and differentiation of everything for everyone.  When one thinks of accommodations the first things that come to mind are standardized testing accommodations.  The general list usually looks something like this:

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