Problems Worthy of Being Solved

In one of my recent talks, Playing the Long Game, I assert that a problem is worth walking away from if it is worthy of being solved.  I’ve been asked a few times to expand upon what I mean by a “problem that is truly worthy of being solved.” 

From my perspective, there are many answers to this question. Sometimes worthy might mean multiple points of entry so that all students can engage in the problem. Other times worthy might mean that the results or strategies that can be generated from the work are really interesting. Worthy might mean that the task is directly connected to an idea or concept that your students are interested in pursuing. And other times worthy may mean that you know a problem will elicit good conversation amongst your students. To me, a problem worthy of being solved can be any or all of these things. It is a problem that is chosen with intention and used to engage your students with important mathematical or social ideas.

As we begin Women’s History Month, I posed the following problem to my first and second-grade students.

34 out of 100 students studying STEM in college are women.
1) How many of those 100 are men?
2) Make a representation of this data in whatever way makes sense to you.

This problem seems worthy to me because it does three of the things I mention above. It allows for multiple points of entry, it’s connected to something happening in our world (that my students are interested in), and has the potential to generate conversation.

First students solved the question, “How many of those 100 were men?” Most students used subtraction to determine how many were men. They used a variety of strategies including subtracting tens and ones and compensation. A few students counted from 34 to 100 by ones and tens to determine the answer. This was a brief but valid conversation. 

From there we had a conversation about how this information made them feel. Here are a few of their ideas.

• Some people think boys are better at math than girls. They are wrong.

• It’s not fair that more boys get to do STEM.

• That is a lot more men than women. That’s not awesome.

Here’s a screenshot of how one remote student felt about this.

The second part of the problem asked the students to represent this new information in whatever way made sense to them. I told them they could draw, use manipulatives, or use BrainingCamp (an online virtual manipulatives site that they are very familiar with). 

Here is work from four students. 

Student A

Student B

Student C

Student D

The use of the base ten blocks (Student A) was an interesting choice—I didn’t know how it might end up, but I love where this student ended. This representation compares the values nicely and shows an understanding that they both equal parts of 100. And the written narrative is brilliant as the student went beyond the task requirements and determines how many more men there are than women represented in this data. 

The two that used the graph with color tiles (Students B and D) found VERY interesting (and different) ways to represent the information. Side note: I love analyzing student work. But, that’s hard to do in a blog format without me just telling you what I notice. So, if you’d rather spend some time on these two pieces before I tell you what I see, pause here. The most remarkable thing about these two pieces to me is how they handled the y-axis. The BrainingCamp template automatically formats the y-axis to be intervals of one. Student B said, “oh no, I can’t use this.” I didn’t react because I wanted to see what she might do next based on this. She was quiet for a minute and then said, “wait, I can just put a zero after each number, and then it will go by tens.” So, she used the text tool to add a zero after each number. Student D then took her idea and used it as well. 

The hundred chart (Student C) is a startling visual of this data and I love that this student chose to represent it in this way. 

I’m perpetually blown away with what these students can do given problems that are worthy of being solved. They know that they always have the ability to say when it’s too much or when they don’t want to share their thinking. And I believe that this autonomy and choice make them stronger and better mathematicians.