Tag Archives: education

Making Sense of Word Problems

I am sure we have all seen it happen at one time or another in math class. We give a student a story problem to solve and after a quick skim, the student pulls the numbers from the problem, computes them, and writes down an answer.

If the answer is correct, we assume the student has a grasp of the concept. However, if it’s incorrect, we’re left with a laundry list of questions: Do they realize their answer doesn’t make sense? Did they not understand the context? Did they simply pull the numbers and operate to be finished or did they truly not know what to do with them? Most importantly, we ask ourselves, how can I help students make sense of what they are reading and think about the sensibility of their answer in the context of the problem?

If we’re lucky, we can identify a mathematical misconception and work with that. Oftentimes, though, the answer isn’t even reasonable. Then what do we do?

This scenario has me reflecting on the Common Core Standard of Mathematical Practice 1:

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

The best way I’ve found to help students make sense of what a problem is asking is, ironically, to take the question out altogether. Inspired by the wonderful folks at The Math Forum, I do a lot of noticing and wondering with students in this fashion. Most recently, after reading Brian Bushart’s awesome blog post, I have started taking the numbers out as well! Instead of students thinking about how they’re going to solve the problem as they read, they are truly thinking about the situation itself. It’s been an amazing way to give every student entry into a problem and allow me to differentiate for all of the learners in the classroom, while at the same time provide insight into my students’ mathematical understandings.

Recently, I had the opportunity to work with a 3rd grade class. The class recently finished their multiplication and division unit and will soon be starting their work with fractions. In order for their teacher and I to see and hear how students apply the operations, make sense of contexts, and currently think about fractions, I thought it would be interesting to take a story problem from their Student Activity Book and take the question and numbers out.

The Planning

I chose the problem below and thought about what I would learn about a student’s mathematical understandings and sense-making after they answered the questions.

I was curious to observe how students make sense of problems based on the idea of removing the numbers and the question so I changed the problem to this simple statement:

“Webster has boxes of granola bars to share with his class.”

I anticipated the students would wonder about the missing mathematical pieces involved in an open-ended statement like this. I believed their wonderings could lead them to develop questions that could be answered based on the very information they were wondering about. I knew the mathematical ideas of multiplication, division, and/or fractional sharing would arise and that I would learn so much more about their thinking then if I had given them the original problem.

In The Classroom:

I launched the lesson by posting the sentence on the board and recorded things they noticed and wondered.

They noticed:

“We don’t know how many boxes of granola bars.”

“There is not enough information to know what is going on.”

“We don’t know if it is adding, subtracting, multiplying, or dividing.”

“There are multiple people in the class because it says boxes and share.”

They wondered:

“How many granola bars are in each box?”

“How many boxes he bought?”

“How many kids are in his class?”

“What kind of granola bars are they?”

Based on their noticings and wonderings, I felt everyone had a strong grasp of the context and sense of where this was going. Based on their noticing that there is not enough information to know what is going on, I asked what more they would want to know. They responded that they wanted the answers to the first three of their wonders: bars per box, number of boxes, and number of kids in the class.

I asked them what questions they could answer if I gave them those pieces of information and they responded:

How many bars does he have?

How many bars does each kid get?

How many does he have left?

At this point, I could have given them the information they wanted. However, I thought it would be so much cooler to allow them to choose that information for themselves. I was curious: how they would go about choosing their numbers! Would they strategize about the numbers to make it easier for themselves? Would they even think that far ahead? What would they do with the leftovers?

When I told them I was not giving them the information and that instead they were choosing their own numbers along with the question they wanted to answer, they were so excited!

Some partners chose their numbers very strategically to make it easier for themselves. To me, this demonstrated a lot of sense-making and forethought of what was going to happen in their solution path. And as an added bonus, while only asked to answer one question, the group answered all three questions! (Teacher note: if students chose numbers strategically and therefore finished quickly, I gave them extra bars to factor into their problem to see how they dealt with the leftovers.)

Other students chose the opposite route and strategically picked numbers to make it “harder for themselves.” Check out the way these two students showed strong reasoning and perseverance through division of numbers larger than any they’ve ever worked with.

Others chose numbers without much forethought and dealt with some amazing leftovers. This was a great way to formatively assess students’ thinking related to fractions before they began that unit.

And then there are always the surprises. Who would have thought third graders would reason about the leftovers in terms of percentages?

Reflecting on what the students would have been asked to make sense of and the work they would have had to do based on the original problem versus the reasoning and work they did related to this one simple sentence, I’m amazed by the difference. I learned so much more about what each of the students know beyond simply multiplying 5 and 6. Taking out the numbers and question allowed every student to think about the meaning of the sentence, the implied mathematical connections, and plan a solution pathway before jumping into a solution attempt.

I highly recommend everyone try this strategy with a word problem from your current text. It’s a wonderful way to give every student access to the math and freedom to think beyond just getting an answer.

If you know me or have ever read my blog, you know I could talk for days about student math work! You can visit my blog for a more detailed description of the work shown in this post as well as additional work captured from the lesson.

Two Math Routines to Learn About Student Thinking

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Routine 1: Tell Me Everything You Know About ____________

Directions

Choose a word or phrase that is the focus of your first curriculum unit. This could be something like: fractions, addition and subtraction, shapes, data, multiplication, etc. If students are introduced to that concept for the first time during the unit, such as volume in fifth grade, use a term like ‘measurement’ to elicit prior knowledge related to volume.
Write your chosen concept or topic at the top of a piece of chart paper.
Prompt students, “Tell me everything you know about [your chosen topic].”
Give students 1 minute of independent think time and then 1 minute to quickly tell a partner one thing they are going to share with the whole class.
As a whole group, record students’ ideas on the poster as they share.
When they are finished, ask if there are any ideas on the chart paper they have questions about. This is a good opportunity for students to ask clarifying questions of one another, revise their thinking, and agree or disagree with others’ ideas. You do not need to come to a final conclusion on each point of disagreement, especially if it is something they will learn in the unit. Simply just mark that idea with a question mark and revisit it later.
If there is time, you could start another poster with the prompt, “Tell me everything you wonder or have questions about [your chosen topic].” This communicates that sharing things they wonder and asking questions are part of learning. The information you’ll learn about student thinking will be extremely helpful going into the first unit.
As you move through the first unit, refer back to the poster frequently and ask students if they would like to add anything new or revise a previous idea.

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Routine 2: Talking Points

Directions

Arrange students in groups of 3 or 4.
Print a copy of the talking points for each group.
As a class, review how each round works. The first time you do this, it might be helpful to also model the process with a fun talking point such as, “A hot dog is a sandwich.”

ROUND 1 – Read the first talking point aloud. Take turns going around the group, with each person saying in turn whether they AGREE, DISAGREE, or are UNSURE about the statement and why. Even if you are unsure, you must state a reason why you are unsure. As each person shares, no one else comments. You are free to change your mind during Round 2 and/or Round 3.

ROUND 2 – Go around the group a second time, with each person saying whether they AGREE, DISAGREE, or are UNSURE about their own statement OR about someone else’s agreement, disagreement, or uncertainty from Round 1. As each person shares, no one else comments. You are free to change your mind again during Round 3.

ROUND 3 – Go around the group a third time to take a tally of AGREE / DISAGREE /UNSURE votes and record that number on your Talking Points sheet. Then, move on to the next talking point.

Sample Student Handout with Third Grade Talking Points

Talking Point	Agree	Disagree	Unsure
Fractions are always less than 1.
A fraction is a number.
We can locate fractions on a number line.
Fractions tell us a size.
One half is always greater than one third.
We can combine fractions.

Sample Math Mindset Prompts

Being good at math means being able to do math problems quickly.
A person is either good at math or bad at math.
I prefer to work on problems that challenge me rather than ones I find easy.
When working in a small group, if one person knows how to solve the problem, they should show the others in their group how to do it.
There is always one best way to do math.
Getting a problem wrong in math means you failed.
Drawing a picture is always helpful when doing math.

Sample Math Content Prompts

5 is the most important number.
The number 146 only has 4 tens.
Fractions are numbers.
When multiplying, the product is always greater than the factors.

Division of fractions is just like division of whole numbers.
The opposite of a number is always a negative number.
It is easier to work with decimals than with fractions.

For any equation with one variable, there is one best way to solve for the variable.
It is easier to work with degrees than with radians.

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Leveraging Digital Tools for Problem Posing

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I have blogged a few times about problem posing using print materials and lately I’ve become really interested and excited about the potential for digital tools in this work!

If you are new to problem posing, below are a few slides from Jinfa and my NCSM presentation for background – each image is linked to an associated research paper.

What is problem posing?

Many activities can easily be adapted to provide opportunities for problem posing by removing task questions (left) and replacing it with different prompt options (right).

How can digital tools enhance problem-posing experiences?

Being relatively new to both problem posing and digital lessons, I have learned so much trying things out in math classes this year. As always, the more I learn, the more questions/ideas I have. Below are two digital lessons that involve different flavors of problem posing.

Lesson 1: Our Curious Classroom

You can click through the lesson screens to see the full flow, but in a nutshell, students answer questions about themselves and explore different data displays.

After answering the first survey question, we asked students for problems they could answer about their class data and recorded their responses (sorry for the blurry image, I had to screenshot from a video clip:).

The students then worked at their table to answer the questions based on their choice of display.

The lesson continues with more survey questions, data display exploration, and ends with students personalizing their own curioso character (see bottom of post for unrelated, cute idea).

Things I learned:

Student responses can be collected and displayed so quickly with digital which saved us more instructional time for posing and solving problems.
The capability to see data displays dynamically change from one to another enhanced the discussion about which display was most helpful to answer the problems and why.
Students were so motivated to answer questions about themselves, learn about their classmates (audio clip below), and ask and answer questions about their own class, not a fictitious one.

“What did you like about the lesson?”

Things I wonder:

While having the teacher record the questions on the board worked perfectly, I wonder if or how younger students might digitally input their own questions w/o wearing headphones for voice to text or having spelling errors that are challenging for others to interpret? Maybe something like a bank of refrigerator magnets to choose from?
During the lesson, could the teacher input student questions onto cards in the Card Sort in Desmos so they could then sort the problems based on structure before solving?

Lesson 2: Puppy Pile

In this lesson, students generate a class collection of animals, are introduced to scaled bar graphs, and create scaled bar graphs. This one has a different problem-posing structure than the the first lesson which was interesting!

In this lesson, students use the Challenge Creator feature. In order to pose their problem to the class, students create their own set of animals (left) and then select a scale and create a bar graph (right).

After submitting their challenge, students then pick up one another’s problems and solve them.

Things I learned:

Students were extremely motivated to create their own problems and solve the problems of others.
This version of problem posing allowed students to have more control over the situation around which they were formulating problems, which they really enjoyed.
Challenge Creator is an amazing tool for repeated practice that is MUCH more engaging than a worksheet of problems.

Things I wonder:

How could this activity structure support or extend the problem posing experience in Lesson 1?
What other K-5 math concepts would be great candidates for a Challenge Creator problem-posing activity?

Final thoughts

I think problem posing is such an important instructional structure whether done in print, digital, or a hybrid of the two. It is important, however, to also consider the math, student motivation, and amount of time students spend engaging in the problem-posing process when choosing the format we use.

I would love to hear about what you try, learn, and wonder whether you try these lessons or adapt other lessons for problem posing!

Unrelated by Adorable Idea…

After Lesson 1, Katie printed out their personalized Curiosos for the wall;)

Gallery Walks: Engaging Students in Other’s Ideas

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One instructional strategy that I love for collaboration and public sharing of student ideas is a gallery walk. In a gallery walk, students create displays of their thinking on chart paper or white boards and then the small groups walk around the room and visit each other’s posters. And even though students create such beautiful displays of their ideas, it is always challenging for me to structure the walk in a way that actively engages them in one another’s ideas. Like any problem of practice, it takes trying out new ideas to see what works, when, and for whom.

The Lesson

Last week, it was the first 3rd grade lesson about division. We decided to launch by mathematizing Dozens of Doughnuts to set the stage for the subsequent activities. If you haven’t read the book before, it is about a bear named LouAnn who keeps baking 12 doughnuts to share with a different number of guests who arrive at her door. We read the book and did a notice and wonder, anticipating we would hear something about LouAnn sharing doughnuts and the number of doughnuts, friends, or plates, which we did.

Student Displays

We then asked small groups to record all they ways that LouAnn shared her doughnuts. We purposefully didn’t specify the representation so they could look for different ways during the gallery walk.

As we walked around it was great to see the various ways students were representing the situations, but some small groups seemed to have settled on only one way. We had planned for them to look for similar and different ways during the gallery walk, but that can be so passive, with no opportunity for them to connect those new ideas to their work. So, instead of waiting for the gallery walk at the end, we decided to engage them mid-activity with each other’s ideas and allow time for them to use those ideas.

Taking a page from Tracy’s book, Becoming the Math Teacher You Wish You Had, we opted for a Walk-Around to cross pollinate ideas. We asked students to walk around and look for ideas they wanted to add to their poster. These could be new ideas or just a different way of representing an idea they already had.

You would have thought we gave them a chance to ‘cheat’ as they walked around with such intention to other’s posters. I wish I had captured the before and afters of all of their posters, but here are just a few where you can see the new addition of ideas.

After they finished adding to their posters, we paused to discuss the ideas they found from others – both new ideas they hadn’t thought about and ideas they had, but were represented in different ways.

Next Activity

Students then independently solved a few problems. It was great to see the variation we saw on the posters in their work. So many great representations to share and connect in future lessons!