Word Problems and the Problems with Words

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…

  • How much money is there?
  • What kind of bills were in the pile?
  • How many of each bill is there?

We set off to answer our questions. First, Act 2 answered one of our questions, “what kind of bills were in the pile?” The students had not noticed that there were 50 dollar bills in the pile. They added this to the information they were collecting about the context. Then we looked at the piles of bills in act 2. This almost started a math fight! All of the students agreed that the twenty dollar bills was the biggest stack and the fifty dollar bills was the smallest. Which stack was the second biggest got contentious. One student thought the 10’s was the second biggest stack and others thought it was the 1’s. After a quick dialogue, the outlying student agreed with the argument that the bills in the 10’s stack were bent and made it look bigger. We looked at the picture of the bills laid out in rows, which answered our second question, “how many of each bill is there?” The students also noticed that if we multiplied the amount of each bill by its value we could also find the total value. They spent sometime completing this task.

After revealing the total amount in Act 3 and everyone checked their arithmetic, we introduced the idea of comparing the relationships between numbers using ratio language. We rounded the numbers and decided that for every one dollar bill there was about two twenty dollar bills and for every five dollar bill there was about one ten dollar bill.

After class one of my students announced, “That was a fun math class!” Student engagement, check!

In one of the standards listed for Make It Rain, students should “use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.” But in the task there are no words until the end, when they are being used to communicate total amounts. So why is this standard listed if it is not truly a “word problem?” Because enhanced anchored instruction (EAI) has been shown to increase the engagement and access for student with disabilities during problem solving.

Some have proposed anchored instruction as an effective alternative to step-by-step problem-solving strategies. Anchored instruction presents real-life situations as contexts (the anchors). We might consider problem solving within a traditional school store setting to be anchored instruction. Current versions of anchored instruction often use video vignettes, through which any real-life experience with embedded mathematical data might serve as an anchor for problem-solving instruction. For example, some videos focus on discovering measurements embedded in such adventures as fighting a fire or on interpreting medical measurements that emergency teams use. Whereas a typical problem that a teacher or textbook poses depends on the experience and imagination of the students, the videos compensate for different experience levels by giving all students a common picture of the problem situation or setting. The problem situation is reality based and complex enough to incorporate data for several problems. Students can refer to the video at any time to check information (Shih, Speer, Babbitt, 2011).

What this passage doesn’t go into is by presenting the problems in video form (the enhanced part of EAI), it limits the amount of language processing that students must exert before engaging in the problem solving process. Also anchoring the problem in a familiar context, students will be able to more easily engage with the mathematics. Students with disabilities must often tackle various cognitive obstacles (language processing, working memory, executive functioning, etc.) embedded in the problem before being able to tackle the actual mathematics. Using enhanced anchored instruction, of which I am including 3-act tasks, can limit these barriers and increase access for students with disabilities when solving math word problems.

1 thought on “Word Problems and the Problems with Words

  1. Hi Andrew,

    Really fun and easy 3-Act Task for students to do with such big ideas behind it. When reading the standards you had aligned with this 3-Act Task, I was confused about 6.RP.A.1 and the ratios, until I came to this article. I am glad you were able to clarify that as I did not see it initially. It is an interesting way for students to start looking at ratios as well, and putting into money makes it relatable and interesting.I also had the same notice and wonders myself as the ones that your students have, which just goes to show how we do think alike in many ways.

    I like the point that you bring up about this being a “word problem.” I feel that often times word problems do set up many students for failure. I never actually understood this until one of my practicum placements, where I had a student who was a math whiz with numbers, but had severe dyslexia. This would cause him to have issues when it did come to word problems, which is often unfair because math is not about their reading skills, but their ability to analyze a problem and figure out what to do.

    I am curious on how you go about making a 3-Act Task. What do you usually start with to begin your thinking? A standard or a idea of some sorts?

    Thank you for sharing this!


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