Photo credit: Jason Leung on Unsplash

Comprehending Math

What Literacy Instruction Can Show Us About Teaching Math

We have been teaching students how to become better readers and writers for years. With mentor texts, we teach our students about the different genres, text structures, and features that exist within books. Students learn how to identify characters and plots, retell events, or set up a table of contents to reflect the main idea and details of a new writing piece.

The good news is, we can use these same instructional choices during our math instruction. Just as books have a variety of text structures (narrative, informational, biographical) and features (characters, events, language, labels) so too, do math word problems. When we show students how to identify these structures and features within math problems, we increase their ability to comprehend them, solve them, and eventually become the authors of their own math stories. Research shows that readers who can identify the structure of a text are better able to locate the information they need for successful comprehension (Williams, J.P., 2003). This is exactly the result we are looking for when students are solving word problems. We want students to identify the text structure of a math problem, recognize which part is missing, and use questions and known relationships to solve for the missing value.

Problem Structures

There are four basic structures to math word problems, as identified by Thomas Carpenter and his colleagues in their groundbreaking book, Children’s Mathematics, published in 1999 (2nd edition 2013):

Join/Separate: In these problems, there is some action. There is a starting quantity; something is added or taken away; and there is a new quantity at the end. The mathematician must decide which is missing and find its value based on the others.

  • There are 3 kittens in the basket. 1 kitten climbs out. How many kittens are in the basket now? (Separate, result unknown)
  • The teacher sharpened 30 pencils. He put an equal number of pencils into each pencil box. He filled 6 pencil boxes. How many pencils did he put in each box? (Partitive division (similar to separate),amount in each group unknown)
  • Mark has some money in his wallet. He has $520 in the bank. All together he has $137. How much money does Mark have in his wallet? (Join, start unknown)

Part-Part-Whole: In these problems, parts make up a whole, but there is no action or change. One or more parts or the whole may be missing.

  • “There are 100 automobiles on a lot. How many could be cars? How many could be trucks?”
  • “A dealership has 7 rows of cars, with 10 in each row. How many cars are there?

Comparison: In these problems, two quantities are compared.

  • The average giraffe weighs 750 pounds. That is 15 times as much as the average monkey. How much does the average monkey weigh?
  • Joseph has 9 dollars. Margaret has 14 dollars. How much more money does Margaret have than Joseph?

Rate: In these problems, there is a direct relationship between the two quantities. A change in one causes a change in the second.

  • Henry earns $15 per hour. How much will he earn in 8 hours?

Relating Text Structures to Problem Structures: Join/Separate and Narrative

Let’s think about how the basic structures of math word problems mirror those of fiction and non-fiction texts. Part-Part-Whole and Comparison problems have structures similar to those of informational texts. Part-Part-Whole problems are a description of one time and one place, a bit like the main idea and details in a book. Comparison problems have structures similar to compare/contrast informational texts, like a diagram that compares the heights and weights of animals or a book that asks the reader to compare and contrast summer and fall. Rate problems, too, can mirror informational texts.

Right now, though, let’s focus on Join/Separate problems. These problems follow a narrative structure and typically contain characters, a setting, and a plot/action in the form of a verb. There is a flow to the problem with a beginning, middle, and end. Think about the kitten problem above: the characters are kittens, the setting is a basket, and the action is one kitten climbing out.

From a very early age children are exposed to narrative text structures; they learn to follow along with the adventures of characters, wonder about problems, and anticipate endings. This exposure implicitly builds children’s ability to organize stories in their minds, predict what may come next, and ask relevant questions when meaning breaks down.

For example, research shows that children typically learn to solve start-unknown problems through trial and error (Carpenter, 2003). However, understanding structure can make challenging problems more comprehensible for students. By using graphic organizers to explicitly link narrative text structures to word problems, and annotating the features and structures within them, students are better able to determine the unknown variable and use numerical relationships to solve problems. Additionally, many children are able to apply this knowledge of structure and become the authors of their own math problems. (See student work below).

1.Sergio was able to use the graphic organizer (retell boxes) to show the position of the missing value in the problem in this Separate Change Unknown (SCU) problem. He then used the given values to find the difference and proved it with the inverse operation. You’ll notice that there are arrows in place of operation symbols. This is meant to encourage students to think flexibly about the relationships between the numbers.

Sergio’s SCU Problem

2. When given Join Change Unknown (JCU) problems, students often add the two values given. In this case, Chastelyn was able to create her own JCU word problem, with the question mark representing the change.

Chastelyn’s JCU problem

3. April’s problem is sophisticated in that she was able to independently create a two step Separate Result Unknown (SRU) word problem.

April’s two-step SRU problem

4. Natividad uses a question mark to show she understands the missing value is in the middle of the Separate Change Unknown (SCU) problem. Her strategy for solving reflects her understanding of the relationship between the values given. She subtracts 10 from 30 to find a difference of 20.

Natividad’s SCU problem

Over time, just as we expect students to read and write more complex texts, we can also introduce them to and expect them to solve and write more complex math problems. We can use a variety of word problems as mentor texts during interactive read alouds, we can think aloud about the text structures and features that we notice. We can organize small groups during math time and create strategic partnerships that allow students to explain their thinking and gain critical feedback from their peers. We can expose students to more and more complex problem types and watch as their number sense and mathematical reasoning evolves.


References

Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., & Empson, S. (2003) 2nd Edition. Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Williams, J.P. (2203) Teaching text structure to improve reading comprehension. In H.L Swanson, Harris, K.R., & Graham, S. (Eds.), Handbook of learning disabilities, 293–305. NY: Guilford Press.


Danielle Moore

Danielle Moore began her teaching career almost 20 years ago at a Title I school in Long Beach, CA. Her desire to ensure the academic success of her diverse students, under challenging circumstances, continues to drive her work today as an independent consultant. Her experiences as a mentor teacher for the Cotsen Foundation for the Art of Teaching and as a facilitator for the UCLA Math Project have enabled her to support teachers in understanding the diverse needs of their students and the mathematical content and instructional practices needed to ensure access and equity for all learners.

Danielle Moore is a committee member for the California Mathematics Council- South and the founder and co-creator of the website www.TeachingOneMoore.org.

Heinemann Publishing

Written by

We publish books for teachers and provide professional development services. We are dedicated to teachers.

Welcome to a place where words matter. On Medium, smart voices and original ideas take center stage - with no ads in sight. Watch
Follow all the topics you care about, and we’ll deliver the best stories for you to your homepage and inbox. Explore
Get unlimited access to the best stories on Medium — and support writers while you’re at it. Just $5/month. Upgrade