the language of mathematics

THE LANGUAGE OF

MATHEMATICSA Linguistic Approach to Solving Word

Problems

John J. Gaines

Agenda

Introduction Background Motivations

Reading Word Problems Student Approach vs. Linguistic Approach Implications of the Linguistic Approach

Writing Word Problems Linguistic Approach

Questions/Thoughts/Suggestions

Student Approach to Word Problems

How do students approach word problems? Numerical cues Vocabulary cues (Tier 2 and Tier 3)

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Sample Student Approach

Students are running in a relay race. Each team will run a total of 2 miles. Each member of a team will run 1/5 of a mile. How many students will a team need to complete the race?



What do you initially see?



What do you initially see? The numbers (i.e. 2 and 1/5)

Are there any words or phrases that stand out?



What do you initially see? The numbers (i.e. 2 and 1/5)

Are there any words or phrases that stand out? total, miles, of a mile, how many students



Why do those words or phrases stand out to you? “They sound like math words.”

Using what you know about this problem, how would you solve it?



I asked a different student to try solving the problem while making a connection to the words used.

Linguistic Analysis


Linguistic Analysis


“Each team” - More than one team

Team 1 Team 3Team 2

Linguistic Analysis


“will run” – implies motion, distance, length

Team 1

Linguistic Analysis


“will run” – Implies motion, distance, length

“a total of 2 miles” – Total distance travelled = 2 miles

Team 12 miles

Linguistic Analysis


“Each member” – There are multiple members or people.

Linguistic Analysis


“of a team” – A certain amount of people are a part of a team. Team 1

Linguistic Analysis


“will run” – Implies motion, distance, lengthTeam 1

2 miles

Linguistic Analysis


“1/5 of a mile” – Implies motion, distance, lengthTeam 1

2 miles

15

15

15

15 1

5

15

15

15

15

Linguistic Analysis


“1/5 of a mile” – Implies motion, distance, length2 miles

15

15

15

15 1

5

15

15

15

15

Team 1

Linguistic Analysis


“How many students” – There is some exact number of students, but it is unknown.

“will a team need” – This unknown number of students belong to a single team.

“to complete the race” – Completing the race means running 2 miles.

Linguistic Analysis


Which operation should we use to solve this?2 miles

15

15

15

15 1

5

15

15

15

15

Team 1

What is the difference?

Linguistic Analysis


VS.Sample Student Analysis


SAMPLE LESSON

Writing Your Own Word Problem

Lesson Sequence

Part 1 – Word Phrase Start with a word that is familiar to the students (e.g.

“each”). Focus on Tier 1 (Basic Vocabulary) and Tier 2 (High

Frequency/Multiple Meaning Vocabulary). Activate background knowledge by having students share

what they know about the word and how they have heard this word used before.

Use graphic organizers to help the students discern different examples that they offer to develop descriptive meanings.

Developing word problems from a single word emphasizes the impact that language has on understanding word problems.

Lesson Sequence

Part 1 – Word Phrase Brainstorm simple phrases that could contain the

given word (e.g. “each box”). After the students brainstorm their own simple phrases,

have them share them with the rest of the class and discuss any patterns.

For example, in “each box,” “each car,” and “each house,” “each” is positioned before a noun. In this case, “each” acts as an adjective, implying some sort of distributive meaning.

“each” “each box”

Lesson Sequence

Part 2 – Phrase Sentence Create the context for a phrase within the

framework of a sentence. Using one of the phrases that the students listed, have

them write a sentence that incorporates numerical values.

“each box” “John placed 4 marbles in each box.”

Lesson Sequence

Part 3 – Sentence Word Problem Analyze the basic structure of a word problem.

Once the students have created their sentence, they need to understand its function within the structure of a word problem.

Setting – Introductory Sentence(s)

Information – Core Sentence

Direction – Question

Basic Structure of a Word Problem


Information – Core SentenceDirection – Question

Lesson Sequence

• Low amount of (or no) information • Creates a setting/environment, providing context

for the information given in the core sentence


• High amount of information• Provides information through numerical and

linguistic cues

Information – Core Sentence

• Low amount of information• Specifies the outcome

Direction – Question

Lesson Sequence


The sentence that they have created contains some important information for solving the word problem, but there is usually some setting that provides the environmental context for the information given. For example, “John placed 4 marbles in each box” depicts 4 marbles

having been distributed equally in a certain number of boxes, but why is John placing marbles in each box?

In this same example, we could create the setting with the introductory sentence, “John found a whole tray of marbles in his closet and wanted to give them to his students.”

There might be missing information with only one introductory sentence. Fortunately, this could lead to a class discussion on the need for further clarification.

Lesson Sequence


Based on the information provided in the Introductory Sentence and the Core Sentence, create a question that requires mathematical calculation.

For example, in “John found a whole tray of marbles in his closet and wanted to give them to his students. John placed 4 marbles in each box.” I would like to find out how many marbles he passed out in all. So, I could ask, “If there are 28 students in his classroom, how many marbles did he pass out in all?”

Lesson Sequence

Part 4 – Create Word Problem Individually and in small groups, have the

students create their own word problems using the word you have given them.

Provide them with the opportunity to: Create their own word problems Collaborate with their peers Critically think about the structure of the word problem

and the information provided Communicate their reasoning for creating their word

problem

The 4Cs

Lesson Sequence

Part 5 – Share Word Problems and Analyze Share word problems with the class and analyze

other possible approaches to creating a word problem based on the word given. Focus on the structure of the word problem, whether

enough information is provided, in what part the most/least information is provided, and key vocabulary (Tier 1-3). The focus should be on the effectiveness of the language used to communicate the desired outcome.

Encourage students to ask questions. For example, is it obvious what the question is asking for? Is there enough information provided to discern the best procedure to use?

Connection to CCSS

MP 1 – Make sense of problems and persevere in solving them Students have to understand the structure of a

word problem and persevere in creating one of their own. In essence, students are solving a number of word problems through the process of creating one of their own.

MP 2 – Reason abstractly and quantitatively Students practice contextualizing and

decontextualizing as they create their own word problems.

Connection to CCSS

MP 3 – Construct viable arguments and critique the reasoning of others Students have the opportunity to work

collaboratively with their peers on creating their own word problems. In Part 5, they are given the opportunity to evaluate the effectiveness of word problems created by other groups.

MP 4 – Model with mathematics Students create real-world problems by

contextualizing a word provided by their teacher.

Connection to CCSS

MP 5 – Use appropriate tools strategically Students have the opportunity to strategically

use certain linguistic structures and cues to foster greater comprehension of their word problem.

MP 6 – Attend to precision Students must use precise and exact language

to convey the information necessary for properly understanding and solving the problem.

Connection to CCSS

MP 7 – Look for and make use of structure Students analyze the language used in word

problems for specific patterns to aid them in producing more effective word problems.

MP 8 – Look for and express regularity in repeated reasoning Student analyze the language used in word

problems for any regularity in the expected solution.

Contact Information

Questions/Thoughts/Suggestions

John J. GainesPrimary Email [email protected]

Alternate Email

[email protected]

Facebook https://www.facebook.com/jgaines.glamc

Edmodo https://swhittier.edmodo.com/jgaines

Phone Number

(562) 318-9966

the language of mathematics

Education

relay race

linguistic analysisstudents

miles total distance

word problems student

different student

math words

sense of problems

lengtha total