This presentation will focus on a number of Standards for Mathematical Practice, particularly MP 1, 6, and 7. Participants will analyze the linguistic features of word problems and gain a better understanding of the strategies necessary for promoting mathematical literacy. They will use these strategies to rethink their approach to teaching students about problem solving, providing students with greater access to the language used within a word problem. Thus, participants will learn how to explore word problems with greater precision (MP 6), making using of their linguistic structure (MP 7), allowing them to persevere in not only solving word problems, but understanding them at a much deeper level. After attending this presentation, participants will have developed a greater appreciation for the richness of the language used in mathematics. They will learn a unique approach to problem solving that focuses on the linguistic features of word problems. Through this approach, they will learn to address the Standards for Mathematical Practices while fostering metacognitive awareness and mathematical fluency.

1. THE LANGUAGE
OF MATHEMATICS
A Linguistic Approach to Solving Word
Problems
John J. Gaines

2. 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

3. Student Approach to Word
Problems
 How do students approach word problems?
 Numerical cues
 Vocabulary cues (Tier 2 and Tier 3)

4. 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?

5. 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?

6. 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?
 The numbers (i.e. 2 and 1/5)
Are there any words or phrases that stand out?

7. 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?
 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

8. 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?
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?

9. 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?
I asked a different student to try solving the problem
while making a connection to the words used.

10. Linguistic Analysis
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?

11. Linguistic Analysis
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?
“Each team” - More than one team
Team 1 Team 2 Team 3

12. Linguistic Analysis
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?
“will run” – implies motion, distance, length
Team 1

13. Linguistic Analysis
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?
“will run” – Implies motion, distance, length
“a total of 2 miles” – Total distance travelled = 2 miles
Team 1
2 miles

14. Linguistic Analysis
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?
“Each member” – There are multiple members or people.

15. Linguistic Analysis
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?
“of a team” – A certain amount of people are a part of a
team. Team 1

16. Linguistic Analysis
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?
“will run” – Implies motion, distance, length
Team 1
2 miles

17. Linguistic Analysis
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?
“1/5 of a mile” – Implies distance, length 2 miles
1
5
1
5
1
5
1
5 1
5
1
5
1
5
1
5
1
5
1
5
Team 1

18. Linguistic Analysis
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?
“1/5 of a mile” – Implies motion, distance, length 2 miles
1
5
1
5
1
5
1
5 1
5
1
5
1
5
1
5
1
5
1
5
Team 1

19. Linguistic Analysis
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?
“How many students” – There is some exact number of
students, but it is unknown.
“will a team need” – This unknown number of students
belongs to a single team.
“to complete the race” – Completing the race means
running the length of 2 miles.

20. Linguistic Analysis
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?
Which operation should we use to solve this? 2 miles
1
5
1
5
1
5
1
5 1
5
1
5
1
5
1
5
1
5
1
5
Team 1

21. What is the difference?
Linguistic Analysis
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?
VS.
Sample Student Analysis
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?

22. 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.

23. 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”

24. 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.”

25. 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
Setting – Introductory Sentence(s)
Information – Core Sentence
Direction – Question

26. Lesson Sequence
Setting – Introductory Sentence(s)
• Low amount of (or no) information
• Creates a setting/environment, providing context for the
information given in the core sentence
Information – Core Sentence
• High amount of information
• Provides information through numerical and linguistic cues
Direction – Question
• Low amount of information
• Specifies the outcome

27. Lesson Sequence
Part 3 – Sentence  Word Problem
 Analyze the basic structure of a word problem.
 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.

28. Lesson Sequence
Part 3 – Sentence  Word Problem
 Analyze the basic structure of a word problem.
 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?”

29. 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
The
4Cs
 Critically think about the structure of the word problem and the
information provided
 Communicate their reasoning for creating their word problem

30. 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?

31. 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.

32. 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.

33. 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.

34. 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.

35. SAMPLE LESSON
Writing Your Own Word Problem

36. Contact Information
Questions/Thoughts/Suggestions
John J. Gaines
Primary Email jgaines.glamc@gmail.com
Alternate
Email
jgaines@glamc.org
Facebook https://www.facebook.com/jgaines.glamc
WordPress http://mathematicsleadership.wordpress.com
/
Phone
Number
(562) 318-9966