# THREE-M IN WORD PROBLEM SOLVING.

We describe three activities that help undergraduates (pre-service teachers) to develop scientific vocabulary on measurable attributes and units of measurement. Measurable attributes arc important features in understanding a word problem and solving the problem. These activities help students comprehend word problems better by identifying measurable attributes and units of measurement. The third activity prompts students to identify symbols as well as deepens students' understanding of symbols used to decontextualize word problems.

Keywords: measurable attributes, units of measurement, quantity, word problem solving.

Introduction

The Common Core State Standards for Mathematics (CCSSM) (National Governors Association for Best Practices, Council of Chief State School Officers [NGA& CCSSO], 2010) emphasizes the importance of developing algebraic reasoning. Algebraic reasoning goes hand-in-hand with quantitative reasoning, which is another objective of mathematics education listed by the Standards.

Quantitative reasoning stands behind the ability to create word problem models using visual and mathematical representations. Quantitative reasoning also helps to use these models to solve challenging word problems. On the other hand, creating these models requires superior understanding of the measurable attributes of the quantities involved in word problems, as well as proficiency with units of measurement (Thompson, 2011).

Every quantity in a word problem has three characteristics--measurable attribute. unit of measurement, and magnitude.

MMM of a Quantity Measurable attribute Measuring Unit Magnitude

Measurable attribute refers to the "measurable quality of something" (Thompson, 1988, p. 164). For example, consider the following word problem:
```   Measurable attribute presented explicitly:
Chest Problem- The weight of
a silver chest is 210 pounds. The weight
of a gold chest is 36 pounds more than
the weight of the silver chest. What is
the weight of the gold chest?
```

In the Chest Problem, the Three-M's are the following:
```MMM of a Quantity, 210 pounds

Measurable    Magnitude of   Measuring unit
attribute     the quantity

Weight            210            Pound
```

In the Chest Problem, the measurable attribute is explicitly mentioned in the word problem. In many other problems, however, the attributes are not stated directly but implied in the context. For example, in the following Grass Problem, the measurable attribute must be derived from the context.
```   Measurable attribute presented implicitly:
Grass Problem- Elephant Cleo ate
6 tons of grass. Elephant Nadine ate 7
tons of grass. Elephants Diana and Roti
ate 5 tons of grass together. How much
grass did the elephants eat in all?
```

In this problem, the Three-M's are the following:
```MMM of a Quantity, 6 tons

Measurable    Magnitude of   Measuring unit
attribute     the quantity

Weight             6              Ton
```

In the Grass Problem, 6 refers to a weight, but this attribute is not stated explicitly (the word 'weight' is missing). When measurable attributes are not stated explicitly, we have observed students (children and pre-service teachers) have difficulties to define them. The Common Core State Standards on Measurement and Data (K-3) emphasize the importance of teaching children to discuss, measure, and compare attributes of objects. By Grade 3, children should be solving word problems involving time, volume, and weight (National Governors Association for Best Practices, Council of Chief State School Officers [NGA & CCSSO], 2010). Therefore, the need to develop fluency with defining measurable attributes and units of measurement in word problems is undisputable.

As educators, we need to ensure our teacher candidates (pre-service teachers) are fluent with this material because, according to Patton et al. (2008), pre-service teachers "possess naive conceptions believing that teaching mathematics is only about delivering facts and memorizing mathematics procedures" (p. 494). Not only teacher candidates but as study (MacGregor & Stacey, 1997) suggests, many other college students also display difficulties with algebraic reasoning, particularly with interpreting variables when solving math problems.

In this article, we propose three activities that, after assessing students' knowledge regarding measurable attributes, help them deepen their understanding of quantities by categorizing measurable attributes and unit of measurement.

In the first activity, students demonstrate their level of fluency and concurrently foster their vocabulary by matching measurable attributes with measuring units. In the second activity, students practice identifying measurable attributes and measuring units in word problems. In the third activity, students learn to name measurable attributes when defining symbols used in alternative representations of word problems. These activities can serve as a great tool for educating pre- and in-service teachers, since researchers (Sulentic-Dowell et al., 2006) claim that teachers' ability to meaningfully discuss word problems depends on their own literacy.

Activity 1: MMM of a Quantity- Vocabulary

Materials needed: 3"x3" Post-it notes, construction papers, and scissors (Figure 1).

Prompt cards: Use construction paper to make 4"x4" squares. Write various measurable attributes and measuring units on post-it notes and stick them on top of the squares (Figure 2).

Attribute Categories: Print (write) 5-10 names of the most popular measurable attributes on separate cards (Figure 3). For example, weight (mass), distance, length, volume, area, temperature, time, age, height, and speed (velocity).

The Activity: Students identify the units of measurement used to measure or calculate the given attributes. Particularly, students work to find the prompt cards with measuring units to match the attribute categories.

As an example, we asked one pre-service teacher (Karen) to identify the units of measurement for weight. She put 'Ounces', 'Tons', and 'Pounds' cards under the measurable attribute Weight (Figure 4). Then, Karen wanted to add more cards.

Teacher: Which ones?

Karen: Umm, like Gallon, ... Cup.

Teacher: So you think that's [a fit for] weight?

Karen: I guess is.

Teacher: So I'll give you the next one [Teacher places the card with a measurable attribute 'Volume' on the table]. Does that help?

Karen: Oh, like volume. Ok, Yeah.

Then Karen placed 'Gallon', 'Pint', 'Quart', 'Liter', and 'Cup' as units of measurement under the Volume Prompt card. The teacher asked Karen whether she wants to leave the three cards she had chosen for the measurable attribute, weight. Karen confirmed her choices.

Below lists the measurable attributes and units, which can be used within the proposed activity.

Reversing the order can extend the activity--students can be prompted to find the measuring attributes matching the given unit of measurement (Figure 5).

Activity 2: MMM of Quantities-MMM in Word Problems

Materials needed: Prompt cards, several categories of word problems, and T-structure.

Prompt cards: One can use the cards described in Activity I. The set must include attributes and units of measurement taken from the word problems used for this activity. In addition to measurable attributes and units of measurement, the cards must have words students frequently mistaken to explain quantities. We call them fake attributes.

Categories of Word Problems: Three sets of colored construction paper have been used (Figure 6). The first set (pink paper) contains word problems where measurable attributes are explicitly named. The next set (yellow paper) contains word problems with measurable attributes not listed but implied in the text. The third set (blue paper) contains challenging multi-step word problems--some with explicitly and some with implicitly mentioned attributes (problems are shown in table 3).

T-structure: Make a T-shape from construction paper with two titles: Measurable Attribute and Unit of Measurement (Figure 7).

The Activity: Students identify the measurable attributes for each word problem, starting from simple problems and progressing toward challenging ones. When students demonstrate difficulties, we propose the following scaffolding (Figure 8):

1. Teacher places the T-structure on a paper with the word problem.

2. Students place prompt cards under each title of the T-shaped structure.

Particularly, we ask students to first identify the unit of measurement and then the measurable attribute. As we observed, such scaffolding fosters students' abilities to differentiate between the units of measurement and the measurable attributes.

To demonstrate this activity, we present the following conversation between the teacher and a pre-service teacher (Amanda). Amanda was asked to identify the measurable attribute and the unit of measurement in the Temperature Problem.

Temperature Problem- On January first, the temperature in Chicago was 12 degrees Fahrenheit. On the same date, the temperature in New York City was 8 degrees warmer. What was the temperature in New York City?

Amanda: Okay. Well, the quantity [measurable attribute] is the number degrees, then the unit is Fahrenheit, degrees Fahrenheit.

Next, the teacher asked Amanda to use the Prompt cards. She picked the words temperature and degree, and placed the temperature under the title Units of Measurement and Degree under Measurable Attributes (Figure 9). Then, the teacher asked again about the unit of measurement.

Amanda: ... Unit, maybe it is the other way around [Amanda changed the position of the Prompt cards (Figure 9)].

By the time Amanda reached the second set of word problems, she was identifying measurable attributes in these problems with confidence.

Activity 3: MMIVI of Quantities: MMM and Symbols' Definition

Materials needed: Word problems from Activity 2 and the set of Prompt Cards to provide options for describing the meaning of the symbol, which can be used to decontextualize word problems. For scaffolding, the Prompt Cards from Activity # 1 and the T-shape construction from Activity #2 can be added.

The Activity: Students are provided with symbols, which can be used in algebraic models to represent problems. The symbols are chosen in a form of parameter pointers. Unlike simple symbols, such as A, B, C, x, y, z, the parameter pointers remind readers about a key word or/and measurable attribute connected with the quantity they represent. For example, 'C' points to the word 'Chicago' and means 'the temperature in Chicago', while 'N' means 'the temperature in New-York' in the Temperature Problem; 'S' reminds the word silver and denotes 'the weight of the silver chest', while 'G' denotes 'the weight of the gold chest' in the Chest Problem. Parameter pointers are widely used in higher mathematics and science classes. For example, [v.sub.car], [v.sub.train], and [d.sub.train] can be used to mark 'the velocity of a car', 'the velocity of a train', and 'the distance covered by a train', respectively.

During this activity, students define the meaning associated with the symbols. To help students with this task, prompts were created. Each prompt lists several options including the correct ones (Figure 10). Additional scaffolding (contrasting units and measurable attributes) is provided when needed.

Students are guided to include in each definition of a symbol, the name of the measuring attribute of a quantity and a key word associated with this symbol.

We found most students include key word in definitions, but frequently forget to name a measurable attribute. Below, we demonstrate one example of the activity. We asked Amanda to define a symbol 'C' in the Temperature Problem using the options listed below (Table 4).

Measurable Attribute + Key Word/ Symbol Definition

Teacher: ... Can you identity the correct definition of parameter [symbol]? [Amanda reads the options] Which one will be?

Amanda: 'C' is Chicago. That will be for the parameter.

Then the teacher prompted her by asking if there is any measurable attribute that 'C' is referencing. After some thinking, Amanda points to the phrases (c) and (d).

Amanda: ... maybe it's this one [pointing at (d)], 'C' is the degrees in Chicago maybe. Yes, I am gonna say this one [pointing at (d)]. [Amanda did not consider other options at this point. ]

Next, the teacher asked her to define parameter 'S' in the Chest Problem. Observing Amanda's difficulties with defining parameter 'C' in the Temperature Problem, the teacher began with discussing the measurable attribute and units of measurement.

Amanda quickly identified the measurable attribute as weight and the unit as pound. Then, using the prompts below (Table 5), she found a correct definition (c) for symbol 'S.'

As soon as Amanda finished analyzing the Chest Problem, she decided to change her definition for parameter 'C' from the Temperature Problem.

Teacher: So what do you want to change?

Amanda: So, I feel like this make more sense, so then that means 'C' is the temperature in Chicago, not the degrees of temperature, because we are measuring the temperature.

As follows from this excerpt, Amanda not only found a correct definition for the symbol 'S', but also corrected her previous response for the Temperature Problem.

Once students grasp the difference between the measurable attributes and the units of measurement as well as begin understanding how to define symbols, we ask them to identify the measurable attributes in word problems without using the prompt cards and the T-shaped structure. Students also define parameters without prompts.

What Pre-service Teachers say about these Activities

Providing a feedback on the activities, some pre-service teachers informed us the activities helped them deepen their knowledge regarding the relation between measurable attributes and units of measurement. Others mentioned the importance of these activities for their future students.

[The activities are useful for children] "Because it gives kids not only the unit of measurement ... for a kid to put these together [pointing at the T-structure with the prompt cards having the attribute and the unit of measurement] that would be a lot more helpful. "

Contrasting helps understand difference beetween measurable attribute and units of measurement

One pre-service teacher mentioned how she was nervous in the beginning, but the scaffolding helped her to understand how to correctly define symbols.

"....Umm, I was really nervous up here [at the beginning], because I didn't want to be wrong. And so I felt though, as you started prompting me, because these [choices for the definition of parameters] are all the same .... you had to remind me of this [showing T-structure] like 12 times, the more I started to walk through it and get hang of it, I felt more confident."

Contrasting measurable attribute and units of measurement helps define symbols

Another pre-service teacher mentioned the activities were helpful, because "it helps to solve the problem when you can correctly define like what the parameters are."

Defining symbols helps with solving words problems

Conclusions

These presented activities challenge students' thought processes and intuitive knowledge regarding units of measurement, measurable attribute, and symbol meanings.

We believe, these activities are efficient because they provide the necessary conditions for learning (Marton & Pang, 2006)--teaching one learning dimension at a time, while providing contrast and variations. The method received considerable attention during last years and shows considerable potential (Marton, 2015).

The activities also involve a lot of scaffolding (Vygotsky, 1978)--it allows using the activities with students of any level. Providing students with several options and varieties of measurable attributes helps students rethink their prior experiences and significantly improve their awareness and performance in terms of understanding quantities in word problems. Moreover, the activities improve students' ability to define symbols used in word-problems' models.

Author's Note

The first author, Sayonita Ghosh Hajra, is now at Hamline University. She is an assistant professor of mathematics at Hamline University.

SAYONITA GHOSH HAJRA

University of Utah

VICTORIA KOFMAN

References

MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 35(1), 1-19.

Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of the Learning Sciences, 15(2), 193-220.

Marton, F. (2015). Necessary conditions of learning. New York, NY and London, England: Routledge Taylor & Francis Group.

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington, DC: Authors. Retrieved from http:// www.corestandards.org/the-standards

Patton, B. A., Fry, J. & Klages, C. (2008). Teacher candidates and master math teachers personal concepts about teaching mathematics. Education, 128(3), 486-497.

Sulentic-Dowell, M. M., Beal, G. & Capraro, R. M. (2006). How do literacy experiences affect the teaching propensities of elementary pre-service teachers? Reading Psychology, 27, 235-255.

Thompson, P. W. (1988). Quantitative concepts as a foundation for algebra. In M. Behr (Ed). Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education Vol. 1 (pp. 163-170). Dekalb, IL.

Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education WISDOMe Monographs (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming Press.

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Cambridge, MA: Harvard University Press.

Caption: Figure 1. Materials for the activities.

Caption: Figure 2. Prompt cards.

Caption: Figure 3. Attribute categories.

Caption: Figure 4. Karen's work.

Caption: Figure 5. Extension: matching attributes.

Caption: Figure 6. Categories of word problems.

Caption: Figure 7. T-structure.

Caption: Figure 8. Teacher demonstrates an example of the activity.

Caption: Figure 9. Pictures of Amanda's work.

Caption: Figure 10. Prompts for defining the symbols.
```Table 1. List of measurable attributes
and units of measurement

Measurable
Attributes         Units of Measurement

Length/        Inch      Foot         Yard
Distance       Mile      Meter        Kilometer

Weight         Ounce     Pound        Gram
Ton       Kilogram     Grain

Volume         Cup       Gallon       Pint
Barrel    Cubic
Centimeter   Quart

Time           Second    Minute       Hour
Day       Week         Year

Temperature    Degrees   Degrees      Degrees
Celsius   Fahrenheit   Kelvin

Area           Square    Acre         Hectare
foot

Square    Square       Square
inch      unit         meter

Table 2. List of attributes and fake
attributes used as prompt cards

Measurable       Fake Attributes
attributes

Height           Long, Inches, Tall

Weight (mass)    How much

Distance         Far, Number of miles

Temperature      Degrees, Warmer, Colder

Volume           Often mistaken with weight,
Liters, Number of liters

Age              Old, Years, Number of years

Number of CDs    How many

Table 3. Three categories of word problems

Types of problems          Word problems

One-step word              * The weight of a silver chest
problem with explicit        is 210 pounds. The weight of a golden
measurable attribute or      chest is 436 pounds more. What is the
the simplest attribute:      weight of the golden chest?
"number of objects"
* The number of CDs Rob has is 180.
The number of Kate's CDs is
greater by 60.
What is the total number of CDs
do they have altogether?

* There are 5 pieces of cucumber on a
plate. There are also 8 pieces of tomato
on the same plate. How many pieces of
vegetables are there altogether?

One-step word              * Anita bought 435 g of potatoes and
problem with measurable      256 g of tomatoes. How much more
attribute implied in         potatoes than tomatoes did Anita buy?
the text.
* A 82 in. pole is painted green and
yellow. If 67 in. of the pole is
painted green, how much more of
it is painted green than yellow?

* Della Rae collected 46 liters of
rainwater. Paula collected 24 liters
of rainwater less than Della Rae.
Gail collected 25 liters of rainwater
more than Paula. How much rainwater
did Gail collect?

* Loraine finished her project in two
days. Monday, she worked on her project
for 2 hours. Tuesday, she worked on it
for 4 hours. How much time did she
spend on her project in total?

* Cindy has a 6-foot long ribbon.
Her ribbon is shorter than Celina's
ribbon by 2 feet.
How long is Celina's ribbon?

* Tony was four years old when he met Pam.
Also, he was 40 inches tall. After two
years, he met Pam again. At this time,
Tony was 46 inches tall. How much did
Tony grow in the two years?

* Concetta is 6 years older than Melissa.
Concetta also is 7 years older than
Brigitte. How old is Brigitte,
if Melissa is 14 years old?

* John ran 10 miles. Sam ran 5 miles
fewer than John. How far did Sam run?

* Cleo ate 6 tons of grass. Nadine ate
7 tons of grass. Diana and Roti ate
5 tons of grass altogether. How much
grass did they eat in all?

Multi-step word            * It took a purple dragon 3 days to cross
problems with explicit       Trinity Lake. It took him 2 days more
and implicit measurable      than it took a golden dragon to fly
attributes.                  over Forbidden Forest. Silver dragon
was flying over Sestina Mountains
4 days longer than the golden dragon
was flying over the forest. How long
did it take the silver dragon to fly
over Sestina Mountains?

Superman cut off 4 heads. Then, the
next day, 6 new heads grew. When
superman came to the dragon's cave,
did the dragon have the first day,
before Superman removed any of

* The comfortable hopping speed for
Kangaroo Danita is about 15 miles per
hour (mph). Her highest speed is 43 mph,
but she can use it only on very short
distances. A fast speed that she can run
with for up to one mile is only 10 mph
higher that her comfortable hopping
speed. When Danita has to run fast, how
much faster her highest speed from her
maximum speed that she can run about
a mile?

* Dondo-Huks is looking at circling
around bugs. To eat or not to eat,
it is the question that Dondo-Huks asks
himself again and again. If he will eat
12 bugs, then the same number of bugs
will fly away, and only three bugs will
stay. If he will waif more bugs might
come. Let Dondo-Huks solve his problem,
while you solve yours. Find the number
of the bugs that are circling around
Dondo-Huks.

Table 4. Multiple choice list for definition
for a parameter for the Temperature
Problem

Choose the correct definition for a parameter
(symbol).

a) 'C' is Chicago.

b) 'C' is the temperature
in Chicago.

c) 'C' is how much
warm/cold in
Chicago.

d) 'C' is the degrees in
Chicago.

Table 5. Multiple choice list for definition
for a parameter for the Chest Problem

Choose the correct definition for a parameter
(symbol).

a) 'S' is the silver chest.

b) 'S' is silver.

c) 'S' is the weight of the silver chest.

d) 'S' is how many pounds the silver chest weighs.

e) 'S' is how much the silver chest weighs.
```