What is 3/5? Interpreting Mathematical Symbols in Problem Solving

Spring 2026
P. Mark Taylor (Roane State Com. Col.)

Teacher Understanding of the Complexity of Reading Mathematics 

When solving mathematics problems, the first step is to read the problem (Krawitz, et al, 2021). While many studies have broached the topic of reading word problems, reading mathematics is a much broader issue that must be addressed in mathematics teaching as well as mathematics teacher education. “The issue of reading, recognizing, and understanding symbols underpins all mathematics topics” (Bardini & Pierce, 2015, p. 2). It has been posited that mathematics qualifies as its own language (Wakefield, 2000; Whitin & Whitin, 2000) which makes it a second language for most students. As such, students and their teachers need to develop an awareness of the complexities of reading mathematics as a second language.  Specifically, deciphering the meaning of mathematical symbols and combinations of those symbols requires “metalinguistic awareness, their ability to reflect on and analyze the language” (Adams, 2003, p. 786). 

Preparing pre-service teachers to teach mathematical problem solving requires teacher educators to provide experience for the preservice teachers to read, write, and discuss mathematics problems and solutions as well as posing problems (Krulik & Rudnick, 1982; Crespo, 2003). These problem-solving discussions afford the opportunity to experience, analyze, and discuss the complexities of reading mathematics problems allowing them to reflect on their own metalinguistic awareness.

Addressing the Complexity of Reading “fraction problems” with the Activity: What is 3/5?

When encountering the title question, readers must interpret the meaning of three symbols arranged together: 3, /, and 5. In the mind of the reader, it can sound like any of these:

  • three-fifths

  • three out of five

  • three to five

  • three divided by five

A deeper understanding of students’ difficulties with fractions, rates, and ratios can come from a discussion of this simple question. This discussion can be productive to varying extents with students at various levels of K-12 mathematics. The power is increased greatly when the discussion is held with preservice or inservice teachers.  This activity aligns well with Steele & Hillen’s (2012) design principles for content-focused methods courses:

“1. Focuses on a narrow slice of mathematical content or process central to developing mathematical proficiency in secondary school.

2. Uses a guiding inquiry to frame and motivate the course and provide a unifying thread.

3. Organizes content and pedagogical activities into sequences that engage teachers across the continuum from learner to teacher”  (p. 54).

The narrow slice of mathematics in this activity is the process of interpreting the meaning of symbols related to fractions, rates, and ratios. Our students must be able to see multiple possibilities for the meaning of 3/5 and similar representations.  When the symbols appear, the context must be considered.  

The guiding inquiry starts without context, which leaves open all the possibilities for the meaning of 3/5. The discussion without context can continue for several minutes and can include several related forms of representation including pictures of five objects (or some multiple of five), linear graphs, pictograms, and fraction representations such as fraction bars or circles.  As the preservice teacher answers with their interpretation of the meaning of 3/5, this prompt is given:  “How would we draw that?”  Pairing the symbols to a specific representation enables the student to be more specific in defining the meaning they are trying to convey.  

The next stage of inquiry is the development of possible contexts for each interpretation of the meaning of 3/5. Some prompts to encourage discussion include things like:

  • What kinds of problems would use the symbols 3/5 with each particular meaning?

  • Are we doing operations with fractions in a “naked-number” scenario? 

  • Are we using 3/5 as the slope of an equation that models a real-world application?  What is the application?  

  • Is this a ratio problem that requires an extra step of interpreting the context and multiplying or dividing? What would that problem look like?

The third stage of this discussion is typically searching for examples of multiple problems that correspond to each interpretation.  The teachers can also be challenges to write some of their own problems.  This provides an opportunity to check for their understanding.

Finally, the discussion turns towards how to support students as they navigate the multiple possibilities of 3/5 and similar representations as they read, interpret, and solve mathematical problems.  

Results of One Implementation

This activity is implemented each semester in a mathematics for elementary teachers course. In the most recent iteration of this implementation, dialogue, drawings, and student work during the discussion were analyzed to document the preservice teachers’ journey of discovery during this lesson.

Starting with the initial prompt of “What is 3/5?”, each response for the preservice teachers was followed with the prompt: “How would we draw that?” This move from symbols to drawings requires the reader to attribute a specific meaning to the symbols, choosing just one of several possible meanings.  Figure 1 shows the results of the preservice teachers’ ideas of the meaning of 3/5 in this brainstorming. Each drawing represents a different mathematical model or at least a new way to represent a model. 

 

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Figure 1. Students suggestions for how to draw the meaning of 3/5.

The circle has three out of five wedges marked to represent the preservice teacher’s interpretation of 3/5 as three out of five in a part/whole fraction model. Three out of five squares are marked to represent another student’s interpretation of 3/5 as a ratio but not a fraction.  Six out of ten squares were marked also marked as a ratio of 3/5. The line was drawn to show a slope of 3/5. The division problem 3 divided by five was suggested by another preservice teacher as a separate meaning since it represents an operation instead of a ratio or a number.  The number line was intended to show 3/5 as a location on that number line. This distinction highlights the multiple conceptual layers students may associate with the same symbolic form.

Students then brainstormed a list of mathematics problems using 3/5 and sorted them by meaning.  Here are the groups of problems generated by this group of preservice teachers:

3/5 as a Fraction/Number

  • 3/5 + 1/2

  • 3/5 – 1/4

  • 3/5 x 2/3

  • 3/5 ÷ 1/10 

  • Solve for x: 3/5 + x = 7/10 

3/5 as a Ratio

  • Sarah ate 3/5 of a pizza. If the pizza had 10 slices, how many slices did she eat? 

  • Shade 3/5 of a rectangle. 

  • What is the equivalent fraction of 3/5 with a denominator of 10? 

  • If 3/5 of a number is 12, what is the whole number? 

  • Graph:   y = 3/5x - 2

3/5 as a Place on a Number Line or Measuring Tool

  • A recipe calls for 3/5 cup of flour. If you want to double the recipe, how much flour do you need? 

  • On a number line, mark the point that represents 3/5. 

  • Put the following fractions in order from least to greatest: 1/2, 3/5, 2/5. 

It can be argued that there are not enough categories and that several of these problems could have been placed in another category. Pedagogically, working to perfect the list may have distracted the students from the main goal of the lesson. The point of the activity was metalinguistic awareness: to give the teachers a model of how to sort out meaning while also modeling how to get the students to begin sorting out the meaning depending on context of the problem.

Implications for Mathematics Teacher Education

In a post-activity discussion, preservice teachers often express their surprise at the complexity of reading that these three symbols brought with them. They reached an awareness that reading mathematics requires openness to multiple meanings and therefore language skills such as using context to decide the intended meaning. This initial metalinguistic awareness requires revisiting throughout the course to reinforce and expand the lessons learned. 

While the “What is 3/5?” activity represents a starting point for this kind of thinking, it bears reinforcing that reading even elementary mathematics problems can be a process fraught with complexities and difficulties.  In this case, three symbols combined in a certain way can mean many different things.  Those things are related, but each meaning has different implications. as one group of three symbols. 

Perhaps the greatest effect of this activity is in giving the teachers, at any stage of development, an insight into the complexities of student thinking about mathematics. Removing the context from “3/5” highlights the many meanings that are all possible interpretations happening in the minds of their own students while reading and interpreting a problem.  The straightforward approach of “when you see this, do this” begins to crumble and the journey into the depths of students’ thinking begins. With complexities such as this simple example, reading and interpreting mathematics problems is a skill that must be taught, supported, and reinforced whether the student in question is a K-12 student, a preservice teacher, or an inservice teacher.

References

Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786-795.

Bardini, C. & Pierce, R. (2015) Assumed mathematics knowledge: the challenge of symbols. International Journal of Science and Mathematics Education, 23(1), 1-9.

Crespo, S. (2003) Learning to pose mathematical problems: Exploring changes in preservice teachers' practices. Educational Studies in Mathematics 52, 243–270. https://doi.org/10.1023/A:1024364304664

Krawitz, J., Chang, Y., Yang, K. & Schukajlow (2022) The role of reading comprehension in mathematical modelling: improving the construction of a real-world model and interest in Germany and Taiwan. Educational Studies in Mathematis,109, 337–359. https://doi.org/10.1007/s10649-021-10058-9

Krulik, S. & Rudnick, J (1982) Teaching problem solving to preservice teachers. The Arithmetic Teacher, 29(6), 42-45. https://www.jstor.org/stable/41192013 

Steele, M.D., & Hillen, A.F. (2012). The content-focused methods course: A model for integrating pedagogy and mathematics content. Mathematics Teacher Educator, 1 (1), 53-68. https://doi.org/10.5951/mathteaceduc.1.1.0053

Wakefield, D. V. (2000). Math as a second language. The Educational Forum, 64, 272-279. https://doi.org/10.1080/00131720008984764 

Whitin, D.J. & Whitin, P. (2000). Math is Language Too: Talking and Writing in the Mathematics Classroom. Reston, VA: National Council of Teachers of Mathematics and National Council of Teachers of English.