I do these more for myself than anyone else, but here, I am quoting the most useful parts of this ASCD book (click links to read online for free). Basically, I'm editing out the boring. You're welcome.

I'm also just doing a couple of chapters at a time because it's kind of dry and I never know how much time I will have to read. So consider it a series if you like.

Joan M. Kenney

Chapter 4. Graphic Representations in the Mathematics Classroom

The following aspects of mathematical language are particularly confusing to students:

The following aspects of mathematical language are particularly confusing to students:

- Technical symbols such as ∑, ≤ , or Δ. These signs, also known as logograms, stand for whole words but have no sound-symbol relationship for students to decode.
- Technical vocabulary- words such as
*rhombus*,*hypotenuse*, and*integer*, which are rarely used in everyday conversation. - The assignment of special definitions to familiar words such as
*similar*and*prime*. - Subtle morphology (one
*hundred*,*hundreds*-place,*hundredths*) and the use of "little words" (prepositions, pronouns, articles, and conjunctions) in a technical syntax so precise that meaning is often obscured rather than clarified.

**Scenario #1: Measuring Cups Activity***A teacher models drawing a one cup and one-fourth cup four times to quadruple a recipe. When asked how many cups are needed, students responded 'eight' because there were literally eight cups in the drawing. They counted the one-fourth cup the same as the one cup.

- In what other ways might the students have attempted to model the problem if the teacher hadn't offered the initial suggestion?
*Let students try first, then offer suggestions?* - How might using actual three-dimensional
*models*(i.e., a set of nested measuring cups) before using two-dimensional*representations*alter the transfer of learning?*Always teach from concrete to abstract?*

**Scenario #2: A Round Pizza in a Square Whole**

*When solving a problem dealing with rectangular pieces of pizza, William drew the problem with a circle pizza. To him, fractional parts

*had*to be round.- How can the teacher help [William] move beyond a single conceptual image and experiment with new metaphors and visual models?
- When are models useful, and when do they get in the way of new learning?

**Scenario #3: An Uphill Struggle (Slow on a Slippery Slope)**

*When discussing slope from points on a graph and connecting points to create intervals, a student took the graph literally, thinking the the intervals were literally hills and valleys.

- For how long, and to what depth, should a teacher continue to probe in order to get to the logical and intuitive root of confusion?

**Scenario #4: Breaking Even**

*Students looking for the break even interval on a graph of monthly profits picked the flat interval because they were thinking break even literally meant to find the interval that is even.

Note that the phrase

*breaking even*is not a formal mathematical term but rather a conversational idiom with mathematical implications. Pimm (1987) calls these types of phrases*locutions*- "certain whole expressions whose meanings cannot necessarily be understood merely by knowing the means of the individual words, that is, the expressions function as semantic units on their own" (p.88).- How can teachers become more aware of the mathematical locutions embedded in their classroom conversations?
*Pre-teach all mathematical terms, especially if they are also used in a nonmathematical context? Think literally.*

**Scenario #6: The Wordsmiths**

*Students related exponential decay to exponential growth but couldn't figure out the precise terminology: "undoubling, doubling down, divided in half, taking half of it, dividing by two, halving".

- Would this particular mathematical conversation have occurred if the students didn't have a graphic "prop"?
- How did the visual representation act as a catalyst for student discourse?
- How might conversations about a graphic display encourage students to put their mathematical perceptions into writing?
- Now that a conversation had begun on the topic of exponential decay, how might the teacher build on the richly descriptive terms that the students created to press for more precise mathematical terminology?

**Scenario #7 Where's the Fourth Fourth?**

*Benjamin folded a paper strip into fourths and wrote on each of the folds 1/4, 2/4, and 3/4. When asked what his strip created, he said thirds. He was judging each strip based on the last numerator instead of the denominator. When asked about his strip that was folded in half, he wanted to say halves but according to his system, it would be one whole, based on the one in the numerator of one-half.

- What distinction was Benjamin making between the terms thirds, three folds, and dividing in to three parts?
- How might students conceptualize
*fourths*differently, depending on whether they are asked to label the strip's*segments*(1/4, 2/4, 3/4, 4/4) or its*folds*(1/4, 2/4, 3/4)? - What consideration might the teacher need to give to precision of language when providing directions for this task?
- What role might peer-to-peer discourse have played in helping Benjamin test his conjecture?
- What questions might a teacher ask to check more deeply for understanding if, at first glance, a student's thoughtfully done work is apparently correct?
*Ask them how they would explain it to their grandma or ask them to create a new example?*

**Scenario #8: Three Three/Ten Combos**

*Benjamin and his class were instructed to divide a square pan of brownies into 30 equal squares. Students drew 5 rows of 6, 3 rows of 5, and 2 rows of 15. Later they were given a sheet with a giant square divided into 10 vertical strips. When asked to model three-tenths, Benjamin drew three horizontal lines across the vertical strips instead of shading in 3 of the 10 strips.

- From what assumptions might Benjamin have been working when he approached this new concept?
- Where was there a language-based component in his confusion?
- What do the phrases "three-tenths" "three and tenths", and "thirds and tenths" mean mathematically? What do they seem to mean to Benjamin?
- In a roomful of 20 students, how difficult is it to hear the subtleties of different word forms?
- What might Benjamin's drawing imply about what he heard?
- How could the teacher use Benjamin's drawing to encourage him to express his personal understanding in words?
- How might making additional sketches have helped Benjamin communicate what he head and understood?

**Scenario #9: Holes in Her Logic**

*Sarah was dividing donated food into boxes for 24 families. She drew 24 squares and drew dots in each box until she ran out so that they would be evenly distributed. For 12 pounds of cheddar cheese, Sarah divided 12 by 24 in the calculator and got .5, so she then drew 5 dots in each box instead of half of a dot.

- What windows to Sarah's thinking do her cocoa, milk, and cheese drawings provide?
- How did Sarah seem to understand division in general? What about dividing in situations when the quotient is less than one?
- What did Sarah believe about the decimal point?
- What was Sarah hearing, and how did this influence the manner in which she conceptualized decimal fractions?
- What is appropriate calculator use for students beginning to work with fractions and decimals?
- Would language-based cues, such as the teacher's asking how to divide a 12-pound wheel or chunk of cheese among 24 families, suggest useful visual models? Or might metaphors create further confusion?
*If I was Sarah and the words wheel or chunk were mentioned, I would have drawn wheels or chunks instead of dots but I still would have drawn 5 wheels or chunks in each box.* - What might the teacher do ti reinforce the use of proper terminology when students are working with decimal fractions?

I don't feel like typing Scenarios #5 and #10.

When teachers are asked to reflect on how student drawings can inform their practice, three themes emerge. Teachers feel that the drawings:

- Make the students more aware that they're "speaking mathematics" in class,
- Show a need for greater precision in the students' use of mathematical language, and
- Suggest areas in which directions and explanations should be more clearly phrased.

**Suggestions from Teachers**

- Combine verbal with visual.
- Monitor your language for words with double meanings.
- Assume positive intent to understand even from silly questions or offhand remarks.
- Ask students to move from drawing detailed pictures to simpler shapes, another step toward abstraction.
- Consider the sequence of representations students select. Does it promote depth of mathematical thinking?
- Actively point out connections to other representations so that students become fluent in translation from one representation to another.

**Suggested Practices**

- Articulate and enunciate!
- Keep writing utensils available if you expect students to draw.
- Extra time: putting ideas on paper takes more time than talkingg.
- For open-ended questions, suggest drawing a diagram first in order to have something concrete to write about.
- Make copies of important textbook pages so that students can highlight, draw on, underline, and write notes in the margin to better interact with the text.
- Have students use graphic organizers that incorporate both words and pictures.
- Designate a section of notes/worksheets for drawing, showing that you value and expect to see visual thinking.

Drawing is a device to capture the language of mathematics in order to make it visible to themselves.

Drawing slows students down and allows them to self-correct their thoughts while their hands are sketching; it also helps them to keep track of and record their solutions.

*Paraphrased

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