## A – Abstract

In the abstract stage, students translate their two-dimensional drawings into **conventional mathematics notation** to solve the problem. The manipulations in the concrete and representational stages build students’ conceptual understanding, which contributes to fluency and automaticity in the abstract phase. However, for students with LDs, this represents the greatest challenge [i]. Therefore, it is recommended that instruction supports the transition from Representational to Abstract through strategy instruction, such as heuristics.

In this section of the module, you will learn about **heuristics** as a tool for solving mathematical problems at the abstract stage.

## Math Heuristics

At its most basic, a heuristic is a **short cut**** or rule** for reducing the cognitive load of information-processing. A general heuristic might be something like “Read the problem. Highlight the key words. Solve the problems. Check your work”. Other examples might include using a rule of thumb, an educated guess, an intuitive judgement, or common sense. Sometimes heuristics are taught by teachers, sometimes students come across them on their own [ii].

The use of heuristics in mathematics can have a profound impact on a student’s ability to **quickly and accurately solve** a math fact or word problem. Students with LDs, whether math-specific or not, will especially benefit from the **structure and sequence** a heuristic provides [iii].

### The Key-Word Strategy: A Word of Caution

The **most basic strategy** suggested for students with LDs is the key-word strategy. The key-word strategy involves **associating common words with the operation they represent** [iv].

For example, they might associate “gave away” to mean the question involves subtraction. In the word problem “*Amy had 10 pencils. She gave away 4 to Meaghan. How many pencils does Amy have left?*” this strategy would work.

However, while this heuristic is helpful for students with LDs in mathematics, it **can actually be** **too narrow** to be applied to all problems. For example, “gave away” takes on a different meaning in the problem “*Amy gave away 4 pencils to Meaghan. She gave away 6 pencils to Mino. How many pencils did Amy give away?*” as the problem does not involve subtraction. So, this strategy should be taught to students as a last resort if no other strategy works [v].

### Paraphrase, Visualize, Hypothesize, Estimate

Students with LDs are more likely to rely on checking and computing strategies rather than higher-order problem representation strategies [vi]. Therefore, Landi [vii] proposes a 4-step problem representation strategy for students with LDs.

#### Step 1: Paraphrase (put in your own words)

**Skill 1. ****Underline the important information:** Many word problems contain extraneous information, which can be distracting, particularly for students with LDs. Underlining or highlighting the information relevant to solving the problem can help simplify the process. M**odel** several word problems for the class, underlining the important information, and discussing why certain parts were highlighted and certain parts were not.

**Skill 2. Put the problem in your own words**: When students rephrase the problem using their own words, it helps them process the information to make sense of the problem [viii]. Discuss several examples of good paraphrasing as well as ineffective paraphrasing.

#### Step 2: Visualize (a diagram or a picture)

**Skill 1. Make a drawing or diagram**: Converting a word problem into a diagram helps students grasp the structure of the problem, and make connections between the visual representation and abstract mathematical symbols [ix]. To practice, provide students with example problems that do not ask them to solve anything, but simply provide information that can be converted into a diagram [x].

#### Step 3: Hypothesize (number of steps, operation, equation)

**Skill 1. Decide how many steps are needed**: To help students identify multi-step problems, consider colour-coding some examples that associate a colour with a particular step or operation. This helps to highlight the similarities and differences between problems, and to associate mathematical symbols to words in the problem [xi].

**Skill 2. Decide which operation is needed**: To practice, provide students with a series of example problems, and have them circle the ones that would be solved with the same operation. Then discuss the similarities among those problems to help students identify patterns in problem types [xii].

**Skill 3. Write the equation**: To practice, provide students with example problems that do not ask them to solve anything, but simply provide information that can be converted into an equation [xiii].

#### Step 4: Estimate (predict the answer)

**Skill 1. Estimate the solution**: According to the Ontario elementary mathematics curriculum, “Knowing how to estimate, and knowing when it is useful to estimate and when it is necessary to have an exact answer, are important mathematical skills … Estimation should not be taught as an isolated skill or a set of isolated rules and techniques. Knowing about calculations that are easy to perform and developing fluency in performing basic operations contribute to successful estimation” [xiv]. Provide students with models and examples of estimating in a variety of problem contexts.

The following printable resource can be used to help students work through paraphrasing, visualizing, hypothesizing, and estimating in preparation for solving word problem.

### Mnemonic Devices

Researchers Miller and Mercer [xv] present two mnemonic devices, DRAW and SOLVE, to bridge the gap between the representational and abstract stages of the CRA approach. Both mnemonics cue students to retrieve math facts from memory or use strategies to calculate those that are not yet memorized.

**DRAW**was developed for students in the primary years, and has been used to solve addition, subtraction, multiplication, and division problems. Click here to access LD@school’s template for the DRAW mnemonic.**SOLVE**has been used effectively for students in the junior years, particularly in supporting the consolidation of multiplication facts from 0 to 81. Click here to access LD@school’s template for the SOLVE mnemonic.

For **word problems**, Miller and Mercer [xvi] propose FAST DRAW to help students with LDs who often have difficulty selecting the relevant information, determining the appropriate operation, and evaluating their answers. **FAST DRAW **adds four preceding steps to the DRAW mnemonic, and has been used successfully with multiplication word problems. Click here to access LD@school’s template for the FAST DRAW mnemonic.

While research supports the use of mnemonic devices for some students with LDs, it is important to note that they **rely on verbal skills and memory, and therefore may not be best suited for students with deficits in these areas**. For example, in the LD@school webinar *The SLP in the Math Class*, Sabrina O’Keefe stated:

“I'd like to talk a little bit about mnemonic as a tool. Don't get me wrong, I love a mnemonic. But this particular one, this FAST DRAW mnemonic, is eight steps and it's a strategy that relies on phonemic awareness for students who may have impairments with phonemic awareness. And it also has a heavy load on language, literacy, and memory. And there's embedded knowledge that describes what to do at each step. So each step is really a multi-step. And if this is going to be successful, it has to be practiced a lot.”

## References

[i] Strickland & Maccini, 2013

[ii] Robinson & Hutchinson, 2014

[iii] Robinson & Hutchinson, 2014

[iv] Kelly & Carnine, 1996

[v] Kelly & Carnine, 1996

[vi] - [xiii] Landi, 2001

[xiv] Ontario Ministry of Education, 2005, p. 16

[xv] Miller & Mercer, 1993

[xvi] Miller & Mercer, 1993