Written by Vanessa Rayner and Joanna McKay
Our second day with Dr. Barbara Dougherty continued to focus on the Six Recommendations from Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades Educator’s Practice Guide. These recommendations are based on high-quality evidence-based research studies focused on Tier 1 mathematics instruction:
The session started off strong with a rich math task called, "Which one does not belong?" Participants found a rule to explain why each of the four numbers does not belong.
The discussion centered around relating numbers based on their characteristics (e.g., even, odd, square), digits, how the numbers are composed (i.e., prime, composite), and alternate ways of expressing these numerals (e.g., prime factors). It was clear that this simple task has an entry point for all students at any grade level. The activity also highlighted the importance of Recommendation 2, or teaching students to use concise and clear mathematical language. Check out the link to find similar problems!
The focus of the day's session was on identifying and understanding multiplicative structures and representing fractions.
We got started with a card sort activity that involved analyzing the structure of word problems (Recommendation 5), to focus our discussion on the difference between multiplicative and additive structures, and understand how to identify these structures in word problems. We had the additional challenge of writing an equation to match the structure of the word problem.
This was not an easy task!
However, it was clear that teaching students how to analyze a word problem in terms of its structure is a strategy that helps avoid using the (ineffective) key word strategy to determine the operation. We learned that,
additive structures involve adding or subtracting quantities with the same unit. Although the type of unit is the same, the units need not be the same in size, meaning an amount can be composed of different-sized parts.
4 apples + 2 apples = 6 apples (unit is apples)
4 apples + 2 pears = 4 apples and 2 pears (unit is fruit)
multiplicative structures involve multiplying or dividing quantities with different units. When multiplied, the different types of units are coordinated to represent groups equal in size (e.g., miles per hour).
6 plates x 4 cookies = 24 cookies
When I coordinate plates (one type of unit) and cookies (a different type of unit), I count 6 units of 4, or cookies per plate (the coordinated unit).
Understanding multiplication as a coordination of units helped us realize that multiplication is not repeated addition. Highlighting the meaning of multiplication is necessary because repeated addition cannot be used to multiply all rational numbers (try using repeated addition to multiply two fractions, for example).
Further, understanding the meaning of multiplication will also help teachers support students’ understanding that a unit designates what 1 is, but does not always have a value of 1 (e.g., 1 unit of 4). So, how do we teach this fundamental mathematical concept?
Indeed, Dr. Dougherty shared that an effective strategy for improving understanding of word problem involves helping students visualize the problem using the Concrete- Representation- Abstract model (CRA; Recommendation 3).
Students first manipulate Concrete materials, and then learn to make connections between Concrete and Semi-Concrete representations (e.g., drawings, diagrams) to ultimately use the Concrete and Semi-Concrete representations in connection with abstract representations, such as numbers and math symbols.
Starting with the concrete, we used paper plates and counters to show 3 x 4 and were asked, "How do students figure out that the answer is 12?" How students determine the product is very telling.
students may count each counter individually (low level, do not see that they are equal sized groups) OR
skip count (counting by equal sized groups and therefore the foundation of multiplying and dividing numbers)
In line with Recommendation 1 we progressively worked through different problem types to focus on representing word problems with the understanding that the visual representation (Recommendation 3) deepens understanding of the structures of word problems (Recommendation 5) and, equally important, helps students “see” the operation to use as well as the relationship between the known and unknown values. Simply put, the Recommendations are clearly interconnected whereby, in this example, progressively building and extending understanding of the meaning and processes for multiplying and dividing numbers (Recommendation 1) is supported by connecting Recommendations 3 and 5. The systematic sequencing (in order of level of difficulty) of problem types was the following:
Equal Grouping (least difficult problem type): product unknown problem, or 3 x 4= ▢
Equal Grouping: group size unknown, or 3 x ▢ = 9
Equal Grouping: number of groups unknown, or ▢ X 5 = 15
Multiplicative Comparison: finding the bigger amount using the referent group (smaller amount in yellow below) and the multiplicative relationship between the amounts compared, or 4 x 3= ▢
Multiplicative Comparison (most difficult problem type): finding the referent amount (or smaller amount) using the bigger amount and the multiplicative relationship between the amounts compared, or 5 x ▢ = 20
Modeling actions and relationships described in the problem, and making connections between symbols and concrete representations helped us focus on the structure of the word problem and understand how these problem types are alike and different. Whether we model the amounts in the word problems as discrete countable objects (such as two-sided tokens), or represent the amounts as a continuous quantity using number lines and Cuisenaire Rods, the Big Idea is that word problems with multiplicative structures illustrate how numbers are composed of and can be decomposed into equal-sized groups, or units. Math is not a spectator sport- students have to be involved to understand the Big Ideas that are fundamental to mathematics!
We also discussed the difference between math fluency, or the fluid use of strategies that are efficient and accurate, and automaticity, or facts that are memorized and therefore easily recalled. Interestingly, and in line with Recommendation 6, both are necessary to support three phases of fluency: (a) Phase 1: promoting the memorization of the foundational facts, (b) Phase 2: teaching strategies that use foundational facts to derive other facts, which ultimately culminates in automaticity, or (c) Phase 3.
How does solving word problems with multiplicative structures help students gain fluency in multiplication facts? Solving word problems promotes
Modeling, drawing, skip counting and looking for patterns, helping students automatize foundational facts
For example, color coding a hundreds chart facilitates looking for and sharing all possible patterns. What patterns do you notice when we skip count by 2s and by 4s along the hundreds chart?
Students can also build derived fact strategies (e.g., finding a near double, doubling, adding a group etc...) with math games like Quad Squad, Get Four Numbers in a Row, because they target fluency, number sense and strategy.
In the afternoon we turned our attention to the teaching and learning of fractions, where the concept of the unit, or the whole, is also fundamental.
Modeling and using the appropriate math vocabulary (Recommendation 2) is often overlooked. How often do we hear students (and teachers) name a fraction as “3 out of 4” instead of “3 fourths”? Or, use language that is both mathematically inaccurate and not widely used. The example below avoids teaching students the meaning of fractions and replaces math vocabulary such as denominator with the word “pants”.
The misuse of language is without a doubt a factor that hinders learning and only leads to confusion and should be avoided.
Learning to represent fractions using well-chosen concrete tools (Recommendation 3) reinforces key concepts, namely partitioning, iterating, the relationship between unit fractions and the unit, as well as the relationship between unit fractions and non-unit fractions.
Each tool affords students the opportunity to understand the meaning of fractions in different. For example, the fractions bars, among other things, help emphasize the relationship between unit fractions and the unit. However, Cuisenaire Rods do this, and much more! For example, fifths can be represented using different-sized units, such as the Orange Rod, the Yellow Rod and a combination of Rods (e.g., 2 Orange and 1 Yellow). Promoting this flexibility, and reinforcing understanding of other key concepts (e.g., composing a fraction using the appropriate unit fraction rod), is integral to comparing fractions, understanding fraction equivalence, simplifying fractions…the list could go on and on!
When using Cuisenaire Rods to compare fractions, the student must understand that both fractions must have the same-sized unit. This concept can be overlooked when comparing fractions with the same denominator and, can be tricky to understand when the denominators are different, especially if the fractions are not represented concretely or semi-concretely. Using the Cuisenaire Rods teaches students to be strategic with the unit you select because your unit must be used to represent both fractions. Not only does this help build procedural knowledge (i.e., finding a common denominator) from conceptual understanding, using the Cuisenaire Rods to compare fractions (for example, fractions with the same numerator but different denominators) brings to light what is being compared. Indeed, once the fractions are represented using rods different in size and colour, it is evident that we are comparing the sizes of unit fractions and not the number of unit fractions (which is the case when comparing fractions with the same denominator).
Before can use Cuisenaire Rods to compare fractions, we need to bear in mind Recommendation 1, and build and connect the part-whole meaning of fractions with comparing fractions. Barb walked us through what this would look like as we engaged in a series of tasks using Cuisenaire Rods to develop our understanding and flexibility using the rods to focus on the unit when making sense of fractions.
Part of integrating Recommendation 3 in the teaching of fractions also depends on exploring which tools can be used with the different models (area, length and set). While Cuisenaire Rods and Fraction Bars make for excellent length models, other tools like pattern blocks, tangrams help show the importance of using area models that go beyond circles (or pizzas) and rectangles (or chocolate bars). Exploring many different tools that represent the area model will only help students transfer this understand to more mathematical and real-life situations!
We were grateful for the content rich and practical session with Barb. What she shared brought the six recommendations to life as we could clearly see how we could implement them in practice and support all of our students but especially those with difficulties in learning math.
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