As I return home from a visit to a middle school, I cannot help but think of how the instructional shifts in Steve Leinwand’s book, "Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement," could make a difference in the performance of students at that school. Students in this middle school are not performing well in math. In the area of fractions, a number of students failed test items that required them to add and subtract fractions with common denominators. Teachers are struggling to cover content for the next assessment that includes topics such as probability, area and perimeter, volume, surface area and interpreting graphs. It's a great deal to cover in one marking period, so work on fractions is being postponed to the next “cycle”. What if teachers applied the 10 Instructional Shifts to reinforce and reteach fraction concepts and skills while moving into other topics? In this post, I will address the first five of the ten instructional shifts.
Instructional shift 1: Incorporate cumulative review every day
Leinwand writes, “skills, terms, and concepts are constantly being forgotten or confused”. Let’s think about the type of review that relates to developing fluency with fractions. It is important that students develop the Big Idea that fractions are division. Concepts such as equivalency, comparisons, estimation and converting between mixed numbers and improper fractions would be valuable examples on a daily review. The following daily review can be completed in less than 10 minutes but can lead to rich discussions, reinforcement of vocabulary and strengthening of skills.
Instructional shift 2: Adapt what we know works in our reading programs and apply it to mathematics instruction.
When teaching reading, teachers often facilitate understanding through questioning. In math we often focus on correct answers and not the underlying reasoning. By asking questions such as “why” and “how do you know”, we would encourage students to think about the meaning behind the math. Introducing concepts that build on background knowledge is another method reading teachers employ. Just because you are moving beyond fractions does not mean you cannot integrate what students have practiced in fractions within to new content. Moving beyond right answers when teaching fractions may look like this:
These examples, while covering new content, provide the students practice with word problems, vocabulary reinforcement and fractions. Follow up can include questions such as: How did you figure that out? How did you know each side was the same length? Why did you add 8 3/6 + 8 3/6? Why did you convert the 3/6 to 1/2?
Instructional shift 3: Use multiple representations of mathematical entities.
The use of models assists students in developing a conceptual understanding of the math, however one model may not reach all students. This is why we need to provide multiple representations.
When teaching fractions we need to move beyond focusing solely on pictures or circle models to include area, set and number line representations. Asking students, for example, to "show 3/6 using different models" can assist in a more flexible understanding. For students who continue to struggle with addition and subtraction of common denominators, use of models such as those in our free tools can allow students to represent an equation using multiple models. When a student creates models for the equation 2/8 +3/8 and displays the sum as 5/16, it becomes apparent that the shaded area of the sum is less than the areas of the addends when one is dragged on top of another.
Instructional shift 4: Create language-rich classroom routines
Pre-teaching, reviewing and including multiple discussions with math vocabulary can be just what many students need to be successful. Sometimes the problem students experience is “not from a lack of mathematical misunderstanding but from serious confusion with the English”. (Leinwand 2009)Terms such as denominator, numerator and equivalence should be discussed and modeled often. Leinwand recommends activities such as “Brain dumps” in which students brainstorm what they know about particular numbers or equations and concepts. In this activity, students generate vocabulary that can lead to follow up questions and discussion. He includes an example in his book in which students generate a list of features of the number 2 1/4.
Instructional shift 5: Take every available opportunity to support the development of number sense.
Even if the topics being covered in class do not specifically include fractions, it is often easy to add an example into the daily review or lesson content in which you can reinforce fraction number sense often referred to as fraction sense. “Fraction sense implies a deep and flexible understanding of fractions that is not dependent on any one context or type of problem. Fraction sense is tied to common sense. Students with fraction sense can reason about fractions.” (McNamara, 2010)
While these students may be working on learning area, perimeter volume and surface area, students can be encouraged to estimate the answers prior to solving an equation. Adding examples with fractions like the ones above now provide an additional opportunity to estimate fractional amounts. Warm up activities can be done using the estimating on a number line tools in which students can estimate individual fractions, sums or differences of fractions to benchmark numbers.
To summarize, expanding students’ opportunities to work with fractions in different contexts will strengthen their fraction sense. This reinforces the fact that mathematics is a language as well as concepts that can be expressed by symbols and or equations. In my next post we will contemplate the value of the Steve Leinwand’s suggested instructional shifts 6-10.