How many of your students are able to tell you what 24 ÷ 4 or ½ x 6 mean? And yet are they able to use their computational skills to solve each expression? How would you assess their proficiency if they are able to provide the answer but are unable to tell what the expression might represent?
If our lessons focus solely on computational strategies to find answers, will our students be able to effectively build an appropriate equation when posed with a problem situation? Will they know when to use the computational skills they have learned? Will they recognize when their solutions are unreasonable?
For many years, isolated computations appeared at the top of math textbook pages and contextual problems appeared at the bottom of the pages. The message seemed to be that computations were most important (the ultimate work of math) and that solving problems came after computational fluency. Our current standards have led us to reconsider that thinking and place greater value on math understanding and application. The ability to solve contextual problems shows that our students understand and can apply the computations they are learning.
When we begin a lesson by posing ¾ – ½ = n, our classroom discussions focus on procedures and answers. One way to help our students make sense of the computations they are learning is simply to start the lesson with a problem. Consider beginning the lesson with the following:
Liam ran ¾ mile and Kellen ran ½ mile. Who ran farther? How much farther?
What will classroom discussions sound like when this problem is posed?
- What is happening in this problem?
- What are we trying to figure out?
- What math operation makes sense with this situation? Why?
- What might this problem look like (model)?
- What do we do mathematically when we compare? Why might it make sense to subtract to compare fractions? How is that like what we do with whole numbers?
- What would the equation look like? Why?
- What do the ¾ and ½ represent? What will the answer represent?
- About what will the answer be? Explain your estimate.
Students are making sense of the problem, discussing possible operations, connecting the fraction problem to previous whole number problems, building an appropriate equation, and estimating the difference based on the problem context. And once the ¾ – ½ = n equation is constructed, the focus transfers to the computational procedures and answers. Win-win! We are able to address the same computational skills as we would if the equation were posed in isolation, but what a benefit to have the opportunity for students to explore a problem context and build the equation together!
When we start our lessons with problems, our students benefit.
- Students are immediately engaged in the lesson through a context rather than being asked to solve an equation that has no meaning.
- Students discuss, and build better understanding, of math operations.
- Students begin to connect real situations to the corresponding math equations.
- Our students are able to identify computational errors because they “don’t make sense” when thinking about the problem context.
- Our students practice solving problems every day and begin to view it as an integral part of mathematics.
- We help our students see that math is not about worksheets and strings of meaningless computations, but that it is meaningful and connected to their lives.
For more ideas on integrating problems into daily lessons, see Heinemann’s Math in Practice series (http://www.heinemann.com/mathinpractice/).