In my current role as the math specialist and curriculum coordinator in a preK through 10th-grade school, one of the questions I am frequently asked by teachers is how to appropriately challenge strong math students. We want all our learners to grow in their curiosity, understanding, and love for math, so how do we do this without simply accelerating through content for students who seem to "get it" more quickly than others?
“Student success occurs when you create an instructional environment that sets high expectations for each student and provides scaffolding without offering excessive help.
The key is to incorporate productive struggle.”
—Barbara R. Blackburn
As Blackburn states in her article
“Productive Struggle Is a Learner's Sweet Spot”, student success is optimized when we set high expectations for our students by incorporating opportunities for all learners to experience productive struggle. At our recent school-wide professional development day, we focused on how using parallel tasks, open-middle tasks, and open-ended tasks can provide all students with the opportunity to be engaged in similar tasks while providing productive struggle at differing levels based on students' readiness.
With parallel tasks, two (or more) different questions are posed that are related to the same mathematical big idea and context. This allows various entry points to meet the needs of students while providing a common experience so everyone can participate in a rich mathematical discussion about the tasks. For example, students may be asked to create a real-world situation that requires counting to 100 or a real-world situation that requires counting to 1000. After having some time to consider scenarios and decide on one they would like to share, the teacher can facilitate a discussion focused on comparing the scenarios students came up with for 100 versus those for 1000 by asking questions like "What do you notice about the difference in these scenarios?" and "Could this scenario for 100 ever require counting to 1000? If so, when? If not, how do you know?"
Mark Chubb also wrote a
blog post on parallel tasks where he gives an example related to algebraic reasoning involving pattern blocks. The most critical component of making parallel tasks valuable learning experiences is that
everyone comes together to share and discuss their problem-solving ideas so we can learn from each other, even if we have different entry points to access the math concepts.
Open-middle tasks are those which have one correct answer but have multiple entry points and possible strategies that can be used to solve. Many problems can be open-middle problems, as long as you
prioritize the time for students to share and compare the different strategies they used to find their solution. When students share their ideas and explain why they have a viable strategy, it is then their peers that should be providing feedback and critiquing their reasoning (
CCSS.MP3) rather than the teacher serving as the judge. For some concrete examples of open-middle tasks, check out Robert Kaplinsky's
Open Middle website or his
problem-based lessons that allow for rich mathematical conversations about problem-solving strategies.
Andrew Stadel's
Estimation 180 and
3-Act Tasks like the ones from
Andrew Stadel,
Graham Fletcher,
Dan Meyer,
Kyle Pearce, and
Dane Ehlert can also be perfect resources for open-middle problems.
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Image made with the
Public Math Sticker app |
Open-ended tasks, on the other hand, have many possible solutions and are further opportunities brimming with potential for students to articulate and justify their mathematical thinking.
Number talks, or math talks, allow for "low entry, high ceiling" so all students can access the mathematics, but it is possible to extend the concepts to accelerated levels as well.
Which One Doesn't Belong and
Fraction Talks have helpful images that can be used for math talks, and you can always use your own images as well. The
Public Math Sticker App is a fun way to add questions like "What do you notice?" "What do you wonder?" "How many?" and "What repeats?" to your images like I did with the gummy bear example on the right.
Brian Bushart explains in
this post how numberless word problems can be another way to promote mathematical thinking through open-ended tasks. When the numbers are removed from a word problem, the context remains.
Asking students questions like "What do you wish the numbers were (what numbers will make it easy)? Why?" and "What numbers will make it challenging? Why?" build students' number sense. Asking in follow-up, "what is another question you could ask about this scenario?" allows students to challenge themselves by thinking about other ways numbers in this context can be manipulated to provide further information.
Marian Small's book
Good Questions: Great Ways to Differentiate Mathematics Instruction in the Standards-Based Classroom is also a wonderful resource to see examples of both parallel tasks and open-ended tasks in different grade bands (K - 2, 3 - 5, 6 - 8) in different math strands (Counting & Cardinality and Operations in Base Tens, Number & Operations with Fractions, Ratios & Proportional Relationships, the Number System, Operations & Algebraic Thinking, Expressions & Equations and Functions, Measurement & Data, Geometry, Statistics & Probability).
In closing, I want to share one quote from a recent professional development experience with
Dr. Yeap Ban Har that I continue to keep in mind. He said, "when enrichment is done well, it leads to natural acceleration." Our goal is to open our students' minds to multiple ways of thinking through incorporating opportunities for all students to experience productive struggle, thus allowing them to make connections among mathematical concepts and think flexibly about math. In what other ways do you do this for your students? I would love to hear from you in the comments!
