As part of CTL’s book study for the FOCUS IN HIGH SCHOOL MATHEMATICS REASONING & SENSE MAKING, this is the seventh in the series of those blog posts. This post expands our thinking to the Middle grades instruction to ask ourselves, how are those youngsters experiencing reasoning and sense making within their instructional day?
For connecting to our book study, I am always searching for pedagogical examples of reasoning and sense making opportunities. I would like to share several of those with you and hopefully we can have some dialogue around those.
Research from NCTM’s Middle Grades Publication:
NCTM’s Mathematics Teaching in the Middle School, Focus Issue on Fostering Mathematical Reasoning, February 2012 is a direct connection to promoting inquiry, risk-taking, and metacognition where teachers and students alike make their thinking public that were discussed in past blogs. The articles in this focused issue gave examples of how to provide opportunities for mathematical reasoning for younger students so that their conceptual understandings can expand to the high school skill set and more importantly to bridge the concrete to the abstract.
From the article “The Value of Debts and Credits,” on page 335, five suggestions with examples for each are given for fostering mathematical reasoning and I quote, “Encourage students to:
- give conceptual explanations.
- provide efficient solutions.
- make conjectures and prove them.
- create different & sophisticated solutions.”
This article also supports tasks that have multiple solutions and provide context to build conceptual understanding of real numbers and their operations.
The best example of providing opportunities for students to engage in reasoning processes comes from my favorite article of the February issue: “Why Don’t We Just Divide Across,” on pp 340. Talk about risk taking and inquiry – “students reasoning, making conjectures, asking why, and justifying conclusions – what mathematics should be.” The questions posed, student work for verification, and explanations are all included in the article. Note that manipulatives and explaining thinking through discourse and writing were used to support and model the meaning of division of fractions. For retention and extension to later sophisticated mathematics, this process certainly is better than, “when dividing two fractions, flip the second number and then multiply.”
The question posed by students was:
How does this way of thinking/process extend to and support the high school curriculum – especially in solutions with equations and simplifying polynomials? Does this experience help to bridge the concrete to the abstract?
Recent Walkthroughs in Kentucky Middle Grades Classrooms:
Recently, after 6 months of GEARUP Walkthroughs visiting in lots of KY Middle School classrooms looking for ACT mathematics indicators, there were only several occasions where I experienced the students reasoning, making conjectures, asking why (both students and teachers), and discussing mathematics.
With the onset of the CCS in mathematics what I did witness was a lot of show and tell – let me show you how to do it and then you repeat after me. This was particularly true where teachers were showing 8th graders how to solve simultaneous equations and solve quadratics using the quadratic formula. In most cases students, very quietly and compliantly took notes and then after the 15 minute or so of lecture, they set about at working their homework. There were lots of students falling through the cracks and for those who “got it” the first time – suspect that they remember the steps long enough to pass the test so that in high school they could tell the teacher – no, I’ve never done/seen that before.
Not only did students not have a voice to ask why, make connections to past experiences, visual the graph for verifying the solution graphically, they never constructed meaning through discourse and/or writing. From my experiences, both concepts are tough to understand for high school students much less for middle grades students.
Within the observed lessons, I was hoping so badly that simple things, yet powerful for student understanding, would happen. For instance, with the quadratic formula I was hoping for students to experience:
- graphs of quadratics first and to conjecture what the graph was telling us about the roots/zeros of the function – that visual representation that our brains need to experience before proceeding to the abstract formula. (Observe possible experiences of graphs and questions that can be posed when observing this video from Kahn Academy)
- reviews with Venn diagrams of the rational, irrational, real, and complex number system – what they look like and how they connect to the formula/graph.
- discussion about the degree of the quadratic versus the linear equations.
- what the degree tells us about the number of possible roots and verification of those roots through investigating the graph of the quadratic.
- differences in the graphs of the quadratic and linear equations – investigating their graphs, tables, and equations using the graphing calculator.
- use of algebra tiles to indicate that the quadratic could/could not be factored through the area model for determining the factors.
- discussion and use of the function notation.
Next blog post will share other research and possibly look at ways to enrich experiences with solving simultaneous equations for middle grades students. Please share with us.
Sharing of Readers’ Experiences:
- What routines in your classroom do you have that support reasoning and sense making? How do these examples connect/impact your routine development?
- What kinds of prior experiences/scaffolding would your students need in order to be successful with these tasks/examples?
- What specific content would a student need to enter into a certain task/example?
- How does reasoning and sense making help students make connections between content domains? Specifically what aspects of task/examples help students make connections?