Welcome Back from the Project for Elementary Mathematics
Greetings!
And welcome to a new school year. I hope you all feel recharged from your summer break and ready to move forward with the practices that allow you to cognitively guide students’ thinking. In the document Principles to Actions published by NCTM in 2014, eight instructional practices are listed that aid in developing strong conceptual and procedural knowledge among students. Below are some “sound bites” that I have used with my students, both elementary students and adult teachers and preservice candidates, to jog their problem-solving choices. Think about how to integrate these ideas into your classroom here at the start of the school year. Also included below are research highlights from the Cognitively Guided Instruction National Biennial Conference held this past June in Orlando. These findings support the need for sustained and vertically integrated professional development as well as the importance of teacher questioning skills in eliciting and supporting students’ mathematical thinking.
- Sound bites reminder
- Research highlights from the CGI National Biennial Conference that you should be alert to
Sound bite reminders
Watch your language!!! — Language development right from the beginning of the school year is critical to get students thinking in value rather than in digits. The other aspect of language development is to help students develop their mathematical voice. Help them first to verbalize their thinking. Students creating written records of their work in words and mathematical notation is its own set of lessons similar to the editing and revision process used in language arts settings. The language functions of explaining, describing, justifying with evidence, comparing and contrasting, analyzing, predicting, summarizing… are all key elements of a language-rich mathematical environment that helps students articulate their mathematical processing.
If you don’t like the numbers I give you… break them apart, make them easier to work with! — Decomposition and reconfiguration of number is a fundamental aspect of mathematics. Helping students to fluently break numbers apart and recombine them to make friendlier combinations develops knowledge of several algebraic properties including the commutative, associative, and identity of properties of addition and multiplication, and the distributive property of multiplication. It also nurtures relational thinking. Fluently organizing around landmarks of ten allows students to add and subtract efficiently as they move back and forth along a number line.
Use what you know to figure out what you don’t know! — Relational thinking involves taking a big picture look at the relationships and information presented in numerical context. Using previous information to then make judgments about which strategy to use in problem solving allows one to be efficient and strategic in one’s work. Deriving and the intentional use of algebraic properties arise out of such thinking. It allows the student to make connections between and among mathematical ideas so that their thinking builds vertically across the grade levels.
Follow the units! — Label, label, label! Even in kindergarten, it is important that students articulate what each quantity represents so that their thinking is grounded in any additional increase or decrease of a set that occurs. When the context is multiplication, the two fours in the expression 4×4 do not have the same meaning. One describes the numbers of groups; the other, the quantity within each group. One of the elements of multiplication and division is that the operations are unit transforming. This leads to being able to coordinate units cognitively and being able to scale up and down. Students need to transition to middle school mathematics more solidly than they historically have. Following the units is foundational to working with rates, ratios, and thinking proportionally.
Keep it equal! — Starting in kindergarten and in each of the succeeding grades having explicit conversation around the meaning of the equal sign [=] is essential to make sure that the misconception that “equals means the answer comes next” is kept in check. In the upper grades, the concept of equivalency is introduced through both the understanding of the concepts of the identity property of multiplication [multiplication by a form of one] and proportionality [thinking in scaled covariance]. It all starts in kindergarten.
Keep the math visible! — As the teacher cognitively guiding your students’ thinking, the representation formats that you choose – physical, pictorial, informal, formal, horizontal, vertical, number line… – aid in students comprehending the mathematical relation being described. Each form of representation captures those relationships slightly differently. Varying the representational forms allows students beyond the social conventions of recording their work to comprehend the underlying mathematical structure.
Research Highlights from the CGI National Biennial Conference
Three of us connected to the Roseville CGI professional development cohorts (Jill Bue, Ben Schwanke, and myself) attended and presented at the CGI Biennial in June. Among the findings presented in the five-year randomized group study comparing 11 CGI based elementary schools and 11 standard instruction based schools were these related to student achievement in Central Florida. (Schoen, et al., 2022)
- Student achievement results at K-2, while positive, were only modestly significant in the CGI-based schools compared to the standard instruction-based schools as measured on the State of Florida required assessments. However, the difference in achievement occurred at grades 3-5. Larger statistically significant effect-sizes occurred in the upper grades compared to primary grades. This augurs for the accumulated growth of student knowledge over time as students vertically progress across classrooms where consistent instructional practices are CGI based.
- Measures of growth in the upper grades were particularly strong for black students compared to all other demographic groups
- The researchers’ thoughts, based on findings from other research, as to why growth was cumulative in the upper grades includes CGI-based primary classrooms infusion of multiplication and division problems into students’ problem solving experiences. This is especially true to develop place value understanding. Additionally, a strong focus on fraction understanding in grades 3-5 may point to content differences that impact achievement compared with those students in standard instruction based schools.
Results from another study, albeit a much smaller study, were reported at the conference (Empson & Jacobs, 2021). This involved assessing teacher questioning techniques over a three-year professional development project on fractions. Impact on student achievement was found to be higher in those classrooms where teacher questioning techniques were more advanced. This supports findings in other studies (Franke, et al., 2009; Webb, et al., 2009) where achievement was higher in those classrooms where student errors were processed constructively and in full.
A goal for you this year: Think about how you wish to focus on your questioning techniques across this year.
Have a good start to your school year.
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James Brickwedde, Ph.D.
Director, Project for Elementary Mathematics
website: www.projectmath.net
exploring how children think and develop their mathematical understandings