Models and Problem Situations
Concrete, Semi-Concrete, Abstract
Concrete, Semi-concrete (Representational), Abstract is an evidence-based instructional plan for teaching mathematics. While it does make sense to initially move from one representation to the other, using Explicit Instruction for each representation, not all students will learn the concept in a strictly linear fashion. Some students may need to move from one representation to another (and back and forth) as needed. A best practice would be to use CSA with all students as new content is introduced, realizing that some students may need to spend more time in concrete or semi-concrete than others.
CSA can be used for Specially Designed Instruction (see IEP and Lesson Plan Development Handbook from KY Dept. of Education), especially in terms of the direct/explicit instruction that some students with IEPs may need above and beyond their same age peers.
Click here to read a brief on the use of CSA with students. This comes from the Evidence Based Intervention Network at the University of Missouri. Then do the Lesson Reconstruction below.
Lesson Reconstruction:
Pick a lesson that you currently introduce in a (mostly or completely) symbolic/abstract way. How could you incorporate concrete and semi-concrete learning experiences to help students develop a more robust conceptual knowledge of that topic. Revise your lesson plan to reflect these experiences. (It may help to consider Mathematics Teaching Practice 4 as you do so.)
Multiplication Models
For each of the models below, there is a video or task sheet (or both). Watch the videos and work through the tasks. Then spend time considering the Reflection Question below.
array and area video.mp4Array and Area
Reflection Question:
For each of the models above, how might the terms factor and product be introduced into instruction?
Two Types of Division Problems
Click on each of the pictures below to see a video on that type of division situation. After watching, consider the Reflection Question below.
Partitive Division
Unknown GROUP SIZE
Measurement Division
Unknown NUMBER OF GROUPS
Reflection Question:
Why might it be important for students to be able to create a context for a plain number problem?
Learning Activities
Click on the link above and make a copy of a Google Slide deck, which is an interactive card sort of context problems for the various problem situations found in the Kentucky Academic Standards for Mathematics (2019), p. 255. The first slide has the different problems in the margins. Move the problems to the cell that best fits the situation. Then check your answers on slide 2 by removing the rectangles over each cell.
Work through each of the tasks below. As you work through each task, examine it from the student perspective and the teacher perspective. It may be helpful to review the MTP, SMP, HLP Handout.
From student perspective:
What math tools might be most helpful in solving each problem?
Which of the 8 Standards for Mathematical Practice could the students hone as they worked on each task?
From the teacher perspective:
Which of the High Leverage Practices for Special Education might you need to incorporate in the facilitation of each task?
What seem to be some purposeful questions you could pose to students to support them in productive struggle with each task (consider questions Before the task, During the task, After the task)?
Revisit (or explore for the first time) the various sets of Multiplication Cards from Making Math Magic found on the Progressions and Fluency page. A different version of the cards that I created is linked above.
Given your new learning about models and problem situations for multiplication and division, how might you use these cards with your students? One idea for consideration might be to give each student (or team) one multiplication fact and have them create a card set for that fact.
Reflect and Write
How might you use the information learned about the various models and problem situations for multiplication and division in various small group instructional settings (e.g., workshop, centers, station teaching, Guided Math, resource rooms, etc.)?
Click here to go to a Padlet and record your thoughts. This is a platform in which you will be able to see the thoughts of others as well and share ideas. Feel free to add your name and district (not required) so that others may reach out to you.
Extended Learning Opportunities
These one hour modules from Northern Kentucky Cooperative for Educational Services (NKCES) have options for elementary and secondary teachers.
This Classroom Challenge (Formative Assessment Lesson) from the Mathematics Assessment Project is a 6th grade lesson. It will help you look ahead to what students will be learning later on and make connections between the work with whole numbers and fractions. Spend time exploring this lesson. For some Formative Assessment Lessons for grades K-5, visit the KDE webpage Elementary Formative Assessment Lessons.
Click on the link above to complete the Exit Slip for this section of the training. After you submit the form, you will be sent a PD certificate for 3 hours of learning. The certificate will come from Certify'Em. Please check your clutter and junk folders if it doesn't appear in your Inbox. If you don't receive a certificate within 24 hours, please email mark.helton@ckec.org.
Continue to the next page to begin work on Strategies!
