Helping children develop conceptual understandings, making math learning relevant, and integrating discovery and skills-based learning are all important.
In the blog post The First Steps in Developing Conceptual Understanding of Place Value I shared my 2nd grade grandson’s experience in learning two and three digit addition and subtraction. He had been learning to add and subtract digits without any understanding of place value, so I introduced the operations using non-proportional objects to teach re-grouping.Earlier this week I shared how 3rd graders used proportional objects, base ten blocks, to help solve division problems despite not receiving formal instruction on division or how to use base ten blocks. See: Relevance: Discovery, Skills-based, and Manipulatives. Obviously, I believe in using concrete objects or ‘manipulatives’ to help children understand math. But what if teachers are unable to schedule the blocks of time required for discovery? In this case use semi-concrete manipulatives. Called representations, illustrations, or visuals, these semi-concrete forms of actual objects or manipulatives can also be effective. (Please note that discovery learning is more difficult for most young learners when concrete objects are NOT used, and kinesthetic learners learn best with a tactile experience.
Lessons M2.1 – 2.5 in the Elevated Math app provide instruction on whole number operations and use the semi-concrete forms of base ten blocks to reinforce place value. This YouTube video is a segment from lesson M2.4 Large Numbers: Division.
The lesson is designed for those who have already mastered basic facts and division with two digits divided by one digit. Before this segment in the lesson is an alternative algorithm using partial quotients. This is part of the lesson that explains the standard algorithm supported with visuals of base ten blocks, illustrating and reinforcing place value concepts. All instruction is followed by two sets of guided practice with additional explanations and feedback. Teacher and students materials support the lessons and are found on the Elevated Math website. (for the M2 lessons download M2.zip)
A teacher needs to guide a student through these lessons since alternative algorithms are used. (this is not the case with all the Elevated Math lessons). Utilizing semi-concrete representations of base ten blocks could help a student who has not understood the traditional or standard operational algorithm.