Join Now
View Cart
NCTM Store

Some Reflections on Connections

  • Some Reflections on Connections

    By Matt Haber, posted August 3, 2015 –

    The teaching and learning of mathematics in America remains fragmented and unconnected in some areas. Many teachers and students have yet to explore the connections that concepts and ideas have to one another. Without connections and meaning, most children do not retain newly taught material. To enter the long-term memory, information must be connected to other pieces of information, or have meaning. 

    2015-08-03 art1

    Let’s put on the lens of a child for a moment. How many ways can you add 39 and 25? Take a moment and record your strategies in your head or on paper.

    Some of you found that 40 + 24 is the same as 39 + 25 and found 64 as the answer. Many of you added 30 + 20 and then 9 + 5 and found 50 + 14 = 64. Some of you added 1 to the 39 to get 40 + 25 = 65, then subtracted the 1 to get 64. A few of you added 39 + 20 = 59, then 59 + 5 more to get 64. Others of you visualized the algorithm/procedure. There are many ways to arrive at 64. It is important to support this type of flexibility in our children; it is the major factor that separates high achievers from low achievers. 

    How can we use these same strategies as we explore fraction addition? Well, let’s try. How many ways can you add 4 3/4 and 1 1/2? 

    Some of you added 1/4 to the 4 3/4 to make a friendlier number of 5, so 5 + 1 1/2 = 6 1/2, then subtracted that 1/4 you had added, to get 6 1/4. Some of you did the same with the 1 1/2 by adding 1/2 to get 4 3/4 + 2 = 6 3/4, then subtracting the 1/2 to get 6 1/4. A few of you took 1/2 out of the 4 3/4 to get 4 1/4, then gave that 1/2 to the 1 1/2 to make it a 2. So, you now have 4 1/4 + 2 = 6 1/4. And again, many of you visualized the algorithm. 

    My research and classroom experience has proven that when students are given opportunities to explore new concepts using the previous strategies, they will begin to construct their own knowledge. When students see the connections between concepts and ideas, math will become meaningful, and mathematics will emerge as a linear story.

    2015-08-03 Haber au-pic

    Matt Haber is the founder of Problem Solved! Innovative Learning for the 21st Century, where he consults with educators, parents, and students with twenty-first–century mathematics, teaching, and learning. For seventeen years, he worked as a teacher and mathematics coordinator for the Los Angeles Unified School District, where he provided districtwide professional development to teachers, principals, and directors.

    Leave Comment

    Please Log In to Comment

    All Comments

    Samantha Birch - 3/9/2020 8:08:33 AM

    A very strong kind of connection and relation should be developed between mathematics teacher and his student. Math is difficult subject for most of students. I prefer to read boomessays.com review first before hiring their services. A good connection of teacher with these students would be very useful.

    Christopher Brownell - 8/5/2015 12:22:35 PM
    Matt, I like and agree with a good deal of what you are saying here. Especially your call to teachers to actively seek to make connections from one algorithm to another, or from one set of conceptualizations to others. The connection between breaking down integers into additive composite parts and fractions or rationals and doing the same is huge, and I would agree under-represented as a viable thinking technique. This connection made can be, as you suggest, extended into heaps of areas and in school mathematics finds its full fruition in Algebra. I have an issue with this phrase though, "mathematics will emerge as a linear story." Mathematics is a rich, human, organic story. Thus, as all organic entities is only at best "locally linear." This richness should not be ignored in school mathematics, it should be touted, put on display, and have its praises sung. I had a professor, who i now consider to be a friend, once tell me that "mathematics is a uniquely human construct, if we were a bit smarter we wouldn't need it, a bit less smart we wouldn't understand it." But human constructs come in fits, flurries, and brain-Fa**s. (sorry the alliteration temptation had me rolling there). We should not be looking all the time for linearity in the development of concepts.