Post # 7 Support for Older Learners

For the teacher working with older students who lack foundation skills, this course should offer ideas for building skills while addressing grade level content. This can be done with even a few minutes dedicated to skill building each day.  More importantly, it can be done by focusing on fewer facts at a time and then using or practicing them to automaticity. 

I reference the NCTM (National Council of Teachers of Mathematics) position paper on interventions.  In this document the organization stressed the need to invest time in hitting the “big picture” concepts rather than relegating an older student to a lower level curriculum.  Later in the video conferences I will refer to aspects of this as the “super powers of math.” 

In a recent presentation I gave at a university STEM conference, I outlined two type of intervention: the intensity of a summer ‘boot camp” or what I call “infusion for inclusion.”  In a summer session, an interventionist can attempt to lay a foundation for math processing by trying to address the core concepts in a hierarchy of math concepts outlined as foundation skills.  In an “infusion” model, an instructor would attempt to direct targeted practice and focus on smaller aspects of these same concepts but on a daily basis in warm ups, practice pages or homework.  Neither is ideal, but core concepts and foundation skills must be addressed if students are to understand the math they are currently studying.

First, you must determine if students have a sense of numeracy.  Are they still counting on fingers?  Are they unable to perform basic calculations without using a calculator?  Do they lack a sense of what operations mean?  You will need to assess your students to find what they know and don’t know. 

In the first day of my summer math camps for adolescents, we do not use pencil and paper.  There are no worksheets, no paper homework, no quizzes.  Students begin by solving problems with manipulatives.  I use the craft sticks, base ten blocks, fraction manipulatives, pattern blocks…anything that can model basic operations.  I like to begin with subtraction with regrouping using the craft sticks and a place value mat.  Sometimes, depending on the age of the students, I include regrouping from the whole to the part so they would need to break the stick to solve for a fractional amount.  They may work independently or in pairs.  In every case, the students go home and say that this is not like any math class they have attended.  They like the nature of the hands on work. 

I usually quickly find that many students do not understand what they have been doing on paper.  They “borrow” and “carry,”  “flip and multiply,” create lattices or divide with partial quotients, but they have poor understanding of what they are doing on paper because it was taught only at the abstract level with few or no quantity representations.  Taking them back to the concrete level is enlightening for many of them.  It is also fun.

Once we establish these quantity representations at the concrete and pictorial level, we can quickly make the links to what they have been doing in the abstract.  Some students will progress in a matter of a day or so.  Others will need more time. 

  

The second key to reaching the students beyond the concrete representations is using explicit language and "friendly" or easy numbers.  Whenever you teach something new, make your language as precise as you can and use easily accessible numbers so the students can focus on the meaning behind the math.  If they must struggle to retrieve number facts in the middle of some sequential algorithm, you have lost them before you start.  Using easy numbers for instruction does not harm any student and it leaves the teacher the possibility of differentiating and adding rigor for those students who need it.  Using “friendly” numbers for instruction and differentiated numbers for practice and applications allows all boats to rise when the curriculum tide comes in.

Thus, one strategy is to present numbers students will use but early in the lesson.  The warm up is perfect.  Lead students to fill in PART of a times table chart rather than giving them one already filled out.  They create their own near point reference.  I like to use the seven times table and the perfect squares. Then USE those numbers in class for all work that day and for several days so that the facts become friendlier and friendlier.  With the seven times table and the perfect squares, you can teach multiplication, division, simplifying fractions, mental math, the distributive property, multi-digit arithmetic and simplifying radicals.  Food for thought.

Finally, I will add that one common deficit among older students is in numeracy- composition and decomposition patterns- for the digits from six through nine.  Early on, students are taught to "count on" to add and to "count back" to subtract.  They are not given sufficient practice in these numeracy patterns and therefore have great difficulty in addition and especially subtraction across a ten.  Provide small incremental practice with these patterns in warm-ups or brief practice sets.  Use no more than one or two patterns at a time and use the patterns across place value.  If you subtract for example 13-7, decompose the 7 as 3 and 4.  Lead students to subtract 3 to get to 10 and then subtract the 4.  Then use the same pattern to subtract 7 from 43, 63, 93 and then to subtract 17 from 83.  Demonstrate math reasoning but with the same pattern in different contexts.  With some students you could even take the pattern to 3-7=-4.