Post # 31 Explicit Instruction in Drawing Solutions

One Added note:  The What Works Practice Guide for Problem Solving was updated as of October of 2018.  It advocates just this approach.  

“You are not Michelangelo and this is not the Sistine Chapel,” I said to a student one day.  He was following his inclination to put more effort into his drawing than into the math reasoning behind it.  He had to draw yet another of those pencil boxes or pouches for a quantity of pencils for a word problem.  I explained the reference and then we discussed the way that pictures can be a way for us to solve problems visually in math, or at least to help us support our reasoning.

In math today we are encouraging young learners to use multiple representations to solve problems in math.  First, it encourages use of multiple regions of the bran to unify thinking across sensory areas.  The more troops you bring to the building, the faster the building will go up. Students can sometimes get bogged down in the art and invest more in the fringe to the pencil pouches than into the numbers of pencils in the group.  

Like the use of manipulatives, the teacher needs to streamline the learning for maximum effect.  We need to encourage efficiency, focus and directed effort.  The goal is the get the child to fluent calculations at the abstract level but those which are supported by sound mathematical thinking.  Visualization of a problem can absolutely support that “reasoning and sense making” as the NCTM calls it.  But we as teachers may need to provide guided practice in the efficient drawing of shapes to solve problems.  

I recall an instance of solving division with an elementary school class.  They were dividing 42 by 6.   It is true that one might begin in division with the dividend and draw 42 objects and begin the division from there.  However, for efficiency’s sake we could also reason that in this case the story problem was asking for groups of 6.  Beginning with the smaller number and making patterns of six, in this case either tally marks + 1 or the dice pattern of six, made absolute sense.  

We were working in groups, some students with me and some with the classroom teacher.  Her small group began by drawing 42 straight lines.  My group began by reasoning out the problem, deciding on groups of 6 and discussing the best …ie, most efficient method of drawing that solution. The teacher’s group arrived at a solution with remainders.  Not a single child achieved an accurate answer independently.  The children in my group all working in pairs, achieved an accurate response.  

How could this happen?  Deficits in attention to detail, fine motor issues, impassivity, and awkward handwriting grip all created problems for her group.  The lines were drawn too haphazardly, close together and at random widths apart that circling groups of 6 was impossible to do with accuracy.  The children could not assess the accuracy of their solutions with clear results.  In addition, we could recall the research on numeracy and invest in making groups of quantities we could recognize by pattern…groupings of smaller quantities which are visually recognizable.  Remember the 9 rhombi in your first power point presentation?  The domino pattern of 9 was instantly recognizable whereas anything over 4 items in a line or not in a pattern must be counted.  

The lesson here is that children need explicit instruction in efficient diagram drawing methods:  dots, tally marks, sets of dots within a circle, boxes, x’s, squiggles…anything that is relatively easy and efficient to draw and group.  This is true even at the middle and high school level when calculating surface area.  It helps to draw the shapes for which one must calculate area and determine how many of them are a part of the shape.  

Venn diagrams are easy for illustrating the two meanings of division.  I created an elaborate one for my professional development seminars but simple dots and circle work especially well with students.  In any case, we need to encourage, supervise and praise students as they think mathematically, reason through story problems and represent their thinking in a multitude of ways.  This includes efficient drawing to support their reasoning.  

Here is an example of a child's work for solving a multiplication with fractions problem.  Note the wording:  Five "groups" of the quantity 3/4.  The problem would look differently if it were 3/4 of five.  Note that in this representation he can see the total number of fourths as well as the mixed number result. 

And another based on his work with fraction division.  In my field we say that one never gives up teaching private students because they teach us as much as we teach them.

One final representation, allows student to define terms with pictures.  Think: Visual Dictionary.  In this file, teachers can create a student worksheet, warm up activity, and exam study sheet all in one file.