had developed an understanding of the area available in the empty spaces inside the puzzles. He was able to give up using the shapes he wanted to fit for those he knew would fit. He had constructed spatial and geometric knowledge! This was a great leap in Isaiah’ s spatial reasoning. Upon completion, he smiled as his knees bounced up and down under the table, and he gently patted his finished puzzle.
Later that day, Isaiah chose a large, difficult puzzle. Figure 3, on page 84, is a detailed set of drawings documenting each block placement Isaiah made to fill the puzzle. He took 2 minutes 41 seconds to work, made 35 block placements, and filled the puzzle with 22 blocks. Four times he brought a block close to the puzzle and rejected it without attempting to insert it. With two exceptions, every block he inserted stayed and fit without any gaps as he filled the entire puzzle with confidence.
Isaiah’ s development happened over time and through independent exploration. When the same challenges are given to other children, it may take some longer than others; however, most of the children I have worked with and studied appear to construct spatial and geometric reasoning as they explore pattern blocks and puzzles.
Early geometry research suggests that young children look only at whole shapes( Van Hiele 1986). However, it appears that using geometric shapes to fill a confined space dictated by the angles of its perimeter forces children to analyze the different sizes of angles and different area sizes and shapes. It is the challenge of these puzzles that inspires interest; after all, if a child has nothing to figure out, what will they learn? Joy and confidence come from figuring out solutions.
The Teacher’ s Role
Educators play an important role in creating and supporting children’ s spatial and geometric reasoning as they work with pattern blocks and pattern block puzzles. As they encourage children’ s construction of shapes and spatial relationships, educators can consider the following strategies:
› Experiment. When teachers experiment with pattern block puzzles before giving them to children, they will experience some of the same challenges, problems, and satisfactions as children will.
› Consider placement. Educators should place pattern blocks and puzzles on a table( or on the floor) with enough room for several children and a teacher to work alongside each other. Teachers can foster interest by sitting at the table and experimenting themselves. Children will become curious and join the activity.
› Offer variety. Providing puzzles with various levels of difficulty will encourage children’ s autonomy and agency— even on the first day of the activity. If a puzzle is too difficult, children can choose another. If a puzzle poses no challenge, they can search for a“ trickier” one.( The word tricky is more exciting and less scary than the word difficult.) Educators can also encourage children to invent designs that they can convert to puzzles.
› Encourage reflection. When children appear to be stuck, teachers can use reflective questions to offer hints. For example, they can try to change a child’ s focus by pointing to a different space in the puzzle and asking,“ I wonder what might fit there?” Other possible questions are“ Can you find a block that will fill that whole corner?”;“ What can you change, so you can get rid of those little spaces?”;“ What might fit into that really little space?”( This question is particularly effective because it helps children notice impossible spaces.) It is important to give a cue or hint but not take over during child-directed explorations.
› Observe. Educators can use the information they gather as they watch children work to create learning experiences and materials that promote spatial reasoning and other areas of geometry.
› Think out loud. Educators can make thinking visible through comments such as“ I wonder if I should try a square here? Maybe that would work.”;“ Hmm, I think I’ ll try this skinny rhombus in that little angle.”;“ I am stuck. I think I will take all the pieces out and start all over.” They might even say,“ I’ m having trouble. Can you help me?” Sometimes, a child will take over.
› Offer encouragement. When supported to solve their own problems, children are likely to try out their ideas, make errors, observe the results, and try again. If children ignore questions or comments,
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