What’s Up with Kate? (Part 1)

Last week’s blog was our last installment of our book-inspired series. We received quite a few thoughtful and inspiring comments, and we gave away five free copies of Schools for All Kinds of Minds! We hope you enjoyed the sneak peeks into some of the ideas in the book, and we hope the series inspired you to pick up a copy if you hadn’t done so already.

Up Next …

This week we’re trying something a little different – a case study of Kate, a 6th grader with a puzzling array of learning challenges. Read Kate’s story and let us know what you think is going on with her and how you’d approach her challenges. Then, tune in next week for our explanation and recommendations!

Nothing’s Easy for Kate

Kate, a popular 6th grader, earns good grades and participates regularly in class. But Kate always has to work really hard to succeed. Nothing seems to come easy, but once Kate knows something, she appears to know it well and apply it effectively.

Occasionally, Kate’s dad helps her with her homework and studying – but by both accounts, these sessions are painstaking and don’t seem very productive. Kate can go over a list of spelling or vocabulary words repeatedly for more than an hour yet retain only a few of the items. The same goes for reading – she can read a passage easily but remembers only bits and pieces.

What Kate’s Teacher Sees

Kate’s teacher is puzzled by Kate’s constellation of challenges in the classroom. She’s noticed that Kate often needs to have explanations repeated and that she has a lot trouble complying with multi-step instructions of any type. It also takes Kate a long time to copy from the board; her classmates finish when she is barely halfway there!

Kate’s teacher has also observed that Kate does much better in day-to-day class work than she does on tests.

Reading and Math: A Mixed Bag

In the last year, reading has started to be a problem for Kate, especially in social studies and science. She has a particularly hard time summarizing what she’s read, despite her general ability to express herself well verbally.

While Kate is good at understanding math concepts, it’s been hard for her to master math facts, so she needs more time to complete math assignments and quizzes.

What’s Going Right

Kate seems to have a knack for graphic design. She looks forward to her computer class and has talked about being an architect one day. She loves animals and has a very special fondness for cats and has written several very perceptive reports about cats.

What do you think?

What areas are strengths for Kate? Weaknesses? How could you leverage Kate’s strengths to help her improve in other areas? What would you say to Kate?

Share your ideas with us, and next week, we’ll share our thoughts about Kate with you!

Summer Blog Series Post #5: The Role of Spatial Ordering in Understanding Math Symbols

The results of our recent poll are in!  You, our readers, expressed a strong interest in hearing about learning challenges related to math … so in response, this week’s blog is about the spatial ordering demands involved in understanding math symbols. Thank you to everyone who participated in our poll.  We love the feedback.

In developing an understanding of mathematical concepts, students must engage their nonverbal thinking skills. Nonverbal thinking involves the use of spatial and visual processes to learn or think about a problem or concept.

One mathematical concept that involves nonverbal thinking is the use of symbols, such as numbers. The number 6, for example, is a symbol that represents a quantity. Another common math symbol is “=”, often referred to as an “equals sign,” that represents the concept that quantities on each side of the symbol are the same, or equal (e.g., 3+3 is the same as 6).  Students use and manipulate symbols when doing operations ranging from basic addition to algebraic equations.

Understanding and using math symbols taps into a student’s higher order cognition and spatial ordering abilities.  In this post, we’re going to focus on the role of spatial ordering

Neurodevelopmental factors:

Nonverbal thinking involves visual or spatial representations of math processes and relationships. Students must be able to interpret visual and spatial information (as when looking at a graph or geometric shape), and to form and understand visual and spatial concepts (as when interpreting information from a graph or describing attributes of shapes).

Some concepts lend themselves to “visualization,” creating a mental image to represent a mathematical relationship. The concept of proportion is a good example. A student may have a difficult time interpreting proportion through words and verbal explanation, but being able to visualize the relationship (e.g., the number of boys to girls in the class, the ratio of eaten slices in a pizza) may greatly enhance his/her understanding of proportion as a concept.

Here are some possible signs that a student is succeeding with the spatial ordering demands of math:

The student …

  • understands mathematical symbols and can visualize patterns, math concepts, and the parts of a problem in his/her head
  • uses visual analogies successfully (e.g., determines how two symbols relate and applies that understanding to link other symbols)
  • quickly learns new science and math concepts (e.g., place value, perimeter, equations, resistance in a wire)

Here are some possible signs that a student is struggling with the spatial ordering demands of math:

The student …

  • has trouble associating math symbols with the concepts they represent
  • is unable to recognize the systematic organization of charts, diagrams, tables, or maps
  • is slow to master the alphabet and numbers because of difficulty recognizing symbols
  • has trouble forming concepts and solving problems without substantial use of language

Strategies to help students struggling with understanding and using mathematical symbols:

  • Integrate hands-on activities and verbal explanations into the learning of spatially based concepts. For example, have students use pattern blocks to make geometric shapes, then discuss and write down the characteristics of the shapes, such as number of sides, types of angles, etc.
  • Use examples of familiar situations, or analogies, to talk and think about math concepts. This helps students link the concepts to a visual image. For example, the concept of ratio may be illustrated by asking students to imagine two brothers sharing a pizza, and the amount of pizza left over after the big brother takes his portion.
  • Guide students in visualizing patterns. For example, talk students through ‘seeing’ a geometric shape in their minds, “picturing” a math process taking place, such as 1/3 of a pizza being taken away, and 2/3 of the pizza remaining, etc.

We’d love to hear what strategies or activities you’ve used to help promote understanding of math symbols in your classroom.  Leave a comment below with your ideas!

Related links:

Learn more about our summer series

  1. More information and strategies on understanding math concepts
  2. Related research on spatial ordering (check out the section on Higher Spatial Thinking)
  3. All Kinds of Minds’ “Thinking Mathematically” podcast
  4. Mathematics section of the All Kinds of Minds Parent Toolkit
  5. Interactive spatial ordering activity