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
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Summer Blog Series Post #4: The Role of Attention and Temporal-Sequential Ordering in Time Management

When in school, students are expected to follow routines and complete assignments within certain time frames. Children must follow these same guidelines when continuing the learning process at home, managing their time and effort to complete homework assignments and projects on time.  

Time management is critical to many of the expectations placed on students, including initiating assignments, taking the appropriate amount of time to complete tasks, meeting deadlines, and maintaining a busy schedule. 

Neurodevelopmental factors: 

Time management involves several neurodevelopmental functions, including attention and temporal-sequential ordering.  

Getting started on assignments requires students to engage their attention. Students must be alert to the task at hand, possibly shifting focus to a new activity, and have the mental effort necessary to initiate the task.  The ability to preview, or think about the outcomes of a task before beginning, can help students conceptualize what a report will be like once a topic is selected, what materials will be necessary to do an assignment, etc.  Previewing is an aspect of attention. 

Taking the appropriate amount of time for a task involves both temporal-sequential ordering and attention.  Temporal-sequential abilities help us understand the order of steps, events, or other sequences; generate products in a meaningful order; and organize work, time, and schedules.  These skills are related to a student’s ability to appreciate time in general and estimate time appropriately. 

Tempo control, a facet of attention, helps students allocate the appropriate amount of time to the task at hand and predict the time required for an upcoming task. Tempo control also instills a sense of “step-wisdom,” the knowledge that it is more effective to undertake activities in a series of steps, rather than all at once. Tempo control allows a student to match his/her pace to the demands of a given task, e.g., to take the right amount of time to finish an essay test, to do a homework assignment thoroughly yet efficiently, etc. 

Here are some possible signs that a student is competent in time management:

 The student …

  • Is able to get started on homework assignments, reports, or projects on his own
  • Takes an appropriate amount of time to complete the task at hand, such as doing a homework assignment or studying for a test
  • Is able to meet deadlines related to schoolwork and follow established schedules
  • Comprehends time-related vocabulary (e.g., first, last, when, before, after, next)

Here are some possible signs that a student is struggling with time management: 

The student …

  • Either rushes through work, not taking the time necessary for a thorough job, or takes an excessive amount of time to complete a task/assignment
  • Has difficulty meeting deadlines and/or following an established schedule
  • Is often tardy, frequently not realizing when he or she is running behind
  • Has trouble with long-term assignments

Strategies to help students struggling with time management: 

  • To help students get started on an assignment, encourage them to start a homework session or study period by planning what will be accomplished during the session. If necessary, help students develop objectives that are clear, specific, and measurable (e.g., how long they will work, how long the report will be, how many problems they will do, etc.).
  • To help students understand the appropriate amount of time to allot to tasks, require students to plan for a designated number of minutes, work for a designated number of minutes, review for a designated number of minutes, etc.
  • Have students practice estimating and managing their time. For example, have students keep track of activities in a log, first recording the estimated time they think the activity will take, and then documenting the actual time it took to complete the activity.
  • Create a large classroom wall calendar that shows an outline of the stages and time frame for completing long-term projects. Note important steps and dates with color cues. Review the calendar regularly.
  • Allow students to practice managing time by being a “project manager” when working in cooperative groups, making sure activities lead to products on schedule.

We’d love to hear what strategies or activities you’ve used to help promote effective time management in your classroom or at home.  Leave a comment below with your ideas! 

Related links:

Learn more about our summer series

  1. More strategies on getting started on assignments
  2. More strategies on taking the appropriate amount of time for tasks
  3. More strategies on meeting deadlines and keeping schedules
  4. Related research on temporal organization

 

Summer Blog Series Post #3: Higher Order Thinking, Creativity, and Brainstorming

The ability to come up with ideas, to elaborate, and to think about objects or topics in a new way all involve what we refer to as “creativity.” At All Kinds of Minds, we believe that students should be encouraged – both at home and in school – to find areas in which they can discover forms of creative output that are meaningful to them. Providing activities in which students engage in brainstorming and creative thinking may help to uncover unrealized strengths in a struggling student and to provide a successful form of expression for a student in need of recognition.

Neurodevelopmental factors:

Creativity and brainstorming are two aspects of higher order cognition. Brainstorming involves the generation of original ideas or perspectives. Creativity involves the process of thinking in a new or innovative way. Brainstorming and thinking creatively are important components of our ability to generate ideas.

Students who are skilled in brainstorming and thinking creatively will find these abilities beneficial to many other endeavors in school including problem solving, decision-making, and understanding concepts.

Here are some possible signs that a student is competent in the areas of creativity and brainstorming:

The student …

  • Comes up with his/her own ideas during activities
  • Writes imaginative stories or draws original cartoons
  • Takes risks and is willing to get out on the fringes
  • Finds new or unique ways of solving problems

Here are some possible signs that a student is struggling in the areas of creativity and brainstorming:

The student …

  • Has difficulty generating new ideas
  • Wants to be told what to do
  • Has trouble choosing topics or using imagination in class
  • Is unwilling to engage in active thought
  • Has difficulty in art, music, or dance classes

Strategies to help students struggling in these areas:

  • Help students generate ideas necessary for doing an assignment, such as providing prompts to help in the selection of a topic or help them get started on a brainstorm. Providing a few cues or prompts may give such students the initial support, or scaffolding, they need to succeed in the assignment. For example:
    • Provide the first sentence of a paragraph.
    • Start one or more math problems.
    • Read the first paragraph of text.
    • Have cue cards handy, for example listing the steps of writing a paragraph, etc.
  • Incorporate guided higher order thinking activities in order to promote students’ creativity, brainstorming, and critical thinking. For example, an English teacher might ask, “Why do you think E. B. White called his book Charlotte’s Web instead of Wilbur or Zuckerman’s Farm?” while a Social Studies or History teacher might ask, “In social studies, “How might America’s history have been changed if Lincoln had not been assassinated?”
  • Develop activities that promote students’ ability to think ahead, or predict possible outcomes. For example, implement collaborative activities in which students start with the same beginning and work in teams to predict outcomes, or all students start with the same outcome and work in teams to determine what led to the outcome, etc.

We’d love to hear what strategies or activities you’ve used to help promote brainstorming and creativity in your classroom or at home.  Leave a comment below with your ideas!

Related links:

Learn more about our summer series

  1. More strategies around brainstorming and creativity
  2. General information about the neurodevelopmental functions related to creativity
  3. Related research on creativity
  4. Sir Ken Robinson’s TED talk on the importance of fostering creativity in schools

Summer Blog Series Post #2: The Role of Memory in Reading Comprehension

Reading comprehension is one of the most complex academic skills. Skilled readers construct meaning by synchronizing a bottom-up approach to reading (decoding words fluently and accurately) with a top-down approach (using prior knowledge and experience during reading).

Neurodevelopmental factors:

Reading comprehension involves a variety of neurodevelopmental functions, including attention, memory, language, and higher order cognition. In this post, we’re going to focus on the role of memory.

Memory:

While reading, we must hold important information and concepts in our minds. We must process words, sentences and paragraphs together in order to gain full meaning of what we’re reading. In addition, we must call up relevant information we already know. Memory is essential in helping us comprehend as we read, make associations between prior knowledge and new information, and remember that same information at a later time, such as during a test.

Here are some possible signs that a student is succeeding with the memory demands of reading:

The student …

  • Is able to pick out main ideas
  • Paraphrases/summarizes well
  • Holds onto the beginning of a story while reading the end
  • Keeps in mind the plot of a story while working on a single part of a paragraph
  • Easily learns new vocabulary words and definitions

Here are some possible signs that a student is struggling with the memory demands of reading:

The student …

  • Feels overwhelmed by the number of ideas presented
  • Retains only fragments of what was read
  • Can restate the gist of ideas, concepts, or directions, but not the details
  • Loses the meaning of a passage when looking up the definition of an unknown word

Strategies to help students struggling in this area:

  • Have students read in pairs, alternating between passages and then switching parts to re-read the text.
  • Have students take quick notes that describe the main idea of what they are reading. For example, have students stop to summarize what they’ve read after each paragraph. This approach will help ensure that students are recording important information in their minds.
  • Stress self-monitoring of comprehension while reading, by encouraging students to ask themselves: “Is this passage about what I thought it was going to be about?” “Have I linked what I just read to the parts I read earlier?,” etc.
  • Teach students how to create useful notes that reinforce understanding and help to trigger information recall at a later time. For example, teach students how to create concept maps based on their reading, as one technique for consolidating and organizing what they’ve read. Have students save their maps, and use them as study tools for upcoming tests.

We’d love to hear what strategies you’ve used to help students struggling with the memory demands of reading.  Leave a comment below with your ideas!

Related links:

Learn more about our summer series

  1. More information and strategies about the role of memory in reading comprehension (See “Tips to Help” links for more strategies!)
  2. General information about the neurodevelopmental demands of reading
  3. Research on active working memory and reading
  4. Reading resources on the web