Forming a Healthy Self-Concept of Math

This week I learned to think about math in a healthier way. Before students can develop a healthy self-concept of Math, they must be given the opportunity to share their thoughts, beliefs, ideas, and perceptions on what math is and how they use math in their daily life. It is easy to take for granted how much mathematical reasoning goes into everyday activities.

Through open discussion and social context, the below mini-lesson from Leatham, K. (2010), helps students to recognize different patterns and to re-examine their definition of math. Simple everyday language is mathematically categorized and discussed. Examples: driving a car, quilting, playing the drums, hanging  a picture on the wall. In a second group lesson today, I was introduced a similar theme when I was asked to use a variety of atypical measuring tools (paper clip, post-it, poker chip), to measure a specific rectangular shape.

After plotting the length and width data points on an x, y graph, I came to realize that the slope was constant, forming a linear line. y = mx + b.  I found this to be surprising, as all data points were in different units of measurement. The implications for classroom practice in this 2nd exercise revealed a new way of seeing the x, y data when applying the concept of slope. Connecting the relationship to the content through experience, is a healthier way to see mathematics than memorizing and plugging in a formula without context.

I am reminded to continue asking what do the students know? What do I want them to know? How do I get them there? Did they get there and where do we go next?

The Largest Container

A vod presentation by Annenberg Media, Learner.org, by Suzanne Duncan a 7th and 8th grade class discusses length, width, and height of volume of a rectangular prism and area in a container.  shows a lesson “The Largest Container.”

Moving her arms in the air, Ms. Duncan defines the volume as a cylinder as a circle moving through space (height). Maximizing the idea of surface area and volume she asked the students to design the largest area possible out of a sheet of paper. What elements come to mind? Volume. Surface area. Once students had a possible solution they were encouraged to flatten out their shape, and write out the dimensions, and their computation of surface area and volume (so their thinking doesn’t get lost).

Causes for intervention in helping students correct for computational errors: 1) running out of time in the lesson, (stop where you are and move to the next phase), 2) adjusting to individual needs and learning dispositions. 3) redirect after students have had the opportunity to explore and discover on their own, preferably through small group engagement and whole class sharing.

As the topic of volume and formulas is too advanced for a 2/3 split classroom, I can restructure this lesson for my students by modeling and constructing a cube and a rectangle from a blank sheet of paper. An essential question may offer, “How many sides are in a cube?” Can you prove it? Can you build a cube? Sharing the many different possibilities in a class poster can encourage discussion and knowledge transfer. I can introduce vocabulary of length, width and height and touch upon concept of surface area.

Ms. Duncan holds a strong belief that mathematics is for all students. Moving from teaching a select few, who either understood the concepts or didn’t, to reaching and teaching all students, gave Ms. Duncan a sense of pride and accomplishment. Re-charging her batteries at the end of a 15 year career, she believes today that all students are naturally gifted learners. According to Gardner (1993), student intelligences can be categorized as follows: Verbal-linguistic (word smart), Visual-spatial (picture smart), Musical-rhythmic (music smart), Body-kinesthetic (body smart),  Interpersonal (people smart), Intrapersonal (self smart), Naturalist (nature smart). Considering the multiplicities of how all people learn, an important goal of education is to apply knowledge outside of school in the social and cultural settings of our greater life.

How can we make math exciting?

Ted Talk - Adora Svitak

Thinking in 3-D

A vod presentation by Annenberg Media, Learner.org,  shows a lesson called “Building Viewpoints.” Seventh-graders learn about spatial sense and geometry from a blueprint of 
ancient buildings. They then create their own three-dimensional models and draw them from different viewpoints.

One questioning strategy Ms. Hardaway uses is to follow the students’ lead and ask a probing questioning to build on the story being shared. “Can anyone tell me what this is a picture of?” After one student commented that they are blueprints, she continued along the same path of questioning, “What are blueprints used for?” This formed a natural segue to the objective to build and draw a 3-D model from the left, right, front and back. A second questioning strategy helped students to consider how their views were alike or different from each other. A comparative-based question such as comparing all four views falls within the analysis stage of Bloom’s Taxonomy of learning.

The content of this activity is important in middle grade because it builds on a visual and special sense of geometric problem solving. Students were actively engaged in sculpting 3-D figures and translating this information into 2-D drawings. Leading up to an activity that will involve analyzing a 3-D model and transferring their new knowledge into 3-D drawings. This newly acquired prior knowledge will be useful to apply to future projects.


When constructing 3-D buildings from manipulatives, tactile learners are engaged as they move it, feel it, turn it around and flip it upside down. Through small group sharing, I would enrich this activity with writing a conclusion paragraph to restate the main idea, include supporting details, and build on higher-order questioning. Bloom’s Taxonomy continues to open minds by moving into a stage of synthesis. At this stage, students may contrast, categorize and discriminate the various views observed. Evaluation can encourage students to reconstruct, reorganize, summarize and validate their ideas as they explore their understanding along an investigative path.

Cluless about Tactile Clues?

This week I learned that alternate interior angles (from a bisected parallel line) are equal to each other. I was lead to draw this conclusion by the use of a simple paper manipulative. Folding the points of a square in on themselves to form another square, and then folding in half to create parallel lines and perpendicular bisectors. I was asked to prove what I know through open class conversation. According to Marshall (2000), “Saying out loud what one has just learned is an excellent reflective strategy to improve learning.” Whole class discussion solidified this Math manipulative experience.

There is no substitute for direct experience. Yet, when might a mathematical manipulative not be appropriate as a teaching tool?

According to Freerweiss, when we use manipulatives we can reach up to 10% more students who naturally gravitate toward kinesthetic and tactile learning. Indeed, anytime I incorporate a wider range of teaching with the senses I routinely see greater engagement. In addition, manipulatives lower anxiety in a historically anxiety rich content area that is frequently plagued with memorizing facts and formulas.

Learning through experience first, and teaching for conceptual understanding can help students to solve problems they have never seen before. To successfully take a mathematical formula out of a text book, requires a teaching strategy that incorporates real world context. If I introduce a tactile experience along the way, and connect this to prior learning, I can help students’ understanding take hold with permanence.

Lasting Impressions with iTouch apps

My impressions of using iTouch applications in the classroom as a learning tool are evolving with hit or miss results. I continue to experiment with hope and optimism – this week with the free ap Story Builder. Of note, this is not to be confused with the Mobile Education store ap by the same name. This week, after completing a reading conference with a 4th grade student, I gave him the opportunity to build his literacy in a different manner. One example: “A drunken jewel thief with OCD Must save a city from a swarm of killer bees.”

A word of CAUTION: Sincere discretion should be used with respect to grade level vocabulary. Word choice such as: cross-dresser, and gigolo may be contrasted with options to stop global warming and gain the respect of his/her peers.

On a positive note, changing the genre feature in this micro-ap. to an uplifting documentary can deliver, “A superhero werewolf stranded on an island must go back to grade school to save a secret world.” Toggling between genres can offer surprising results. I would prefer to see a more extensive G-rated vocabulary list. Given the sophistication of the genres and word choice options, this ap is better suited for a young adult audience. My point…be mindful of the audience and resources when connecting the student to literacy at his/her just right level.

From Assessment to Instruction

What have you learned about your buddy’s needs, abilities and interests?

Alan is interested in tetherball, soccer and ping-pong. His scientific mind embraces future potential writing topics of liquid, solid and gas, water vapor, salinity, and the life cycle of insects. He willingly embraced new reading content of electricty and magnetism while showing joyful expression and a sense of humor in his writing.

Alan is a curious, self-motivated reader and a pleasure to collaborate with. Scoring at a 65-70 percentile with his oral reading fluency, Alan is on target for where he should be during the beginning of the winter trimester. Alan performed with 97% accuracy on the fluency scale at 4th grade level for words correct per minute. Strong suits: reading motivation, fluency and decoding for meaning.

Occasionally, Alan misses critical details of the story. While Alan reads at a steady pace with efficiency and three-four word phrase groups, there is room for improvement in comprehension. Alan should continue to read 4th level material while focusing on gaining strategies for comprehension. He can benefit from reading more carefully and asking questions along the way to more fully capture details of the main idea. 


As a result of this knowledge, what learning objectives and materials are you considering using for your lesson?

A reading lesson for Alan will focus on teaching a strategy for recognizing main ideas with the purpose of summarizing a challenging work of non-fiction. In addition, scoring well into a 5th grade frustration level, shows that Alan can move beyond common word use to work with new vocabulary and more challenging text.  Importantly, laying the groundwork of prior knowledge within the context of interdisciplinary instruction (his intrinsic motivation is Science) can guide toward greater engagement, ownership and comprehension.

The process for this lesson includes creating an “I wonder” poem to preload for conceptual understanding of doing what good readers do — asking questions. Modeling, I will read a short passage, and think aloud as I ask predictive questions. Following this exercise, together we will read a grade level text while placing sticky notes and writing Alan’s questions in the margins. Materials: journal notebook for poem, pencil, grade level text (2 copies), question prompts on cards, sticky notes.