Interpreting Standards as Sense-Making Opportunities

Interpreting Standards as Sense-Making Opportunities

Sue Pawula

Summary

Algorithms created by students can be one tool to use when teaching them to make sense of math. In this article, this approach is initiated through in lessons targeted at teaching students: 

“Understand subtraction of rational numbers as adding the additive inverse, p-q=p+(-q). Show the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. (7.NS.c)

Instead of teaching students the “keep-change-change” method, it is suggested that teachers should model the procedure and then ask students to work independently.  The old method could hinder student sense-making of subtraction with rational numbers or connections to related subtraction ideas.  When students can think about subtraction in terms of distance or vector length they are provided with chances to connect their actions in a visual space to operations on numbers.  This also should allow the teacher to elicit student reasoning that will indicate their connections and meaning that they assign to mathematical procedures.  In this instance it allows students to develop their own algorithms for rational number addition and subtraction. This takes more time and requires teacher commitment to content coverage in order to arrive at the student goal.  However, it requires students to have conversations about the patterns they see related to adding and subtracting rational numbers in a variety of different problem contexts.  The teacher will observe that this activates students to form conjectures about the patterns they have noticed and discussions will allow students to express these conjectures to their peers.  During the discussions, students will need to use precise mathematical language, establish standards of evidence, and focus on patterns that will be useful when they later develop algorithms.  The teacher will also notice that some of their conjectures are ambiguous or incorrect because some students are unable to distinguish between relative and absolute values of the numbers.  Teachers can use this opportunity to formally define absolute value.  Students can acquire deeper understanding of operations on rational numbers, by proceeding through these steps and working as a group to generate a collective algorithm, which they decide, is accurate, sensible or efficient.  Following this student should justify the accuracy of the algorithms in different ways such as number lines, chipboard models, or a few examples that the algorithm works. The student understanding developed in this manner is important since it allows students to flexibly experiment and use rough-draft mathematical language to convey their final outcome. 

Reflection

Although this was a time consuming activity, the outcome allowed the students to build flexibility in thinking.  It gives them control of the learning process and allowes the teacher to facilitate while allowing the students to experiment and reflect, discuss, and discover different ways to reach the goal.  This kind of learning is very important to expansion of individual cognitive development.  It also encourages students to work in teams and share information, rationalize, and debate the different ideas, and come to provable conclusions.  Teachers need to plan how they would implement this type of student learning, being mindful of the time constraints.  Even so, this is an excellent way to move student thinking onward and upward and let them work on projects, individually or in teams, the same as they would have to in the real world

Choppin, J. M., Callard, C. H., & Kruger, J. S. (2014). Interpreting standards as sense-making opportunities. Mathematics Teaching in the Middle School, 20(1), 24-29.

A Tool for Rethinking Questioning

A Tool for Rethinking Questioning

Sue Pawula

Summary

NCTM wants teachers to create classrooms where mathematical discourse elicits students’ thinking and reasoning to form viable arguments and judge the reasoning of their peers.  A tool to assist in this is the Cognitive Rigor Matrix exhibited and explained in this article.  The CRM should help math teachers analyze and reflect on the type of questions they pose to students.  The CRM is designed to promote and develop higher-order thinking and inquiry in students through use of purposeful questioning and mathematical task design.

This matrix is composed of a two-dimensional structure that incorporated the cognitive dimension of the revised Bloom’s Taxonomy on one axis and Webb’s Depth of Knowledge (DOK) on the other axis.  The Bloom’s levels are rated in difficulty from the basic to the higher level as: remember, understand, apply, analyze, evaluate, and create.  The DOK’s levels are rated in difficulty from the basic to the higher level as: recall and reproduction, basic skills and concepts, strategic thinking and reasoning, and extended thinking.  The authors suggest that the process of question placement within the matrix is a practice that mathematics teachers should deliberate and think deeply about, in so far as their intent and implementation of the questioning practices.  They further suggest that to be effective, it is advisable that teachers work with a colleague, math team, or instructional coach in order to facilitate the process before proceeding individually.  Further suggestions were: 1) peer observations or video recoding lesson with questioning activities, 2) writing out questions, before teaching, that target a variety of cells in the matrix and having that matrix on hand during the lesson, 3) role-playing with colleagues different scenarios involving this type of questioning, and 4) providing feedback to student responses and solutions that apply to the process and not the product or person.  This two-dimensional matrix incorporates a range of depth to help view and focus question design to improve and engage students to think rigorously mathematically.

Reflection

This was a very thought provoking article about how to elevate questioning in the classroom.  The matrix is an awesome tool that will help teachers become more adept in creating higher-order questioning in their classrooms.  The levels on it allow the teacher to increase and elevate students thinking levels as they see the need and the competence in their students.  The CRM not only targets student growth but also facilitates teacher growth since teachers will increase their thinking and knowledge of their students as they proceed through the question creation process.  This is a very important article and needs to be shared not only in the math community but also across the educational spectrum.  The suggestions by the authors for teacher interaction would make excellent professional development interactions in our schools.  I can’t wait to use it and share it with my colleagues.

Simpson, A., Mokalled, S., Elllenburg, L. A., & Che, S. M. (2014/2015).  A tool for rethinking questioning. Mathematics Teaching in the middle school, 20(5), 294-301.

Assessing for Learning

Assessing for Learning

Sue Pawula

Summary

This is a report on research-based strategies using levels of math cognition to create and improve assessment design and practice in the classroom.  The strategies were applied to warm-up formative assessments, informal questioning, classroom discussions, practice problem sets, evaluation and feedback, and some summative assessments, such as unit tests, projects, and quizzes.  The design of these assessments was based on Cangelosi’s seven cognitive learning levels: construct a concept, discover a relationship, simple knowledge, comprehension and communication, algorithmic skill, application, and creative thinking.   These are designed with consideration for typical mathematical thinking and are ordered according to levels of perceived learning progression.  Students usually construct a concept and discover relationships before proceeding to simple knowledge and algorithmic skills.  As students progress in understanding they must also be able to express this knowledge through formal notation and vocabulary as their comprehension deepens.  Instructional integration of comprehension and communication learning will prepare students for the deductive thinking required for application and allow them to acquire the knowledge to become creative mathematical problem solvers.  Creating a rubric will make teacher evaluation of student progress easier and provide the student with meaningful and useful feedback.  When measuring the “construct a concept” learning level on the assessment, teacher could require students to sort or categorize since that can help display concepts and connections they have acquired. Another way to show this could be asking students to describe concept attribute as well as creating example of them or non-examples to show acquisition.  To evaluate their “discover a relationship” learning level, it is best to prompt them to recall something similar and write about it, since that will indicate that the thoughtfully doing a discovery activity is valued as much as knowing the answer.  “Simple knowledge” should be measured through a simple recall question and can be modified as they progress to become a “comprehension and communication” level item by asking them to explain the method.  “Algorithmic skill” can be evaluated through requirements to recall the steps and follow the mathematical procedures. “Application” can be evaluated through students’ demonstration in their ability in deciding how to solve problems. Evaluating “creative thinking” will require some synectic work where the teacher, through little spontaneous and playful exchanges with the students, sparks their curiosity and creativity.  It can also be assessed through open-ended application prompts.  When teachers design tests with rubrics that explicitly tell the students the expectations, they will actually learn from their errors.

Reflection

Designing meaningful assessment for students is equally as important as the daily instruction.  When they are evaluated in a format that gives them meaningful feedback, they are more inclined and able to learn from their mistakes since they will know what is expected and how to improve their level of learning.  However, we always need to individualize not only our instruction but also our evaluation process.  We cannot evaluate every student the same even when using the same rubric.  We always have to keep in mind: “What did the student know at the start?  Have they made progress?  How much progress is enough?  What level should be their next goal?”  When we keep these questions in mind then we truly will make the rubrics meaningful to each and every student. Cangelosi’s seven cognitive learning levels are great building blocks to use as we make rubrics for our students because they can work as the base from which to build these meaningful rubrics.

Kohler, B., & Alibegovic, E. (2015).  Assessing for learning. Mathematics Teaching in the Middle School, 20(7), 424-433.

Mardi Gras Math

Mardi Gras Math

Sue Pawula

Summary

Mardi Gras Math is an article about an activity created to support the eighth grade algebra curriculum use of linear equations and functions.  It also was targeted to help students develop mathematical habits of mind and demonstrate how to use mathematics in the real world.  Students watched a film about the Mardi Gras celebration and discussed some issue that arise in the planning phase. The students needed to “analyze givens, constraints, relationships, and goals” in order to make linear functions and coordinate points based on certain necessities in the project. Students job was to decide on the route the Madri Gras parade would take using linear equations and points on an x-y grid that identified streets and attractions. It is designed to develop mind habits that involve algebraic reasoning in areas of: visualizing math, using functions, performing good calculations, using algebraic representations and algorithms, mixing deduction and experiment, and breaking things into parts.  Student discussion developed along the lines of some students wanting do the linear equations and graph, while other thought that the linear equations were the most efficient and effective way.  Misconceptions surrounding the accuracy of using estimation to determine certain points were addressed and students recommended that they graph to verify their estimations.  Students started formulating algorithms for points of intersection for linear functions.  This scenario contained many issues, including mathematical as well as social, and engaged student interest because they were working on real world problems.  Students liked transforming linear equations to slope-intersect form even though it took longer.  They also commented that math was easier to understand when it was tied to something fun and seemed to have more purpose in real life.  This was a modeling activity that gave students connections to careers related to urban planning and engaged them in finding reasonable and logical solutions.  They enjoyed being in charge of the final outcomes and felt that it elevated their confidence in personal algebraic manipulation skills.

Reflection

This article shows the difference student engagement and interest in an educational task can make.  Being presented with the project of setting up the parade route seemed like a fun exercise at first glance but proved to be a little more complex than students initially thought it would be.  Through the use of linear equations, students were tasked with locating streets and attractions on an x-y grid to help the parade committee plan the parade route.  Some students tried to take the easy way out after doing some of the beginning graphing and only use estimation to get the correct answer which did not work out accurately for them.  Because the students were so involved they were willing to do the extra work necessary to complete the different problems in the project. Motivated student involvement in real life situations is the goal of classroom instruction and this was a very good example of how that can take place.  The project took life in the students eyes and because of that they worked more intensely on the math portion since it would be successful in the endeavor.  

Eubanks-Turner, C., & Hajj, N. (2015). Mardi gras math. Mathematics Teaching in the Middle School, 20(8), 492-498.

Gone Fishing: Science, Proportions, and Probability

Gone Fishing: Science, Proportions, and Probability

Sue Pawula

Summary

CCSS and NCTM promote mathematical practices leading to reasoning and connections of ideas to contexts outside of mathematics.  Students are led through an interactive experience where they record fish sample data on an activity sheet and transform it to a connection between rationalizing ratios based on the proportions they observe and connect that to different population ratios of fish in specific ponds.  Teachers should encourage them to use precise language when describing their rationale for the relationship between the proportions of fish from the sample to what the corresponding pond population possibilities it could be determined to be.  Further discussion should be encouraged in other ways that students could determine correspondence between pond population and sample sizes.  Students discover that it is difficult to determine the probability between two pond samples without additional information.  Important ideas on sampling variability, matching proportions, and how this would be dealt with in a real life situation are discussed as well as other strategies and concerns.  Students are led to understand that they can develop proportional reasoning without making calculations through understanding that the proportions can be correlated through similarity predictions.  Questioning by the teacher can lead to the basic understand of the probability continuum and lead to differentiation of other possibilities, probabilities, and the degree of probability.  Students should further understand how different proportions could affect ability to make choices based on statistical data.  The second activity they participated in was to determine fish population increase due to the research facility’s need to determined the best conditions of increasing fish population and estimation of the current fish population from samplings taken after time has passed.  Once again students use fish samples as a basis for the data collected.  This time they are asked to record only the number of the original fish sample (tagged) and the new (untagged) progeny.   From this they are required to use more advanced proportional reasoning than in the previous activity and will require extra guidance from the teacher as they work on strategies.  When sharing strategies the teacher will need to carefully allow teams with more basic strategies to share first in order for them to not feel insecure as other groups share their more advanced strategies.  Student language will use references to part-to-part and part-to-whole and they will need to be exposed to a whole new range of more advance strategies.  These motivating activities allow students to work through the steps that increase their understanding of ratios, proportions, statistical variability, and fractions.

Reflection

These were excellent, interactive mathematical exercises for students to participate in and see their relevance to the world around them.  The creation of the graphs from which to base their proportional predictions and the discussions that were provoked by them caused students to think on a higher level and share with each other.   This was a scenario for rich use of mathematical language, reasoning, and strategy.  Ratio creation helped student identify some proportional correlations easily but the other more difficult relationships caused brainstorming.  The second exercise brought this activity to a higher plain in that students really had to strive and learn new strategies to apply to be able to develop reasoning for their predictions.  This was a very interesting and really great way to get students talking.

Cochran, J. A. (2014). Gone fishing: Science, proportions, and probability. Mathematics Teaching in the Middle School, 20(1), 16-22.

Tasks to Develop Language for Ratio Relationships

Tasks to Develop Language for Ratio Relationships

Sue Pawula

Summary

Pre-service teachers build core concepts on ratio relationships by making connections through context, language, and drawings for better conceptual understanding in this article.  Two key ideas they will address with students in the classroom are: 1) “recognizing and expressing the inherent multiplicative comparisons in ratios”; and 2) “coordinating fraction language with ratio relationships.” In small groups the PSTs were asked to make comparisons in sentence form using precise language to compare two ribbons lengths.  They were asked to illustrate their sentences through diagrams, labeled units, and equations. They were prompted to produce statements that correctly compared the lengths in a reciprocal relationship. They discovered that through teaching learner to label the comparisons as additive or multiplicative, they could be referenced in two different ways of thought. Labeled diagrams particular helped language development of the multiplicative relationship when neither quantity was a whole-number multiple of the other. The tape or strip diagram model used in Asian curricula was introduced because of its simplicity, flexibility, and generalizability, and transferability to number lines.  Also, when labels were provided to distinguish the two way of thinking the additive comparison and multiplicative comparison became easier to discern. Learners need to use elements of the meaning of a fraction to understand ratios: 1) what is the whole, 2) identify the unit fraction, and 3) iterate the unit fraction to name the new amount in terms of the whole.  When learners experience language work this way it extends and solidifies fractions meaning and previews the multiplicative language needed for ratios.  PSTs then worked to identify part:part vs part:whole ratio language. Students need to learn that a critical idea of ratios is that there is a constant multiplicative relationship between two amounts. Appropriate tasks will provide student language development to be supported by the teacher and concept clarification.  When student comprehension is scaffolded through discussions like these, it establishes a firm base for language needed for scales, proportions, and probability.

Reflection

The diagrams produced by the PSD’s made the comparisons and language used much more relatable to ratios.  When you are addressing ratios with your students, different diagrams would greatly enhance student discussion and understanding.  Through drawing their diagrams and explaining their thoughts with their peers, students are allowed to use the math language necessary to build understanding of ratios.  The teacher should always be carefully listening to the discussions in order to direct student attention to incorrect conclusions or language usage while at the same time allowing the learners to teach each other.  This is an excellent introductory exercise for students and the Asian curricula model gives an easy added degree of support to students visualizing the language.

Rathouz, M., Cengiz, N., Krebs, A, & Rubenstein, R. N. (2014). Tasks to develop language for ratio relationships.  Mathematics Teaching in the Middle School, 20(1), 38-44.

If Only Clairaut had Dynamic Geometric Tools

If Only Clairaut had Dynamic Geometric Tools

Sue Pawula

Summary

      This discussion centered on Clairaut’s historic-genetic approach and how it can be combined with technology through using dynamic geometry software to better afford student development of conceptual foundation based on exploration, discovery, and explanation before they are exposed to formal geometry proof.  CCSSM expects middle school students to use informal debate to establish facts about the angle sum and exterior angle of triangles, angles created when parallel lines are cut by a transversal, and angle-angle criterion for similarity of triangles.  Clairaut believed that students new to geometry should focus on big ideas instead of propositions whose truth may be discovered by concentrating on the details. The three ideas focused on were 1) ”in any triangle, the greater side subtends the great angle”, 2) “a circle does not cut another circle at more than two points”, and 3) “if two circles touch one another then they will not have the same center.”  Teachers could pose questions to challenge students such as “it has been said that the sum of the angles of a triangle (any triangle) is equal to the sum of the angles of any other triangle…is this true?”  Following that students should be expected to use the dynamic geometry tools to help them visualize that a particular theorem holds even when factors change. The software tools will allow them to recognize, explain, and generalize geometric properties through interaction and exploration on the computer where they can drag the vertices of a triangle.  Teachers should coordinate and organize the flow of student thought, prompting students to recognize invariants. Student discussion and reflection should follow as to what they observed and rationally decided about geometric properties. Students are better able to become dynamic learners that discriminate between variants and invariants when they are allowed to explore and identify geometric ideas through the use of the dynamic software.  All of this requires the teacher to guide student reflection and encourage them to think back on the geometric meaning they have garnered from this exercise.

Reflection

Students can begin building a much better foundational knowledge base through hands on activities such as this.  The new geometric software is amazing and can really allow students to experience the changes that occur and yet factors that remain the same, through the simple movement of a cursor.  Reading the article and looking at the diagrams made me understand more fully what they were requiring students to experience and the understanding they would walk away with after the exercise.  As always, the teacher has to be mindful of any misconceptions that may have taken place and guide the discussion to make sure student understanding is complete and accurate.  This is a great way for students to construct knowledge on which to scaffold.

Chang, H., & Reys, B. (2013/2014). If only clairaut had dynamic geometric tools. Mathematics Teaching in the Middle School, 19(5), 280-287.