135: Assessing Understanding

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  1. Minute Paper
    • Purpose:
      • Assess how well students are understanding content
      • Help make mid-course adjustments
      • Feedback on minute papers helps students distinguish between how experts and beginners distinguish what’s important
    • Description: 
      • Students take 1 minute to answer following questions:
        1. What was the most important thing you learned in class?
        2. What important questions remain unanswered?
    • Suggestions for Use:
      • Assess what students learned from a variety of learning activities
      • Wrap-up or warm-up activities
      • Good for courses that present a great deal of new information
      • Well suited for large classes because it is easy to analyze
    • Example of Implementation:
      • European History
        • Two questions were:
          • What is the single most significant reason why Italy was the center of the Renaissance?
          • What one question puzzles you most about the role of Italy in the Renaissance?
        • Analyzed responses and found that some students were confusing cause & effect
        • Reshaped outline for future activities using student questions
        • Categorized responses by: Major Causes, Minor Causes, Effects, Actors, TBD
        • Answered popular questions
      • Statistics
        • Questions
          • What were the 5 most important points from that session
          • Top 2 questions about the session
        • Compiled lists of responses.  Categorized and tallied similar responses.
        • Responses revealed that students struggles to sort the wheat from the chaff.
        • Showed students top 10-12 responses and discussed their relative importance to each other and the course.
        • Learned to be more explicit in his instruction – example – started providing key points at start of lectures
    • Step-by-Step Procedure:
      1. Decide what to focus on and choose a time to administer assessment that takes place soon after learning activity related to focus item(s)
      2. Write 2 questions – use 2 questions in the Description as a template.  Try out assessment.
      3. Set aside 5-10 minutes to do minute paper and debrief time.
      4. Prepare visual related to 2 questions.
      5. At appropriate time, hand out paper for 1 minute papers.
      6. Suggested – let papers be anonymous.
      7. Let students know their time limits, format of acceptable responses and when the results will be debriefed.
    • Analysis Tips:
      • Tabulate responses and make related useful comments
      • Compare results over time.
    • Extension Tips:
      • Use Half Minute papers – only one question.
      • Make prompt more specific – example – most illustrative example, most compelling charcter, etc.
      • Let students compare and discuss their responses in pairs or small groups
      • Let students in small groups invent own minute paper questions and let members of the group analyze and present results to the class.
    • Pros:
      • Immediate teacher / course feedback
      • Gather questions while time is fresh (and limited) to address them
      • Data can be analyzed and summarized quickly
      • Encourage active listening
      • Shows how teachers value student feedback
      • Feedback on minute paper allow students to compare their responses with the rest of the class
    • Cons:
      • If overused, may seem gimmicky
      • Tricky to come up with questions that can be quickly comprehended and answered
    • Caveats:
      • Technique is flexible but not universally applicable
      • Accept students’ starting points even when they are troubling or irritating.  Don’t develop responses to their paper until frustration (if it arises) subsides.
      • Set flexible time limits for feedback.
      • Promise less feedback than you plan to deliver.
  2. Muddiest Point
    • Purpose:
      • Assesses what students misunderstand
      • Identify which topics need more explanations
      • Requires some higher order thinking
    • Description:
      • Students answer question: What was the muddiest point in ______________?
    • Suggestions for Use:
      • Good for large class sizes because it is easy to analyze
      • Use frequently in classes that present a lot of new information (muddy points accumulate quickly)
    • Example of Implementation:
      • Chemistry:
        • Question: What was the muddiest point in enthalpy versus entropy?
        • Results revealed students had trouble distinguishing between 2 concepts.
        • Showed need for more explicit instruction of each concept in isolation.
    • Step-by-Step Procedure:
      1. Determine which learning activity (or part of learning activity) you want feedback on.
      2. Allow time for students to respond to question at the appropriate time.
      3. Let students know time limit and how responses will be used.
      4. Pass out paper for students to write on.  Collect papers.
      5. Present feedback in next class or soon after.
    • Analysis Tips:
      • Find common muddy points.
      • Divide responses into common categories of muddy points and one miscellaneous pile.
      • Tally responses in each pile.
      • Classify piles as concepts, facts, and skills.
    • Extension Tips:
      • Assign muddiest point to homework assignments
      • Students read each other’s drafts and list muddiest points
      • Ask different groups of students to categorize and summarize responses
      • Followup with other assessment – Directed Paraphrasing, Memory Matrix, Concepts Maps to see if muddy points remain muddy
      • Relate muddy points to upcoming exam questions
    • Pros:
      • Very little prep – can be spontaneous
      • Can be safe outlet for student reluctant to ask question in class
      • Can help teachers identify what students find hard to learn – help set different focus for future learning activities
      • Get in students’ shoes
      • Can help students become more metacognitive
    • Cons:
      • Can undermine motivate and sense of self efficacy – can combat that this by teaching students the value of struggles and mistakes
      • Can be discouraging to know what students misunderstand in a well-prepped learning activities
      • Students may struggle to identify and describe their struggles
      • Students may raise challenging questions that are hard to answer on the spot
    • Caveats:
      • Don’t express anger or disappointment when students list muddy points that you thought you explained well
      • Don’t spend too much time responding to muddy points – may lose course momentum
      • Don’t convey that all muddy points can be resolved quickly – some are landslides that take a lot of time to uncover
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The Minute Paper and Muddiest Points are easy-to-analyze assessments of student (mis) understanding.  These can be used to give teachers immediate feedback on the effectiveness of learning activities and insights on how to refine upcoming activities.

 

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Preparation Steps
  • Decide opportune times for students to summarize what they know (or don’t know) using Minute Paper or Muddiest Point.
  • Decide how you will quickly analyze and summarize the data and how you will use that data summary.
Early Implementation Steps
  • Assign Minute Paper or Muddiest Point
  • Analyze and summarize the assessments.  Decide how to make adjustments that highlight student understandings and resolve student misunderstandings.
  • Share results with students with students and how these will impact instruction and student learning.
Advanced Implementation Steps
  • Involve students in the writing of Minute Paper & Muddiest Point questions and in the analyze and summarization of results.
  • Have students reflect on how their understanding of muddy points is improving (or not) over time.

 

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134: Assessing Recall

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  1. Focused Listing
    • Purpose:
      • Assessing what students recall in regards to a specific topic
      • Improve students’ focus and recall
    • What It Is: 
      • Students list several ideas that relate to a focus topic
    • Suggestions for Use:
      • Can be used before, during, or after a lesson
      • Simple assessment that can be used in many classes
      • Good for courses that involve a lot of new information
    • Example of Implementation:
      • Physics:
        • Used to assess students’ understanding of vocabulary such as “work” (2 min exercise per list)
        • Divide into 3 piles: mostly current, confused with everyday meanings of words, the rest
        • Includes concepts and student wording into lesson on work – especially when differentiating between everyday and physics definitions of work
      • Finance:
        • List 5-7 fundamental concepts related to “stocks”
        • Briefly define each concept (10 min exercise)
        • Analyzed sheets to see what was present and missing from students’ lists
        • Following class meeting handed out printed sheet with concepts and definitions and reviewed 3 fundamental concepts not found in sheets
        • From then on, prior to lessons – listed few topics to key in on during upcoming lesson
      • Political Science
        • Prior to a lesson on Federalism handed out 3×5 index cards
        • Wrote BEFORE on one side of cards and listed topics related to Federalism
        • Near end of lesson, wrote AFTER on back of cards and listed more topics related Federalism
        • Gathered top 3 topics from each students
        • To respond to wide variation in topics, at start of next lesson showed student visual that included
          • all 23 topics mentioned by students organized into 6 categories – top 5 fundamental ideas and Other
          • Also organized topics in a concept map
    • Step-by-Step Procedure:
      1. Select important topic currently being studied in class.
      2. Write topic at top of a blank answer sheet.
      3. Set time and/or list item number limits.
      4. Following own time limits, create sample focus list.
      5. Revise list, add items as needed.
      6. If list is well defined and worth discussing, run the same exercise with your class.
    • Analysis Tips:
      • Compare students’ lists to your list and divide into piles: Appropriate, Inappropriate Or Related / Unrelated
      • Categorize responses by relationship to the focus topic – examples: definitions, examples, descriptions, illustrations, primary, secondary, tertiary relationships to focus topic
    • Extension Tips:
      • Allow students to work in small groups to develop collective focus lists
      • Make your focused lists available for comparison and discussion in class
      • Have students in small groups create compiled lists that contained best items from their lists and your list
      • Ask students to define terms on their lists
      • Have students convert into paragraph(s) that relates terms to each other and focus topic
      • Use Focus List strategy at regular intervals to increase recall and prioritization of content
      • Follow-up this strategy with Empty Outlines activity – see below
    • Pros:
      • Simple, quick and flexible way to measure student recall
      • Identify terms students recall and don’t
      • Time limits have students list what they perceive to be the key terms, not what they think the teacher wants
      • If used before instruction, can be used to prime the pump, ready the brain for new learning
    • Cons:
      • Basic form only assesses low cognitive skill, recall
      • Some students can product reasonable lists without really understanding terms
      • Focuses on one idea at time – some key knowledge focuses on relationships among several key concepts
    • Caveats:
      • Create your own master focused list to trial key topic prior to assigning focused list.
      • Focus lists on key big ideas (enduring understandings).
      • Pick a topic that is not too broad or too narrow to create a somewhat convergent variety of lists.
      • Add specifics about relationship between focus list words and focus topic – examples, defining words, synonyms, examples, etc
  2. Empty Outlines
    • Purpose:
      • Assess students’ recall and note-taking skills
      • Emphasizes key topics and their sub-topics
    • What It Is:
      • Students complete an empty or partially completed outline – if partially completed, include key headings and empty spaces for sub-topics
    • Suggestions for Use:
      • Good for courses that have a lot of detailed information that are highly structures
    • Examples of Implementation:
      • Nursing Course
        • Provided students with outline with 4 major topics and empty slots for 5-7 subtopics for each major topic
        • Students completed outlines using their notes
        • Teacher compared outlines to her lecture outline
        • Uses disparities in the prioritization in her notes and her students’ notes to learn how to better emphasize key points in future lectures
      • Child Development Course
        • Prior to showing a video to a class, teacher watched the video and created outline of video contents
        • Created empty outline by deleting sub-headings and keeping major headings
        • After students watched the video, gave students 5 minutes to complete the empty outlines in pairs
        • Found that their notes coincided with his notes at the beginning and end of the video and deviated most near the middle of the video
        • In the future, paused video in the middle to give students time to take notes
    • Step-by-Step Procedure:
      1. Create an outline for an upcoming learning activity.
      2. Decide what info you want students to provide – major topics, sub-topics, supporting details etc.  Let that inform empty outline design.
      3. Limit number of blank items for students to complete on empty outlines to less than 10. (if you want them to complete it from memory)
      4. Communicate expectations – time limits and types of things to put in empty outlines
      5. Convey purpose of assignments, when results will be shared, and how results will be used.
    • Analysis Tips:
      • Compare student outlines with your outline and learn from agreements and disagreements
      • Look at range of responses and notice patterns in responses
    • Extension Tips:
      • If students struggle to complete the outline, provide a word/phrase bank.
      • Vary between providing major subtopics and asking for supporting details and providing supporting details and asking for major topics.
      • For advanced students, provide guidelines only
      • Use focused listing activity prior to completing outlines – use focused list items in outlines
      • Do outline as a warmup to see student expectations for a lesson
    • Pros:
      • Repeated use can improve student listening and note taking.
      • Feedback on outlines gives students better models for note taking.
      • Can help students better organize knowledge in their notes and in their brains.
      • Can make organizing ideas of a subject more explicit
    • Cons:
      • May feel constrained by the master empty outline
      • Not all information is best organized in hierarchical structure of outlines
      • Unless students make outlines from scratch, little higher order thinking is required
    • Caveats:
      • Students’ varied readiness levels will lead to variation in their empty outlines
      • Limit amount of info captured in empty outlines (less than 10 points)
  3. Memory Matrix
    • Purpose: 
      • Assess recall and organization of important information
    • What It Is:
      • Students complete a chart that has row and column labels that emphasize key relationships between ideas
    • Suggestions for Use:
      • Works well with subjects with a lot of detailed information that relates to each other
      • Can assess recall of information after a variety of learning activities
      • Can be used as a pre-assessment
    • Example of Implementation:
      • Spanish:
        • Gave students a matrix to complete that had types of verbs (-ar, -er, ir) as the column headings and (regular/irregular) as the row headings
        • Found that students sometimes classified regular verbs as irregular verbs and confused -er and -ir verbs
        • Info help teacher decide to review in upcoming classes
      • Art History:
        • Gave students a matrix with column headings: France, US, and Britain and rowing headings: Neoclassicism, Impressionism, Postimpressionism, Expressionism.
        • Students completed matrix in groups of 5; then transferred responses to a whole class chart
        • Students could categorize artists by country but struggled to separate them by time period
        • Use their misconceptions as a starting point for upcoming lectures
      • Nursing class:
        • Gave students a matrix with column headings: structure, functions, enzymes and row headings: mouth, esophagus, etc (other digestive system organs)
        • Students completed matrix in teams of 5
        • While students watched a video on enzymes he analyzed the students’ matrices.
        • Using model matrix, he reinforced what students got right and discussed what students got wrong.
        • Then gave individuals another blank matrix and had them complete it at class end.
    • Step-by-Step Procedure:
      1. Draw simple matrix with row and column labels that represent key topics in the course.
      2. Create a key based on learning activities.
      3. Revise Memory Matrix if needed – check for fit of row and column labels to key ideas.
      4. Create a blank version of Memory Matrix that only has row and column labels.  Make blank cells large enough to record several ideas.
      5. Give students time to complete the matrix.  Set a lower limit for the number of items in each cell.
      6. Analyze matrices for correctness.
    • Analysis Tips:
      • Analyze correctness of matrices and look for trends in correctness and incorrectness to identify student strengths and gaps.
      • Analyze errors and look for patterns in the errors.  Use this diagnose amount of learning time & type of activities associated to each topic.
    • Extension Tips:
      • Complete Memory Matrix as part of a class discussion.
      • Allow students to work in groups on the matrix.
      • Fill in some of the middle cells and have students guess the rowing headings or column headings.
      • Give students a Work/Phrase Bank and have them create the Memory Matrix (complete with original row and column headings) that organizes the terms in the bank
    • Pros:
      • Assess recall of information and how well students can relate information
      • Simplicity of format makes it easy to analyze
      • Graphic format may appeal to visual learners
      • Can improve memory organization and retrieval
    • Cons:
      • Row and column headings impose organization formats that may hide organizational relationships students are using to relate content
      • Basic format of assessment may not separate prior knowledge and current knowledge well
      • Can obscure flexibility and complexity of the actual relationships among content
    • Caveats:
      • Start with small matrices (2×2) for students unfamiliar with this strategy.
      • May obscure relationships that are flexible / blurred.  Need to point out these nuances in learning activities.
      • Recognize matrices as a convenient simplification of a more complex reality
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All 3 of the strategies in this article assess student recall and assess relationships students see among the information items they recall.  Providing feedback on these items can provide opportunities to make priorities and relationships among content more explicit.  Varying the format for the recall assessments can emphasize different relationships.  The Focused List assessment shows how one central idea relates to sub-ideas or supporting evidence.  The Empty Outline can show how several central ideas relate to other pieces of information.  The Memory Matrix shows how 2 groups of ideas (categories) can be used to show relationships between detailed pieces of information.

 

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Preparation Steps
  • Analyze the knowledge and skills in the standards in the upcoming standards.  Look at the key relationships you’d like to teach to students to students among major and supporting ideas.
  • Decide which type of recall assessments best illustrates the relationships you’d like students to use to organize ideas in upcoming learning activities.
  • Prepare samples / keys for the selected strategies you will use.
Early Implementation Steps
  • Ask students to work on assessment(s) individually or in teams.
  • Analyze assessments.  See above for tips.  Try to learn student strengths and gaps and what are the patterns in their strengths and gaps.
  • Use what is learned from assessments to modify instruction.
  • Share results with students and how results will impact upcoming instruction and student learning.
Advanced Implementation Steps
  • After students have had practice with the recall strategies – try extension version of the activities.  See above.
  • Incorporate a recall strategy into classroom routines – pick one that emphasizes skills and organization styles that fit well with your course.
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133: Assessing Prior Knowledge

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  1. Background Knowledge Probe
    • Purpose
      • gather detailed information on students’ prior knowledge related to most important materials to be studied
      • determine more effective starting points for scaffolding
      • determine range of readiness / preparation levels in a class
      • serve as pre- and post- assessments
        • administer before scaffolding for pre-assessment
        • administer after scaffolding to post-assess what was learned during instruction
    • What it is:
      • short, simple questionnaires used to assess knowledge at the start of a course or the start of a unit
    • Suggestions for Use:
      • Focus on key concepts needed to be successful in upcoming material
      • At least one low floor question – most can get it right
      • At least one high ceiling question – most will get it wrong
      • Use to introduce important concepts about to be uncovered
      • Use to assess knowledge midway and after related scaffolding
      • Share enough data related to assessments that students can determine how they performed relative to the class
    • Examples of Implementation:
      • ELA – List all Shakespeare’s plays you’ve been exposed to.  Check off if you read the play, listened to the play, watched a live performance, watched an adaptation, etc.  After analysis – point out which plays they will see again in the course and how they can help students who haven’t had prior exposure to the plays.
      • Electrical Engineering – Showed pictures of 5 pieces of electrical equipment and asked students to determine readings on the illustrations.  Observe what students can read or not and the format (standard notation, engineering notation, with units or without) of their responses.  After 1st individual probe, combined students into heterogeneous groups and had them try again with the expectation that all students needed to make sense of the correct answers.  Gave tips to more experiences students on how to help less experienced students
    • Step-by-step procedure:
      1. Consider what students may know at the start of a course or a unit (and to what degree they may know it).  Try to identify at least one point most will know and have that lead off into related, less familiar points.
      2. Prepare probe.  Formats could include
        • 2-3 open ended questions (see ELA example)
        • Handful of short answers
        • 10-20 multiple choice questions
        • Use student-friendly language that will make it easier to access what they already know
      3. Give students the probes.  Emphasize the need for thoughtful responses and the fact these pre-assessments will not be graded.
      4. In a timely manner, report the results and discuss implications:
        • how results will affect the way the course will be taught
        • how results will affect what they do as learners
    • Analysis Tips:
      • Affinity group responses by different levels of preparedness
      • Assign scores to each pile of preparedness
        • +2 significantly prepared
        • +1 some relevant background knowledge
        • 0 no background knowledge
        • -1 wrong background knowledge
      • Total the scores to determine overall level of preparedness
      • Fast method – Divide responses into prepared and not prepared piles prior to teaching related concepts.  This related to Clustering student needs for more efficient planning.
    • Extension tips:
      • After individually completing groups, have students work in small groups to come up with mutually acceptable answers.
      • Ask students in small groups to rate and sort answers from other groups – see Analysis Tips above
      • Ask students to interview each other and annotate responses and sense-making related to probe questions.
      • Use version of probes as a post-assessment
    • Pros:
      • Provide info on students’ content and communication skills
      • Provide specific baseline data that can inform instructional decisions
      • Can provide opportunities to hook students in by tying things back to their prior knowledge
      • Can prime the pump, i.e. prepare students to take in new information by making them aware of connection to things they already know
    • Cons:
      • Feedback might demoralize teacher
      • Responding to probe can be frustrating for unprepared students
      • Classifying responses may create hard-to-change false impressions of students which may impact expectations later in the course
    • Caveats:
      • Can show big holes in course-long sequence due to knowledge gaps in students – only do this if you have time and energy to make significant revisions to course materials
      • Do not generalize too much from one assessment
      • Plan response for both prepared and underprepared students
  2. Misconception / Preconception Check
    • Purpose: 
      • Identify incorrect or incomplete knowledge that can interfere with new knowledge
      • Help students identify “early on” beliefs that may hinder their understanding of new knowledge so they have a better chance of revising and transforming their knowledge structures to accommodate new info
    • What It Is:
      • Pre-assessments that uncover prior knowledge or beliefs that my hinder or block further learning
    • Suggestions for Use:
      • Use it to uncover common sense knowledge that is counteracts content information
      • Use it to uncover beliefs that allow some facts to get through but block out deeper understanding of method or worldview
    • Example of Implementation:
      • History –
        • Anonymous responses to 3 questions: 1) How many people lived in North America in 1491? 2) About how long had they been on this continent in 1491? 3) What significant achievements had they made at that time? (5 minutes to respond).
        • Shuffled papers and handed back.
        • For (1) and (2) collected highest and lowest answers.
        • Teacher analyzed question (3) at home.
        • Then added question (4), where did you get your answers for 1-3.  Class realized that most of their guesses were grounded on thin ice.
        • As homework students paired up and conducted research to find acceptable ranges of answers to 1-3.
      • Health –
        • Prior to a unit on STDs, used a probe that had 10 statements that represented true facts or common misconceptions about the symptoms, treatment and transmission of STD’s.  Students answered in Likert scale from this is certainly true, mid range (I have no idea) to this certainly false.
        • Analyzed responses and found that many students were clinging more (or less tightly) to several misconceptions.
        • Tied their response results to specific lessons related to the prompts.
      • Astronomy  –
        • On large blank sheets of paper asked students to respond to question – What makes the season change on Earth” and said any answer was acceptable except “I don’t know”.
        • Divided responses in 4 piles – correct, distance, weather, others piles.
        • Picked best response from each pile and created another assessment – same as before with 4 responses as multiple choice question.
        • Asked proponents of each model to explain their answers to the class.
        • For homework, students had to identify and verify the correct response.
        • After discussing their research with some specific feedback, concluded lesson by explanation why some incorrect model were reasonable and reminded students how long it took to figure out what actually caused the seasons.
    • Step-by-Step Procedure:
      1. Identify most troublesome common misconceptions or preconceptions students bring to the course.
      2. Focus Misconception/Preconception check on most common or harmful preconceptions
      3. Create simple questionnaire to elicit beliefs related to list in step 2.
      4. Get another teacher to check questionnaire to make sure it’s helpful and not patronizing or threatening.
      5. Think of how you will respond to several outcomes of questions – strike out questions that you are comfortable responding to.
      6. Explain purpose of pre-assessment.  Let it be anonymous.  Explain how and when you will use the feedback gathered from it.
    • Analysis Tips:
      • Use responses to answer these questions
        1. What specific misconceptions do my have about course material that may interfere with their knowledge?
        2. How many students have these beliefs?
        3. How deeply embedded are these problematic beliefs?
      • Affinity group responses by type of response / misconception
      • Use Likert responses to get at question 3.
      • Also group Likert responses to identify –
        • strongly held correct ideas
        • loosely held correct ideas
        • strongly held incorrect ideas
        • loosely held incorrect ideas
    • Extension Tips:
      • Prior to answer questions, have students identify common misconceptions held by other people related to topic or field.
      • Have students come up with reasonable explanations for misconceptions.
      • Use same questionnaire as a post-assessment later in the term
    • Pros:
      • uncover likely barriers to learning early on
      • anonymity of responses will make students more likely to be truthful about what they know and don’t know
      • can generate sense of relief for not being the only one to have a misconception
      • develop students’ metacognition skills
    • Cons:
      • unlearning errors can be challenging, unpleasant
      • changes in ideas take time
    • Caveats:
      • thread lightly around sensitive issues to help students open up about their opinions
      • don’t use this strategy until a positive safe classroom culture is established

 

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Knowing how to effectively gather and analyze assessment data related to students’ prior knowledge can help teachers tailor course materials to match students’ readiness levels.  Uncovering unhelpful beliefs can help students and teachers investigate the rationale for the truth (and falsehood) of these beliefs and create room in students’ knowledge structures for new ideas.

 

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Preparation Steps
  • Analyze upcoming standards.  Identify key big ideas and possible misconceptions related to these.
  • Design questionnaires that are design to elicit ideas related to big ideas and misconceptions.
  • Build a positive safe culture that values using student feedback to make better instructional / learning decisions.
Early Implementation Steps
  • Implement prior knowledge questionnaires at the start of a course or unit.  Do NOT grade them.  Tip: Keep them anonymous.  Also see Step-by-Step Procedures above.
  • Analyze responses and use patterns in their responses to inform future instructional decisions.  See Analysis Tips above.
  • Share responses with students and explain how these will be used to change instructions and tips to improve student learning.
Advanced Implementation Steps
  • Use student data and feedback to design better questionnaires and refine upcoming projects and scaffolding activities.
  • Implement some of the Extension Tips above.
  • Have students conduct research on questionnaire questions and discuss their findings in class.

 

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132: Mathematics & the Path to Equity

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The Elitist Structure of Mathematics
  • Elitist views place math as subject harder than other subjects that can only be accessed by a select few.
  • Math is taught as a performance subject that weeds out people with & without the math gene
  • Some people enjoy sorting mechanism of math because they have been sorted into the side of the limited Have’s
  • Some people enjoy thinking that their math ability is due to genetic superiority
  • Sometimes math teachers feel like they are superior to teachers who teach other subjects
    • these same teachers may feel justified in failing many students because they feel like they are the guardians of math success and only stars can move to higher levels
  • Some university math departments lower grades of students who display hard work habits such as attending office hours
The Myth of the Mathematically Gifted Child
  • Even math geniuses had to work hard to be able to produce relevant work
  • “Gifted” status awarded to students who can do things quickly, not necessarily kids who work hard and are persistent
    • Myth of genetic difference can make “gifted” students intellectually brittle because they may end of devoted a lot of energy to protecting their gifted identities
  • Valuing “giftedness” over hard work may cause:
    • high achievers to hide or underemphasize the effort they exerted to achieve
    • hard workers to feel like imposters because they had to work hard to achieve
  • Elitist math views + stereotypes of who can be good at math create large equity gaps in math
    • in 2014 – 73% math doctorates were male, 94% were white or Asian
    • the more a field values giftedness, the less likely are women and minorities to enter the field
  • Rushing students to higher levels of math can dilute the depth at which they understand fundamental concepts and processes
    • could lead to students who are procedurally fast, but can’t explain rationale for procedures
Equitable Strategies
  1. Offer all students high-level content.  
  2. Work to change ideas about who can do mathematics.
  3. Encourage students to think deeply about mathematics.
    • The desire to think and understand deeply is more critical to math achievement than the ability to perform procedures quickly.
    • Include experiences that are
      • hands-on
      • project-based
      • tied to real life applications
      • allow for collaboration
  4. Teach students to work together.
    • Shared struggles make challenges less intimidating
    • Discussing math helps people make sense of it
  5. Give lots of encouragement to people who are normally left out (women and minorities).
    • Do not comfort kids by buying into their “I’m just not a math person” fixed mindsets
    • Anyone can perform poorer when they are on the under-side of a stereotype of performance
  6. Eliminate (or at least change the nature of) homework.
    • Homework spreads low income equity achievement gap because low income students have less time and less resources while completing homework
      • inequities are magnified when class starts with homework review
    • Instead of practice problems, offer reflection questions such as
      • what was the main idea learned today?
      • what is something you are struggling or have questions about?
      • how could lessons from today be applied in real life?
    • Instead of practice problems, offer inquiry problems that have students seek out examples of current concepts in their lives

 

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Uncovering the elitist structures embedded into the structures of math curricula and the attitudes it promotes can help teachers be more aware of how to revise their practices to close equity gaps.  Equitable teaching practices have been shown to have a greater impact than minority role models.  This empowers any teacher to practice strategies that can close achievement gaps.

 

4-nowwhat
Preparation Steps
  • Research specific strategies related to equitable practices listed above.
  • Examine practices and attitudes critically to see if any are directly or indirectly elitist.
  • Develop strategies, visuals, and lesson plans that eliminate elitist views of math and replace them with growth mindset views of math.
Early Implementation Steps
  • Implement policies, visuals, scaffolding and assessments that combat elitist views of math and promote growth mindset views of math.
  • Teach students skills related to math achievement:
    • brainstorming
    • communicating
    • sense making
    • drawing to understand
    • reflecting
    • collaborating, etc
  • Use student feedback to fine tune policies, scaffolding and assessments
Advanced Implementation Steps
  • Assess students attitudes over time to see if their views of math and their place in it is changing over time
  • Research and implement strategies that set high expectations and also offer high levels of support to all math learners
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131: Rich Mathematical Tasks

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5 C’s of Mathematical Engagement:
  1. Curiosity
  2. Connection making
  3. Challenge
  4. Creativity
  5. Collaboration
Why Rich Mathematical Tasks:
  • expose people to openness and flexibility of mathematics
  • generate excitement
  • create more opportunities for understanding
  • provide support for high challenge tasks
Characteristics of Engaging Lessons with Rich Mathematical Tasks
  • Task is challenging (high ceiling), but accessible (low floor).
  • Students view task as a puzzle
  • Visual thinking builds understanding
  • Classroom culture values mistakes
  • Students respect each other’s thinking
  • Students use own ideas (not blindly follow given procedures)
  • Students collaborate
  • Students have different things to offer to team effort to solve problem
  • Students don’t think they are finding a standard answer – think they are finding own solutions to a problem
  • Good Timing (Just in Time Teaching)
    • Let students explore applied problems first
    • Observe students
    • Introduce methods when they reach a stuck point that can be resolved by the method
    • Helps students learn value of methods and develop intuitive sense for methods
 
Examples of Rich Mathematical Tasks:
  • 18×5
    • Solve 18×5 using mental math
    • Display different solutions using simple thumbnail sketches
  • Growing geometric patterns
    • Explain visually how pattern is growing
    • Share different growth pattern ideas
    • Use growth pattern ideas to find 100th and nth iterations of patterns
  • Build a fence out of 36 1-m planks
    • Use 36 1-meter planks to build fence that encloses the most land
  • Lemon volume
    • Find the volume of a melon (precursor to teaching integration methods)
  • Cuisenaire Rod Train Task
    • Use Cuisenaire Rod to determine how many different trains can be made from a rod of prescribed length
    • Find connection between Pascal’s triangle and numbers of trains in rods
  • Negative Space Task
    • Find patterns in geometric patterns that go up in positive indexes and then down in negative indexes
    • Figure out how to represent negative index results visually and numerically
  • How Close to 100?
    • Roll pair of dice – use pair of dice to draw an array on a 10×10 grid.  Write number sentence that related dice numbers to array area,
    • Take turns rolling dice, drawing arrays and writing related number sentences
    • Place until no more arrays can be fit unto 10×10 grid
  • 1 Divided by 2/3
    • Don’t apply the fraction rule of division
    • Use visuals to make sense of answer
  • Four 4’s
    • Make all numbers from 1 to 20 using four 4’s and any operation
    • For tips on how to stage this lesson, go here
  • Paper Folding – Use paper folding of a square piece of paper to do the following
    • Construct a square with exactly 1/4 the area of original square – convince partner that it is 1/4 the area
    • Construct a triangle with exactly 1/4 the area of original square – convince partner that it is 1/4 the area
    • Construct another triangle with exactly 1/4 the area of original square that is not congruent to the first triangle – convince partner that it is 1/4 the area
 
Developing Rich Mathematical Tasks: Questions to Consider:
  • Open the task encourage multiple methods, pathways and representations.
    • Add visual requirement
    • Ask students to make sense of solutions
  • Make it an inquiry task.
    • Ask students to come up with ideas, not follow a procedure
    • Create products about solutions (brochure, newsletter, etc)
      • examples
        • write a book about y = mx + b
        • create a coffee table book about similarity
  • Ask the problem before teaching the method.
    • Introduce methods after students have developed methods that relate or approximate new methods or after students reach a stuck point that can be unlocked with new method
  • Add a visual component.
    • Use drawings, math manipulatives, etc to represent solutions and ideas
    • Use color coding to feature common features in solutions such as “x” or to highlight relationships
  • Extend the task to make it low floor and high ceiling.
    • Low floor – ask students how they see the problem
    • High ceiling – ask students to write a new question that is similar but more difficult
  • Add the requirement to convince and reason
    • Explain methods and why they make sense
    • Reasoning is practicing mathematics
    • Reasoning gives access to understandings that can close equity gaps
    • 3 levels of convincing
      1. Convince yourself
      2. Convince a friend
      3. Convince a skeptic
 
Suggested Reading & Related Resources
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Rich mathematical tasks can be used to engage all students in mathematical solving and sense making.  The principles behind good design for these tasks can be used by teachers to frame problems and lessons that invite students to apply their own ideas to problems and to relate these ideas to new concepts and methods.

 

4-nowwhat
Preparation Steps
  • Analyze standards and develop learning targets related to concepts and problem solving methods
  • Use resources above to find rich and engaging mathematical tasks that relate to learning targets
  • Build scaffolding and assessments around rich mathematical tasks that align to learning targets
Early Implementation Steps
  • Use rich mathematical tasks to:
    • create engagement
    • introduce new concepts
    • help students see need-to-knows in new content
    • make sense of new content
    • to get students to discuss their reasoning
  • Use assessments to fine tune activities to improve student learning
  • Ask students to reflect on how activities are improving their learning
Advanced Implementation Steps
  • Build a bank of rich mathematical tasks that goes with a course’s scope and sequence
  • Find rich mathematical tasks inspired by nature and the real world
  • Recruit stakeholders to serve as clients for students solving rick mathematical tasks

 

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130: Creating Mathematical Mindsets

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Mathematical Mindsets – What They Are:
  • Present naturally in children who like to inquire, build things, solve puzzles, notice/make patterns, etc
  • Seeing math as a conceptual subject that they can grow to make sense out of
  • Stepping forward deliberately and deeply in math while making sure each step makes intuitive sense before moving more forward
How to Stunt Development of Mathematical Mindsets:
  • Presenting math as a dry set of methods can stop development of mathematical mindsets
    • this is especially true if methods do not make intuitive sense to students
  • Presenting math as seires of short questions obscures growth opportunities – math is something you get or you don’t, instead of something to make sense of
  • Assigning large homework sets with simple isolated problems
  • Valuing rote memorization and speed over deep thinking and conceptual understanding
How to Develop Mathematical Mindsets:
  • Encourage students to play with numbers, shapes and puzzles
  • Present math as a broad landscape of unexplored puzzles that create opportunities for wandering around, asking questions, thinking of relationships, …
  • Present math as a flexible conceptual subject that is about thinking and sense making
  • Be mindful when designing practice set because mindless practice does not lead to brain growth, thoughtful practice does – mindful practice involves applying same strategy to many different situations
  • Assign less homework that requires more reflection – example 5 carefully selected problems and one student chosen reflection question such as:
    • What are the main mathematical ideas we discussed in class today?
    • What questions do you have about ________?
    • Describe a mistake or misconception you or a student had in class today.  What did you learn from this mistake or misconception?
    • How did you approach your practice set? Was your approach successful? What did you learn from your approach?
  • Cultivating Number Sense:
    • Approach arithmetic operations flexibly and conceptually:
      • concept of sum -> counting on
      • concept of product -> repeated addition
    • Try to help students make sense of concepts and patterns so that their brain can more readily go from compression – more efficient storage of concepts (not rules)
    • Math facts are stored in working area of brain – this area can be blocked when students are stressed
    • Avoid techniques that value speed of knowing math facts (example – timed tests)
    • Do NOT emphasize rote knowledge and speed – gets in the way of thinking about numbers and their relationships to each other
    • Teach strategies instead of memorization of facts
      • example: 17 x 8
        • strategy – 17 x 10 – 17 x 2 = 170 – 34 = 136
        • memorize 17 x 8 = 136
    • play math games that  activate both sides of brain by using visual and intuitive math thinking:
      • example: grid multiplication game
        • object of game – fill as many grid squares as possible in a 10 x10 grid
        • roll 2 number dice – color in area that corresponds to product of 2 numbers rolled and write number sentence
        • partners take turns rolling dice, coloring in areas and writing related number sentences until no more arrays can be added to the grid
      • example: multi rep matching game
        • players take turns picking pairs of equivalent cards and explaining why they are equivalent
        • find cards and more cool strategies here
    • Do “Number Talks” as warmups
    • Recommend math games that emphasize concepts over drill & kill:

 

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Developing mathematical mindsets will help student approach mathematics with a growth mindset.  Mathematical mindsets help students understand math concepts more deeply and apply them more flexibly.  Valuing conceptual understanding over speedy rote memorization is one way to cultivate mathematical mindsets.

 

4-nowwhat
Preparation Steps
  • Research more strategies for developing mathematical mindsets.  See Mathematics articles for ideas.
  • Develop lesson plan components (Warmups, practice sets, discussions, activities, etc) that promote mathematical mindsets
Early Implementation Steps
  • Regularly use scaffolding and assessments that promote mathematical mindsets
  • Have students reflect often on what they are learning
    • about concepts
    • about how concepts are applied to problem solving
    • from mistakes
    • from different problem solving approaches
Advanced Implementation Steps
  • Have students interact with mathematicians and professionals who apply mathematical reasoning often and learn about their problem solving approaches
  • Develop bank of Number Talk problems and games that promote mathematical mindsets and incorporate these into classroom routines
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129: Creativity & Beauty of Mathematics

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Traditional School Math & Math Misconceptions
  • Math is hated/feared because it is taught and thought of in ways that are different from other subjects.
  • Primary role of students in traditional math classrooms is to perform and get questions right.
  • Performing takes precedence over learning.
  • Testing culture promotes idea that math is about finding short answers to narrow questions under pressure.
  • Math is a subject of procedures, calculations and rules.
  • Math is a dead subject that is only present in textbook calculations.
  • Math is a series of answers to questions that nobody asks in the real world.
  • Math in classrooms focuses primary on Stage 3 of math stages (see below)
  • Student shouldn’t have to show work if their answer is right.
  • Math is done by individuals.
  • People who are good at math perform calculations QUICKLY.  Math is a speed race.
 
Real Nature of Mathematics
  • Besides getting question right, doing math involves:
    • appreciating beauty of math
    • thinking deeply
    • exploring math connections
    • applying math to different situations
    • exploring patterns
    • using math to create and analyze new technology and strategies
    • formulating questions
  • Math exists throughout nature, art, and the world.
  • Nature contains many examples of mathematics
    • spiders are experts in spirals
    • dolphins use a form of algebra to interpret echolocation signals
  • Instead of study of procedures, calculations and rules, math is
    • study of patterns (aesthetic, beauty of subject)
    • subject of visual images, creativity and connections
    • subject that is full of uncertainty – answers can be explorations, interpretations, conjectures
    • set of ideas, connections and relationships that we can use to make sense of the world
  • 4 Stages of Math Work
    1. Posing a question
    2. Going from real world to a mathematical model
    3. Performing a calculation
    4. Going from model back to real world to see if original question was answered
  • Real math is often done collaboratively
  • Speed of calculations has nothing to do with math fluency.
 
How to Align Math Schooling with Real Mathematics
  • Give students opportunities to consider situation and formulate math problems to investigate these situations
  • Give students opportunities to use all 4 math stages
  • Require students to show work because displaying logical mathematical lines of reasoning is the main part of doing math
  • Facilitate math discussions about mathematical reasoning (what it is, how to critique and justify it)
  • Pose open-end problems and allow students to develop methods and pathways to solutions
 
Workforce Implications
  • Employers need people who can ask good questions, set up models, analyze results and interpret mathematical answers.
  • Employers no longer need people to calculate; now they need people to think and reason
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Math misconceptions have crept into the design of math curricula.  These math misconceptions have made the subject appear uninteresting and unappealing to some.  Knowing the true nature of mathematics can help teachers design learning experiences for students that are engaging, challenging and relevant to the real world.

 

4-nowwhat
Preparation Steps
  • Research more methods for designing learning experiences that are more true to the math discipline.  See Mathematics articles.
  • Research strategies for 4 Phases of Math (see above).
  • Create a culture that values mistakes.  See this article for ideas.
  • Design scaffolding that includes elements such as:
    • balance of 4 phases of math
    • facilitated math discussions about mathematical reasoning
    • pattern recognition
    • students posing questions and possible solutions
    • student creating models for the real world
Early Implementation Steps
  • Implement scaffolding that provides students with many opportunities to appreciate and practice real mathematical thinking
  • Have students reflect on how their math attitudes are changing as a result of math activities that deliberately mimic the math discipline
Advanced Implementation Steps
  • Provide students with opportunities to solve real world problems using math
  • Provide students with opportunities to interact with real world stakeholders in order to pose better questions, formulate better models, learn better calculation methods and compare/interpret their results to real situations
  • Try to brainstorm what math looks like when mapped unto the 6 Facets of Understanding and Bloom’s taxonomy

 

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128: The Power of Mistakes & Struggle

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Mistakes & the Brain:
  • Mistakes grown synapses.
  • Mistakes generate more brain activity than correct responses.
  • 2 brain responses to mistakes:
    1. ERN responses – increased electrical activity due to conflict between correct response and an error
    2. Pe responses – brain signal due to recognition of error
  • Brain sparks can occur even when people are unaware that mistakes were made
  • People with growth mindset show more brain activity in response to mistakes and are more likely to recognize errors
Mistakes & Life
  • More successful people make more mistakes than less successful people
  • Making mistakes is key to creative, entrepreneurial thinking
  • Successful people tend to:
    • feel comfortable being wrong
    • try wild ideas
    • are open to different experiences
    • play with ideas without judging them
    • persist through difficulties
    • willing to go against tradition
  • Practicing the attitudes above can help people learn math (or probably anything)
How Can We Change How Students View Mistakes?
  • Teach students about the positive impacts of mistakes on the brain
  • Crumble paper with mistakes, throw it against something to let out frustration.  Then open it and smooth it out and trace over crumple lines with marker to remind oneself of brain growth as result of mistake.  Then keep paper as a record of mistakes.
  • Teach and display positive brain messages.
  • Have teachers and students select and highlight “favorite mistakes”.
  • Have class discussions about mistakes.
  • Do not downgrade assignments for mistakes – upgrade assignments for mistakes.
  • Avoid over-testing and over-grading.
  • Display positive attitudes towards mistakes in group and individual settings.
  • Remind students repeatedly about brain growth that goes with mistakes and lack of brain growth that goes with correct responses
  • Teach students to appreciate & be aware of disequilibrium (Piaget) – state of disequilibrium occurs when students try to incorporate new information into existing mental maps – states of disequilibrium are uncomfortable but lead to wisdom
  • Expose students to math experiences that create disequilibrium
  • Value work with mistakes more than correct work
  • Make showing of mistakes a common occurrence in classroom and discussing how to think through the mistake

 

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Knowing about the impact of mistakes on the brain can teach students and teachers to value mistakes more and leverage them better to grow.  Knowing strategies for creating cultures that value mistakes will help students develop growth mindsets and help them to approach mistakes creatively and constructively.

 

4-nowwhat
Preparation Steps
  • Research strategies for creating classroom cultures that value mistakes.  See above.
  • Develop scaffolding activities and strategies that will be used to teach & remind students of the value of mistakes.
Early Implementation Steps
  • Implement policies, strategies, and scaffolding lessons that value student mistakes such as:
    • Presenting (teachers & students) mistakes and hold classroom discussion around them
    • Crumple paper strategy (see above)
    • Creating situations that will place students in disequilibrium and funnel students towards learning targets
    • Teaching students about the relationship between brain activity and mistakes
    • Selecting favorite mistakes and why they are so helpful
    • Reflections on how new attitudes towards mistakes impact learning
    • Using grading policies that value errors
Advanced Implementation Steps
  • Create bank of problems that create disequilibrium that explore big ideas in mathematics
  • Create bank of discussion and question prompts that highlight and analyze mistakes

 

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127: Differentiated Curriculum Charts

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Offer students choices for learning learning targets by creating Differentiated Curriculum Charts:
  • The chart provides learning mode & extension activities for each learning target.  See below for example.
  • Students get to choose which activity to perform to explore learning target
  • For extension ideas, see the Analysis, Evaluation, & Synthesis products & trigger words in this article.
  • For learning mode ideas, see below.
chart
  1. Auditory Products 
    • audio recording, autobiography, commentary, crossword puzzle, debate, dialogue, documentary, editorial, experiment, fact file, finding patterns, glossary, interview, journal, newspaper, oral report, petition, position paper, reading, scavenger hunt, simulation game, song lyrics, speech, story, survey, teach a lesson, video, written report
  2. Visual Products 
    • advertisement, art piece, brochure, collage, comic strip, diagram, diorama, drawing, filmstrip, flow chart, graphic organizer, greeting card, multimedia presentations, illustrated manual, magazine, map, photo essay, picture dictionary, poster, slide show, video, website
  3. Tactile – Kinesthetic Products
    • acting things out, activity plan, animated movie, dance, demonstration, dramatization, experiment, field experience, flip chart, game show, how-to book, jigsaw puzzle, manipulative, mobile, model, museum exhibit, play or skit, rap, scale drawing, sculpture, simulation game, survey, TV broadcast, video

 

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Differentiated curriculum charts create options for students that fit their learning styles and readiness levels.  Charts like these can be used as tools to create scaffolding that fits the needs of diverse groups of students.

 

4-nowwhat
Preparation Steps
  • Recruit teacher team to help gather all the scaffolding.
  • Analyze standards and rewrite in terms of long term and supporting learning targets
  • Develop activities for each learning target that go with each learning mode – see above for example.  Use suggested products above and here for ideas.
  • Create learning centers to house the activities for the different learning modes.  If many resources are posted online, this can be as simple as different wall segments (1 per learning mode) that house QR codes to activities.
  • Create a grading system for crediting students’ different choices – a simple way to do this is to require 1 activity per learning target and assign a daily grade to each
Early Implementation Steps
  • During scaffolding days, allow students to select 1 activity per targeted learning target.  Explain how to get to resources and how to get feedback on work.
  • Provide a lot of formative feedback on the work and (if possible) grade student work in class in conjunction with formative discussions with students.
  • Use other formative assessments to ensure that ALL students are developing an understanding of learning targets.
Advanced Implementation Steps
  • Develop a bank of short rubrics for assessing various types of products that appear frequently in differentiated curriculum charts.
  • Use formative feedback data to determine which learning activities are the most engaging and effective and incorporate similar activities into upcoming differentiated scaffolding
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126: Taxonomy of Thinking

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  1. Knowledge
    • recall, remember
    • Trigger words: tell, recite, list, remember, memorize, define
    • Products: worksheets, quizzes, tests, skills work, vocabulary work, facts in isolation
  2. Comprehension
    • restate concepts in own words
    • Trigger words: restate in own words, give examples, explain, summarize, translate, summarize, translate
    • Products: drawings, diagrams, responses to questions, revisions, translations
  3. Application
    • transfer knowledge from one context to the next
    • Trigger words: demonstrate, use guides, maps, charts, etc., build, cook
    • Products: recipe, model, artwork, demonstration, craft
  4. Analysis
    • understand how parts relate to a whole
    • trouble shoot
    • understand structure and motive
    • Trigger words: investigate, classify, categorize, compare, contrast, solve
    • Products: survey, questionnaire, plan, solution to problem report, prospects
  5. Evaluation
    • judge value of something using criteria
    • support judgement
    • Trigger words: judge, evaluate, give opinion, give viewpoint, prioritize, recommend, critique
    • Products: decision, rating/grades, editorial, debate, critique, defense, verdict, judgement
  6. Synthesis
    • reform individual parts to make a new whole
    • Trigger words: compose, design, invent, create, hypothesize, construct, forecast, rearrange, imagine
    • Products: lesson plan, song, poem, story, advertisement, invention, other creative products

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The Bloom’s taxonomy levels can be used to create questions and activities at different levels of thinking.  The varied products can be used develop menus of products that match the same learning targets to differentiate instruction.  The top 3 levels can serve as extension activities for gifted students.

 

4-nowwhat
Preparation Steps
  • Analyze standards and write aligned long term and supporting learning targets
  • Determine which cognitive levels match a range of thinking that is appropriate to the learning targets
  • Use range of cognitive levels to design different options for scaffolding learning targets that can be used to differentiate instruction and offer student choice
  • Use trigger words to design good questions sequences that explore range of cognitive levels for each learning target
Early Implementation Steps
  • Initiate discussions that involve ALL students using questions sequences designed by using learning targets and thinking trigger words.  See this article for ideas on how to increase student participation.
Advanced Implementation Steps
  • Teach students Thinking Levels and associated trigger words and products.  Use this as a tool for students to ask better questions and to create alternative product choices for project.
  • Incorporate thinking level activities and learning targets into scaffolding that uses differentiated curriculum charts to offer students choices on how to learn and demonstrate mastery of learning targets

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