Notes/Comments

NCTM (2000). An Overview: Principles & Standards for School Mathematics. Reston, VA: NCTM. [pp. 1-6].

Principles for School Mathematics:

·         The Equity Principle – Excellence in mathematics education requires equity – high expectations and strong support for all students. Too many students are victims of low expectations in mathematics.

·         The Curriculum Principle – A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.  Students should see the relationships among important mathematical ideas.

·         The Teaching Principle – Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. There is no one “right way” to teach.

·         The Learning Principle – Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.  Conceptual understanding enables students to deal with novel problems and settings; opposed to procedural knowledge. 

·         The Assessment Principle – Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.  Teachers must ensure that all students are given an opportunity to demonstrate their mathematics learning. 

·         The Technology Principle – Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and embraces students’ learning. Technology cannot replace the math teacher – the math teacher decides when to use it to enhance mathematical thinking.

Manouchehri, A. (2007). Inquiry-Discourse: Mathematics Instruction.  Mathematics Teacher, 101(4). 290-300,

·         Unexpected results and even false solutions can be used to enrich student learning

·         Teachers should be reflexive and responsive allowing students to direct themselves towards some learning outcomes

·         Allowing discussion time ensures that mathematical ideas become solidified in students’ minds

·         When teaching math, teachers should find a balance between teaching math facts and processes, and allowing students to develop problem solving skills, make predictions, and work collaboratively

Sullivan, P. & Liburn, P. (1997). Good questions for math teaching: Why ask them and what to ask [K-6]. Sausalito, CA: Math Solutions Publications. [pp. 3-15, 40-45].

Three features of good questions: 

·         They require more than remembering a fact or reproducing a skill – higher on Bloom’s taxonomy

·         Students can learn by answering the questions, and the teacher learns about each student from the attempt – assessment for learning

·         There may be several acceptable answers – stimulates higher level thinking

Creating good questions:

·         Working Backward – like the Understanding by Design model

o   Identify a topic

o   Think of a closed question and write down the answer

o   Make up a question that includes or addresses the answer

·         Adapting a Standard Question

o   Identify a topic

o   Think of a standard question

o   Adapt it to make a good question

Using good questions:

·         Pose the good Question – explain the question and the directions

·         Students work on the question – be a coach, don’t give unnecessary hints

·         Whole class discussion – draw conclusions, form connections

·         Teacher summary – wrap it up, finish the learning

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Weame, D., Murray, H., Olivier, A., & Human, P. (1997). The nature of classroom tasks (pp. 17-27). In Making sense: Teaching and learning mathematics with understanding.  Portsmouth, NH: Hinemann.

·         Mathematics is about problem solving so teachers should engage students in problem solving tasks not memorization

·         Tasks should encourage reflection and communication – students should buy into what they are doing, teachers should try and make tasks interesting, fun, and challenging – set the proper context

·         Tasks should allow students to use tools – a tool is like a math fact or procedure but put to use in a way that helps solve the task.  This makes the student better at using the tool because of the ‘genuine’ application

·         Tasks should leave behind important residue – some sort of learning has to be the result of the task!

Neyland, J. (1994).  Designing rich mathematical activities.  In J. Neyland (Ed.), Mathematics education: A handbook for teachers, Vol. 1. (pp. 106-122). Wellington: Wellington College of Education.

Rich Mathematical Activities:

·         Must be accessible to everyone at the start

·         Need to allow further challenges and be extendable

·         Should invite children to make decisions

·         Should involve children in speculating, hypothesis making and testing, proving, explaining, reflecting, interpreting

·         Should not restrict pupils from searching in other directions

·         Should promote discussion and communication

·         Should encourage originality and invention

·         Should encourage ‘what if’ and ‘what if not’ questions

·         Should have an element of surprise

·         Should be enjoyable

Adapting rich math tasks is a process of taking current, not-so-rich tasks and reworking them so they are higher on Bloom’s taxonomy.

Teacher involvement is a conscious decision when starting rich math tasks and can be measured on a scale:

Teacher withdrawal – Interactive teaching – Teacher intervention – Hinting – Explanation

Flewelling, G. & Higginson, W. (2000). How am I doing? Aseesing rich learning tasks (pp. 45-60, 112, 114-115). In: A Handbook on Rich Learning Tasks. Kingston, ON: Centre for Mathematics, Science, Technology and Education.

Assessment – gathering learning evidence from students

Evaluation – attaching value to the evidence

Assessments for learning give feedback to teachers and students.

Using level, rubrics, and dimensions are good ways of assessing rich tasks.  A more complicated assessment method is necessary because of the higher level of questioning.

Reys, B., & Long, V. (1995).  Teacher as Architect of Mathematical Tasks.  Teaching Children Mathematics. 296-299.

Good mathematical tasks:

·         Are authentic – students can relate to them

·         Are challenging – at an appropriate level

·         Pique curiosity – are aimed at student interests, come from student discussions

·         Encourage students to make sense of mathematical ideas – tasks require students to use math facts and procedures in more real situations

·         Encourage multiple perspectives – work in groups, share ideas

·         Nest skill development in the context of problem solving – skills are used to solve problems, not just for memorization and replication

Fuchs, L., Fuchs, D., Powell, S., Seethaler, P., Cirino, P., & Fletcher, J. (2008). Intensive Intervention for Students with Mathematics Disabilities: Seven Principles of Effective Practice. Learning Disability Quarterly 31(2). 79-92.

Seven Principles:

·         Instructional explicitness – clear, laid out instructions are more helpful than discovery for students with disabilities

·         Instructional design to minimize the learning challenge – carefully plan your instruction so that you answer questions before they arise

·         Strong conceptual basis – always start with the curriculum

·         Drill and practice – flash cards, computers, review

·         Cumulative review – use varying types

·         Motivators – extrinsic motivators help kids stay focused and hopefully lead to intrinsic motivation

·         Ongoing progress monitoring – make sure your methods are working!

Baroody, A. (2006). Why Children Have Difficulties Mastering the Basic Number Combinations and How to Help Them.  Teaching Children Mathematics, August. 22-31.

Process of Mastery:

·         Phase 1 – counting strategies – using fingers, manipulatives

·         Phase 2 – reasoning strategies – using facts and information

·         Phase 3 – mastery – fast and accurate recall

How Children Learn Basic Combinations

·         Conventional wisdom – mastery grows out of memorizing individual facts by rote through repeated practice and reinforcement

·         Number-sense view – mastery that underlies conceptual fluency grows out of discovering the numerous patterns and relationships that interconnect the basic combinations

Reasons for Children’s Difficulties

·         Conventional wisdom – difficulties are due to deficits in the learner

·         Number-sense view – difficulties are due to defects inherent in conventional instruction

Helping Children Master Basic Combinations

·         Conventional wisdom – mastery can best be achieved by well-designed drill

·         Number-sense view – mastery can best be achieved by purposeful, meaningful, inquiry-based instruction – instruction that improves number sense

To sum it up: teach kids how to understand numbers and combinations instead of teaching them to memorize them for later.

Strong, R., Thomas, E., Perini, M., & Silver, H. (2004). Creating a Differentiated Mathematics Classroom. Educational Leadership, February. 73-78.

Four Mathematical Learning Styles:

·         Mastery style – work step-by-step

·         Understanding style – search for patterns, categories, reasons

·         Interpersonal style – discussion and relationships

·         Self-expressive style – visualization and multiple approaches

Teachers should:

·         Rotate strategies – try to consistently hit on all styles

·         Use flexible grouping – put similar styles together

·         Personalize learning – encourage students to use their own style

·         Be cognizant of testing style bias – many tests favour mastery and understanding

Holden, B. (2008). Preparing for Problem Solving. Teaching Children Mathematics, January. 290-295.

Mathematical Developmental Framework:

·         Direct modeling – similar to phase one above

·         Derived fact – similar to phase 2

·         Recall – similar to phase 3

Differentiated instruction must target all three phases.  Some ideas to help achieve this in a Grade one classroom are:

·         Math Read-Alouds

·         Read-Alouds with manipulatives

·         Materials protocol – classroom management techniques

·         Describing – talking about what you’re doing!

·         Speaking and listening

·         Recording

This article is all about reflective practice: constantly ask yourself “could I be doing this better?”

Moyer, P. (2002). Are we Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics.  Educational Studies in Mathematics, 47. 175-197,

Fun-math vs. real-math – teachers consistently misunderstand the importance of including manipulatives in their math lessons.  To help students first master the concrete stage of their mathematical learning, manipulatives are necessary.  They are not simply toys that get to be played with once the pencil and paper tasks have been completed.  They are not toys, they are tools.  Also, teachers are most likely to revert to a teaching style that they experienced in their own classrooms, likely one void of manipulatives.

Richardson, K. (2004). Designing Math Trails for the Elementary School.  Teaching Children Mathematics, August. 8-14.

Math Trail – a series of stops along a pre-planned route on which students examine mathematics in the environment

EDEL 415 Math Trail:

·         Go to the cafeteria and estimate the number of litres of canned drinks in all the coolers

·         Estimate the number of floor tiles in the S wing hallways.  What is the hallway area in the S wing based on this?

·         How far is it from our room to F2003?

·         If the front security desk is your starting point, go 80 paces west, then 4 paces south, then got at a 50 degree angle up 20 paces, 8 paces east…. What is there?

·         What is the mean, median, and mode age of the people in the weight room?
·         Calculate the width to height ratio of the bulletin board

Fraivillig, J. (2001). Strategies for advancing children’s mathematical thinking. Teaching Children Mathematics, 7(8). 454-459.

Advancing Children’s Thinking Framework:

·         Eliciting – getting students to speak, explain, elaborate, collaborate, discuss, etc.

·         Supporting – giving background knowledge, coaching, provoking questions, etc.

·         Extending – holding high standards, promoting metacognition, reflection, efficiency, motivation

In a nutshell – teach kids how to think!


Vanderhye, C., & Demers C. (2008). Assessing Students’ Understanding Through Conversations. Teaching Children Mathematics, January. 260-264.

Conversations are one of the best forms of formative assessment.  Students and teachers easily discover which concepts are clear and which need more time to be developed.  Also, the act of speaking solidifies one’s learning just as much, or even more so, than the act of listening.

“Tell me and I’ll forget, show me and I may remember, involve me and I’ll understand” – Chinese Proverb

Silver, E., & Cai, J. (2005). Assessing Students’ Mathematical Problem Posing.  Teaching Children Mathematics, October. 129-135.

Problem posing, as opposed to problem solving, is the act in which students come up with questions to proposed answers or come up with questions of their own.  These can be assessed by considering their quantity, originality, and complexity.  These types of tasks are much higher on Bloom’s Taxonomy.

Example: Write three different questions that can be answered from the following information.  Jerome, Elliott, and Arturo took turns driving home from a trip.  Arturo drove 80 miles more than Elliot.  Elliot drove twice as many miles as Jerome.  Jerome drove 50 miles.

Glanfield, F., Bush, W., & Stenmark, J. (2003). Ch. 1: How do I get started? And Ch. 2: What do I assess? (pp. 4-46).  In Mathematics Assessment: A Practical Handbook for Grades K-2.

Why we assess: the more comprehensive our assessment, the better able we are to make appropriate adjustments in our teaching and to communicate students’ thinking, abilities, and accomplishments to others.

How we assess:

·         From the Latin, “to sit beside” – so sit beside your students and see what they’re doing!

·         Observation

·         Conversation

·         Looking at work; analysis

Standards:

·         Criterion referenced – program of studies

·         Norm referenced – compared to others

Begin with the end in mind! Consider what you want your students to know then design assessments and lessons from there.

Hendrickson, S., Siebert, D., Smith, S., Kunzler, H., & Christensen, S. (2004).  Addressing Parents’ Concerns about Mathematics Reform.  Teaching Children Mathematics, August. 18-23.

If things aren’t done the way they used to be done then people get uncomfortable.  Parents do not understand new math curricula so part of being a teacher is not only education your students, but also your students’ parents.  This can be done by having a presentation, a handout, have parents visit, and being open to questions.  Supporting yourself with research is important and backs your teaching style objectively.

Fagan, N. (2008). Identifying Opportunities to Connect Parents, Students, and Mathematics. Teaching Children Mathematics, August. 6-9.

Educating parents about children’s mathematical learning can be aided by the following recommendations when having a parent information night:

·         Be creative

·         Showcase students in action

·         Work with colleagues

·         FOOD!

·         Provide childcare

Rigelman, N. (2007). Fostering Mathematical Thinking and Problem Solving: The Teacher’s Role.  Teaching Children Mathematics, February. 308-314.

·         Problem performers – students who follow the rules looking for an answer without really understanding what they are doing

·         Problem solvers – understanding what the question is and developing a way to answer it using understood tools and techniques

True problem solvers have:

·         Flexible understandings of mathematical concepts

·         Confidence and eagerness to approach unknown situations

·         Metacognitive skills

·         Oral and written communication skills

·         Acceptance and exploration of multiple solution strategies