K – 12 Mathematics Numeracy Research and Standards at a Glance

Common Core State Standards for Mathematics Content / Common Core State Standards for Mathematics
Practice 1 - 4 / Common Core State Standards for Mathematics
Practice 5 - 8 / Helpful Websites for Technology-Enhanced and Performance Based Assessments
For information on Evidence-Based information refer to the websites in this document including the one below on the Common Core.
The Common Core State Standards (CCSS) define what students should understand and be able to do in the study of mathematics. The CCSS include a focused and coherent set of standards that provide students the opportunity to achieve proficiency in key topics that are introduced in early grades and built upon in successive years. By focusing on central concepts necessary for the study of more advanced mathematics in later years, students gain greater depth of understanding.
Standards for Mathematical Content
  K-8 standards presented by grade level
  Organized into domains that progress over several grades
  Grade introductions give 2–4 focal points at each grade level
  High school standards presented by conceptual theme (Number & Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability)
Standards for Mathematical Practice (SEE NEXT TWO COLUMNS)
  Carry across all grade levels
  Describe habits of mind of a mathematically expert student
Smarter Balanced Assessment Consortium (SBAC) is writing the new state assessment to be implemented in the 2014 – 2015 academic school year. Information on the SBAC can be found at the website below.
Elementary and Middle School Items will consist of 19 Selected Response Items (22%), 3 Extended Constructed Response Items (14%), 18 Technology Enhanced Items (41%) and 2 Performance Events (23%). High School Items will consist of 19 Selected Response Items (22%), 3 Extended Constructed Response Items (14%), 18 Technology Enhanced Items (41%) and up to 6 Performance Events (24%) by grade 11. This is a preliminary blueprint of the assessment.
The average gain in learning provided by teachers’ use of formative assessment is marginally significant. Results suggest that use of formative assessments benefited students at all ability levels. Integration of real world problems, and applications of technology can have a positive impact.
The following instructional strategies for mathematics have a minimum effect size of .32 and a maximum effect size of 1.19.
·  Visual and graphic depictions of problems (.50)
·  Systematic and explicit instruction (1.19)
·  Student think-alouds (.98)
·  Use of structured peer-assisted learning activities involving heterogeneous ability groupings (.42)
·  Formative assessment data provided to teachers (.32)
·  Formative assessment data provided directly to students (.33)
(NCTM Research Brief 2007 – “Effective Strategies for Teaching Students with Difficulties in Mathematics”).
Our Mathematics and Science Center at Wayne RESA supports the research on the relationship between teachers’ mathematical knowledge and students’ achievement. We provide a mathematics institute structure in grade spans for teachers to participate in as school teams including general and special education teachers at pre-school, K – 1, 2 – 3, 4 – 5, 6 – 8 and high school. The institutes focus on content and pedagogical knowledge, creating professional learning communities and coaching support for implementation of strategies.
We have noted a clear positive impact of teachers using a document camera and projector in their classrooms to engage students in discussions by sharing their work, describing their strategies and listening to others critique their work. We also support the use of graphing calculator technology at the middle and high school levels. / 1.  Make sense of problems and persevere in solving them.
Mathematically proficient students:
·  explain to themselves the meaning of a problem and looking for entry points to its solution.
·  analyze givens, constraints, relationships, and goals.
·  draw diagrams of important features and relationships, graph data, and search for regularity or trends.
·  use concrete objects or pictures to help conceptualize and solve a problem.
·  check their answers to problems using a different method.
2.  Reason abstractly and quantitatively.
Mathematically proficient students:
·  make sense of quantities and their relationships in problem situations.
·  use quantitative reasoning that entails creating a coherent representation of quantities, not just how to compute them
·  know and flexibly use different properties of operations and objects.
3.  Construct viable arguments and critique the reasoning of others.
Mathematically proficient students:
·  understand and use stated assumptions, definitions, and previously established results in constructing arguments.
·  make conjectures and build a logical progression of statements to explore the truth of their conjectures.
·  recognize and use counterexamples.
·  justify their conclusions, communicate them to others, and respond to the arguments of others.
4.  Model with mathematics.
Mathematically proficient students:
·  apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.
·  simplify a complicated situation, realizing that these may need revision later.
·  map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas.
·  analyze those relationships mathematically to draw conclusions.
·  interpret their mathematical results in the context of the situation.
As noted on page one, the new state mathematics assessment will differ in the types of items on the new test. While the old test was completely multiple choice, the new test will include three new types of items, Extended Constructed Response Items, Technology Enhanced Items and Performance Events. Students will need practice preparing for these three new types of items. Below are some findings about curricula for which research has found a positive impact.
The United States Department of Education (USDOE) Clearing house research lists Cognitive Tutor Algebra 1 as positive/potentially positive. This is the highest rating they give to any material listed. Cognitive Tutor is one of only three programs listed - the other two are "The Expert Mathematician" and "I CAN Learn Pre-Algebra and Algebra".
The older USDOE lists these as exemplary based upon research evidence:
·  Cognitive Tutor™ Algebra
·  College Preparatory Mathematics (CPM)
·  Core-Plus Mathematics Project
·  Connected Mathematics
·  Interactive Mathematics Program (IMP)
Promising based upon research evidence:
·  Everyday Mathematics
·  MathLand
·  Middle School Mathematics through Application Project (MMAP)
·  The University of Chicago School Mathematics Project (UCSMP)
·  Number Power
Math Reports can be found at the website below.
K – 12 Exemplary Math Programs can be found at the website below.
See the website www.bestevidence.org for information on Evidence-Based Interventions. / 5. Use appropriate tools strategically.
Mathematically proficient students:
·  consider available tools when solving a mathematical problem.
·  are familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools
·  detect possible errors by using estimations and other mathematical knowledge.
·  know that technology can enable them to visualize the results of varying assumptions, and explore consequences.
·  use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students:
·  try to communicate precisely to others.
·  use clear definitions in discussion with others and in their own reasoning.
·  state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.
·  specify units of measure and label axes to clarify the correspondence with quantities in a problem.
·  calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the context.
7. Look for and make use of structure.
Mathematically proficient students:
·  look closely to discern a pattern or structure.
·  step back for an overview and can shift perspective.
·  see complicated things, such as some algebraic expressions, as single objects or composed of several objects.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students:
·  notice if calculations are repeated
·  look both for general methods and for shortcuts
·  maintain oversight of the process, while attending to the details. continually evaluate the reasonableness of intermediate results.
MARZANO (2000)
1. Vocabulary. Research indicates that student achievement will increase by 12 percentile points when students are taught 10-12 words a week; 33 percentile points when vocabulary is focused on specific words important to what students are learning. (Effect size=0.95 )
2. Comparing, contrasting, classifying, analogies, and metaphors. These processes are connected as each requires students to analyze two or more elements in terms of their similarities and differences in one or more characteristics. (Effect size=1.61)
3. Summarizing and note-taking. To summarize is to fill in missing information and translate information into a synthesized, brief form. Note-taking is the process of students’ using notes as a work in progress and/or teachers’ preparing notes to guide instruction. (Effect size=1.0 )
4. Reinforcing effort and giving praise. Simply teaching many students that added effort will pay off in terms of achievement actually increases student achievement more than techniques for time management and comprehension of new material. Praise, when recognizing students for legitimate achievements, is also effective. (Effect size=0.8)
5. Homework and practice. These provide students with opportunities to deepen their understanding and skills relative to presented content. Effectiveness depends on quality and frequency of teacher feedback, among other factors. (Effect size=0.77)
6. Nonlinguistic representation. Knowledge is generally stored in two forms—linguistic form and imagery. Simple yet powerful non-linguistic instructional techniques such as graphic organizers, pictures and pictographs, concrete representations, and creating mental images improve learning. (Effect size=0.75)
7. Cooperative learning. Effective when used right; ineffective when overused. Students still need time to practice skills and processes independently. (Effect size=0.74)
8. Setting objectives and providing feedback. Goal setting is the process of establishing direction and purpose. Providing frequent and specific feedback related to learning objectives is one of the most effective strategies to increase student achievement. (Effect size=0.61)
9. Generating and testing hypotheses. Involves students directly in applying knowledge to a specific situation. Deductive thinking (making a prediction about a future action or event) is more effective than inductive thinking (drawing conclusions based on information known or presented.) Both are valuable. (Effect size=0.61 )
10. Cues, questions, and advanced organizers. These strategies help students retrieve what they already know on a topic (PRIOR KNOWLEDGE). Cues are straight-forward ways of activating prior knowledge; questions help students to identify missing information; advanced organizers are organizational frameworks presented in advance of learning. (Effect size=0.59) / Balanced Assessment in Mathematics
MAISA Collaboration Project
Michigan’s Assessment Consortium
Michigan Department of Education – Mathematics Website
Michigan Department of Education – Educational Technology Standards and Expectations
Michigan Live Binder – A Knowledge Sharing Place for the Common Core
Michigan Mathematics Assessment Items
Michigan Mathematics and Science Centers Network (MMSCN) Mathematics Website
Michigan’s Teaching for Learning Framework
NCTM’S Illuminations Website
Smarter Balanced Assessment Consortium Resources
Wayne RESA’s Mathematics Website
What is a meta-analysis?
Meta-analysis is a particularly powerful way of summarizing large bodies of research, as it aggregates conceptually similar quantitative measures by calculating an effect size for each study. The strength of meta-analysis is that it allows consideration of both the strength and the consistency of a treatment’s effects.
What is an effect size?
An effect size reports the average difference between a group who receives an intervention and another group that does not.
Effect size is related to the standard deviations difference between the two groups. An effect size of 1.0 reports that the mean for the group receiving the intervention has moved one whole standard deviation to the right. Since 95% of the scores are within two standard deviations of the mean, this a very large effect.
The following guidelines provide a benchmark for interpreting the magnitude of an effect:
.20 = small or mild effect
.50 = medium or moderate effect
.80 = large or strong effect
Formative Assessment
Formative assessment has an effect size of .4 to .7.