
Comprehension of mathematical concepts, operations, and relations 


Skill in carrying out procedures flexibly, accurately, efficiently, and appropriately 


Ability to formulate, represent, and solve mathematical problems 


Capacity for logical thought, reflection, explanation, and justification 


Habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy 


Learners are creators of their own learning, give meaning to things they perceive or think about 

Instrumental understanding 

Ideas have been memorized, unlikely to be useful for constructing new ideas. (Knowing the skill) 


Each new concept is not only learned but is also connected to many existing ideas, so there is a rich set of connections 

How much to tell and not to tell 

Introduce mathematical conventions: symbolism and terminology should be introduced after concepts have been developed and then specifically as a means of expressing or labeling ideas Discuss alternative methods: propose method and identify it as “another” way not the only or the preferred way Clarify students’ methods and make connections 

Threephase lesson format 

Before: preparing students to work on the problem. When planning, analyze the problem to anticipate possible misconceptions and plan questions During: students explore the problem. Opportunity to find out what your students know, how they think, and how they are approaching the task you have given them After: students work as a community of learners, discussing, justifying, and challenging various solutions to the problem. Teacher is reinforcing precise terminology, definitions, or symbols 


Planning lessons around meaningful content, grounded in authenticity, recognizing each student’s readiness, interest, and approach to learning and connecting content and learners by modifying content, process, product, and the learning environment 


Student’s readiness typically informs the level of complexity or depth at which the content is initially presented for different groups of students. Interest and learning profiles tend to inform differentiation geared towards breadth. 


Teachers can use different strategies or encourage students to take different “roads” to increase access to the essential information, ideas, and skills embedded in a lesson 


Allows a variety of ways for students to demonstrate their understanding of essential content 

Differentiating Learning Environment 

Attending to students’ needs which affects the seating arrangement, specific grouping strategies, access to materials, and other aspects of the classroom environment 


Two or three tasks that focus on the same big idea but offer different levels of difficulty 


A question that can be solved in a variety of ways or can have different answers 


You set the same goal for all students, but different pathways are provided to reach those learning goals. You must first decide which category to tier: content, process, or product. Characteristics: address the same learning goals, require students to use reasoning, and are equally interesting to students (content examples: parallel tasks and open questions) 


Degree of assistance, Structure of the task, complexity of the task(s), and complexity of process 

Mathematics as a Language 

Conceptual knowledge (what division is) is universal, however, procedures (how you divide or factor) and symbols are culturally determined and are not universal 

Culturally Responsive Instruction 

For all students! Includes consideration for content, relationships, cultural knowledge, flexibility in approaches, use of accessible learning contexts, a responsive learning community, and working in crosscultural partnerships. Strategies for differentiation: communicate high expectations, make content relevant, communicate the value in students’ identities, and model shared power. 

Response to Intervention Tier 1 

Represents the core instruction for all students based on highquality mathematics curriculum, instructional practices, and progressmonitoring assessments (8090%) 

Response to Intervention Tier 2 

Represents students who did not reach the level of achievement expected during tier 1 instructional activities. Receive supplemental targeted instruction outside the core mathematics lessons that uses more explicit strategies, with systematic teaching of critical skills and concepts, more intensive and frequent instructional opportunities, and more supportive and precise prompts (510%) 

Response to Intervention Tier 3 

For students who need more intensive periods of instruction, sometimes with oneonone attentions, which may include comprehensive mathematics instruction or a referral for special education evaluation or special education services (15%) 

Oneway communication strategies 

letters sharing the goals of a unit, websites where resources and curriculum information are posted, and newsletters 

Twoway communication strategies 

log of student work (signed or commented on by parent), PTA meetings/open houses, and oneonone meetings, class or home visits 

Three (or more)way communication strategies 

family math nights, conferences (with parent and child), and log/journal of student learning with input from student, parent, and teacher 

Fraction Misconception: students think that the numerator and denominator are separate values and have difficulty seeing them as a single value 

Help:Find fractions on a number line, measure with inches to various levels of precision, avoid the phrase “three out of four” or “three over four”; instead say “threefourths” 

Fraction Misconception: students do not understand that 2/3 means two equalsize parts 

Help: students need to create their own representations of fractions across various types of models, provide problems where all of the partitions are not already drawn 

Fraction Misconception: students think that a fraction such as 1/5 is smaller than a fraction such as 1/10 because 5 is less than 10. Conversely, students may be told that the bigger the denominator, the smaller the fractions leading to the belief that 1/5 is more than 7/10 

Help: many visuals and contexts that show parts of the whole, ask students to provide their own example to justify which fraction is larger 

Fraction Misconception: students mistakenly use the operation “rules” for whole numbers to compute with fractions (1/2 + 1/2 = 2/4) 

Help: ask students to connect to a meaningful visual or example, include estimation in teaching operations with fractions 

Fraction Constructs: PartWhole 

part of a whole: shading a region, part of a group of people, part of a length (how many parts) 

Fraction Constructs: Measure 

involves identifying a length and then using that length as a measuring unit to determine the length of an object (how much or how long) 

Fraction Constructs: Division 

the idea of sharing one whole with another whole (sharing $10 with 4 people). Students should understand and feel comfortable with the various written representations of the division of a fraction 

Fraction Concepts: Operator 

used to indicate an operation such as a fraction of a whole number (2/3 of the audience was holding banners). requires mental math to determine the answer 


PartWhole and PartPart constructs. students have to use the context to make sense of the partpart and partwhole relationships 


emphasizes the concept of partwhole and avoids the confusion with division. 


“repeating a process”; counting fractional parts to see how multiple parts compare with the whole helps students understand the relationship between parts and the whole (length models) 

Connecting Fractions and Decimals 

A decimal is a special case of a fraction in which the denominator is part of the baseten number system, so it can be written in this was as a convention. For example 0.03 is “threehundredths”, which can be written as 3/100 or as 0.03. Ways to ensure connection: say decimal forms correctly and use fraction models 


Longer is Larger (more digits is largest) Shorter is Larger (because the digits far to the right represents very small numbers, longer numbers are smaller) Internal zero (confusion with zero in tenths position, zero has no impact) Less than zero (since 0 is a whole number it is greater than a decimal fraction) Reciprocal thinking (incorrectly converting to fractions) Equality (don’t integrate the idea of regrouping decimals) 


A ratio relating one part of a whole to another part of the same whole 


Ratio expressing a comparison of a part to a whole 


Ex: if you can buy 4 kiwis for $1.00, the ratio of money for kiwis is $1.00 to 4 kiwis 


Involve two different units and how they relate to each other. Miles per gallon, square yards of wall coverage per gallon of paint, passengers per busload, and roses per bouquet.
*A rate represents an infinite set of equivalent ratios 

Proportional Reasoning Strategy: Reasoning 

Unit rate and scale factor can be used to solve many proportional situations mentally 

Proportional Reasoning Strategy: Ratio Table 

Show how two variable quantities are related. Serve as tools for applying buildup strategy but can also be used to determine unit rate. Neither variables nor equations are needed so it is less abstract than using proportions 

Proportional Reasoning Strategy: DoubleLine Comparison 

A line segment is partitioned in two different ways or increments. 

Proportional Reasoning Strategy: Percents 

All percents can be set up as equivalent fractions. Using a line segment, represent the number or measures in the problem on one side of the line and indicate the corresponding values in terms of percents on the other 

Proportional Reasoning Strategy: Cross Products 

When using cross products, students should be encouraged to reason in order to find the missing value, rather than just to apply the crossproducts algorithm. (ex. create a visual, solve the proportion) 

Van Hiele Levels of Geometric Thought 

Level 0: Visualization: shapes and what they look like to groupings of shapes that seem to be “alike” Level 1: Analysis: classes of shapes rather than individual shapes to properties of shapes Level 2: Informal Deduction: Properties of shapes to the relationships among properties of geometric objects Level 3: Deduction: relationships among properties of geometric objects to deductive axiomatic systems for geometry Level 4: Rigor: deductive axiomatic systems for geometry to comparisons and contrasts among different axiomatic systems of geometry 




1. decide on the attribute to be measured 2. select a unit that has that attribute 3. compare the units – by filling, covering, matching, or using some other method – with the attribute of the object being measured. The number of units required to match the object is the measure 

Tips for Teaching Measurement Estimation 

1. create lists of visible benchmarks for students to use 2. discuss how different students made their estimates 3. accept a range of estimates 4. encourage children to give a range of estimates that they believe includes the actual measure 5. make measurement estimation an ongoing activity 6. be precise with your language, and do not use the word “measure” interchangeably with the word “estimate” 


Bar Graphs and Pie Charts are the only graphs that can be used for nonnumerical data 


Bar Graphs, Pie Charts, StemandLeaf Plots, Line Plots and Dot Plots, Histograms, and Box Plots 

Mean: Leveling Interpretation 

A number that represents what all of the data items would be if they were leveled (size of leveled bars is the mean) 

Mean: Balance Point Interpretation 

measure of the “center” of the data (move data points in toward the center or balance point without changing the balance around that point. When you have all points at the same value, that is the balance, or the mean) 

Probability: Likely or Not Likely 

Focus on impossible, possible, and certain to have students make predictions about how likely a particular occurrence is 


a measure of the chance that the event will occur 


the likeliness of an event happening based on all possible outomes 


Some probabilities cannot be determined by the analysis of possible outcomes of an event; instead, they can be determined only through an experiment, with a sufficiently large number of trials conducted to become confident that the resulting relative frequency is an approximation of the theoretical probability 


the occurrence or nonoccurrence of one event has no effect on the other 


a dependent event is a second event whose result depends on the result of a first event 

Probability Misconceptions 

1. commutativity confusion: the combination of two girls and one boy can be written as three events, not one. 2. gambler’s fallacy: the notion that what has already happened influences the event 3. law of small numbers: students expect small samples to be like the greater population 
