Foundationsof Math

AND

Handbook for Implementation Sites

Alt+Shift, encompassing Michigan’s Integrated Mathematics Initiative, is an Individuals with Disabilities Education Act (IDEA) Grant Funded Initiative through the Michigan Department of Education, Office of Special Education.

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Table of Contents

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• What is Foundations of Math?• Why Foundations of Math?• Why something different?• What are the “Components of Number Sense”?• What is the “Prototype for Lesson Construction”?• Frequently Asked Questions: General�Who else is using Foundations of Math?� How does Foundations of Math work with our current textbooks,

math programs, and/or textbook?� How do I get others (administrators, colleagues, parents, etc.)

on board?�What kind of data will you be collecting, or have you collected, to

study the implementation of new practices?• Frequently Asked Questions: Teachers� How can I “go slow to go fast” when I have a required amount of

content to cover in a required amount of time?� The training was a good start, but how do I learn more?�Why are we teaching “academic” math when student(s) most

need to know functional and daily living skills, or at least mathfor daily living?

• Frequently Asked Questions: Parents�Why are we doing something new?�Why can’t the students just have worksheets?� If worksheets and homework are not coming home, how can I

track the progress of my child?� How will you still challenge my student who’s always been

advanced in math?�Why are you teaching “academic” math when my child most

needs to know functional and daily living skills, or at least mathfor daily living?

• Supporting Resources• References 20

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What is Foundations of Math?

Foundations of Math is a research-based training course for all educators of students with special needs (K-12), including general and special education teachers. The course is designed to develop participants’ knowledge of the mathematics they teach by seeing mathematics through the lens of a well-delineated number sense.

Participants will:

• build deep foundational content and pedagogical knowledge• learn how to make solid instructional choices that positively impact students• connect procedures used in mathematics to conceptual understanding• build mathematical understanding and accurately assess learning for a range of learners

Foundations of Math was developed at the North Carolina Department of Public Instruction by two secondary mathematics educators and two special educators. The course is intended to provide meaningful professional development to teachers that results in stronger classroom practices for those who instruct students with special needs, including an emphasis on instruction that goes deeper than techniques and strategies, and presents content in a way that increases the coherence of mathematics.

At the heart of the training is the Components of Number Sense wheel (see page 7), and the Prototype for Lesson Construction (see page 8). Practitioners implementing skills and knowledge gained in the training should be making multiple connections among the components of number sense, and should approach lesson planning using quantity, mathematical structure, and symbols as presented in the Prototype for Lesson Construction.

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Why Foundations of Math?

Student math achievement has been linked to the teacher’s own understanding of mathematics. A teacher cannot teach a student what the teacher does not know. Mathematical knowledge for teaching goes beyond passing college level math courses or an ability to calculate. Teachers of mathematics also need to “know how to use pictures or diagrams to represent mathematics concepts and procedures to students, provide students with explanations for common rules and mathematical procedures, and analyze students’ solutions and explanations” (Hill, et. al. 2005).

Certified K-12 special education teachers and K-6 general education teachers are generalists, which means they do not necessarily specialize in a subject area, and have similar mathematics methods training. Despite “mounting evidence regarding the importance of mathematical knowledge in teachers, little or no attention has been paid to developing the mathematical knowledge of special educators and their general education colleagues who work with low-perorming students" (Faulkner and Cain 2013).

In fact, certified K-12 special education teachers often have taken the same mathematics methods courses as K-6 general education teachers. This is problematic for two reasons: (a) These teachers are expected to help the struggling math student using the same sets of tools as the general education teacher and (b) they are often dealing with a higher level of content in middle and high school than what their teacher training has prepared them to teach” (Faulkner and Cain 2013). Even general education secondary math teachers may find that knowledge of upper level college mathematics is minimally helpful when working with students struggling with basic concepts and skills.

Feedback from Foundations of Math reflects this phenomenon. Teachers say the most useful information gained from the course was “Finding out what I don’t know about math,” “The different technique for solving math problems,” “[Looking at math examples] because they really helped me understand where my thinking/teaching is going wrong.” Teachers also report “I am looking forward to a deeper understanding of math concepts.”

Gaining needed content understanding, then, leads to a shift in mindset and approach. Teachers understand why, and how, to address needs of students struggling in mathematics: “I need to change my thinking on how I teach math,” “The most useful part of the day was helping me rethink how I think about math,” and “I enjoyed seeing the misconceptions that we have had/taught in math and why they are wrong.”

Foundations of Math improves teachers’ mathematical knowledge and understanding of number sense. Participating teachers, both in North Carolina and in Michigan, made significant gains in mathematical content knowledge using scores on a measure of mathematical knowledge for teaching including teacher ability to explain terms and concepts, interpret student solutions, and analyze examples of mathematical concepts (Faulkner and Cain 2013; Hill, et. al. 2005).

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Even students who meet benchmark standards in early grades may see achievement decrease as they attempt to move on to more complex mathematics if, instead of gaining a strong foundational understanding of mathematics, they were taught through memorization, procedures, and rules. International data on performance of students in the United States, national and state data show a consistent, persistent, decline in proficiency for students as grade level increases (Mullis, et. al. 2016a; Mullis, et. al., 2016b; 2015 Mathematics Results Grades 4 and 8 n.d.; 2015 Mathematics Results Grade 12 n.d.; M-STEP data retrieved from mischooldata.org).

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Why something different?

To get different results, we need a different approach. Foundations of Math has a strong research base that builds from what we know about how we learn math to how we teach math and how we have a positive impact on students.

Two factors are at play here: what our students know and can do, and what our teachers know and can do.

In buildings and districts where students, with and without disabilities, are underachieving in math, a different approach to instruction is needed.

A commonly held belief in education is that we teach in the same way we ourselves were taught, which make sense, as teachers were students, observing dozens of teachers, before deciding to become teachers themselves.

We also teach WHAT we were taught, as it is impossible to teach someone something we do not know.

The school environment, and even more specifically, math classrooms in which teachers find themselves is very different from those many may have experienced as students. Inclusion of students with disabilities, equal opportunities to learn, career and college readiness for every student, and “Every Student Succeeds” means that every student needs to meet proficiency in mathematics, not just students who show interest, who plan to engage in post-secondary education, or those who score well on math tests.

What does this mean for teachers?

In her doctoral dissertation which became the book Knowing and Teaching Elementary Mathematics, Liping Ma asked American teachers how they would approach the teaching of four basic math problems that are part of every school curriculum: subtraction with regrouping, multi-digit multiplication, division of a mixed number by a fraction, and the relationship between the area and perimeter of a rectangle.

Ma found that the vast majority of American teachers lacked the “mathematical knowledge for teaching” (as defined in Hill et. al. 2005) to explain the math conceptually. They were reliant on a memorized procedure. If student responses or questions deviated from the procedure, or what was in the textbook, the teachers were unable to offer additional instruction, investigation or assistance (Table 1). For some teachers, they were not even able to access a procedure because they had forgotten it (Ma 1999).

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Table 1: Percent of American teachers who could NOT explain the mathematical concept

Topic Percent

Subtraction with Regrouping 83

Multi-digit multiplication 70

Computation for division by a fraction 57

Representation of division by a fraction 96

Relationship between area and perimeter of a rectangle 96

When teachers were asked, for example, to divide a mixed number by a fraction, teachers who could neither correctly compute or explain the concepts

• Misapplied a procedure, e.g. convert to common denominators• Cross multiplied• Thought to invert one of the fractions, but could not remember if the dividend or the divisor was

the fraction that needed to be inverted• Admitted he/she did not know how to do the calculation at all.

To get different results we need to try a different approach.

Foundations of Math improves mathematical content knowledge for teaching, and is heavily rooted in the Components of Number Sense, which is the “what” of mathematics, and the Prototype for Lesson Construction, which is the “how” of teaching and learning mathematics.

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What are the “Components of Number Sense”?

Just as reading is not reading if all of the components of reading are not working together (i.e., fluency, vocabulary, phonemic awareness, phonics, and comprehension), math is not math if the student is not engaging in, and making connections between, multiple components of number sense.

The phrase “Number Sense” is commonly used, and means different things to different people. In this course, number sense refers to the integration of these 8 components.

In this training, the components of number sense are presented in a “wheel,” each component connected to every other component through language. Throughout the training, each component is defined, and participants learn classroom applications, diagnosis, and research related to the component.

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What is the Prototype for Lesson Construction?

The developers of Foundations of Math adapted Griffin’s “Core Image of Mathematics”, with her permission, to create the Prototype for Lesson Construction (below) which comes up over and over throughout the course. The math is taught using this prototype so that participants can see it modeled, work through it themselves, and then have time to adapt their own lessons to fit this framework.

In her research, Sharon Griffin identified a “network of knowledge” that is central to children’s mathematics learning and achievement by doing two things: enabling children to make sense of a broad range of quantitative problems in a variety of contexts, and providing the base on which children’s learning of more complex concepts can be built (Griffin 2005b).

Griffin recommends building the relationship between factual, procedural and conceptual understandings by giving students many opportunities to solve oral problems with real quantities before using formal symbols, and use number concepts in a range of contexts to learn the language that is used across contexts to describe quantity (Griffin 2005b).

Griffin (2005a) states:

“Mathematics comprises three worlds: the world of real quantities that exist in space and time; the world of counting numbers (i.e. spoken language); and the world of formal symbols (e.g., written numerals and operation signs).”

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“Math competence rests fundamentally on the construction of a rich set of relationships among these worlds…To teach math competence, it is imperative that these three worlds be continually available in the learning experiences… The development of math competence rests, fundamentally, on the development of cognitive structures that permit a child to interpret the world of quantity and number in increasingly sophisticated ways, to acquire new knowledge in this domain (e.g. to benefit from learning opportunities provided in school), and to solve the range of problems that the domain presents” (Griffin 2005a).

Arrows in the diagram show the need to first build relationships between quantities and counting numbers, and, second, extend that knowledge to build the relationship between quantity, counting numbers and discourse. Arrows between the “worlds” indicate the need to be fluent between the representations, and move among and between the worlds.

Likewise, with the Prototype for Lesson Planning Prototype, first we build the relationship between quantities and mathematical structures, and then we add in the relationship to symbols.

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Frequently Asked Questions: General

Who else is using Foundations of Math?

Alt+Shift has offered state-level trainings since 2014 in

• Lansing• Marquette• St. Johns• Bellaire• Traverse City• Novi

In 2016, Alt+Shift began offering regional, as well as state-level trainings, of Foundations of Math and Foundations of Math: Teaching Students with Significant Disabilities. This involved training cohorts of regional and state-level instructors. Since then, six regionally-based training sites have participated.

• Charlevoix-Emmet ISD• Eastern Upper Pennisula ISD• Marquette-Alger RESA• Oakland School• Washtenaw ISD• Wayne ISD

Along with training of regional instructors, Alt+Shift initiated a usability study of implementation supports. The study was conducted collaboratively in each of these regions with at least one local school, Alt+Shift, and an outside evaluator.

The Foundations of Math development team, on behalf of the North Carolina Department of Instruction, has also provided training at the request of other state departments of education in

• Kansas• Utah• Maine

How do I get others (i.e. administrators, colleagues, parents, etc.) on board?

Focusing on implementing the new practices, with fidelity, will help you to use the Foundations of Math approach effectively and have a positive impact on student learning. This is the best way to help others see the value in the approach as well.

How does Foundations of Math work with our current textbooks, math programs, and/or textbook?

Foundations of Math is not a curriculum or a program. It is an approach to instruction, and therefore overlays any existing curriculum or program. Foundations of Math is more of a philosophy that informs pedagogical decisions for teachers. When teachers have a firmer grasp of the math content they need for teaching, and understand an approach to lesson planning that follows the natural developmental sequence of how the brain learns math, they can use this information to teach any mathematical content they are asked to teach.

When teachers are able to use consistent and accurate language, respond to student noticings and inquiries in a logical and correct manner, connect content to multiple components of number sense, and represent math concepts in terms of quantity, words, and symbols, learning can happen from any textbook or math program.

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Frequently Asked Questions: Teachers

How can I “go slow to go fast” when I have a required amount of content to cover in a required amount of time?

Foundations of Math teaches participants how to “sameate” instead of “differentiate.” That means that every lesson explains math foundationally, with multiple levels of complexity, so that every student can advance his/her understanding. Going fast to go fast often defaults to utilizing procedures and doing quick calculations, which may result in moving quickly through material, but does not result in learning, especially for students with weaker working memory and slower calculation skills.

Going slow to go fast requires a leap of faith. The point at which the learning curve starts to pick up will vary with each class, teacher and student. Yet, “covering content” and keeping pace are very different than teaching math. Anecdotal evidence from Foundations of Math implementation in Michigan consistently shows that the learning curve does ultimately bend in a positive direction, and Alt+Shift will continue to study this as implementation scales up.

The training was a good start, but how do I learn more?

Sites where regional Foundations of Math training is offered should have an implementation plan in place BEFORE the training takes place. Research is clear on the impact of training. In the absence of ongoing, job-embedded follow up, only about 5% of teachers will accurately transfer what they learned from a training to a classroom. Implementation plans should include provisions for some kind of on going training to support teachers as they try the new approach and refine their practice. This may include coaching, professional learning communities, peer coaching, or some other form of sustained learning. When this is present, that transfer to practice rate goes up to 95% (Joyce and Showers 2002).

Alt+Shift will provide resources to support this ongoing training, including co-construction of implementation plans; templates, slides, and protocols for related trainings; technical assistance; and connections to other Foundations of Math implementation sites across the state.

Regardless of your school’s implementation plan, there are four actions every teacher should take to learn more about how this works with students:

1. Pick a new practice from the training on which to work (e.g., using precise mathematical language,using the Prototype for Lesson Construction)

2. Select content you will use to try out the new practice (choose a lesson that you feel comfortableteaching, content with which you are confident, and, ideally, content with which students arestruggling)

3. Work through the content as a student. What are you going to ask students to do? Can you talkabout the concept using correct language? Can you write answers that you are hoping to get fromstudents? Are you able to write story problems, create models, and explain? Are you able to comeup with multiple solutions to problems?

4. Try the new practice with a student or students.

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The best way to learn how to implement Foundations of Math pedagogy is to do it. There is always more to learn and discuss, but the best way to learn what to do and what not to do is to just try it. And then engage with your regional instructors, your colleagues, your coaches – anyone who can provide feedback, guidance and support - throughout the process.

If you want to learn more about how to adapt lessons, start by adapting a lesson that you feel comfortable with, and ask for feedback from others who have been through Foundations of Math. Use the Prototype for Lesson Construction to adapt a familiar lesson, try it with students, and reflect on the difference the new pedagogy made for you and your students. Alt+Shift has also piloted a one day training focused on developing intervention lessons using Foundations of Math pedagogy, which your regional trainer has access to.

If you want to learn more about how to use the strategies and tools introduced in the training (e.g. algebra manipulatives, fraction tiles, hundreds grids, 1 inch tiles, base ten mats, etc.), revisit the sample problems from the training to see if you can recreate the solutions, and then try out a couple of new sample problems. When you get stuck, engage the help of others who have been through the training, post a question on the listserv, contact your regional trainer, or contact Alt+Shift via email (info@altshift) or the website for assistance.

If you want to learn more about what this looks like in a real classroom, e.g., a classroom with a wide range of abilities, with many students achieving well below their grade level, with many students with disabilities, with co-teachers and paraprofessionals who have not yet been through the training, etc., engage in the four steps listed near the top of this answer, and then post to the Foundations of Math listserv, contact your regional trainer, or contact Alt+Shift directly. There are many ways to connect to others in the Foundations of Math community to find someone with information relevant to your needs. You can also learn more about how it works in your classroom by trying it out. Choose one lesson that you feel comfortable with and try “sameation.” Utilize the Prototype for Lesson Construction and reflect on what went well and where it could be improved.

Why are we teaching “academic” math when the students most need to know functional and daily living skills, or at least math for daily living? (this question also appears in “Frequently Asked Questions: Parents”)

We teach math to students with disabilities, including those with significant disabilities, for the same reasons we teach other people math.

When a distinction is made between “academic” math and “functional” math, the difference often refers to teaching math as an academic subject versus teaching math that is used in every day life. For example, functional math may be considered time, money, calendar, cooking, or keeping a daily schedule. Academic math would encompass, for example, the components of number sense.

To the extent that mathematics is how we make sense of our world, learn to navigate ourselves in it, and describe our surroundings, “academic” math is in itself a functional and daily living skill, and certainly enables the learning of other skills and activities that often fall into those categories.

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Children begin their math education through early sensory experiences, integrate those into understanding of quantity and magnitude, and eventually give names and symbols to those quantities as they learn to use and manipulate those concepts to engage in, and describe their world using, increasingly sophisticated mathematics.

In his book Mathematics for Children with Severe and Profound Learning Difficulties, Staves (2012) provides ample reasons for providing a mathematics education to every student. His reasons apply equally to students who are typically developing as students with significant learning disabilities:

Mathematics is

• Important to communication – Its language expresses the nature and order of things. Wecannot live without descriptions of quantities, space and time.

• Important in our practical lives – Its concepts are the basis of many of our practical skills.• Important in helping us understand relationships – We use it to describe and compare things,

all its parts are interrelated, e.g. addition and subtraction; its concepts help us understand theworld around us.

• Important because it helps us be systematic – With its structures and tools we can record,bring order and remember our observations. Its patterns and rules help us recognize what weknow, and predict what might happen.

• Important because it is a tool for our imagination – We express ourselves and are emotionallyaffected by its patterns, they affect us in music, movement, the visual and tactile arts. A waltz feelsdifferent to rock and roll.

• Important because it fascinates us – Even though many people fear abstract mathematicallanguage and processes, they are fascinated by patterns, comparisons, changes to quantities,etc. and are interested in anticipating and predicting outcomes.

For our students with developmental delays, and those with significant sensory impairments, intellectual stimulation can be a motivating factor that stimulates development. “Feeling an inherent reward from an activity or the content of learning is an intrinsic stimulus that motivates the cycle of development, it generates the desire to repeat, rehearse, practice and advance” (Staves 2012).

When math is taught developmentally, with quantity first, by teachers possessing the mathematical content knowledge needed for teaching, students can progress in their understanding. Students with disabilities learn math the same way their peers without disabilities do, although not necessarily at the same pace, with the same materials, or using the same strategies. For the reasons stated above, giving every student an opportunity to progress in their mathematics understanding every day, week, month, and year is a worthy endeavor, and will support daily living and quality of life.

Teaching students to name numbers from flash cards, match vocabulary to pictures or definitions, read a clock, count money, or say the date from a calendar is not the same as teaching math. Teaching students about meal time, food preparation, to look in a certain direction, to attend to a lesson, or to keep to a routine is not the same as teaching math.

Mathematics is, however, strongly embedded in the content of each of these activities, and it is in the teaching of that mathematics that students will be able to successfully use numbers, vocabulary, money and measurement instruments, along with a myriad of other math skills learned along the way.

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Frequently Asked Questions: Parents

Why are we doing something new?

Referenced in previous sections (see “Why Foundations of Math?” and “Why something different?”) are two important reasons a school would try Foundations of Math training. First, students are underachieving in math, and even where elementary students, as a whole, seem to be achieving proficiency, we know that many of them will fall below proficiency levels by high school. Second, if the current approach is not working, and it would not be if proficiency levels are less than desirable, you need to seek alternative methods to achieve a different result.

Why can’t the students just have worksheets?

Parents may find that teachers who have been through Foundations of Math training send fewer worksheets home with students, which can happen for a variety of reasons.

When worksheets might be used by teachers who have been through Foundations of Math training:

• worksheets allow students to engage with mathematical concepts• worksheets provide opportunities for students to practice newly learned skills• worksheets are used to analyze thinking, address misconceptions, and make instructional

decisions

When worksheets are unlikely to be used by teachers who have been through Foundations of Math training:

• worksheets are used for students to replicate a memorized procedure• worksheets are a way for students to complete work related to a topic• worksheets are used to generate a score that is recorded in a grade book• the turning in of one set of worksheets clears the way for the next round of worksheets to be

handed out

Worksheets support teaching and learning to the extent that they are used for teaching and learning, just like any tool, material, or textbook. In Foundations of Math, participants learn how to use a variety of teaching tools, including discussion, examples, multiple representations, and the use of manipulatives, so the presence of worksheets may be decreased as the presence of other means of teaching and learning increase.

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If worksheets and homework are not coming home, how can I track the progress of my child?

Many teachers use technology to maintain a home-school connection. Classroom websites, district systems like MiStar, and e-newsletters are all ways teachers let families know what is being taught and how their students are progressing.

Another way to monitor is to look at completed work coming home. This will show you what types of problems your child is working on, and how he/she is doing with them.

For students who bring home textbooks, or have online textbooks, you may ask your child to show you where they are in the book and what they have been doing with that material in class. This would be a nice opportunity to have a conversation about what they are learning, at the same time being able to ask some questions to give you an idea of how well they are understanding the information.

Finally, you can contact your child’s teacher directly. A quick email to a teacher letting him/her know that you have not seen anything come home lately and would appreciate a grade check, and a list of any returned work that you should be looking for, could provide the information you need.

How will you still challenge my child who’s always been advanced in math?

Teaching math using the components of number sense and Prototype for Lesson Construction requires teachers to stretch their understanding of mathematics itself, beyond memorized rules and procedures, to be able to provide contexts and quantity-based explanations for increasingly complex concepts, as well as model real world situations using numerals and symbols.

The same is true for students.

In fact, in classrooms where Foundations of Math is implemented, teachers commonly report that the students who usually do well on math assignments are more challenged than their “average” and underachieving peers by having to demonstrate conceptual understanding.

For example, learning three steps to subtracting with negative numbers is much different than knowing contexts when subtracting with negative numbers is needed, or being able to take a problem that involves subtraction with negative numbers and translate it into a meaningful representation. This is even truer as students move into secondary math.

Remember, just as reading is not reading if all of the components of reading are not working together (fluency, vocabulary, phonemic awareness, phonics, and comprehension), math is not math if the student is not engaging in, and making connections between, multiple components of number sense (as referenced in “Components of Number Sense” on page 7).

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If “advanced” status in math is determined by student performance primarily involving calculations, including very sophisticated calculations, the “advanced” status means that the student completed calculations at a rate higher than his peers. The advanced status may qualify students to be placed in classes that teach skills and procedures related to topics usually taught to students in higher grades.

Being skilled at computation will certainly help students in mathematics. Increasingly, however, being college and career ready also means being able to analyze a situation and apply mathematics. In fact, students and workers are more and more unlikely to do calculations themselves. They will more likely use calculators, specialized software, or even internet search engines to do thecalculations for them.

This dichotomy is well described in a report from the 2012 Programme for International Student Assessment (PISA). These results show that the United States “has particular strengths in cognitively less-demanding mathematical skills and abilities” such as “extracting single values from diagrams or handling wellstructured formulae.” But the United States has “particular weaknesses in items with higher cognitive demands, such as taking real-world situations, translating them into mathematical terms, and interpreting mathematical aspects in realworld problems” (2012). While this is good news on some level (the United States is doing well with procedural knowledge), the work force is not in need of people who are good at using formulas. They need people who know when to use those formulas and which models or situations fit those formulas. They need people who are problem solvers, creative thinkers, and able to apply mathematics understanding in ambiguous situations. Teachers will challenge “advanced” students by teaching them how to successfully engage in cognitively demanding mathematical skills and abilities.

Why are you teaching “academic” math when my child most needs to know functional and daily living skills, or at least math for daily living? (this question also appears in “Frequently Asked Questions: Teachers)

We teach math to students with disabilities, including those with significant disabilities, for the same reasons we teach other people math.

When a distinction is made between “academic” math and “functional” math, the difference often refers to teaching math as an academic subject versus teaching math that is used in every day life. For example, functional math may be considered time, money, calendar, cooking, or keeping a daily schedule. Academic math would encompass, for example, the components of numbersense.

To the extent that mathematics is how we make sense of our world, learn to navigate ourselves in it, and describe our surroundings, “academic” math is in itself a functional and daily living skill, and certainly enables the learning of other skills and activities that often fall into those categories.

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Children begin their math education through early sensory experiences, integrate those into understanding of quantity and magnitude, and eventually give names and symbols to those quantities as they learn to use and manipulate those concepts to engage in, and describe their world using, increasingly sophisticated mathematics.

In his book Mathematics for Children with Severe and Profound Learning Difficulties, Staves (2012) provides ample reasons for providing a mathematics education to every student. His reasons apply equally to students who are typically developing as students with significant learning disabilities:

Mathematics is

• Important to communication – Its language expresses the nature and order of things. We cannot live without descriptions of quantities, space and time.

• Important in our practical lives – Its concepts are the basis of many of our practical skills.• Important in helping us understand relationships – We use it to describe and compare things,

all its parts are interrelated, e.g. addition and subtraction; its concepts help us understand the world around us.

• Important because it helps us be systematic – With its structures and tools we can record, bring order and remember our observations. Its patterns and rules help us recognize what we know, and predict what might happen.

• Important because it is a tool for our imagination – We express ourselves and are emotionally affected by its patterns, they affect us in music, movement, the visual and tactile arts. A waltz feels different to rock and roll.

• Important because it fascinates us – Even though many people fear abstract mathematical language and processes, they are fascinated by patterns, comparisons, changes to quantities, etc. and are interested in anticipating and predicting outcomes.

For our students with developmental delays, and those with significant sensory impairments, intellectual stimulation can be a motivating factor that stimulates development. “Feeling an inherent reward from an activity or the content of learning is an intrinsic stimulus that motivates the cycle of development, it generates the desire to repeat, rehearse, practice and advance” (Staves 2012).

When math is taught developmentally, with quantity first, by teachers possessing the mathematical content knowledge needed for teaching, students can progress in their understanding. Students with disabilities learn math the same way their peers without disabilities do, although not necessarily at the same pace, with the same materials, or using the same strategies. For the reasons stated above, giving every student an opportunity to progress in their mathematics understanding every day, week, month, and year is a worthy endeavor, and will support daily living and quality of life.

Teaching students to name numbers from flash cards, match vocabulary to pictures or definitions, read a clock, count money, or say the date from a calendar is not the same as teaching math. Teaching students about meal time, food preparation, to look in a certain direction, to attend to a lesson, or to keep to a routine is not the same as teaching math.

Mathematics is, however, strongly embedded in the content of each of these activities, and it is in the teaching of that mathematics that students will be able to successfully use numbers, vocabulary, money and measurement instruments, along with a myriad of other math skills learned along the way.

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Supporting Resources

For more information on topics in the course, including using correct math language, the Components of Number Sense, and the Prototype of Lesson Construction, visit our Foundations of Math resource page. (tinyurl.com/yy8ru4p6) Articles include:

• The Components of Number Sense: An Instructional Model for Teachers• Nix the Tricks (focused on math language)• 12 Rules that Expire (focused on math language)• 13 Rules that Expire (focused on math language)• Doing the Critical Things First (an interview with Sharon Griffin)

The ANIE: A Math Assessment Tool That Reveals Learning and Informs Teaching is a book by Kevin Bird and Kirk Savage that describes how to use the ANIE to assess student understanding of math concepts in a manner consistent with Foundations of Math pedagogy.

The Kansas Association of Teachers of Mathematics (KATM) “flip books” provide clarification of, and suggested instructional activities to teach, the Common Core State Standards for Mathematics, much of which is consistent with Foundations of Math pedagogy. For educators looking for support in using the Prototype for Lesson Construction in content areas not covered in the course, these flip books may provide helpful information. (tinyurl.com/y4tkl7ze)

References

2015 Mathematics Results Grades 4 and 8. (n.d.). Retrieved November 30, 2016, from http://www.nationsreportcard.gov/reading_math_2015/#mathematics?grade=4

2015 Mathematics Results Grades 12. (n.d.). Retrieved November 30, 2016, from http://www.nationsreportcard.gov/reading_math_g12_2015/#/

Faulkner, V. N., & Cain, C. R. (2013). Improving the Mathematical Content Knowledge of General and Special Educators: Evaluating a Professional Development Module that Focuses on Number Sense. Teacher Education and Special Education, 36(2), 115-131. doi:10.1177/0888406413479613

Fleischman, H.L., Hopstock, P.J., Pelczar, M.P., and Shelley, B.E. (2010). Highlights From PISA 2009: Performance of U.S. 15-YearOld Students in Reading, Mathematics, and Science Literacy in an International Context (NCES 2011-004). U.S. Department of Education, National Center for Education Statistics. Washington, DC: U.S. Government Printing Office.

Griffin, S. (2005a). The development of math competence in the preschool and early school years: Cognitive foundations and instructional strategies. In J.M. Royer (Ed.), Mathematical Cognition. Greenwich, CT: Information Age Pub.

Griffin, S. (2005b). Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades. In How Students Learn: Mathematics in the Classroom. National Research Council.

Hill, H. C., Rowan, B., & Loewenberg Ball, D. (2005, Summer). Effects of Teachers’ Mathematical Knowledge for Teaching on Student Achievement. American Educational Research Journal, 42(2).

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