Middle School Integrated Math Overview:
Updated July 16, 2015
Document Agenda:
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Integrated Math Program Philosophy
What It Is
Individualization:
 This approach will include development of a customized, individualized math learning
approach aligned to each student’s interests and ability.
4. Why a Shift:
 The Observations Driving Motivation to Evolve our Mathematics Learning Approach
towards an Integration Model
5. Mathematical Learning Maps
6. Integrated Math Model
7. Staffing of Math Specialists
8. Challenges with Traditional High School Math Approach
9. Acera Math Approach Advantages
10. Admissions to High School
11. MS Program Overall Evaluation System
Integrated Math Program Philosophy:
We are committed to assure that all students are able to learn math in a way that fits their capacity and
ability without limits because of their age. We believe in deep, contextual and applied learning in
mathematics, and in taking an individualized approach for each student, by knowing where they are in
their math capacity, by clarify additional things to learn, and to provide opportunities for them to learn
based upon their needs, interests, and capacities. At the core, we believe that when you make exciting
learning opportunities available to students, students will readily and joyously engage in them. We will
provide guidance and scaffolding when/where/how it is needed, which will vary for different students.
What it Is:
We plan to evolve our approach to middle school student math learning – by unleashing mathematics as a
topic and focus area for engagement that can occur any time, any day and within the context of interdisciplinary projects. We see a need to no longer contain math discussion, learning, and instruction to a
time block schedule. We plan to morph our approach into a more integrated and applied mathematical
pedagogy. This evolved approach will better reflect our beliefs, promise, and commitment to honor each
students’ abilities to learn math in a way which corresponds to their capacity and needs. Additionally, it
allows for more frequent math learning and application in real scenarios, and it enables students to better
develop and apply their mathematical toolbox as problem solvers, scientists, engineers, computer
scientists, builders, economists, researchers, and artists.
Individualization:
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This approach will include development of a customized, individualized math learning approach aligned
to each student’s interests and ability. This approach includes explicit math instruction, problem sets,
and assignments given by math mentors. This program includes accountability for students to a make
progress against the plan developed at the start of the year built around their interests and around the
advisement of their math mentor. Each student will have a specific math mentor who is on point to
support clarification of each student’s mathematical learning goals and assure evaluation and tracking of
progress through the year. This model includes rich discussion about mathematics that can occur in
context and without time boxed limitations through project based work as well as traditional math
exercises, any time any day or on a regular schedule when necessary. The student will initiate the
creation of their learning goals, and the student-mentor interactions drive the progression.
Why A Shift:
The Observations Driving Motivation to Evolve our Mathematics Learning Approach towards an
Integration Model:
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Students are not as capable of applying the math skills they are being taught in their math block
classes with the agility we’d like to see. We want them to have math tools they “get out” and use
across all learning areas and projects.
We want mathematical problem solving and skills to be able to happen any time and to have
flexibility around a current student sentiment of “It’s 10:00 so math is over. No more math today” –
even if they are engaged and solving a problem. The schedule as it stands in its current form falsely
stops learning based upon a clock. At Acera we want the learning to drive the schedule, not the
schedule to drive the learning.
Acera students exhibit a false barrier about their engagement in mathematical problem solving and
dialogue –they demonstrate a bias and they act as though they should only talk about math in math
class; despite repeated encouragement and strategies employed by MS Core Teachers, they have not
been engaging whole heartedly as much in this important, core area of learning and growth outside
of math class.
Gus and Sean have observed an uptick in math avoidant behaviors over the past 4-6 months and have
felt troubled by it. So they merged their two math groups this spring to try a different approach to
mathematical learning and instruction as a way to assess whether the literature about integrated
mathematics would bear out at Acera with our students. This group of 10 students worked on the
whole class project (soap box derby race) within math block timeframes, using the larger project as a
back drop for real world problem solving, conceptual thinking and reasoning, and math skill
development. The result? This group of students now regularly reference mathematical tools and
reasoning for solving problems across all facets of the school day. Students have gotten charged and
engaged in math for problem solving and engaged in mathematical reasoning in a whole new way,
and now are now generalizing math as a tool for problem solving across other projects and
disciplines.
Mathematical Learning Maps
Next Year we plan that students will have a math menu of options (which can be viewed either in the
form of interconnected mind maps that show relationships between topics, or viewed in spreadsheet/ list
format, showing connections to particular courses (algebra, geometry, etc.). Students will identify their
interests and hopes for math learning, exploration and achievement (particularly if they have personal
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goals to “place out” of a certain level of math upon re-entry to a more traditional math program in Grade
9). They will work with a math mentor to help customize an individual math learning map relative to their
goals for the year. Each students’ particular mentor will be on point for them throughout the year to
support, coach, adapt the math map, and evaluate and track progress.
As currently being envisioned and designed (by secondary math specialist Kate Fractal in concert with
Michel Ohly math specialist and Josh Briggs engineering and math teacher, and Gus and Sean), Acera’s
math map will identify discrete topics typically learned between grade 6 and 12. The topic list is adapted
from the Common Core. The spreadsheet can be converted into an interactive graphic, a mind map that
shows interconnections of topics. This will help identify where there are necessary precursors and
progressions as well as where there are interconnected learning areas that need not be learned in a linear
fashion.
Aspects of the math map will include all different components of math, including
Foundational Middle School Math – proportional reasoning, descriptive statistics, and natural
number properties (factors and multiples, comfort with numbers.) This is the type of
mathematics most important to do and “get” well while in middle school.
Explorative Math – Includes topics that are useful and which can engender love of and
engagement in math, but which may not be typically taught in school. Includes statistics,
probability, discrete math, and more. Everywhere you could possibly go, there are places you
could explore and a student’s math coach can help them find the vistas within their
interdisciplinary projects, with extensions and extra work to deepen understanding using
activities / problem sets / projects listed on the math map.
Course Connections – Based upon a student’s goals, there may be a desire or target to “place
out” of a particular math class upon entrance to high school. There are over 100 distinct topics
on the math map / spreadsheet. Topics will be linked to courses conventionally offered – PreAlgebra, Algebra1, Algebra 2, Geometry, Pre-Calculus, Calculus.
Math engagements, discussions, and projects include emphasis on -- How can we communicate math
ideas and problem solving strategies with someone else? How can we show what we’ve learned? How
can we talk about it? How can we write it down as a proof or series of steps?
Math Map Screenshot
The screen shot below shows one small piece of the Math Map when it is in spreadsheet view. The 100+
topics on this spreadsheet mostly come from (but are not limited to) those listed on the Common Core.
They can be turned into a graphical math which shows the interconnections between math topics. The
spreadsheet scrolls to the right far beyond what is in view in this screenshot -- identifying copious
projects and activities to support learning of a particular topic area.
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Philosophical Viewpoint: Non-Linear Mathematics
Typically, secondary mathematics is taught in a linear way, yet mathematicians know that much of
mathematical thinking and learning is and can be inter related does not need to be learned in a strictly
linear way. Many school districts are evolving to teach more integrated math classes. A point of view
about this is available here: https://prezi.com/aww2hjfyil0u/math-is-not-linear/
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Integrated Math Model:
In the applied, integrated part of learning math in this approach, different students take a different path
to solve problems, based upon their skills. Math group(s) can compare approaches and develop strategies
together. In this application process, students teach and learn from each other in a way that enables
learning to be “owned” and to “stick”. At particular moments of struggle or clarity, math mentors can be
tapped or intervene and engage in explicit instruction either one-on-one or in flexibly created math
groups which can meet once or many times as a cohort. These flexible math groups will be made of
students engaging with a similar concept, idea, or path of study they have come across as part of
discussion about current events, or elective, or other project – anything they have been assigned, which
they happen to be wondering about, or when they are ready for a new, shared mathematical “vista”.
Explicit Math Instruction, Coaching, and Staying On Track to Achieve Progress
Students, MS core teachers, or math mentors are all able to reach out to establish a coaching session or
math group on a particular topic. In periodic fashion, it is expected that students and mentors will reach
out to one another to check on overall math progress towards the student’s goals for the year, to ensure
that students are engaging in the cooperative work and interdisciplinary projects which fit their goals, and
that they are otherwise engaging in a way that fits their ability, math targets, and capacities.
Mentors and MS core teachers alike will keep an eye for math avoidance behaviors to assure that
students remain on track towards meeting their math goals. . . essentially, using the math menu language,
that they are eating enough to have a healthy diet and to grow, but ideally doing so in a way that it can be
built from students’ innate hunger and via their initiative. The strategies for this will vary depending upon
the student and their needs.
Key Components:
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Start of year student initiative around their math learning goals.
Shortly thereafter, math coaching session with math mentor specialist teacher, to define an
appropriately challenging and with the right foundational learning embedded in the goals to
assure clarity on the students motivations and learning goals for the year.
Share Math as part of Listening Conference to share / get parent Input
Regular check-ins with math coach identified for this particular student, to assure they are
making progress against their plan, have problem sets and other support to continue to learn,
and to assure that they are plugging into to temporary math huddle sessions.
Math groups meet. They are built / orchestrated around shared learning goals, struggles,
moments of clarity for reinforcement, or to discover mathematical vistas that math mentors
know a group of students are ready for. A student, math mentor, or MS teacher can initiate a
math group in a spontaneous or regularly occurring fashion.
Opportunistic math group sessions occur regularly. Math groups can be one-offs or mini-courses
over a few days or within a week or two or however long may be needed. Groups will
spontaneously arise or be organized by students and/or mentors around particular topic areas
when students have aligned learning needs. These are times when students can receive direct
math instruction and support within a group of peers working on similar topics.
The best math specialist / coach for that topic will be the lead for each particular math group
session. In this way, the best talents of all specialists are accessible by all students.
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Assignments and projects may be assigned by Math mentors either within one-on-one coaching
sessions or as part of Math groups. Additionally assignments may be given to individuals or
groups
Staffing
Each student will have an assigned math mentor specialist teacher who partners with them, defining
overall math learning goals and evaluation approach through the year. The selection of the mentor who
is the right person for each student will be determined based upon alignment between the teacher the
students’ learning needs. In addition to our math specialist teachers, Gus and Sean are also math mentors
who are available in the classroom at all time. A truly integrated math program can be even stronger
because of the high level of mathematics skills of these two core teachers.
Math groups and individual coaching sessions will include explicit instruction and/or coaching and can
occur any time. Different expertise and math mentor specialists will augment the 100% availability and
math mentoring capacity of Gus and Sean. The time windows when these additional math mentor
specialists are available will be clear to the whole middle school community so that students can seek the
help they need and seek out the best person to help them out with particular needs, and also so that
math mentor specialists can schedule individual and group sessions to work on specific skills, through
particular problem sets, or on particular challenges. Students will also be expected to establish regular
one-on-one math mentor coaching and check-in sessions to assure that they are making progress on the
math learning goals for the year. Each math mentor will have clear office hour timeframes in which
meetings can be scheduled. Specific schedules for math mentors have been clarified and can evolve as
needed during the year.
THESE SCHEDULES AND ROLES ARE CONFIRMED.
Math Mentor Specialist: Kate Fractal BS Electrical Engineering, MEd Secondary Math Instruction (can
readily engage students in learning mathematics through higher levels of calculus, electrical engineering,
computer science). Office Hours for Coaching Sessions and Math Groups or Scheduled Mentoring:
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Monday 9 – 10:30am, Tuesday & Wednesday 1 – 3, Friday 9-10:30am
Kate’s Role As MS Integrated Math Co-Architect and Mentor:
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Co- Architect MS Math Learning Maps. Solicit input and implement input from MS team. Show
interconnections, precursors, and highlight that not all math need be learned in “linear” fashion.
Be a MS Math Mentor for a subset of the MS population – targeting the math students who
uniquely need and stylistically align with Kate’s mathematical thinking and capacity
Give a starting point of input to define how we will evaluate student progress against their
learning math goals for the year. Partner with other MS Math Specialists to define, implement,
and adapt an approach for evaluation of student progress within the context of Math Maps.
Kate will also offer an advanced computer science elective which will be available for both MS1
and MS2 students.
Math Mentor Specialist: Michel Ohly BA in Economics, MS Environmental Management (can readily
engage students in learning mathematics through introductory calculus, economics, environmental
studies, applied analysis) Office Hours for Coaching Sessions and Math Groups or Scheduled Mentoring:
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Mon, Tues, Thurs, Friday 9am – 10:30am
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Michel’s Role with MS Integrated Math Program:
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Give Input to develop MS Math Learning Maps (not the lead role) building topics from the
Common Core enabling Acera students to link to HS programs.
Support development of activities, problem sets and projects which link to topic areas.
Solicit and implement input from MS Core teachers on approaches to be taken.
Be a MS Math Mentor for a subset of the MS population – targeting the math students who
need and align with Michel’s mathematical capacities and applications based
knowledge/experience.
Facilitate regular mathematical learning dialogues for the team to reflect on effective math
learning in general and within the context of Integrated Math + Customized Math Maps +
Math Mentors + Flexible Grouping approach to be used within the new program.
Math & Engineering Mentor Specialist: Josh Briggs MS in Civil Engineering (can handily engage students
in learning mathematics through trigonometry, civil and applied engineering). Office Hours for Coaching
Sessions and Math Groups or Scheduled Mentoring:
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Mon, Tues, Thursday, Friday 9am – 11am
Mon, Thursday, Friday 1-1:30pm
Tuesday 1 – 3pm
Core Teachers Also Teach Mathematics In Context
Our MS Interdisciplinary Teachers will also support math learning in context, and electives offered with
include mathematical content. Core Teachers’ mathematical areas:
Gus Halwani PhD Neuroscience (can engage students in learning mathematics through calculus.
Data science, modeling, and statistics.)
Sean Ingle, emphasis in aeronautical engineering at university (can engage students in learning
mathematics approaching and including introductory calculus.)
Challenges of Traditional High School Math Approach
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It is currently structured and divided into buckets with barriers between algebra, geometry, etc.
There is a perception about the linearity of progression that is not accurate and in many ways is
inaccurate.
Fundamentally, HS math education is often not math – it is computation.
There is a lack of deep conceptual understanding. The deep conceptual understanding is very
valuable, because students can “own it” and can apply the math. The experience of “I get it now!” is
powerful and joyful and meaningful.
In the ideal world (and the one we are inventing at Acera) we do not want to focus on an approach
that is all about going down this road of traditional linear trajectory / math course progression
because we do not need to train kids to be computers; instead we want them to learn how to ask
good questions, apply in-depth knowledge and skills, and we want them to be able to have joy within
their mathematical experience.
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No matter what math class they go into after leaving Acera, if kids go into the math class asking good
questions from the basis of their deep conceptual understanding, they will get more out of their math
program, whatever form they take for their next step.
Acera Math Approach Advantages
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A significant focus on reasoning and problem solving capacity and pattern recognition, which can be
broadly applicable in the world. This is a key focus that will lead to joy in math.
It enables understanding of interconnectedness and applicability of math to all other subjects of
study– between math disciplines and across multiple disciplines out into the world.
Freedom to explore, and find discoveries yourself.
It gives room for math to be art
Ability to study topics you normally wouldn’t see until HS
Ability to work at an individualized pace that fits the students.
Math is not just one subject. There is applied math, pure math, and many ways to leverage and
engage with math.
Students will engage in mathematical study and application for more than the current minimal,
contained 4 hours per week (Standard public school schedules provide 3.75 hours of math instruction
per week. The current Math block schedule only provides 15 minutes more math instruction than the
public school model. We believe this is insufficient for secondary students engaged in mathematical
study.)
Admissions to High School
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A student who has developed deep conceptual understanding and knowledge will do well anywhere.
The SAT level of learning is typically focused on Foundational Math Level.
Any student who says, “I want to place into this class” can verify the appropriateness of this target
with their math mentor, and develop a math map that will enable this goal.
As a result of crafting a customized map, they will care about and own their own learning as well as
drive that process. They can work with their math coach to clarify their learning in an applied
fashion.
The math menu includes traditionally defined math clumps, but it also shows how they are interrelated with other areas.
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