tag:blogger.com,1999:blog-47980394881795836102024-03-21T13:00:08.469-07:00Lesley University Center for Math Achievement Written for and by math educators, this blog is a place where faculty, staff and students at the Center for Mathematics Achievement share their ideas, knowledge, and resources. Our goal is to create an open forum the encourages critical thinking where math educators can work with each other to create classrooms that encourage a love of mathematics.Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.comBlogger19125tag:blogger.com,1999:blog-4798039488179583610.post-39530011346494908792013-08-28T10:05:00.000-07:002013-08-28T10:05:05.292-07:00Incorporating the Mathematical Practices into Formative AssessmentAs the school year gets underway, most teachers are overwhelmed with the looming PARCC assessments coming in 2014-2015 with some inclusion of those types of questions being included in the 2014 MCAS and the emphasis on incorporating the Mathematical Practices in our teaching. To my way of thinking, the best way to prepare for these assessments is by changing the way in which we think about our own teaching strategies, rather than thinking about adding on to what we already do.<br /><br />Research documents the powerful impact the inclusion of effective formative assessment has on student learning. I contend that by incorporating formative assessment strategies as a component of our instructional practices, we will almost naturally foster the inclusion of the Mathematical Practices. Both formative assessments and the Mathematical Practices are processes that will foster better teaching and provide evidence that students are learning. They can and must be included in each and every mathematics lesson as we strive to improve our students’ interaction with mathematics. After all, we are seeking better outcomes for all our students.<br /><br />So, what does this all look like? Every class should begin with posing a <i>range question</i> designed to assess prior student knowledge. Think about the students who demonstrated proficiency in prior mathematics: Do they really need to review what they already know and are able to do, or should they be challenged to apply that knowledge to a more rigorous problem? By posing the range question you will be able to inform your instructional decisions which may require you to tier your instruction or the problems you assign so those students needing more support receive that support and those who need more of a challenge get that opportunity to deepen their understanding.<br /><br />The inclusion of a <i>conjecture board</i> in every mathematics class (a hypothesis board in science class) not only allows students to make conjectures and analyze the reasoning of others (MP3), but more importantly allows you, the instructor, to identify student misconceptions. This conjecture board seamlessly fits into most lessons and does not need to be added on to other protocols you may use. Once conjectures are made, students work to prove or disprove those conjectures using multiple representations and appropriate manipulatives (MP 1, MP 4). As soon as one counterexample or negation is identified, the conjecture is erased, as it is not a mathematical truth. Students as young as kindergarten and first grade are able to and do make conjectures that are appropriate for their level of cognitive development.<br /><br />One of the most important tenets of formative assessment is providing immediate feedback to students. This feedback may be oral, as students work through the problem solving process (MP 1), or written on work that students pass in. Either way, the feedback should address the strengths of the student’s thinking, and also raise questions about whether the procedure used will always work. Feedback should include, “Will that always work?” or “Can you convince me that is true?” or “ What would happen if…?”<br /><br />The inclusion of these formative assessment strategies not only will provide you the time you need to observe, listen, and gather evidence about what the students know and are able to do, but also will empower your students to self-reflect on their own assumptions, misconceptions, and understandings.<br /><br />AnneAnonymoushttp://www.blogger.com/profile/05189631772098082856noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-23631924663643764942013-06-21T08:10:00.002-07:002013-06-21T08:10:09.955-07:00What is effective teaching... Post 2Today, I am sharing the thoughts of two faculty members here at Lesley.<br />
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Steve Yurek, the Associate Director of the Center for Mathematics Achievement, is a former high school teacher. Steve taught at the high school level for 38 years and now teaches in the M.Ed in Mathematics Education (1-8) program here at Lesley University. Here are his thoughts, complete with a video:<br />
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<span style="color: #1f497d;"><span style="color: black;">"There are so many factors that
encompass effective teaching that to mention one may seem to de-emphasize any
of the others, but the reality is that we are all different, so we may favor
one technique or strategy over another, we must not ignore all the other
factors that make us great. I’ve included a youtube video</span><a href="https://www.blogger.com/null" name="_MailEndCompose"> </a></span><span style="mso-bookmark: _MailEndCompose;"></span><a href="http://www.youtube.com/watch?v=orBxCJL8N8Y"><span style="font-size: 12.0pt;">http://www.youtube.com/watch?v=orBxCJL8N8Y</span></a><span style="color: #1f497d;"> <span style="color: black;"> that encompasses this sense of balance more
graphically that I have ever seen. You may have already seen it, but it
drives home the point that all parts of a well-defined system are no more
important than another. Now, regarding my thoughts and experiences, I’ve found
that answering a student’s question with another question is initially
frustrating for the student, but ultimately guides them to the understanding
that is the point of a lesson. And how can I assess that my technique
here is effective? I usually use 2 barometers: The first is visual
- the look on the student’s face during that AHA moment, and the second is when
the student asks a follow-up question that is richer and deeper than the first.</span></span></div>
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Regarding advise for a first
year teacher: When I began teaching, I was told “Never smile before
Christmas”, but that didn’t work for me because it caused me to present a
false/forced image of myself and I was very uncomfortable and quickly realized
that I just had to be myself. <i> </i>I learned to treat my
students as fellow humans beings, in a manner that was appropriate for the grade
level, while maintaining that separation between student and teacher.</div>
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As with effective teaching,
there are so many things that first year teachers should know, but I
think that most important are (a) stay true to yourself, (b) be prepared for each class, (c)
volunteer for stuff (d) continue to learn about mathematics and about good
teaching."</div>
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If you have not seen the video, do watch the whole thing or at until 8 minutes. It all comes together in the last minute!</div>
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Barbara Allen-Lyall, current faculty member here at Lesley and a current K-12 teacher in Connecticut, answered with the following:</div>
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"<span style="color: black;">What is
effective teaching?<span style="mso-spacerun: yes;"></span></span> </div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">When content is artfully
organized and creatively presented, the flow of learning allows students to
naturally reach a succession of learning plateaus.<span style="mso-spacerun: yes;"> </span>These seemingly level places are not a time
for practice or mental rest, but rather an opportunity for super cognitive
challenge through problem solving.<span style="mso-spacerun: yes;"> </span> </span></div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">Done right, this is also
when students–not the teacher–can be heard asking the important and most
interesting questions."</span></div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">I have seen both Barbara and Steve teach in the graduate program and can honestly say that each time I watch them, I learn something new. They each have very different styles, but their passion for the field and effectiveness in their teaching is evident.</span></div>
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<br /></div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">What do you think of their advice? How would you answer the question? </span></div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";">More responses to follow! Happy Friday!</span></div>
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<span style="color: black; font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"><b style="mso-bidi-font-weight: normal;"><span style="color: #1f497d; font-family: Gabriola; font-size: 20.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-themecolor: dark2;">Katie</span></b> </span><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; mso-fareast-font-family: "Times New Roman";"></span></div>
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Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-58401935559654501502013-06-17T09:43:00.000-07:002013-06-17T09:43:25.583-07:00What is effective teaching... Post 1<!--[if gte mso 9]><xml>
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<i><b>What does effective teaching look like?</b></i></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
As the school year begins to wind down, and the summer
begins, we have been discussing how the year went.<span style="mso-spacerun: yes;"> </span>For us, this really begins with whether we
are engaging in effective teaching all year round.<span style="mso-spacerun: yes;"> </span>Effective teaching applies to not only our
courses, but also the workshops and institutes and any professional development
we are a part of.</div>
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<br /></div>
<div class="MsoNormal">
So the question is what is effective teaching?<span style="mso-spacerun: yes;"> </span>Our main cornerstone in the Math Center, is
that effective teaching is about asking the right questions and providing the right
atmosphere in the classroom.<span style="mso-spacerun: yes;"> </span>An
effective teacher knows that in order to engage a student, you cannot just show
them how to answer a problem or to solve an equation, but that you must probe
that students thinking so that they answer the question or solve the equation.</div>
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<br /></div>
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Rather than give my own definition, I asked our faculty to
share what they thought.<span style="mso-spacerun: yes;"> </span>For the next
couple of weeks, I will be posting what they see as being an effective
teacher.<span style="mso-spacerun: yes;"> </span>Today, however, I thought I would
share two important pieces of advice given to me as a new teacher.</div>
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<br /></div>
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<b style="mso-bidi-font-weight: normal;">“The person holding
the pencil is the person doing all the work.”</b><span style="mso-spacerun: yes;"> </span></div>
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All too often, when a student asks me a question, I find
myself reaching for a pencil to work out the problem.<span style="mso-spacerun: yes;"> </span>In my head, I am thinking we are working
together, however as <i style="mso-bidi-font-style: normal;">we </i>work through
the problem, I am the only one writing.<span style="mso-spacerun: yes;">
</span>One time as this was happening; the lead teacher walked by and made this
statement.<span style="mso-spacerun: yes;"> </span>At first, I have to admit, I
was little taken aback.<span style="mso-spacerun: yes;"> </span>However, I put
down my pencil (and sat on my hands) and asked guiding questions instead.<span style="mso-spacerun: yes;"> </span>It was crazy, but the student became more
confident and actually went from a struggling student, to the one who generated
the formula first.<span style="mso-spacerun: yes;"> </span>Every time I pick up
a pencil, I hear this in my head and put it down, and instead ask the student
to work through it.</div>
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<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;">“Never say anything a
student can say.”</b></div>
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This has been said over and over again.<span style="mso-spacerun: yes;"> </span>This might be the most singularly important
piece of advice ever given to me.<span style="mso-spacerun: yes;"> </span>It has
revolutionized the way I think about teaching.<span style="mso-spacerun: yes;">
</span>However, this is the hardest advice to follow.<span style="mso-spacerun: yes;"> </span>I find myself constantly rethinking how to
approach a concept without telling.<span style="mso-spacerun: yes;">
</span>Asking ten questions to a student to have them come up with the answer,
when I just want to say here it is!<span style="mso-spacerun: yes;"> </span>It’s
hard to untrain myself, to question rather than tell.<span style="mso-spacerun: yes;"> </span>To come up with lessons that do not start
with “here is how we…” and change them to “how would you…”<span style="mso-spacerun: yes;"> </span>The more I speak with other teachers, the
more I hear how easy it is to fall back in the “I do, we do, you do” model of
teaching when you are stuck.<span style="mso-spacerun: yes;"> </span>However, I
also hear how much more rewarding it is to have your students develop their own
understanding, it sticks when they get it much more than we they are told.<span style="mso-spacerun: yes;"> </span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Every couple of days, I will post advice from our faculty
and what they see as effective teaching.<span style="mso-spacerun: yes;">
</span>Please post your questions or what you think effective teaching is as
well!</div>
<div class="MsoNormal">
<br /></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-60139117444369765782013-06-04T07:21:00.000-07:002013-06-04T07:21:10.896-07:00Recap of the past year...It's been a while since we posted, and we apologize for that. Over the past few months, we have had seven courses end and three additional courses start. That's over 213 teachers taking mathematics courses this year!! We are so excited that we have so many returning teachers taking more classes and wanted to share some of the feedback we have received over the past couple of months.<br />
<br />
Before we share that feedback, we wanted to share with you some exciting news. This July, Lesley University will co-sponsor with the Association of Teachers of Mathematics in New England (ATMNE) a summer institute <i>Weaving the Mathematical Practices through the K-12 Curriculum</i>. This is a new venture for Lesley, and for the Center for Math Achievement, but we are excited to have some of the most influential mathematics educators of the region lead workshops on the mathematical practices. If you are interested or would like more information, please visit our website: <a href="http://www.lesley.edu/EventDetail.aspx?id=11530" target="_blank">Weaving the Mathematical Practices through the K-12 Curriculum</a>.<br />
<br />
So here is some of the feedback we have received:<br />
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We have had a great year, full of fun and challenging mathematics experiences. We are looking forward to an upcoming year of more mathematical fun! </div>
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If you would like more information about the Center for Mathematics Achievement or have any questions please contact us at <a href="mailto:mathachievement@lesley.edu">mathachievement@lesley.edu</a>. </div>
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Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-29525343058477374922013-04-05T08:34:00.001-07:002013-04-05T08:34:04.344-07:00Ratio, Rate and Proportion Series - Post 3To be posted soon...Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-48742997742529877992013-03-25T07:05:00.000-07:002013-03-25T11:52:05.807-07:00Ratio, Rate and Proportion Series - Post 2<!--[if gte mso 9]><xml>
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Tuesday, we introduced fraction as number and fraction as ratio on the Cartesian
coordinate plane. Both operate differently due to the continuous nature of
fraction as number and the discrete nature of fraction as ratio. What they both
have in common however is the visual representation that allows us to see what
it means to have a common denominator. Too often that is an abstract concept
that students can determine but which many just do not understand. The way in
which we introduce division can set up our students for success or for reliance
on cutesy sayings rather than helping them understand we are simply looking to
see how many groups of something we have or how many items each member of a
group will get.</div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
Using
the graphs to convert ratios to percents is also a valuable tool for students
especially when thinking about slope as percent…think about a 40% grade on a
hill. Students need to be able to move seamlessly within the ratio,
decimal/percent representations and this is yet one more model to use.</div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
On
Thursday, we really worked hard solving problems on the Cartesian coordinate plane. It is
really challenging to use a model for the first few times. None of us were
taught this as students yet the CCSS progressions insist that our students use
it as a problem solving strategy. This model really emphasizes the concept of
ratio as being discrete and answering the question "How Many?" But, what I found
most interesting is how hard it is for some of us to move to a “relative”
representation rather than an exact numerical one. This is really a stretch
since we have not been trained to think that way. What an advantage your
students will have from their exposure to thinking like this!</div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
The
highlight of the week was our field trip to the Science Museum. I was really
impressed with the depth of conversation around the different exhibits
especially the Mt Everest exhibit. I was listening to a news show this morning
and they mentioned that the base camp is 17,000 ft above sea level. Imagine!!!
Tom mentioned how cold it must be, but I was thinking about the thin air. I had
trouble breathing atop Pike’s Peak which is a measly 14,000+ feet above sea
level. Of course today hikers bring portable oxygen tanks, but imagine Sir
Hillary doing it au natural.</div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
The
other interesting point for me was seeing the difference between the pencils
made to a scale of 12:1 versus a scale of 10:1. The end products were amazing. </div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<br /></div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
I
look forward to reading your posts.</div>
<div class="MsoNormal" style="mso-margin-bottom-alt: auto; mso-margin-top-alt: auto;">
<span style="font-family: "Apple Chancery","serif";">Anne</span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com15tag:blogger.com,1999:blog-4798039488179583610.post-41908371030619253152013-03-18T11:57:00.005-07:002013-03-18T11:57:57.804-07:00Functions and Algebra I Series - Post 2<!--[if gte mso 9]><xml>
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Working with the algeblocks again to solve equations
hopefully facilitated an effective way of helping students articulate the
mathematical procedures they are doing when<span style="mso-spacerun: yes;">
</span>solving equations. Students in grade 5 on up should have these kinds of
experiences. I think you agree that if you don’t understand the fact that the
product of any two factors is a rectangle can cause angst in an algebra course
when working with binomials. Obviously, given a trinomial in which like terms
have already been calculated is a lot more difficult then when all the terms
are nicely represented in the rectangle. </div>
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<br /></div>
<div class="MsoNormal">
Factoring a variety of trinomials is
crucial in helping all students understand the differences the signs of the
trinomials represent. For instance if all the signs are positive then the
factors are all positive, x<sup>2 </sup>+ 4x + 4 or (x + 2)(x+2) for example.
Or if the first operator is negative and the second positive then the factors
have two negatives<span style="mso-spacerun: yes;"> </span>x<sup>2</sup> – 4x + 4 or (x – 2)(x - 2). But the tricky part is when you have a trinomial where you
have two negatives such as x<sup>2</sup> – x – 2 which indicates like terms
have been computed from the binomial multiplication (x-2)(x+1) . Not easy to
visualize unless you are using manipulatives.</div>
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<br /></div>
<div class="MsoNormal">
I enjoyed the excitement you displayed with the CBRs when
modeling graphs. I was impressed that many of you made up more and more
challenging graphs to model. This engenders the spirit of inquiry that we are
trying to foster with our students as well as helping to visualize what
distance graphs actually represent.</div>
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<br /></div>
<div class="MsoNormal">
Two hands on activities hopefully broke up the paper and
pencil work we have to do as well. I realize how hard some of us found the
toothpick problem and how important it is to hear from everyone as each time
one of you presented the way you solved the problem it helped others who were
stuck in their own thinking and could not change that vision in their head or
the values in the tables.</div>
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<br /></div>
<div class="MsoNormal">
The inclusion of functions and function notation is a
critical component of algebra and algebraic reasoning. So much so it permeates
the Common Core. The idea of function machines is so appropriate in the lower
grades that the transition to the notation in algebra will be seamless. We did
some work with functions but will solidify it next weekend. I felt that if I
pushed up to compositions of functions Saturday, it would have been overload so we
will begin with functions at our next session.</div>
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<br /></div>
<div class="MsoNormal">
I appreciate the emails some of you send asking for clarifications…always
happy to oblige. Your homework assignment is very clear and Katie is sending
the blogging info for those of you who still need it.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In terms of giving the homework on Sat or Sun, I need to
make sure the homework reflects what we did in class so I do need the extra day
to make it appropriate. On Sun, I am so exhausted and like you work all week
and need some family time I have to give myself permission to get the homework
to you on Monday. I hope you understand.</div>
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Another exhilarating weekend for me.</div>
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<span style="font-size: large;"><span style="font-family: "Blackadder ITC";">Anne</span></span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com19tag:blogger.com,1999:blog-4798039488179583610.post-37935482419034712002013-03-04T08:25:00.001-08:002013-03-04T08:25:20.812-08:00Functions and Algebra I Series - Post 1<!--[if gte mso 9]><xml>
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I don’t know about all of you but I was exhausted at the end
of our marathon Friday evening -all day Saturday immersion into the deeper look
at algebra. Notice that although we were doing algebra we did very little paper
and pencil procedural skills or practice. That was by design. It is my belief
that as we enter an algebra course we discard the notion that algebra is about
manipulating symbols and embrace the fact that algebra is a generalization of
arithmetic and is a logical, sequential, representation of the data that
surrounds us.</div>
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<br /></div>
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If you think about how algebra was invented back in ninth
century by the Arabic mathematician Al-Khwarizmi (whose word al jabr,
describing a common process that we use in algebra, gave us the word algebra) do
you picture a man sitting around simplifying expressions and solving equations,
or do you perceive this gentleman experimenting and investigating the phenomena
he saw around him, a man who tried to make logical sense of those phenomena and
develop a structure that would always hold true? <span style="mso-spacerun: yes;"> </span>He most certainly “played” with the
mathematics as he worked out a systemic process by making conjectures, testing
them, revising them, and testing them again. </div>
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<br /></div>
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What I hope we accomplished this weekend is that need to
model the phenomena whether it was through the water vases or the slinkies. The
inclusion of Hooke’s Law was a more formal method for discovering a general
equation that will always work. The equation y = 0.07x which most of you
discovered illustrates the <i style="mso-bidi-font-style: normal;">k constant
rate of change</i> to be 0.07 which is a ratio, the slope, and as someone in
class stated the rise over run. The problem that we worked on with the texting
options formalized how an equation with a constant rate of change can look in a
table, graph, and equation. Then the big question arose…what do you do with the
data? How do you interpret the information to make the best decision about
which text plan is the best FOR YOUR NEEDS? There really was no right answer
until the question became more specific and asked which plan was the least expensive
plan for 55 text messages?</div>
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<br /></div>
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In order to be successful with integers, it is necessary to
understand how they operate. <span style="mso-spacerun: yes;"> </span>I really
like the algeblocks because they can be used for all for operations, for
solving equations (something to look forward to), modeling multiplication on
the quadrant grid and actually illustrating a binomial times a trinomial. The
“aha” moments students have when they first get that three dimensional object
is heart- warming.<span style="mso-spacerun: yes;"> </span>We will continue
working with the algeblocks next weekend as we work with solving equations and
develop an understanding of function.</div>
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I hope you found the class interesting and engaging.</div>
<div class="MsoNormal">
<br /></div>
<span style="font-size: large;">
</span><div class="MsoNormal">
<span style="font-size: large;"><span style="font-family: "Blackadder ITC";">Anne</span></span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com11tag:blogger.com,1999:blog-4798039488179583610.post-3459132727898526142013-02-28T12:24:00.000-08:002013-02-28T12:24:57.893-08:00Ratio, Rate and Proportion Series - Post 1<!--[if gte mso 9]><xml>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Many of us successfully solve ratio problems partly
because we know how to represent them algebraically and can solve for the
unknown, and some of us because we have innate or developed number sense and
can “see” the relationships between and among quantities that are being
compared. We are lucky again partly because as adults we have been working on
these types of relationships for a long time and have been successful
implementing the algorithms we were taught. </span></div>
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<br /></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">BUT, think back to when you were
learning about ratios and did not understand what it meant to be related to
some other value. I recall being told that the ratio of two things were in a 5
: 7 ratio and there were a total of 763 items. I sat mesmerized when my math
teacher proceeded to write 5x + 7x = 763. It was like magic. At no time was I
shown a visual representation of what was going on nor was I encouraged to
think about the fact that there was a multiplicative relationship happening. </span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">When I finally realized that teaching as telling is totally ineffective I
became a deeper thinker, sought out ways in which to help students visualize,
touch and manipulate the mathematics and relate it to things they are
interested in I discovered my students, no matter how old, started to progress
in their own learning.</span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">I hope that after today’s class the inclusion of the
bar model makes sense to you and it is something that you can bring into your
own battery of teaching strategies. Modeling how the original problem looks and
comparing it to the results after an activity has occurred allows students to
make sense of a given situation and adds sense making to doing the mathematics.</span></div>
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<span style="font-size: large;"><span style="font-family: "Blackadder ITC"; line-height: 115%;">Anne</span></span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com22tag:blogger.com,1999:blog-4798039488179583610.post-30838783658925692372013-02-26T13:10:00.001-08:002013-02-26T13:10:30.600-08:00Upcoming Workshop: Rational NumbersDr. Anne M. Collins will be leading a workshop on Saturday, March 9th on Rational Numbers: Hard to Teach, Harder to Learn.<br />
<br />
The workshop begins at 9am at University Hall, 1815 Massachusetts Ave located in Porter Square.<br />
<br />
In this workshop participants will explore visual representations of
rational numbers that seamlessly develops an understanding of the slope
of a line. We will examine bar diagrams and graphing ratios. Graphing
ratios on the Cartesian coordinate plans allows participants to connect
slope to linear equations. Problems involving proportions can also be
solved using the Cartesian plane. This novel approach is required in the
CCSS and the 2011 Massachusetts Curriculum Frameworks.<br />
<br />
Cost is $50 per participant and there is still room available! <br />
<br />
To register or to get more information, follow this link <a href="http://www.lesley.edu/EventDetail.aspx?id=7995" target="_blank">http://www.lesley.edu/EventDetail.aspx?id=7995</a>.<br />
<br />
We hope to see lots of you there!Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-87564805849899798792013-02-04T13:03:00.002-08:002013-02-14T06:29:16.624-08:00Recognizing Effective Implementation of the StandardsWe hope you all enjoyed the first weekend in February. This past weekend was full of mathematics to bring into the classroom in discussions about Groundhog Day or the Superbowl! In addition, we received the February issues of our NCTM journals and are excited for what's included in those as well.<br />
<br />
Last Thursday, I participated in a class on Ratio, Rate and Proportion being held in one of our partner districts. During this class we had a really lively discussion about the relationship between fractions and ratios. Are ratios fractions? Are fractions ratios? What is the difference? How can we denote them differently? <br />
<br />
This morning, I was speaking with Anne about the class, and it reminded
her of a presentation she did at the MassMATE conference in 2012. Please find a link to it below. It will link to a Scribd site and open in a new window.<br />
<a href="http://www.scribd.com/doc/125476203/Effective-Implementation-of-the-Standards?secret_password=2gl1v8y4wqo5c39lazzr" target="_blank"><br /></a>
<a href="http://www.scribd.com/doc/125476203/Effective-Implementation-of-the-Standards?secret_password=2gl1v8y4wqo5c39lazzr" target="_blank"><span style="background-color: white;"><span style="color: blue;"></span></span></a><span style="background-color: white;"><span style="color: blue;"><a href="http://www.scribd.com/doc/125476203/Effective-Implementation-of-the-Standards?secret_password=2gl1v8y4wqo5c39lazzr" target="_blank"><u>Effective Implementation of the Standards Power Point</u></a></span></span><br />
<br />
<br />
There are some really good problems within the presentation, so I hope you enjoy it!<br />
<br />
<b style="mso-bidi-font-weight: normal;"><span style="color: #1f497d; font-family: Gabriola; font-size: 20.0pt; mso-bidi-font-family: "Times New Roman"; mso-bidi-theme-font: minor-bidi; mso-themecolor: dark2;">Katie </span></b>Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-84194708787546518892013-01-31T06:40:00.000-08:002013-01-31T06:40:19.817-08:00Congratulations Anne!The Center for Mathematics Achievement would like to offer our congratulations to our director, Dr. Anne Collins, on being awarded the Association of Mathematics Teacher Educators (AMTE) 2013 Nadine Bezuk Award for Excellence in Leadership and Service in Mathematics Teacher Education!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJqXTHYUs9t_rZfWp1TJt4KzD5cri4akwDQpRCZRtqnFWHSwpkhMo4MJ-MsOn6UOb1jvlAI8j60ofeNw5hqG9p68h258W-R0MMBmEYsODoX0Iyh1qNyAqMdJABIR65xHoxRhxIVRUh_g/s1600/AnneAMTEaward.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJqXTHYUs9t_rZfWp1TJt4KzD5cri4akwDQpRCZRtqnFWHSwpkhMo4MJ-MsOn6UOb1jvlAI8j60ofeNw5hqG9p68h258W-R0MMBmEYsODoX0Iyh1qNyAqMdJABIR65xHoxRhxIVRUh_g/s320/AnneAMTEaward.jpg" width="313" /></a></div>
From left to right: Dr. Nadine Bezuk, Dr. Anne Collins, <span class="fbPhotosPhotoCaption" id="fbPhotoSnowliftCaption" tabindex="0"><span class="hasCaption">President of AMTE Dr. Marilyn Strutchens, and Awards Committee Chair Dr. Doug Corey.</span></span><br />
<br />
We will be posting a video of Anne's presentation soon. Congratulations Anne! Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-91132678933269596722013-01-14T12:05:00.000-08:002013-01-14T12:05:01.163-08:00Developing Number Sense<!--[if !mso]>
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As our students enter middle school and beyond, too often
they lose opportunities to continue developing their number sense. Students at
this age tend to appreciate more and more the beauty behind numbers, our number
system, and the patterns and relationships among them. For this month’s blog I
am resurrecting some fun number questions from the <i style="mso-bidi-font-style: normal;">Mathematical Digest</i>, Term 1, 1994, Number 105. This mathematical
digest has a wealth of information and challenges to be solved. Enjoy!</div>
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<span style="font-family: "Blackadder ITC"; font-size: 20.0pt; line-height: 115%; mso-bidi-font-family: "Times New Roman";">Anne</span></div>
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<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVnWHoor2i6V1WjTDTvsPHpL5nzyURvavLiz9-QICNctoOsu8X_cIYqDT0b42x251Ec5TwnUx30IV8YtHGKIGJszjEMwhdMW3NtYWbCboSd94wFJQwydLj4LS47VLvBCznQI0OBNq5EA/s1600/jan_blog.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVnWHoor2i6V1WjTDTvsPHpL5nzyURvavLiz9-QICNctoOsu8X_cIYqDT0b42x251Ec5TwnUx30IV8YtHGKIGJszjEMwhdMW3NtYWbCboSd94wFJQwydLj4LS47VLvBCznQI0OBNq5EA/s1600/jan_blog.jpg" /></a></div>
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Match the clues with the numbers in the box.</div>
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<b style="mso-bidi-font-weight: normal;"><span style="font-size: 14.0pt; line-height: 115%;">Clues</span></b></div>
<ol>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">An odd cube</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">The first prime<span style="mso-list: Ignore;"> </span></span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">The fourth triangular number</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">Srinivasa Ramanujan* said that this
number was equal to (9<sup>2 </sup><span style="mso-spacerun: yes;"> </span>+ 19<sup>2
</sup>÷ 22)<sup>25</sup></span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">The second perfect number</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">The smallest odd abundant number</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">6! + 5! + 4! + 3! + 2! + 1!</span></li>
<li>T<span style="font-size: 12.0pt; line-height: 115%;">he first number after 1 to be both a
square and a triangular number</span></li>
<li>T<span style="font-size: 12.0pt; line-height: 115%;">he ninth highly composite number</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">A three digit palindromic square
number</span></li>
<li><span style="font-size: 12.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;"><span style="mso-list: Ignore;"><span style="font: 7.0pt "Times New Roman";"></span></span></span><span style="font-size: 12.0pt; line-height: 115%;">G. H. Hardy’s taxi cab’s number</span><span style="font-size: 12.0pt; line-height: 115%;"> </span></li>
</ol>
<span style="font-size: 12.0pt; line-height: 115%;">There are twelve numbers in the box for the eleven clues.</span><span style="font-size: 12.0pt; line-height: 115%;"> Which number does not have a clue written for it?</span><span style="font-size: 12.0pt; line-height: 115%;"> This number is featured in a very well-known book written in 1726.
What is the name of the book?</span><span style="font-size: 12.0pt; line-height: 115%;"> </span><br />
<br />
<span style="font-size: 12.0pt; line-height: 115%;">*Srinivasa
Ramanujan (1887-1920) has been described as the greatest mathematician India
has produced in the last 1000 years. His work has only just started to be
appreciated and understood. His formulae are being used in areas such as
polymer chemistry, statistical mechanics, computers and even cancer research. </span>
<br />
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-23175118430654004772013-01-07T06:51:00.001-08:002013-01-07T06:51:35.163-08:00Happy New Year!Lesley's Center for Mathematics Achievement has some exciting offerings in the upcoming year. These include monthly Saturday workshops, a Dine and Discuss focusing on the CCSS and PARCC, graduate level math courses in Brockton, Quincy, and Springfield, collaborations with UEI and MoS, and a Summer Institute. We are excited for the upcoming new year and continuing our work with mathematics teachers and education. We hope that you can join us for some of these events. If you want more information, you can find all of it at: <a href="http://www.lesley.edu/PageTemplate.aspx?id=2712">CMA Homepage</a>!<br />
<br />
And to start of the new year...<br />
How many factors does 2013 have? How many of the factors are prime factors?Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-57337989090904407672012-10-10T13:59:00.001-07:002012-10-10T13:59:42.626-07:00Understanding what it means to understand the structure of mathematics (use the language of the mathematical practices)<!--[if gte mso 9]><xml>
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at the vertical progressions of important concepts across the grades. One of
these concepts is how arithmetic is generalized in algebra. The algorithms we
emphasize in arithmetic may ease the transition to algebraic understanding, or
thwart that transition. The Common Core talks about the "traditional"
algorithm but not until students have opportunities and experiences to deeply
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Think, for instance, about the addition of the
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27 + 33</span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">The way in which the computation is presented may
make a difference to students. Some students will see this computation and
round 27 up to 30 and 33 down to 30 to get the sum of 60. Others will see this
computation written horizontally and need to rewrite it vertically before they
can compute:</span></div>
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<br /></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">If they understand place value they may show partial
addends illustrating the following:</span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">60</span></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Or show the regrouping method:</span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 41.4pt;" width="55">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<i style="mso-bidi-font-style: normal;"><span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">1</span></i></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 22.5pt;" width="30">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 41.4pt;" width="55">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">2</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 22.5pt;" width="30">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">7</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 41.4pt;" width="55">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+ 3</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 22.5pt;" width="30">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">3</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 3; mso-yfti-lastrow: yes;">
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 41.4pt;" width="55">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">6</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 22.5pt;" width="30">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">0</span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;"><br />
Of these methods, which do you think best relates to addition in algebra? <br />
<br />
Let's think about using place value representation to compute:<br />
<span style="mso-spacerun: yes;"> </span>2 tens + 7 + 3 tens + 3.</span></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Written this way it is clear that tens will be
combined with tens, and units with units, for a sum of 5 tens + 10. Notice
there is no temptation to mix the tens and the units.<br />
<br />
Written vertically we get</span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 54.9pt;" width="73">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">2 tens</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .5in;" width="48">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+
7</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 54.9pt;" width="73">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+ 3 tens</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .5in;" width="48">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+
3</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 2; mso-yfti-lastrow: yes;">
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 54.9pt;" width="73">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">5 tens</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .5in;" width="48">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+
10</span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;"><br />
It makes perfect sense to combine like terms. This can also be written as<br />
2t + 7 + 3t + 3, which is a standard algebraic representation and would total
5t + 10.<br />
<br />
The partial sum best models the foundational method for addition in algebra.
Now, this is not to say students should always use the partial sums, but they
do help solidify what values are being combined. Once this understanding is
mastered then we expect our students to use a more efficient method, especially
if we are adding three or four columns and rows.<br />
<br />
The same rationalization can be used with multiplication which may also include
additional links or connections to the structure of algebra. Consider the
same two values but this time think of them as factors:</span></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">27 x 33.<br />
<br />
Using partial products we get</span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<br /></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">2</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">7</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">20</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+
7</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 1;">
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">x</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">3</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">3</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">x 30</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">+
3</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 2;">
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<br /></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">2</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">1</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">7 x 3 =</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">21</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 3;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<br /></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">6</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">0</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">20 x 3 =</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">60</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 4;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">2</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">1</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">0</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">7 x 30 =</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">210</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 5;">
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">6</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">0</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">0</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">20 x 30 =</span></div>
</td>
<td style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">600</span></div>
</td>
</tr>
<tr style="mso-yfti-irow: 6; mso-yfti-lastrow: yes;">
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 18.9pt;" width="25">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">8</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">9</span></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: .25in;" width="24">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">1</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 40.5pt;" valign="top" width="54">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<br /></div>
</td>
<td style="border: none; mso-border-top-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">891</span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;"><br />
Or we could think about decomposing 27 into 30 – 3, and 33 into 30 + 3, which
would give partial products of 900 + 90 - 90 – 9, which equals 891.<br style="mso-special-character: line-break;" />
<br style="mso-special-character: line-break;" />
</span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; mso-padding-alt: 0in 5.4pt 0in 5.4pt; mso-yfti-tbllook: 1184;">
<tbody>
<tr style="mso-yfti-firstrow: yes; mso-yfti-irow: 0;">
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 58.5pt;" width="78">
<div align="right" class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in; text-align: right;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">30</span></div>
</td>
<td style="padding: 0in 5.4pt 0in 5.4pt; width: 31.5pt;" width="42">
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">-
3</span></div>
</td>
</tr>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">The mathematical structure for this computation is
the difference of two squares, <i style="mso-bidi-font-style: normal;">a</i><sup>2</sup>
– <i style="mso-bidi-font-style: normal;">b</i><sup>2</sup>. This is an arithmetic
way of showing 27 x 33 = (30 – 3) x (30 + 3) = 30<sup>2</sup> – 3<sup>2</sup>,
which equals 891.</span></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">I propose that when we are teaching arithmetic we
keep in mind the fact that algebra is the generalization of arithmetic. What do think?</span></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">I look forward to your comments,</span></div>
<div class="MsoNormal">
<span style="font-family: "Freestyle Script"; font-size: 20.0pt; line-height: 115%;">Anne</span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-68945209521482315252012-09-01T00:30:00.000-07:002012-09-01T00:30:00.894-07:00Problem of the Week<!--[if gte mso 9]><xml>
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In a week, we will post solutions for this problem. At that time we will also choose one commenter who gave the correct answer to get a copy of <i>Zeroing in on Number and Operations</i> (the commenter who wins can choose from PreK-K, 1-2, 3-4, 5-6, or 7-8). If you want to be entered into the drawing please comment with your solution, and leave your first name last initial.</div>
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<div class="MsoNormal">
We look forward to seeing your responses!</div>
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Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com1tag:blogger.com,1999:blog-4798039488179583610.post-56192706939408525522012-09-01T00:05:00.000-07:002012-09-01T00:05:00.376-07:00Student Misconceptions...<!--[if gte mso 9]><xml>
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I often wonder how effective we can be as teachers if we
don’t first get an indication of what students misunderstand. I did a lot of
work with <i style="mso-bidi-font-style: normal;">Dynamic Classroom Assessment</i>,
George Bright and Jeane Joyner’s work and they talk about four ways to think
about students’ written responses. They differentiate among “Communicating an Understanding”,
“Communicating a Misunderstanding”, “Miscommunicating an Understanding”, and “Miscommunicating
a Misunderstanding”<i style="mso-bidi-font-style: normal;">. </i>If we think
about student work in terms of these categories then I believe it also means we
need to have multiple representations for which we can determine whether there
is understanding or not, and if it is computational or procedural, conceptual
or representational.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Think about the students who submitted the following work (I
am recreating it here but have not changed any of the students’ notations):</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
Which of the following procedures results in a larger
numerical answer?</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-left: .5in; text-indent: .5in;">
<b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 115%;">1 ½ </span></b><b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 115%; mso-bidi-font-family: Calibri; mso-bidi-theme-font: minor-latin;">÷</span></b><b style="mso-bidi-font-weight: normal;"><span style="font-size: 16.0pt; line-height: 115%;"> ¾</span></b><span style="font-size: 16.0pt; line-height: 115%;"><span style="mso-spacerun: yes;">
</span>or <b style="mso-bidi-font-weight: normal;">1 ½ x ¾</b></span></div>
<div class="MsoNormal" style="margin-left: .5in; text-indent: .5in;">
<br /></div>
<div class="MsoNormal">
Student A: <i style="mso-bidi-font-style: normal;">They are
the same because division is just the opposite of multiplication.</i></div>
<div class="MsoNormal">
Student B: <i style="mso-bidi-font-style: normal;">1 ½ x ¾
because multiplication always gives a bigger answer.</i></div>
<div class="MsoNormal">
Student C: <i style="mso-bidi-font-style: normal;">They are
the same because in division you just change the sign and multiply.</i></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
What was interesting is the fact that none of the students
actually computed the quotient or product to allow them to check their
thinking.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
In talking with these students I asked them what it means to
divide. None of the three students was able to articulate that division is used
to determine how many equivalent groups into which a dividend can be arranged,
nor given a number of groups what makes a fair share for each of the groups.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
I am convinced that conceptual understanding must be
developed rather than giving students “cutesy” sayings like “keep, change,
flip”.<span style="mso-spacerun: yes;"> </span>How might we, as teachers,
develop a stronger conceptual model to deepen student understanding of division
and then division by a fraction?<span style="mso-spacerun: yes;"> </span>Please
post your thoughts and I will be reading them and posting a follow-up to this
in the next couple of weeks.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-family: "Blackadder ITC"; font-size: 18.0pt; line-height: 115%;">Anne</span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0tag:blogger.com,1999:blog-4798039488179583610.post-44984735295331769032012-08-29T00:05:00.000-07:002012-08-29T06:18:51.136-07:00Welcome to our blog!<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The
Center for Mathematics Achievement director, Anne M. Collins and staff Steve
Yurek, Katie Aspell, and Matt Rogers welcome you to our center, our blog and
our inaugural post. We are dedicated to transforming mathematics from rote
recitation, symbolic manipulation, and procedural prowess to engaging
student-centered experiences that build conceptual understanding, emphasize the
inclusion of multiple representations and knowing which representation best
represents a situation, and computations which can be justified, and through
which the reasonableness of the solutions is discussed.<o:p></o:p></span></div>
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;"><span style="mso-tab-count: 1;">S</span>o,
let’s look at an example. In my research, I have queried thousands of students
to solve the following equation:<o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt 2in; text-indent: 0.5in;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">3
+ 4 = ∆ + 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">The most common responses include<span style="mso-spacerun: yes;"> </span>3 + 4 = 7 + 5; 3 + 4 = 7 + 5 = 12; and 3 + 4
= 7 + 5 = 17. One, thankfully the only one, actually neglected to honor the
equal sign altogether and illustrated their thinking with something along the
lines of the following:<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">3
+ 4 = 7 + 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">7
+ 11 +12 + 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">18
+ 12 + 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">30
+ 5<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">35<o:p></o:p></span></div>
<br />
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">All
of the solutions are incorrect but for different reasons. I contend that if,
when equations were first introduced they were done so with balance beams, pan
balances, counters, number lines, or algeblocks the misconceptions would not
have occurred and intervention or remediation would be unnecessary. <o:p></o:p></span><br />
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Students who understand that an
equation is simply two equivalent expressions set equal to each other would
have little difficulty in solving such a simple equation. If taught, however,
as a procedure then difficulties and misconceptions arise.<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;">Let’s
examine how parallel number lines might have prevented the expressed
misconceptions.</span></div>
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<br /></div>
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt;"><span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%; mso-ansi-language: EN-US; mso-bidi-language: AR-SA; mso-fareast-font-family: Calibri; mso-fareast-language: EN-US; mso-fareast-theme-font: minor-latin;"> 3 + 4 = ∆ + 5</span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS94TMmQgyBXqxvqJRY0Cwp38e08NeOdL5-guUx0xlpctdR7CI5-oz4H9bEtHVZ5m15HbhUArenIOA8xHe0-98WXnZF0Hbszjxt_ZJAD7p3YQzTl129JQ8vAzAV7lyNZdi7OgHRwHgKQ/s1600/Number+Line+Blog.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS94TMmQgyBXqxvqJRY0Cwp38e08NeOdL5-guUx0xlpctdR7CI5-oz4H9bEtHVZ5m15HbhUArenIOA8xHe0-98WXnZF0Hbszjxt_ZJAD7p3YQzTl129JQ8vAzAV7lyNZdi7OgHRwHgKQ/s640/Number+Line+Blog.jpg" width="640" /></a></div>
<div class="MsoNormal" style="line-height: normal; margin: 0in 0in 0pt;">
<br /></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-family: "Times New Roman","serif"; font-size: 12pt; line-height: 115%;">So, as you can see, this blog intends to discuss
mathematical concepts, student misconceptions and ways in which to avoid them,
share interesting problems, and discuss any and all mathematics issues of
interest to our followers. We welcome you to comment on this post and/or to
begin a new thread. We are looking forward to our on-going discussions and
learning together as a community!<o:p></o:p></span></div>
<br />
<div class="MsoNormal" style="margin: 0in 0in 10pt;">
<span style="font-family: "Brush Script MT"; font-size: 18pt; line-height: 115%; mso-bidi-font-family: "Times New Roman";">Anne<o:p></o:p></span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com1Cambridge, MA, USA42.3736158 -71.109733542.3266938 -71.1886975 42.420537800000005 -71.0307695tag:blogger.com,1999:blog-4798039488179583610.post-56391350448078355382012-08-29T00:00:00.002-07:002012-08-29T06:21:24.759-07:00Common Core State Standards, MA Frameworks and the Work of the Center<!--[if gte mso 9]><xml>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Since it became evident that Massachusetts, as well
as 45 other states, was adopting the Common Core State Standards (CCSS) we have
struggled with how best to serve the needs of the teachers with whom we work. The
staff of the Center for Math Achievement (CMA) at Lesley University is committed to supporting the
mathematical needs of teachers in any way possible. We will continue to offer
courses on site in districts and weekend workshops, participate in dine-and-discuss
meetings with the Association of Teachers of Mathematics in Massachusetts (ATMIM), mentor/coach teachers, and publish materials that will
promote effective teaching and learning.</span></div>
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<br /></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">Sol Garfunkel, mathematician, author, and
professional development provider, in an email delivered through Jerry Becker’s
list serve describes himself as “schizophrenic” when it comes time to deal with
the CCSS. His dilemma, as with many mathematics educators, is how to reconcile
an untested set of standards which he does not support with his commitment to
best support teachers who have to implement them. I am in agreement with Sol,
who went on to articulate how well-written the 1989 National Council of Teachers of Mathematics (NCTM) standards were and how
we, as a nation, never truly implemented those standards as articulated. I
would add that the revision of those standards as articulated in <i style="mso-bidi-font-style: normal;">Principles and Standards for School
Mathematics 2000</i> have also been largely ignored. Yet, after spending time
at the International Congress of Mathematics Educators-12 (ICME-12) in Korea, I
heard from many educators from around the world how much they value the work of
NCTM and use their standards and publications religiously. In fact, many
presenters from countries which out-perform the US on international assessments
stated that the reason their students do so well is because their curriculum is
based on the NCTM standards. </span></div>
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<br /></div>
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<span style="font-family: "Times New Roman","serif"; font-size: 12.0pt; line-height: 115%;">More information on the PARCC assessment will be
released in the coming weeks and we will pass on that updated information as
soon as it becomes available. Until then, feel free to ask any questions you
might have, or to provide us with information you may have that we are missing.</span></div>
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<br /></div>
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<span style="font-family: "Blackadder ITC"; font-size: 20.0pt; line-height: 115%; mso-bidi-font-family: "Times New Roman";">Anne</span></div>
Center for Mathematics Achievementhttp://www.blogger.com/profile/09911351777477723258noreply@blogger.com0