Wednesday, August 28, 2013

Incorporating the Mathematical Practices into Formative Assessment

As 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.

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.

So, what does this all look like? Every class should begin with posing a range question 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.

The inclusion of a conjecture board 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.

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…?”

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.


Friday, June 21, 2013

What is effective teaching... Post 2

Today, I am sharing the thoughts of two faculty members here at Lesley.

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:

 "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  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.

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 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.

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."

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!

Barbara Allen-Lyall, current faculty member here at Lesley and a current K-12 teacher in Connecticut, answered with the following:

"What is effective teaching? 

When content is artfully organized and creatively presented, the flow of learning allows students to naturally reach a succession of learning plateaus.  These seemingly level places are not a time for practice or mental rest, but rather an opportunity for super cognitive challenge through problem solving.   

Done right, this is also when students–not the teacher–can be heard asking the important and most interesting questions."

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.

What do you think of their advice?  How would you answer the question?  

More responses to follow!  Happy Friday!

Monday, June 17, 2013

What is effective teaching... Post 1

What does effective teaching look like?

As the school year begins to wind down, and the summer begins, we have been discussing how the year went.  For us, this really begins with whether we are engaging in effective teaching all year round.  Effective teaching applies to not only our courses, but also the workshops and institutes and any professional development we are a part of.

So the question is what is effective teaching?  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.  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.

Rather than give my own definition, I asked our faculty to share what they thought.  For the next couple of weeks, I will be posting what they see as being an effective teacher.  Today, however, I thought I would share two important pieces of advice given to me as a new teacher.

“The person holding the pencil is the person doing all the work.” 
All too often, when a student asks me a question, I find myself reaching for a pencil to work out the problem.  In my head, I am thinking we are working together, however as we work through the problem, I am the only one writing.  One time as this was happening; the lead teacher walked by and made this statement.  At first, I have to admit, I was little taken aback.  However, I put down my pencil (and sat on my hands) and asked guiding questions instead.  It was crazy, but the student became more confident and actually went from a struggling student, to the one who generated the formula first.  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.

“Never say anything a student can say.”
This has been said over and over again.  This might be the most singularly important piece of advice ever given to me.  It has revolutionized the way I think about teaching.  However, this is the hardest advice to follow.  I find myself constantly rethinking how to approach a concept without telling.  Asking ten questions to a student to have them come up with the answer, when I just want to say here it is!  It’s hard to untrain myself, to question rather than tell.  To come up with lessons that do not start with “here is how we…” and change them to “how would you…”  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.  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.  

Every couple of days, I will post advice from our faculty and what they see as effective teaching.  Please post your questions or what you think effective teaching is as well!

Tuesday, June 4, 2013

Recap 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.

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 Weaving the Mathematical Practices through the K-12 Curriculum.  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: Weaving the Mathematical Practices through the K-12 Curriculum.

So here is some of the feedback we have received:

"Each week we were given fabulous hands-on activities that could immediately be used in the classroom.  The similar triangles exercise where we used mirrors and went outside to try and estimate the height of the light post was fantastic.  It was so much better than drawing the usual flag pole and stick figure to compare their shadows.  The giant balance scales were also a big hit and even though they are not readily available in our schools, Anne had a way to do the same thing will a ruler and a couple of paper clips." - D.P.

"I really like the double number line diagram. It is a great visual tool for thinking. I introduced the double line model (diagram) to my 6th graders, they were impressed with it. We solved about 9 problems using the double number line diagram and the students fell in love with it. It is a good tool to have to solve problems involving rates and ratios. I can see myself using next year. I can say I learn some great key mathematics skills on rate, ratio, and proportion and I am ready to pass on these skills to my students. It is my belief that my students will learn more from me and become more self-sufficient learners." - A.S.

"I have begun using the Algebolcks with this class. I told them that these were new to me and we were going to figure out how to use these together. We have only added and subtracted integers using the basic mat, but a few of my students have actually said “OHHHHHH, I really get this now.” This made me so happy!!!" - E.J.

"The making of the pencil was a very fun activity – that is what learning should be about – fun and learning." - C.L.

"People dont want to travel, and they are missing out!" -Anonymous

"It was interesting, engaging, and gave us lots of good problems to take back to the classroom." - Anonymous

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!  

If you would like more information about the Center for Mathematics Achievement or have any questions please contact us at

Monday, March 25, 2013

Ratio, Rate and Proportion Series - Post 2

I wrote an epistle reflecting on our past three sessions but somehow it did not publish. Now, I am trying to reconstruct what I wrote.

On 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.

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.

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!

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.

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.

I look forward to reading your posts.

Monday, March 18, 2013

Functions and Algebra I Series - Post 2

Working with the algeblocks again to solve equations hopefully facilitated an effective way of helping students articulate the mathematical procedures they are doing when  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. 

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,  x2 + 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  x2 – 4x + 4 or (x – 2)(x - 2). But the tricky part is when you have a trinomial where you have two negatives such as x2 – 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.

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.

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.

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.

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.

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.

Another exhilarating weekend for me.