Problem of the Week

A number of children are standing equally spaced around the edge of a large circle and are numbered consecutively.  Abby is 14^th and Emma is 32^nd and are standing directly across from each other.  How many children are there?

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 Zeroing in on Number and Operations (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.

We look forward to seeing your responses!

Katie

 

 

Student Misconceptions...

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 Dynamic Classroom Assessment, 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”. 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.

Think about the students who submitted the following work (I am recreating it here but have not changed any of the students’ notations):

Which of the following procedures results in a larger numerical answer?

1 ½ ÷ ¾   or 1 ½ x ¾

Student A: They are the same because division is just the opposite of multiplication.

Student B: 1 ½ x ¾ because multiplication always gives a bigger answer.

Student C: They are the same because in division you just change the sign and multiply.

What was interesting is the fact that none of the students actually computed the quotient or product to allow them to check their thinking.

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.

I am convinced that conceptual understanding must be developed rather than giving students “cutesy” sayings like “keep, change, flip”.  How might we, as teachers, develop a stronger conceptual model to deepen student understanding of division and then division by a fraction?  Please post your thoughts and I will be reading them and posting a follow-up to this in the next couple of weeks.

Anne

Welcome to our blog!

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.

So, let’s look at an example. In my research, I have queried thousands of students to solve the following equation:

3 + 4 = ∆ + 5

The most common responses include   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:

3 + 4 = 7 + 5

7 + 11 +12 + 5

18 + 12 + 5

30 + 5

35

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.

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.

Let’s examine how parallel number lines might have prevented the expressed misconceptions.

                                                          3 + 4 = ∆ + 5

 

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!

Anne

Common Core State Standards, MA Frameworks and the Work of the Center

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.

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 Principles and Standards for School Mathematics 2000 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.

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.

Anne