Math Interview Assessments

April 4th, 2009

Recently I had the pleasure of being asked to present at the 34th ANNUAL CONFERENCE sponsored by ORBIDA (Oregon Branch of the Dyslexia Association of America). My audience was made up primarily of special educators from local area schools. I was asked a great question. I thought I would share it here with the readers of my Blog and also tell you what my response was. The questions was whether I had identified any key concepts children needed to have in place in order to be successful in math. If I thought that there were these concepts, the question then became whether I used any of them as assessments to identify students who might need targeted interventions.

My answer was affirmative that I believe there were key concepts and that that they could be used to identify students who were likely candidates for intervention. I then went on to describe examples for my second grade students. This is the list I shared with that group:

1) The first concept is number sequence and seriation. I assess this concept by asking my students to count for me as high as they can go. Any of my students who cannot count to 100 raise a level of concern. If they can count to at least 100, I then ask the student to tell me what comes before or after targeted 2 digit numbers. Difficulty with this task also raises concern. Either students are having language difficulties or haven’t picked up the patterns present in our number system. Handing students a set of counters and asking them to count them backwards helps me check to see if it is a language issue or difficulty with the patterns of counting. This one also gives me some information about the child’s innate problem solving abilities. Sometimes the child will stare and me and not know how to proceed, other students will start with any number that comes to mind and count backwards from there.

2) Next, I check to see if students are able to subitize. For those of you for whom subitize might be a new word, the following definition comes from Wikipedia. “Subitizing, coined in 1949 by E.L. Kaufman et al. refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus (meaning sudden) and captures a feeling of immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range.”

To assess this concept, I show students random pictures of sets of objects representing the numbers from 1 to 20. My object is to see if they are able to instantly recognize and identify the totals for these sets. The items represented in the sets have been arranged in ways that make it easy for students to group subsets and quickly calculate the total. For example, they may be grouped in chunks of fives or tens plus some extras. Any students having trouble instantly recognizing the total amounts in these sets would raise a red flag.

3) Next on the list is a check to see whether students have the ability to conserve numbers. For this check, I hand students a handful of around 12 two-colored markers. Students shake them up and drop them. I ask them to tell me how many of each color there are and how many markers there are in total. I then repeat this exercise. What I’m looking for here are those students who have to count the total each time even though nothing has been added or taken away from the original set of counters.

4) My next check centers on place value understandings. First, I find out whether students have 1 to 1 correspondence. I hand them a set of about 35 objects and ask them to count them for me. I observe whether they have some kind of organizing strategy like touching or grouping items to help them keep track of what has already been counted. If they count the set accurately, I ask students to write the numeral that represents the total items in the set. So if a student accurately records a total such as 35, I would point to the 3 in the tens digit and ask the student where he or she could find that 3 in the set of objects that has just been counted. For this assessment I am looking for those students who pull out 3 items from the set or say “thirty” but don’t express an understanding that the 3 represents 3 sets of ten.

There are more interview questions I use related to geometry, measurement, time, and money but I won’t list those here to keep the length of this blog to a manageable level. What are your thoughts? Have you found any others you would include? Would this change depending on the grade level?

Math Station Activities

March 15th, 2009

Recently a visitor to this blog wrote, “I have not used manipulatives with instruction for quite some time. However, I am trying to build up my use of them once again. We really run our math class as centers and I want one to be an exploration of concepts via manipulatives. The only problem is I don’t have enough activity resources to support this. Any thoughts.”  I thought this was another issue related to manipulatives that would be useful to explore. See my response below.

I am unsure what grade level you teach or how many children might be accessing these centers at any one time. However, I will take a stab at your question and describe how I’ve done this while working in a variety of K-5 teaching assignments. Math stations are a part of my math block about 2- 3 times a week. Their purpose is to reinforce and practice math concepts I have introduced in whole group lessons. The whole class participates in stations during this time, so I need enough activities to give some choice to a class of 24 students on average. I have found 10 stations to be a good number. Students organize their completion of the different station activities using math station contracts. They consist of games or activities students participate in that utilize manipulatives in many of them.   Go to: http://www.octm.org/jcooke/GBSD%20Align/CORE.htm to see examples of math station contracts I’ve used in the past. These station contracts are aligned to our state core standards for each grade level (K-5). 4th and 5th grade and literature lists aligned to these standards are found on this site as well. The page is not complete yet but I hope to have it done sometime this summer.

Time Spent on Problem Solving

March 6th, 2009

Very often when I work with a group of teachers, I hear the question, “How much time do you spend doing Problem Solving?” I have two different answers for this question. I will share both answers below.

For the first response, I remind teachers that problem solving is one of the process standards clearly discussed in the Principles and Standards of School Mathematics (PSSM), NCTM 2000. As is stated in the National Council of Teachers of Mathematics’ Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (NCTM, 2006),

“Organizing a curriculum around these described focal points, with a clear emphasis on the processes that Principles and Standards addresses in the Process Standards—communication, reasoning, representation, connections, and, particularly, problem solving—can provide students with a connected, coherent, ever expanding body of mathematical knowledge and ways of thinking. Such a comprehensive mathematics experience can prepare students for whatever career or professional path they may choose as well as equip them to solve many problems that they will face in the future.”

Therefore, the answer to the question would be, “I would ‘do’ problem solving as often as I possibly can.” For example, whenever I introduce a new concept, I create a problem solving task as part of the introductory Warm Up. Asking students to reflect on their understandings, communicate their thinking, utilize a variety of strategies and so on are the foundations of a problem-based curriculum and are all part of my instructional standard operating procedures.

However, I believe what the questioner really intended to ask was, ” How do you structure your daily and weekly math times? Then, the answer would be the following. My main block of math time runs about 50 minutes. As mentioned above, if the lesson focus is on developing a new concept, the structure will definitely be problem-based. If we are at a point where I want students to practice and reinforce skills, that block will likely consist of small group station activities. Often these games and activities have a problem solving strategy component built into them, as well. At other times of the day, I have a calendar time for about 20 minutes I use to develop math vocabulary and reinforce and practice skills previously taught and a math journaling time that usually runs for another 15 minutes or so, where students reflect and self-evaluate around the concepts they have been learning. Therefore, 90 minutes a day are usually devoted to math in my usual daily routine.

My answer would be a little different if on the other hand, the question being asked is really, “How often do you use the Making Sense of Problem Solving (MSPS) materials?” In that case, the answer would be that approximately once a week I use the MSPS materials. Because there are 15 units, one week I focus on introductory Warm Up activities followed the next week with Problem Solving and Enrichment tasks assigned to the appropriate level students. I may carry over to a second day, student presentations of their solutions. How about you, my readers? How much time do you spend doing math and how is it structured?

Organizing Math Manipulatives

February 11th, 2009

In my last blog entry, I talked about a common problem teachers are faced with when using math manipulatives — the issue of stealing.  This time I want to talk a little more about another common issue that confronts teachers when trying to incorporate a variety of math manipulatives as part of their instruction.  I often am asked a variation on the following question:

“How  can I better organize all these math manipulatives? I’m tired of finding plastic coins all over the floor at the end of a lesson on money.”

Ultimately, you will have to find the best answer that works for you, but I thought I’d share a couple ideas on this topic. Teachers I know utilize three different strategies for how to disburse manipulatives to use in classroom lessons. Some teachers pass out kits where they give each student a gallon bag containing a variety of the most often used math manipulatives. Other teachers organize table group buckets or baskets and have table group representatives come up and get the supplies for their group at the start of a lesson. The third strategy is to keep each type of manipulative in its own container. I’ve seen containers like plastic dishpans, baskets, or boxes as examples of storage containers.  Again, the teacher asks students to come up a few at a time to help themselves to the number of manipulatives needed per person or for partner  or small group work.

Teachers have shared lots of different strategies for helping to return manipulatives that have fallen on the floor to the place they belong. If the manipulatives are sorted and handed out by shape or color it is that much easier to return a lost manipulative to its proper home. Teachers use “sharpies” to number or letter manipulatives or place small stickers on them as a way to help in the sorting process. When a particular kind of manipulative has a large variety of small pieces, I like to sort them into the kinds of drawer systems found at hardward stores. (You know, the kind used for housing nails, screws, nuts, bolts, etc.) Every year I have a few students who love to help with sorting these math materials. They are often even willing to give up a recess to fill “student orders” for manipulatives.

Here’s a fun variation of this idea used for money manipulatives. Turn one of these drawer sets into the class bank. Identify a few students to be the bankers. They can even wear green eyeshades (cut out from old baseball caps) as they assume their Banker role.

Teacher to Teacher’s Making Sense of Problem Solving Books A and B include information about managing manipulatives, too. Click on the following file: A_Note_About_the_Use_of_Manipulatives to read the excerpt from these books.

What strategies have you come up with to help organize all those math manipulatives? We’d love to hear your comments.

Theft of Math Materials

January 31st, 2009

1/30/09

I was working with a group of teachers the other day. We had just completed a workshop that had focused on looking at the elements of a quality problem solving environment. As part of our agenda, teachers reflected on the use of manipulatives for deepening student understanding and communication of their thinking around math concepts presented in problem solving tasks. Veronica, a second grade teacher, in a suburban school outside of Portland, OR came up to me after the workshop and said, ” The biggest challenge I have around the use of manipulatives, is student theft. I don’t even want to use white boards anymore because I keep having to replace the dry erase markers. And I have about half the pattern block I started out with. How do you deal with this issue in your classroom?” I explained to Veronica that I use storytelling a lot in my classroom to help with a variety of different issues, stealing being one of them. (As a matter of fact, I have written a book that contains a whole collection of these types of stories.)

Here’s how it works. First, I introduce the topic of stealing by telling my whole class the story called “Finders Keepers.” Next, I get them to discuss and write a reflective response in their creative writing journals. We have a process in my classroom for students who would like to share what they have written, where they put a star at the top of the entry. The next day anyone who puts the star at the top will get to take the author’s chair and share what they have written. The clas, as the audience, gives the student writer feedback in the form of “Stars” and “Wishes”. They have to give 3 stars for every wish. “Stars” are things they like about the student’s writing and “Wishes” are things they wish the person would have included in their journal entry. I also give a “Star” and a “Wish” to each student author.  Sometimes I will approach a student and request that he or she allow me to share their story if I think there is something in it that would be beneficial to the class as a whole.

Finally, if I know that a particular student is responsible for a theft, I pull that student aside and talk to him or her about having a chance to earn whatever item it is that he or she wants so desperately. In my 28 years, I’ve had only a few children request that they be allowed to earn something like this and have actually had very little theft. Just calling students’ attention to the fact that it is normal to want items that belongs to someone else, hearing about how I myself struggled with that issue as a child, and helping students to see that they harm the whole class both now and into the future if they aren’t respectful of classroom property seems to be enough to have made this a non-issue throughout my years of teaching.

If you have any other ideas of ways to help children with this issue of stealing or just want to comment on something I’ve written here, I’d love to hear from you.

Greetings

January 24th, 2009

Jan. 24, 2009

Wow! My own blog. That’s not something I would have seen coming even a couple of years ago. Truthfully, writing has been a newer development in my life. Guess you really can teach an old dog new tricks. In 1999, when I was approached to consider training to become a co-editor for The Oregon Mathematics Teacher (TOMT) journal, writing was not even a blip on the radar screen. I remember being startled when I was asked to take over the editor’s position from Anne McEnerny-Ogle.  Here I am 10 years later still going strong with TOMT and loving every minute of it.  Now I’m setting out on this brave new adventure as a bold blogger.

As a part of the team of Oregon classroom teachers who have worked to revise the math problem solving series  called Making Sense of  Problem Solving (MSPS) published by Teacher to Teacher Publications, I’d like to use this space as a place to address any issues regarding the teaching of MSPS lessons. Feel free to send me your questions. I’d also like to invite feedback on any of the MSPS lessons. What lesson did you try out? How did it go?

I’ll use this space to write about my thoughts as a teacher of young children for over 28 years. I’ll pull from my experiences as a Scoring Director for Oregon’s Mathematics Problem Solving Assessment. The insights shared here will also be influenced by my experience as a math coach working with teams of teachers to revitalize their math instruction. And of course, these days my world is totally colored by my three granddaughters and my grandson, so I’ll probably talk about them some, too. I hope you the reader will find something of value here, and will come back time and again to consider  what has been posted and share your own thoughts and experiences related to teaching young children to be mathematical problem solvers.