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?
Tags: Add new tag, Formative Assessment, Math Interventions, Math Interviews
Thanks for sharing this with our math community, Karen. I think the more we all share what is working with our students, the better off we’ll all be. I’d love to hear from others about what they are finding as effective resources for intervention.
Hi everyone. I just wanted to comment on the Marilyn Burns materials for intervention. I used her addition, subtraction and now I am doing multiplication with kids in my school. I feel it is going very well. Today, one student said to me, ” I know why we are in your class. It is so we will stop counting by ones with our fingers, and use our strategies instead. I have learned some new strategies already.” Boy was that a great comment from a 4th grader. She is so right as Brent is. Too many of our programs only show kids how to deal in ones. When we show them ways to think in tens and hundreds, they understand our number system better, and use these strategies to solve problems. I see my students becoming more comfortable with math after just six weeks.
Great question. Educators are so bad about using the jargon without explaining what they mean. When students are good at understanding our base ten number system, they have the skill of composing and decomposing by ten. It means they would understand that 2 tens and 19 ones is equivalent to 1 ten 29 ones, 3 tens and 9 ones and so on. Composing means joining sets together and decomposing means breaking them apart so the examples show breaking 39 apart or decomposing it into sets of equivalent value. In the classroom this concept is usually examined using some type of base ten manipulative.
Can you explain what you mean by composing and decomposing by ten?
Hi Jackie,
My project started out with examining high school students who, when the new graduation requirements are put in place, would be in danger of not receiving a regular diploma. These are the kids that start out so low in the high school math sequence that they won’t pass, or maybe even take 3 years of high school math at Alg. 1 or above. I interviewed (2) 9th graders and (2) 10th graders to get an idea of where their number sense was. I found out that they were largely operating by ones. They didn’t decompose or chunk numbers when performing simple arithmetic tasks. More specifically, they didn’t compose and decompose 10. Moving through decades, whether adding or subtracting was consequently difficult and cumbersome (they often resorted to counting up or down by ones. ).
I decided to continue my interviews, this time with 4-6 sixth, seventh and eighth graders. The results and issues were pretty much the same. So, I continued the research down through 5th, 4th, 3rd, and 2nd grades. At each level, I found the same deficiencies regarding the composition and decomposition of 10 and in moving through decades while adding or subtracting. I should note that none of these students were special ed students. I asked the teachers to provide me with their lowest, non-special ed kids.
I shared these findings with our curriculum director and our 3 elementary principals, and convinced them to pilot a unit of the new Marilyn Burns materials for one month with some of these students. We used the 2nd level of the series, which focuses on moving through decades while subtracting. If necessary, we did a little prerequisite work with composing and decomposing 10 before starting the unit. The pilot was conducted in 3 schools. One school worked with 3rd graders, one with 4th graders, and one with 5th graders. All of the students had similar number sense issues as I described earlier. This was a brief pilot, but we were encouraged with the results. The students made good gains in the areas we were targeting, increased in confidence, and became more active participants in their math classes. Most of them met standard on the state test.
I’m convinced that this issue surrounding “10″ is an indicator of future success, or trouble, in math. These students needed a direct intervention that provided them with good conceptually-based strategies and a greatly increased number of opportunities to work with these strategies.
They may need other interventions in the future, but participating in just this one intervention allowed them far better access to the math they were currently studying. As a case-in-point, I had an opportunity to work with two 4th graders who were struggling with learning multiplication combinations. I worked for about an hour with each student, using techniques and strategies from Van de Walle’s book. The boy I was working with had good number sense, especially around decomposing 10 to move through decades. He made remarkable gains in just that hour. I came back a week later he still had it. On the other hand, the young lady I worked with did not have a good sense of how to decompose 10 to move through decades. Given the same access to the same multiplication strategies, her gains were far less.
Brent Freeman, Ashland School District Mathematics TOSA
First I would use the information to do classroom interventions. As I mentioned in a previous entry, I use math station or center time as a structured time in my weekly schedule that allows me to create flexible groupings of students based on who needs to practice what skills. The readiness activities listed in Volumes A and B of Making Sense of Problem Solving are examples of a source I might look to for activities to do with each flexible grouping to reinforce low skills in a particular area. I would also share this information and suggested practice activities with parents so they might be able to give their child extra practice time at home. If I have parent volunteers, they might be assigned activities to do one on one or with a small skill grouping for extra practice and reinforcement. Depending on the results of all those informal interventions, I would then consider referring the child on for a more formal intervention through our school’s Multidisciplinary Team .
Jackie- I very much like your list as each of the questions/conditions you describe are critical parts of “number sense”. I believe students that struggle with mathematics lack number sense. You have offered clear examples of what is critical and how you might discover the strengths and/or weaknesses of a student related to number sense.
Interesting questions to ask. If you see this kind of “gap” do you refer for intervention right away, or do you try doing classroom intervention first. If you try classroom intervention first, what resources do you like to reference for instructional ideas for each of these “core” ideas?