Introduction

Number sense is the basis of numeracy, serving as the foundation on which all other areas of mathematics understanding are built.  British Columbia’s curriculum emphasizes the development of student competencies across all subject areas as well as encouraging competency-based assessment. Looking at my own professional context, building student numeracy skills has been a school goal in our action plan for learning for the past 5 years with seemingly little growth in students District Numeracy Assessment (DNA) and Foundation Skills Assessment (FSA) results. Our FSA results hover around 50% of students meeting or exceeding grade level expectations. Through prior courses in the Math Post Graduate Certificate program and the PME program I have begun to explore reasons for our consistently lower than expected numeracy scores on both large-scale assessments. The more I explored different avenues for building students numeracy skills, fact practice, problem solving, using multiple strategies, manipulatives, rich tasks and low threshold high ceiling tasks the more I began to notice the same gap. My grade 4’s didn’t understand what the numbers meant or how they related to each other, some even struggled to count objects beyond 50! I began to research number sense, reading the works of multiple authors and realized that this was our gap, my students lacked number sense. This has lead to a three year long deep dive into exploring number sense and it’s role in student’s ability to learn and engage in mathematics. The more I read about number sense and took the time to teach it as explicitly as phonics is taught the more, I realized I needed to be able to accurately assess number sense.

I began searching for number sense assessments tools that fit my needs or could be adapted to fit what I wanted. The SNAP assessment from Chilliwack had potential, it focused on representing a single number in multiple ways and some number relationships like putting the number on a number line but didn’t include counting. From there I went to the source referenced in the SNAP assessment and purchased the ANIE book, and found a wonderful resource for getting a snapshot of student progress and a great once a week check in that could work for any math topic but still not what I was searching for. I then looked at the Island Numeracy Network and their Island Net Assessment which covered most of the topics in each grade for math but lacked specific number sense content. I began to broaden my search outside of BC and while I found number sense resources there were no available assessments that looked at all facets of number sense in a developmental manner

The lack of a developmental number sense assessment tool is the problem that this capstone project aims to address. By creating a formative assessment and accompanying learner and class profile I will be able to identify gaps in my students understanding and focus my instruction on strengthening those areas.

Format and Audience

https://numbers.missyounger.com

I will be presenting my work as a resource website linked to my existing website, that contains a variety of resources and media types. The website will be geared towards teachers working with students in the elementary and middle school years. The assessment will support both classroom instruction and resource, learning assistance or special education supports for students. The resources created for this capstone will allow teachers to accurately assess their students as well as to monitor their growth throughout the school year. This capstone project will contain all of the intermediate (grade 3-5) assessment pages for number sense as well as the student and class learner profiles. If appropriate I will include short 4–5-minute videos on each strand of number sense as well as how to utilize the assessment.

Website Sections

The 5 Strands Of Number Sense

• Detailed explanations of each of the five strands
  • Counting and subitizing
  • Types of numbers
  • Number representations
  • Place value and base 10 numeration
  • Number relationships
• Developmental progressions of each strand
  • Research based on age
    • Learning Trajectories
    • First Steps in Mathematics
  • Related to the British Columbia curriculum
• Practical examples of each of the three strands

The Assessment

• Competency based
• Related to the BC curriculum
• Number sense specific
• Written assessment
• Oral assessment / interview
• Broken into levels
  • Not grade specific
  • Correlated to BC grades
  • Students can have skills at multiple levels
  • Not all students will accomplish every skill in a level
• Grouped into 4 level groupings

Grouping	Early Years	Primary	Intermediate	Middle Years
Ages	2-5	4-8	7-11	10-15
BC Grades	Preschool	K-2	3-5	6-9

•  

Learner profiles

• Individual
  • One for each group of levels
  • A section for each of the five strands of number sense
  • Indicators for skills that are above or below the level range
  • Skill progressions with in the level range
  • Room for observation of student number sense behaviors
  • Included BC performance standards-based rubric
• Class
  • Space to record levels for all students in a class
  • Allows for instructional groups to be formed based on data
  • See the whole class data at a glance
  • Supports instructional planning
  • Accommodates data from all levels

Instructional supports

• Videos on each of the number sense strands
• Links to supporting videos and resources
• Sample activities

Research Methodology

While my capstone project will require a longer-term strategy for full implementation in a school setting I am able to do three sections of it during this course.

To be completed in this course

Literature review

Academic sources

Working with the education librarian, library database searches and Google Scholar I will access relevant articles and books to support developing the assessment and creating a resource which is meaningful and useful to educators in schools. I will be looking for resources surrounding assessing number sense, competency-based assessment and the development of assessments. These searches will also look at validity and fairness when creating assessments.

Websites, Videos and Podcasts

Much of the newer and more accessible information comes from other educators who publish their ideas in videos and podcasts some of the resources I will be accessing include:

• The Debate Math Podcast
• Recovering Traditionalist
• Build Math Minds
• Make Number Sense Make Sense
• You Cubed
• NRICH

Assessment Development

My primary goal this term is to develop the assessment portion, so it is ready for pilot testing when school resumes in the fall. The assessment will consist of one larger version for each level group as well as smaller level specific tasks. Either version could be used in a classroom to assess students and monitor their progress as well as to plan for assessment. The assessment will be constructed based on the information gathered in the literature review as well as working in alignment with the BC curriculum and existing research on how number sense develops. There will need to be emphasis on meeting Universal Design for Learning goals in the assessment design and having items that all or most of the students can be successful on. Effort will also need to be made that the assessment has minimal language requirements, so it is a true test of number sense and not reading skills. It will also be necessary to develop implementation resources for teachers on how to use the assessments with students and how to score them. Significant research will be needed to determine how to create a valid and reliable assessment that is specific to number sense.

Learner Profile Development

To accompany the assessment, it will be necessary to develop the learner profile for each level grouping. Allow for skills to be noted as well as room for teachers to add observations about student work or attitudes towards number sense. Research will be needed to determine the best format for the Learner Profile as well as the class profile so the data is visible to teachers at a glance and can be quickly used for planning instruction. The learner profile documents should be presented in a way that allows them to be used for the teacher’s records and planning as well as for sharing with parents when discussing student progress.

Future steps

• Needs assessment – use surveys developed in PME 894
  • Use surveys to determine teacher needs for a number sense assessment
  • Use surveys to gather input from families on what they need to know about their student’s number sense
• Pilot testing- use Logic Model developed in PME 894
  • Invite teachers to test the assessment, aiming for one per grade level
  • Test the assessment and make needed changes
• Implementation and launch – See Logic Model from PME 894

Conclusion

My capstone project aims to fill a gap in the assessment of numeracy skills for students in grades 3-5 in relation to BC’s curriculum. By creating a detailed number sense specific assessment as well as the accompanying resource website I am able to provide teachers with the tools they need to assess student number sense. Having a number sense specific assessment as well as the learner profile will allow teachers to address gaps in student understanding and provide extensions in areas of strength.  By being able to target number sense specifically, teachers can increase students over all numeracy skills. The resources provided on my website will also give teachers the opportunity to explore the five strands of number sense and access written and video resources discussing both the five strands of number sense and their progressions. My goal is to ultimately fill the gap that I discovered when I was looking for a number sense assessment. In speaking with colleagues about my idea several were interested in having a number sense specific assessment, I hope that this assessment will be a positive tool for my school as we work towards our numeracy goal.

Bibliography

BC Numeracy Network ; J. Novakoski ; D. Lightbody. (2021, September). Langley School District Prioritized Essential Learning Standards in Math K-7. Retrieved from Langley School District Instructional Services: https://instructionalservices.sd35.bc.ca/wp-content/uploads/sites/76/2021/10/Prioritized-Essential-Learning-Standards-in-Math-K-7-2021Sep.pdf

Released in 2021 this document published by the BC Numeracy Network and the Langley School District highlights the key curricular content and competencies students need to master in each grade level. Designed as a tool to aid teacher planning this document highlights number sense outcomes at every grade level. This document is unusual for BC as it highlights curricular content as the focus of the learning over curricular competencies or big ideas. This document provides guidance for the order in which skills are mastered as well as allowing for correlation between research and the BC curriculum.

Bird, K., & Savage, K. (2014). The ANIE. Pembroke Publishers.

Created by two BC teachers The ANIE is a one-page assessment for mathematics skills. The assessment has students solve a single problem in multiple ways to demonstrate their understanding of a topic. This assessment is a formative assessment tool that allows teachers to get a snapshot of students’ understanding and adapt instruction accordingly. The ANIE is designed to be used on a regular basis in classrooms and can provide strong evidence of learning to be stored in student portfolios. The assessment does not use word problems to decrease student language demands and increase the fairness of the assessment.

Clark, L. A., & Watson, D. (2019). Constructing Validity: New Developments in Creating Objective Measuring Instraments . Psychological Assessment, 1412-1477.

In this article, Clark and Watson discuss ways to improve construct validity when creating assessments including having clear constructs, a large item pool, clear item wording, checking for similarities with other topics, measuring one area of understanding.  They also discuss creating alternative forms for test items to tailor them to the unique features of different groups that are being presented with the assessment. While the article discusses test development in relation to psychology and conducting personality tests much of the information on test item creation can be extended to other forms of assessment in mathematics.

Clements, D. H., & Sarama, J. (2014). Learning and Teaching Early Math The Learning Trajectories Approach (Second ed.). New York: Routledge.

Clements and Sarama are two of the leading voices in early mathematics learning. The Learning Trajectories provide an age referenced progression of multiple mathematics skills from birth to age eight.  The Learning Trajectories are meant to provide a map for determining students current level of understanding and guide teachers in planning for next steps. The trajectories break skills into manageable steps laid out in sequence for a variety of mathematics topics including counting and subitizing as well as place value. The incremental progress of the trajectories allows teachers to clearly see what students have mastered and where growth or additional teaching is needed.

Fergus, S., & Smith, C. P. (2022). Characteristics of Proficency-Based Learning and Their Impacts on Math Anxiety in the Middle Grades. Research in Middle Level Education.

Fergus and Smith address the issue of math anxiety by looking at the use of proficiency or competency-based assessments where students are aiming to reach a specified level of understanding rather than a numerical or letter grade. The authors also give an overview of the causes of math anxiety and how it can impact students’ perceptions of themselves. In the article the authors advocate teachers presenting clear success criteria as well as utilizing formative assessment to give feedback. In addition, the idea of reassessment was addressed with the authors being in favour of having students retake or resubmit an assignment or assessment in order to achieve proficiency.

Greisen, M., Hornung, C., Baudson, T. G., Muller, C., Martin, R., & Schiltz, C. (2018). Taking Language Out of the Equation: The Assessment of Basic Math. Frontiers in Psychology.

In this article the authors address the need to limit the use of language in mathematics assessments to ensure that the test is fair for all learners. In the article two groups of students were compared, one receiving verbal directions and the other receiving non-verbal video directions. The non-verbal tasks reminded me of three act math tasks, which are a good low threshold high ceiling task but challenging to assess. The authors also address the role of language proficiency in being able to communicate about mathematics and how students that struggle with language are at a disadvantage when it comes to math assessments.

Huynh, T. (2023, July 28). Making Assessments More Equitable for Multilingual Learners. Retrieved from Edutopia: https://www.edutopia.org/article/making-assessments-equitable-multilingual-students

This article from July 2023 addresses strategies for scaffolding assessments to ensure ELL students as well as those requiring language support. One of the key ideas in this article is that scaffolding and providing language support does not need to simplify the content of an assessment. This article advocates including supports that would address UDL goals for all learners in assessments with out reducing the content requirements.

International Test Commission. (2001 ). International Guidelines for Test Use. INTERNATIONAL JOURNAL OF TESTING. https://www.intestcom.org/files/guideline_test_use.pdf

This document from the International Test Commission provided guidelines for how large scale assessments should be carried out as well as around some ethical considerations for testing. The guidelines do not cover the content of the test instead looking at how the assessment is conducted ensuring that it provides every opportunity for the participants to be successful and that the human rights of those taking tests are respected. The guidelines aim to ensure that assessment is handled professionally in all aspects. These guidelines will help me in developing not only an assessment that is respectful of the students but also the directions for test administration and scoring for teachers.